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IIH1ING LIST DEC 1 1920
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o
PERIODIC ORBITS
>
BY
F. R. MOULTON
IN COLLABORATION WITH
DANIEL BUCHANAN, THOMAS BUCK, FRANK L. GRIFFIN,
WILLIAM R. LONGLEY and WILLIAM D. MacMILLAN
f,
>?
Published by the Carnegie Institution of Washington
Washington, 1920
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CARNEGIE INSTITUTION OF WASHINGTONPublication No. 161
OB
PRESS OF GIBSON BROTHERS, INC.
WASHINGTON, D. C.
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Hi
INTRODUCTION.The problem of three bodies received a great impetus in 1878, when
Hill published his celebrated researches upon the lunar theory. His inves-
tigations were carried out with practical objects in mind, and compara-tively little attention was given to the underlying logic of the processeswhich he invented. For example, the legitimacy of the use of infinite
determinants was assumed, the validity of the solution of infinite systems of
non-linear equations was not questioned, and the conditions for the conver-
gence of the infinite series which he used were stated to be quite unknown.These deficiencies in the logic of his work do not detract from the brilliancy
and value of his ideas, and his skill in carrying them out excites only the
highest admiration.
The work of Hill was followed in the early nineties by the epoch-makingresearches of Poincare, which were published in detail in his Les MethodesNouvelles de la Mecanique Celeste. Poincare brought to bear on the prob-lem all the resources of modern analysis. The new methods of treating the
difficult problem of three bodies which he invented were so numerous and
powerful as to be positively bewildering. They opened so many new fields
that a generation will be required for their complete exploration. On the
one hand, the results were in the direction of purely theoretical considera-
tions, in which Birkoff has recently made noteworthy extensions; on the
other hand, they foreshadowed somewhat dimly methods which will doubt-
less be of great importance in practical applications in celestial mechanics.
The researches of Poincare are scarcely less revolutionary in character than
were those of Newton when he discovered the law of gravitation and laid
the foundations of celestial mechanics.
In 1896 Sir George Darwin published an extensive paper on the problemof three bodies in Acta Maihematica. In mathematical spirit it was similar
to the work of Hill; indeed, the methods used were essentially those of Hill,
but the problem treated was considerably different. For a ratio of the finite
masses of ten to one, Darwin undertook to discover by numerical processes
all the periodic orbits of certain types and to follow their changes with
varying values of the Jacobian constant of integration. This program was
excellently carried out at the cost of a great amount of labor. It gave
specific numerical results for many orbits in a particular example.The investigations contained in this volume were begun in 1900 and,
with the exception of the last chapter, they were completed by 1912. Those
not made by myself were carried out by students who made their doctorates
under my direction.
in
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iv Introduction.
The following chapters have been heretofore published in substance:
I. Sections III and IV American Journal of Mathematics, vol. xxxm (1911).
II. Astronomical Journal, vol. xxv (1907).
III. Hcudiconti Matematico di Palermo, vol. xxxn (1911).
IV. Transactions of the American Mathematical Society, vol. xi (1910).
VII. Mathcinatisilie Annalen, vol. lxxiii (1913).
VIII. Annals of Mathematics, 2d Series, vol. 12 (1910)
XI. Transactions of the American Mathematical Society, vol. vu (1906).
XII. Transactions of the American Mathematical Society, vol. xin (1912).
XIII. Transact ions of the American Mathematical Society, vol. vin (1907).
XIV. Transactions of the American Mathematical Society, vol. ix (1908).
XV. Proceedings of the London Mathematical Society, Series II, vol. 2 (1912).
The investigations and computations contained in the last chapterwere completed in 1917.
It was originally intended to publish only the first fifteen chapters,
and if that program had been carried out they would have appeared in 1912.
But as the work of printing progressed the ideas contained in the last chapterwere being developed and the computations were begun. It was thoughtthat an even more nearly complete and certain idea of the evolution of peri-
odic orbits with changing parameters could be obtained in a year than were
obtained in five years. The difficulties and enormous amount of labor
involved were not foreseen. No one can now read with better appreciationthan I the following words from Darwin's introduction to his paper:
"As far as I can see, the search resolves itself into the discussion of particularcases by numerical processes, and such a search necessarily involves a prodigiousamount of work. It is not for me to say whether the enormous amount of labor I
have undertaken was justifiable in the first instance; but I may remark that I havebeen led on by the interest of my results, step by step, to investigate more, andagain more, cases."
The results which now appear had all been obtained when service in
the army made it necessary to lay them aside before the final chapter could
be put into form for publication. After they had been laid aside for abouttwo years it was not easy to gather up the details again and to arrange themin a systematic order. This explains the long delay in the appearance of
this volume. It is clear that it was in no wise due to the Carnegie Institu-
tion of Washington. Indeed, the patience of President Woodward with
long and expensive delays has been far beyond what could reasonably havebeen expected.
In the greater part of this work complete mathematical rigor has beeninsisted upon. On the other hand, the developments have been in a formapplicable to practical problems in celestial mechanics. For example,sections III and IV of Chapter I treat non-homogeneous equations of the
types which arise in practical problems; Chapter II is devoted to questionswhich have long been classic in celestial mechanics; Chapter III contains,
among other things, a new and rigorous treatment of Hill's differential
equation with periodic coefficients; Chapter IV treats a problem that arises,
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Introduction. v
at least approximately, in the solar system; Chapter V is developed in a formsuitable for numerical applications; Chapter VI is an alternative treatmentof the same problem, and in Chapter VII an extension of the probleminvolving entirely new types of difficulties is found and more powerful
methods of treatment are required; Chapter XI covers the same ground asHill's work on the moon's variational orbit and Brown's work on the paral-lactic terms; Chapter XII contains the corresponding discussion for superior
planets; and Chapter XIV treats a problem similar to that presented by the
satellite systems of Jupiter and Saturn or by the planetary system. ChapterXV contains a discussion of limiting cases of periodic orbits, namely, closed
orbits of ejection. It forms a basis for part of the work of the last chapter,and it may be found to have practical applications in the escape of atmos-
pheres. The last chapter is an attempt to trace out the evolution of periodic
orbits as the parameters on which they depend are varied. In spite of the
fact that infinitely many families of periodic orbits were found, whereas onlya few such families were previously known, the discussion remains in cer-
tain respects incomplete. It should be stated that many results were found
which have not been included because they did not contribute to the solu-
tion of the particular question under consideration. For example, a series
of orbits asymptotic to the collinear equilibrium points was computed.The amount of labor the last chapter cost can scarcely be overestimated.
I should not be true to my own feelings if I did not express the appre-ciation of the assistance of my collaborators. The association with themhas been a deep source of satisfaction and inspiration. They are to be held
accountable only for those chapters which appear under their names. Mostof the computations on which many of the results of the last chapterare based were made by Dr. W. L. Hart and Dr. I. A. Barnett. Without
assistance of such a high order the laborious computations could not have
been carried out.
F. R. Moulton.
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CONTENTS.
Chapter I. Certain Theorems on Implicit Functions and Differential
Equations. (By F. R. Moulton and W. D. McMillan.)I. Jolution of Implicit Functions. page
1. Formal solution of simultaneous equations when the functional determinantis distinct from zero at the origin 1
2. Proof of convergence of the solutions 23. Generalization to many parameters 54. The functional determinant zero but not all its first minors zero at origin 55. Case where P (a, a) = a* P t (a, fi) p
x P 2 (a, /i) 66. A second simple case 67. General case of power series in two variables G
II. Solutions of Differential Equations as Power Series in Parameters.
8. The types of equations treated 109. Formal solution of differential equations of Type 1 10
10. Determination of the constants of integration in Type 1 1111. Proof of the convergence of the solutions of Type 1 11
12. Generalization to many parameters 1413. Generalization of the parameter 1414. Formal solution of differential equations of Type II 1515. Determination of the constants of integration in Type II 1616. Proof of the convergence of the solutions of Type II 1617. Case of homogeneous linear equations 19
III. Homogeneous Linear Differential Equations with Periodic Coefficients.18. The determinant of a fundamental set of solutions 2119. The character of the solutions of a set of linear homogeneous differential
equations with uniform periodic coefficients 2220. Solutions when the roots of the fundamental equation are all distinct .... 2421. Solutions when the fundamental equation has multiple roots 2622. The characteristic equation when the coefficients of the differential equa-
tions are expansible as power series in a parameter n 3023. Solutions when a l
\ a^, ... , a,^0) are distinct and their differences are
not congruent to zero mod. V 1 3224. Solutions when no two a^
0) are equal but when a^0)
o['
is congruentto zero mod. V 1 34
25. Solutions when, for m= 0, the characteristic equation has a multiple root . . 3526. Construction of the solutions when, for n = 0, the roots of the character-
istic equation are distinct and their differences are not congruent to
zero mod. V 1 3627. Construction of the solutions when the difference of two roots of the
characteristic equation is congruent to zero mod. y/ 1 4028. Construction of the solutions when two roots of the characteristic equation
are equal 43IV. Non-Homogeneous Linear Differential Equations.
29. Case where the right members are periodic with the period 2ir and the a;
are distinct 4630. Case where the right members are periodic terms multiplied by an exponen-
tial, and the a;are distinct 47
31. Case where two characteristic exponents are equal and the right membersare periodic 48
V. Equations of Variation and their Characteristic Exponents.32. The equations of variations 50
33. Theorems on the characteristic exponents 51
VII
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Mil CONTENTS.
Chapter II. Elliptic Motion. page.
34. The differential equations of motion 55
35. Form of the solution 56
36- Direct construction of the solution 58
37. Additional pro|>ertie8 of the solution 59
38. Problem of the rotating ellipse 60
39. The circular solution 61
40. Existence of the non-circular solutions 61
41. Direct construction of the non-circular solution 64
42. Properties of the solution 66
Chapter III. The Spherical Pendulum.
I. Solution of the Z-Equation.43. The differential equations 67
44. Transformation of the z-equation 68
45. First demonstration that the solution of (12) is periodic and that n and the
period are expansible as power series in n 69
46. Second demonstration that the solution of equation (12) is periodic 71
47. Third proof tha t the solution of (12) is periodic 73
48. Direct construction of the solution 73
49. Construction of the solution from the integral 76
II. Digression on Hill's Equation.50. The x-equation 78
51. The characteristic equation 78
52. The form of the solution 81
53. Direct construction of the solutions in Case 1 83
54. Direct construction of the solutions in Case II 85
55. Direct construction of the solutions in Case III 87
III. Solution of the X- and Y-Equations for the Spherical Pendulum.56.
Applicationto the
spherical pendulum88
57. Application to the simple pendulum 9058. Application of the integral relations 93
Chapter IV. Periodic Orbits about an Oblate Spheroid.
(By William Duncan MacMillan.)
59. Introduction 9960. The differential equations 10061. Surfaces of zero velocity 101
I. Orbits Re-entrant after one Revohdion.62. Symmetry 10263. Existence of periodic orbits in the equatorial plane 103
64. Existence of periodic orbits which are inclined to the equatorial plane . . . 10465. Existence of orbits in a meridian plane 10666. Construction of periodic solutions in the equatorial plane 107
67. Construction of periodic solutions for orbits inclined to the equatorial plane. Ill
68. Construction of periodic solutions in a meridian plane 114II. Orbits Re-eniraid after many Revolutions.
69. The differential equations 11670. The equations of variation 11871. Special theorems for the non-homogeneous equations 12772. Integration of the differential equations 13273. Existence of periodic orbits having the period 2kw 14074. Construction of the solutions with the period 2kt 143
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CONTENTS. IX
Chapter V. Oscillating Satellites about the Straight-line Equi-librium Points. (First Method.) page.
75. Statement of problem 15176. The differential equations of motion 152
77. Regions of convergence of the series P u P 2 , P 3 . . 15478. Introduction of the parameters and 5 15579. Jacobi's integral 15g80. The symmetry theorem 15681. Outline of steps for proving the existence of periodic solutions of equa-
tions 15782. General solutions of equations (12) for 5 = e = 15883. Periodic solutions when 5 = e = *
16084. Normal form for the differential equations 1G185. Existence of periodic orbits of Class A 16386. Some properties of solutions of Class A 16687. Direct construction of the solutions for Class A 16888. Additional properties of orbits of Class A 17389. Application of Jacobi's integral to the orbits of Class A 17690. Numerical examples of orbits of Class A 17991. Construction of a prescribed orbit of Class A 18092. Existence of orbits of Class B 18093. Direct construction of the solutions for Class B 18394. Additional properties of the orbits of Class B 18995. Numerical example of orbits of Class B 19096. On the existence of orbits of Class C 191
Chapter VI. Oscillating Satellites. (Second Method.)
97. Outline of method 19998. The differential equations 19999. Integration of the differential equations 201
100. Existence of periodic solutions 202101. Direct construction of the solutions for Class A 210102. Direct construction of the solutions for Class B 213
Chapter VII. Oscillating Satellites when the Finite MassesDescribe Elliptical Orbits.
103. The differential equations of motion 217104. The elliptical solution 218105. Equations for the oscillations 220106. The symmetry theorem 222
107. Integration of equations (13) 223108. The terms of the second degree 227109. The terms of the third degree 228110. General properties of the solutions 230111. Conditions for the existence of symmetrical periodic orbits 230112. The existence of three-dimensional symmetrical periodic orbits 233113. Case I, n even, N odd 234114. Case II, n odd 236115. Convergence 237116. The existence of two-dimensional symmetrical periodic orbits 239117. Case I, n even 240118. Case II, n odd 241
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\ CONTENTS.
Chapter VII Continued.
Construction of Three-Dimensional Periodic Solutions.page.
1 19. Defining properties of the solutions 241
120. Coefficients of X 241
121. Coefficients of X 243
122. Coefficients of \1 245123. Coefficients of X2 248
124. Coefficients of X* 248
125. The general step for the x and j/-equations 250126. The general step for the z-equation 251
Application of the Integral. .
127. Form of the integral 253
128. The integral in case of the periodic solutions 254129. Determination of the coefficients of Z 2 j+\ when e = 257
Determination of the Coefficients of Z2/+1 when e = 0.
130. Case when e ?* 0. General equations for Z, 260131. Coefficients of e 262132. Coefficients of e 263133. Coefficients of e
2 and e* . . 264134. General equation for v= 1 265135. Terms independent of e 265136. Coefficients of e 266
Direct Construction of the Two-Dimensional Symmetrical Periodic Solutions.
137. Terms in Xl 267138. Coefficients of X 269139. Coefficients of Xi 271140. General proof that the constant parts of the right members of the first two
equations of (148) are identical 278141. Form of the periodic solution of the coefficients of Xi 280142. Coefficients of X2 280
143. Induction to the general term of the solution 282
Chapter VIII. The Straight-line Solutions of the Problem op n Bodies.
144. Statement of problem 285I. Determination of the Positions when the Masses are given.
145. The equations defining the solutions 286146. Outline of the method of solution 288147. Proof of theorem (B) 289148. Proof of theorem (C) 290149. Computation of the solutions of equations (6) 294
II. Determination of the Masses when the Positions are given.150. Determination of the masses when n is even 296151. Determination of the masses when n is odd 296152. Discussion of case n = 3 297
Chapter IX. Oscillating Satellites near the Lagrangian EquilateralTriangle Points. (By Thomas Buck.)
153. Introduction 299154. The differential equations 299155. The characteristic exponents 301156. The generating solutions 303157. General periodicity equations 305
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CONTENTS. XI
Chapter IX Continued. PAGE
158. The first generating solution 307159. The third generating solution 309160. The fourth generating solution 309161. The fifth
generatingsolution
311162. The seventh generating solution 312163. Construction of the solutions in the plane 313164. Construction of the solution with period 2ir 319
Chapter X. Isosceles-Triangle Solutions of the Problem of Three Bodies.
(By Daniel Buchanan.)
165. Introduction 32
I. Periodic Orbits when the Finite Bodies move in a Circle and the Third Body is
Infinitesimal.
166. The differential equation of motion 325167. Proof of existence of a periodic solution of equation (1) 326168. Direct construction of the periodic solution of equation (1) 328
II. Symmetrical Periodic Orbits when the Finite Bodies move in Ellipses and the Third
Body is Infinitesimal.
169. The differential equation 330170. Determination of the period by a necessary condition for a periodic solu-
tion of (23) 331171. Existence of symmetrical periodic orbits 333172. Direct construction of symmetrical periodic solutions of (23) 335
III. Periodic Orbits when the Three Bodies are Finite.
173. The differential equations , 340174. Proof of existence of periodic solutions of equations (70) 341175. Proof that all the periodic orbits are symmetrical 346176. Direct construction of the periodic solutions of (70) 348177. The periodic solution of equation (69) 353178. The character of the periodic solutions 355
Chapter XI. Periodic Orbits of Infinitesimal Satellites and Inferior Planets.
179. Introduction 357180. The differential equations 358181. Proof of the existence of the periodic solutions 360182. Properties of the periodic solutions 363
Direct Construction of the Periodic Solutions.
183. General considerations 364184. Coefficients of m2
365185. Coefficients of to' 365186. Coefficients of to
4 366187. Induction to the general step of the integration 369188. Application of Jacobi's integral 372
189. The solutions as functions of the Jacobian constant 375190. Applications to the lunar theory 376
191. Applications to Darwin's periodic orbits 377
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XII CONTENTS.
Chapter XII. Periodic Orbits of Superior Planets.PAGE.
192. Introduction 379
193. The differential equation 379
194. Proof of the existence of {>criodic solutions 381
195. Practical construction of the periodic solutions385
196. Application of the integral 388
Chapter XIII. A Class of Periodic Orbits of a Particle Subject to the Attractionof n Spheres having Prescribed Motion. (By W. R. Longley.)
197. Introduction 389
198. Existence of periodic orbits having no line of symmetry 390
199. Construction of periodic orbits having no line of symmetry 396
200. Numerical Example 1 402
201 . Some particular solutions of the problem of n bodies 404
202. Existence of symmetrical periodic orbits 410
203. Construction of symmetrical periodic orbits 414
204. NumericalExample
2 415
205. Numerical Example 3 417206. The undisturbed orbit must be circular 419207. More general types of motion for the finite bodies 420208. Numerical Example 4 422
Chapter XIV. Certain Periodic Orbits of k Finite Bodies Revolvingabout a Relatively Large Central Mass. (By F. L. Griffin.)
209. The problem 425210. The differential equations 42721 1. Symmetry theorem 427212. Conditions for periodic solutions 428213. Nature of the periodicity conditions 429214. Integration of the differential equations as power series in parameters. . . . 430215. Existence of periodic orbits of Type 1 434216. Method of construction of solutions. Type 1 438217. Concerning orbits of Type II 442218. Concerning orbits of Type III .' 447219. Concerning lacunary spaces 450220. Jupiter's satellites I, II, and III 452221. Orbits about an oblate central body 455
Chapter XV. Closed Orbits of Ejection and Related Periodic Orbits.
222. Introduction 457223. Ejectional orbits in the two-body problem 457224. The integral 461225. Orbits of ejection in rotating axes 461226. Ejectional orbits in the problem of three bodies 462227. Construction of the solutions of ejection 463228. Recursion formulas for solutions 465229. The conditions for existence of closed orbits of ejection 469230. Proof of the existence of closed orbits of ejection 471231. Conditions at an arbitrary point for an orbit of ejection 473232. Closed orbits of ejection for large values of n 477233. Periodic orbits related to closed orbits of ejection 479234. Periodic orbits having many near approaches 483
i
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CONTENTS. XIII
Chapter XVI. Synthesis of Periodic Orbits in the Restricted Problemof Three Bodies. page.
235. Statement of problem 485230. Periodic satellite and planetary orbits 486
*237. The non-existence of isolated periodic orbits 487238. The persistence of double orbits with changing mass-ratio of the finite
bodies 490239. Cusps on periodic orbits 495
240. Periodic orbits having loops which are related to cusps 497241. The persistence of cusps with changing mass-ratio of the finite bodies. . . . 499242. Some properties of the periodic oscillating satellites near the equilateral
triangular points '. 501
243. The analytic continuity of the orbits about the equilateral triangular points 504244. The existence of periodic orbits about the equilateral triangular points for
large values of n 505245. Numerical periodic orbits about equilateral triangular points 507246. Closed orbits of ejection for large values of n 510247. Orbits of ejection from 1 ft and collision with n 515
248. Proof of the existence of an infinite number of closed orbits of ejection andof orbits of ejection and collision when ju
= J^ 517
249. On the evolution of periodic orbits about equilibrium points 519
250. On the evolution of direct periodic satellite orbits 520251. On the evolution of retrograde periodic satellite orbits 522
252. On the evolution of periodic orbits of superior planets 524
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-
PERIODIC ORBITS
BY
F. R. MOULTON
IN COLLABOHATION WITH
DANIEL BUCHANAN, THOMAS BUCK, FRANK L. GRIFFIN,
WILLIAM R. LONGLEY and WILLIAM D. MacMILLAN
U
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2 PERIODIC ORBITS.
From the coefficients of the first power of m we get
<
a<" = dP,tf-1, ,n). (4)
Since by hypothesis the functional determinant
A =
dP,
3a,
dPdcii
da
3a,
is distinct from zero for a! = = a = n =0, equations (4) can be solved
uniquely for the a"'. If not all of the dP t /dn are zero, then not all of the a"'
are zero; but if all of the dP t /dn are zero, then all of the a( " are zero.
Equating the coefficients of the second power of /z in (3) to zero, we get
2|?- -*?>(?, . . ., tfo 0-i,** ,n),
the right members of which are completely known. The determinant of
the coefficients of the af is A, and the af are therefore uniquely determined,
being all zero or not all zero according as the Ff are all zero or not all zero.
And it is seen from (3) that the treatment of the general term is entirely simi-
lar and depends upon the same determinant A. Hence, under the hypothesisas to A, a formal solution is possible, and it is unique.
2. Proof of Convergence of the Solutions. In order to prove the con-
vergence of the series (2), consider the solution of the comparison equations
Q. (ft , ,/3 B ; M)=0 (*-l n) (!')
for /3j , . . .,
in terms of n, where the Q t are power series in the #, and n,
vanishing for #, = 0, m = 0, and convergent for| fij \
< r > 0, | n |< p' > 0; and
where also the coefficients of all terms beyond the linear in each Q t are real,
positive, independent of i, and greater than the moduli of the correspondingcoefficients* in the expansions of any of the P t . The character of the coeffi-
cients of the linear terms will be specified when they are used.
Suppose the solutions of (10 have the form
(i-1, ...,>. (20
*For proof of the possibility of satisfying these conditions see Picard's Traite d'Analyse, edition of
1905, vol, II, pp. 255-260.
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IMPLICIT FUNCTIONS.
On substituting (2') in (1') and arranging in powers of n, there results a
system of equations similar to (3) . The 0f are determined by
Wft ft
15 =(-l, , n). (4')
It is necessary now to specify the properties of the linear terms of
the Qt . It will be supposed first that the dQ f /dn are real and positive, and
that dQ l /dfi= =dQ n /dn = dQ/d/j,. It follows that for fixed values of
the dQi/dfis the values of the /3( "
satisfying (4') are proportional to dQ/d/j..
The dQi/dj3j will now be so determined that when dQi/d/i is replaced by the
greatest \dP t /dn\ the ftj' determined by (4') shall be equal, positive, and
at least as great as the greatest |al
"| for all values of dPi/d/x such that
! dP,/dn J^ dQ/dn. This must be done in such a way that the determinant
A' =
aft'
dQn
aQi
dQn
aft
shall be distinct from zero for $ s= n = 0. These conditions can be satisfied in
infinitely many ways. A simple way is to choose dQ,/d^= 1 if j?i and
dQ t /dp t = -(1+c). Then the determinant of (4') is A' = (-1)" cn ~ l
(c+n),which can vanish only if c = or c = -
n, and the solutions of (4') are
(i) _fl" = (i)
dQdfx (5)
(c + n)
For any n we can give c such a value that the P? shall be positive and at least
as great as the greatest |a
( "|
.
The Pf are determined by equations of the form
2^> i/3<2)= _ GW> . . .
, {%>) (t-i, ...,). (6)
It follows from the properties of the Q t , together with the explicit structure
of the G'f and the values of the 0?', that
G< 2) = G? ^ | F? |(i=
l, . . . , ),
and therefore that
jw = =** ^ I
a<;()
(<-l, , n).
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4 PERIODIC ORBITS.
It is very easily shown by induction that for every value of the index k
GP- = G ^ | FT I 0-1, ...,), 1
tf- -tf fc|,"| 0-1 n).J
( }
Therefore if (2') converge for| p | ^ p'> then also (2) converge for
| m I ^ P
All the conditions imposed upon the Q< can be satisfied by functions
of the form*
Ql =-c j8 l -(l+Jlf)((8 I + +.) + ,
M(^t' '
',
+ ^" + ^, =0
0-5) (1
ft+ ' +/3nr /
(t'=l, . . . ,n),
(8)
where Mis a real positive constant. Adding these n equations and solving
for 0i + + /3, it is found that
0,+ +j8 =n p-p-yj\ ^n p-/x/ p-m
PM
(9)
0+5+*)Since each /3, , and therefore the sum &+ +/3 ,
is zero for p = 0,
the negative sign must be taken before the radical in (9). The right
member of (9) can be expanded as a converging power series in p. if| p |
is
taken so small that |p| <p and |/Gu)| < |<p(p)|, where
conditions which can always be satisfied, whatever may be the values of n, r,
p, c, and M. Moreover, the coefficients of all powers of p. in the expansionof (9) are real and positive. Hence it follows that
&+ +(3 n = pR(p), (10)
where R(p.) is a power series in p whose coefficients are all positive. It
follows from (8) that all the ft are equal. Hence
n (11)
For |p| sufficiently small the right members of these equations are converging
power series in p; moreover, they identically satisfy (8). It follows fromthis result and the second set of (7) that p" > exists such that the series (2)
converge for| p |
< p".
*Picard's Trailt d' Analyse, loc. cit.
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IMPLICIT FUNCTIONS. 5
3. Generalization to Many Parameters. Suppose the equations to besolved are
P t (a, ,. . .
, a. ; /*, ,. . .
, Mt ) =0 (i=
l, . . ., n ), (12)
and that the functional determinant with respect to the a., is distinct fromzero for aj= = a = ii 1
= = Mt = 0. Then the problem can be reducedto that discussed in 1 and 2 by letting fij
=Cjix. After the solutions have
been obtained the c } n everywhere can be replaced by n for the Cj m will occur
only in integral powers.
4. The Functional Determinant Zero, but not All of its First MinorsZero at the Origin.* Consider the equations
P t (a, ,. . .
, a; p) ={i=
\, . . ., ), (13)
where the P, have the properties imposed upon the P, of 1. Suppose thatthe determinant of the linear terms in the a, is zero for a! = = a = n =
0,
but that not all of its first minors are zero. It may be supposed without
any loss of generality that the determinant of the terms remaining after
deleting the last row and column of the linear terms is distinct from zero.
Hence, as a consequence of the theorems proved in 1, 2, 3, the first n 1
equations can be solved uniquely for a 1; . . ., a_i as power series in a a and
H, vanishing for a =fj.= 0.
Suppose the solutions of the first n 1 equations for o t ., . . ., a_i are
substituted in the lastequation.
It will become a function of a n andnwhich
may be written, omitting the now useless subscript on the a,
P(a, M)= S 2c w aV = (k + j>0). (14)
i=0 )=Q
Since the determinant of the linear terms of (13) is zero, this equation carries
no linear term in a. Suppose the term of lowest degree in a alone is c to a\
Then, for each value of n whose modulus is sufficiently small there are k
values of a satisfying (14) and, moreover, the modulus of /x can be taken so
small that the moduli of the solutions for a shall be as small as one pleases, f
Also,for each set of values of a = a and /x whose moduli are sufficiently small
there is one set of values a! ,. . .
, a_, satisfying the first n 1 equations of
(13). Consequently, for each /x whose modulus is sufficiently small there are
precisely k sets of values of a,, . . .,a B satisfying (13). A special discussion
is required to determine the character of these solutions and the method of
finding them. These questions are taken up in the immediately following
articles.
*The more difficult case, in which all the first minors of the functional determinant vanish, does not arise
in this work. It has only recently (in 1911) been completely solved by MacMillan, in a paper which will
appear in Maihemalische Annalen.
fWeierstrass, Abhandlungen aus der Functionenlehre, p. 107. Picard, Traite d' Analyse, vol. II, chap. 9,
{7, and chap. 13. Harkness and Morley, Treatise on the Theory of Functions, chap. 4.
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6 PERIODIC ORBITS.
5. Case where P(a, p) =a*P,(o, m)~mxPj(a, p). Suppose the P(a, p)
of (14) has the form
a*P,(a,p) - ^PA*,n) =0,
wherePi
and P 2 arepower
series in a and p which do not vanish for a = p =
and which converge for|
a|
< r > 0, | p |< p > 0. Upon extracting the k' h
root, this equation gives
a Qi (a, p) V M* Qa (a, p) 0,
where Qi and Q2 are power series in a and p which do not vanish for a = p =
and which converge for|a
|< r' > 0, | p |
< p' > 0, and where r\ is a A;" 1 root of
unity. If we let p = vk
, this equation takes the form of those treated in 1
and 2 and can be solved uniquely for a in terms of v for each 77. The k
solutions arc obtained by taking for 17 the k roots of unity.
6. A Second Simple Case. Suppose P (a, p) has the form
P (a, p) = 2 C, a*-' p' + Q (a, p) = 0, (15)
where c ^ and C> contains no term of degree less than k + 1 in a and p.
It can be supposed without loss of generality that c = 1. Then1
2c,a*- y= P (a,n) = (a
-6, p) (a
- 6 2 p) (a b k n). (16)(-0
Suppose (a bjfi) is a simple factor of the homogeneous polynomialP (a, p), and exclude the trivial case in which it is also a factor of Q (a, p).
Then dP /da ^ for a = bj p. Now let
B-&,M + 0M. (17)
After this transformation both P and Q are divisible by p*. After p* is
divided out, P carries a term in P to the first degree whose coefficient is
not zero, and Q carries no term independent of p, but has at least one term
in p alone, for otherwise P (a, p) would be divisible by (a b } y). Conse-
quently by 1 and 2 the equation in /S and p can be solved for /3 as a con-
verging power ser ies in p, vanishing for p = 0. Therefore a can be expandedas a converging power series in p, vanishing with u, for each of the simpleroots of the polynomial P (a, p) = 0. If fe, ,
. . .,
b k are all distinct, the
expansions for the k branches of the function P(a, p) which pass through the
origin can be found by this process. The actual determination of the coeffi-
cients is by the method of 1 in the simple case n = 1 .
7. General Case of Power Series in two Variables.* The method of
treatment consists in reducing the equation, by suitable transformations,
to forms of a standard type from which the solutions can be found. In cer-
tain special cases successive transformations are required. The analysis in
*This problem has been treated by Puiscux, Nother, etc. For references and discussion see Harknessand Morley's T realize on the Theory of Functions, chap. 4, and Crystal's Algebra, part 2, chap. 30.
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IMPLICIT FUNCTIONS.
general, as well as the particular transformations required in any special case
to reduce the equation to the standard forms, is indicated most simply byNewton's Parallelogram.
In constructing Newton's parallelogram it is sufficient to consider onlythose terms c y aV of P(a, n) for which i <! &, j ^ X, cw a* being the term of
lowest degree in a alone and c ox mx the term of lowest degree in n alone.
Take a set of rectangular axes and for each of the terms c w anJ
,c tj 9^ 0, lay
down a degree point whose coordinates are i and j. Then the line passing
through the origin and the point (k,0) is rotated around (k,0) as a pole so that
it moves along the ./-axis in the positive direction until it strikes at least one
other degree point (it may, of course, strike several simultaneously). Let the
one of those which it first strikes having the greatest j be (ii,ji). Then the
line is rotated around (t'i, j x ) in the same direction until it strikes at least one
other degree point. Letting the one of these having the greatest j be (i 2 ,j 2 ),
the line is rotated around (i 2 , j 2 ) until another is encountered. This process is
continued until the point (0, X) is reached. The number of steps in the process
evidently can not be greater than k or X. The part of Newton's parallelo-
gram needed in discussing the character of the function near the origin is made
up of the segments (k, 0) to (tiji), OwO to (i 2 , j 2 ), , (w,) to (0, X). For the
terms of P (a, n) corresponding to each one of these segments there is, as will
be shown, a transformation which throws P(a, n) into a standard form.
In order to illustrate Newton's parallelogram, consider the example
P(a, n) = CM a5-\-C 3i a
3fi + C22 a
2fj.
2 + C l3 a n3 + Cu a fi' + Qx //+Q (a, n),
where Q(a, m) contains only terms of the seventh and higher degrees in a
and m- The coordinates of the points in Newton's Parallelogram are (5, 0),
(3,1), (2,2), (1,3), and (0,6), and it
consists of three segments which are
shown in Fig. 1.
Consider the segment (t,, j t ) to
(i 2 ,j 2 ) and make the transformation
a =j3 M%
_ J2-J1 - mti x 2 n (18)
(5,0)
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8 PERIODIC ORBITS.
It follows from the position of the segment (%,j,) to (i,jt) with reference
to every other degree point, that in the case of any term c ti a y' of P (a, y)
whose degree point is not on this segment the exponents i and j satisfy the
inequality
i U2-J1) +j (h - H) + itjt - hh > 0.
Consequently, after the transformation (18) the term c,jayJ becomes c^py'",
where
; : .> <T .*" =
This discussion proves that after the transformation (18) the terms
belonging to the segment (ii,ji) to (i 2 ,j 2 ) contain y"' as a factor, and that every
other term contains y to a higher power than a'. Since a' is not necessarily an
integer the series will not be, in general, a series in integral powers of y, but
it will be in integral powers of y' = MB Hence, dividing out y"' the series
becomes= c uh (}" + + c hh 0* + y' P, (0, m')- (20)
For n' = equation (20) becomes
c, P [>-''+ +}]- *? (0-cd- (/3-c fl _ (! )
= 0. (21)
The solution j8''= is not to be considered for this transformation because
it belongs to the solutions obtained from the segments having smaller values
of i. Suppose (8 c,, is a simple factor of (21) and let
= c, + t.. (22)
Then the right member of (20) becomes a power series in 7, and fi', van-
ishing with y y= n' =
0, and the coefficient of y y to the first power is distinct
from zero. Therefore, by 1, the equation can be solved for 7,, uniquely as a
converging power series in yJ , vanishing for ju'= 0. Then, on substituting
back in (22) and (18), a is expressed in integral powers of /*'. This is an
integral ser ies in n only if a is an integer. If c, ,. . .
, c,,_ ( . are all distinct weobtain at this step i, i 2 solutions, and if a is an integer the number of them
is precisely i\ i 2 . Since when a is not an integer y has more than onedetermination, and since the series obtained by the transformation (18)after removing the factor y,"' is not in integral powers of y, it would seemthat the number of solutions for the segment is greater than ii i 2 - Butit will now be shown that the number of distinct solutions is ?', i 2 , whether<r is an integer or not.
Now <t= (J2 ji)/(ii h)=m/n, where m and n are relatively primeintegers. It is clear that t, -i 2 equals n, or is greater than n, according as^- j t
and i, i 2 do not have, or have, a common integral divisor greater than unity.Consider first the case where i x i 2
= n. There can be no degree point
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IMPLICIT FUNCTIONS. 9
(i',f) on the segment between (t'i , j x ) and (ia,jt), for its coordinates wouldhave to satisfy the relation (j 2 j')/{i' i 2 )
= m/n, which is impossible wheni' i 2 <n. Therefore equation (21) becomes
0*+ + A = 0,
whose solution gives i x i 2 values of /3, differing only by the i x %2 roots of
unity. Hence the same final results are obtained by using the principal value
of y." in (18) and all of these it it values of /3 as are obtained if any other
determination of // is used.
Suppose now i x i 2= qn, where q is an integer. There can not be more
than q1 degree points on the segment (i x , j x ) to (i 2 , j 2 ), i. e. satisfying the
relation (j 2 -j')/(i'-i 2 ) =m/n. Therefore in this case (21) is a polynomial in 0"
of degree q, and for each of the solutions for ft" there are n values of /3 obtained
one from the other by multiplying by the nth
rootsof
unity. Hencein
thiscase also the final results are the same as are obtained by using any other
of the n determinations of n".
It follows that in every case all the distinct solutions are obtained bytaking the principal value of y! ', and that the number of them for the
segment (ii,ji) to (i*,ji) is precisely ii i 2 . In a similar manner the
solutions associated with each of the other segments can be obtained. Thewhole number of solutions found in this way is
N = (k-h) + & - i 2 ) + + (i-
0) - *, (23)
which is the number of solutions of (P a, fj.) = for a which vanish with n = 0.
In this case the problem is completely solved.
Suppose, however, that in treating the terms belonging to the segment
(i\,ji) to (12,^2) it is found that d = = c . The analysis above fails to
give the solutions for these roots. In this case the transformation
= c, + 7 (24)
is made, after which the right member of (20) is a power series in 7 and n';
and for /x'= 0, 7" = is a solution. That is, the equation is of the same form
as P(a, n) =0, only in place of having k zero roots for ju= there are now
only p such roots. This number p is always less than k except when ii = k,
j l= 0, i 2
= 0,j 2=
\, and d = c 2= -*-*. But whatever the value of p
a new Newton's parallelogram for the 7 /-equation is to be constructed.
It will depend upon terms of higher degree in the original a^-equation
because the terms which gave rise to the p equal roots, c, ,. . .
, c, have been
concentrated, so to speak, by the transformations into the single one 7",
and the parallelogram depends upon the term in // alone of lowest degree.
By this step, or some succeeding one, the solutions will all become distinct
unless, indeed, the original P(a, m) =0 has two or more solutions for a which
are identical in n.
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10 PERIODIC ORBITS.
II. SOLUTIONS OF DIFFERENTIAL EQUATIONS AS POWERSERIES IN PARAMETERS.
8. The Types of Equations Treated. In the course of this work certain
types of differential equations will arise and they will be solved by processes
adapted to attaining their solutions in convenient forms. It will tend to
clearness and brevity of exposition of the actual dynamical problems to set
down in advance those methods of solving differential equations which will
be used, and to state, the conditions under which the results obtained bj r
them are valid. Consequently, this section will be devoted to these ques-tions without making here any applications to physical problems.
The equations which will be treated are characterized chiefly by being
analytic in the independent and dependent variables and in certain parameters
upon which they depend; and the solutions are considered only for those
values of the variables and parameters for which the equations are all regular.
In the case where the differential equations are linear, their coefficients are
either constants or periodic functions of the independent variable.
9. Formal Solution of Differential Equations of Type I.* The dif-
ferential equationsdx-jt - M/(*i ,
x n , m; t) 0'=l ,. . .
, n) (25)
will be said to be of the Type I when the right members have n as a factor
and when all the /, are analytic in x x ,. . .,x,n and t, and are regular at
the point x, = a, , m ,for all to^t^T. Then the /, (x x ,
. . ., x, n; t)
can be expanded as power series in (x t a t ) and /x which will converge if
|x,-a,| <r>0 and |/x| <p>0 for t ^.t^T.
Suppose x t a ( &tt =to, whatever be the value of p. That is, suppose
XtiQ^a,, (26)
in which the letter under the identity sign = indicates the parameter in
which the identity is defined.
Equations (25) can be solved formally as power series in n which havethe form
x,=
S x{
? M'.
(27)
where the x\nare functions of t. On substituting (27) in (25) and arranging
in powers of n, it is found that
+(28)
*8ee Moulton's Introduction to Celestial Mechanics, pp. 264-272.
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NON-LINEAR DIFFERENTIAL EQUATIONS. HIf these series are convergent the coefficients of corresponding powers of /*
in the right and left members are equal. On assuming for the moment thatthey are convergent, the identity relations become
-^- = (*!,...,), (29)
^=f t (x?,0;t) (j=i,...,n), (30)
dt dx kXt +
dn' ( 31 )
^f = y dJ 4+i y y ay t (1) > y ay (1) i ay rq2.
These sets of equations can be integrated sequentially. From (29) we get
x? = a (33)
where the a,'" are constants of integration. The right members of (30) arenow known functions of t, and their solutions can be written
*? =//. (*?, 0; t) dt + ? - F? (0 + a (
;>, (34)
where F*1
'^) is the primitive of/, fa 0; t). Then (31), (32), give in
order, similarly,
xf = F?\t) + a?, x? = Ff(t) + a?, (35)
In this manner the process can be continued as far as may be desired.
10. Determination of the Constants of Integration in Type I. At each
step n additive constants of integration are obtained, and they must bedetermined in terms of the initial values of the x { . From (26), (27), (33), (34),
(35) ,. . .
,it follows that
af +[*?(fc) + afltt'ta,.
Thereforea (
(
0) =a, ,
<* - - K(U) (i-1, ... oo). (36)
By these equations all of the constants of integration are uniquely determinedin terms of the constants of the differential equations and of the initial values
of the dependent variables.
11. Proof of the Convergence of the Solutions of Type I. Themethod of integrating differential equations as power series in parametershas been in use in more or less explicit form since almost the beginningsof celestial mechanics. For example, in the year 1772 Euler published his
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12 PERIODIC ORBITS.
second Lunar Theory, in which he used a process quite analogous to this;*
and the method of computing the absolute perturbations of the elements of
the planetary orbits is virtually that of developing the solutions as powerseries in the masses. But the actual determination of the conditions for the
validity of the process was not made until Cauchy published his celebratedmemoirs on differential equations in 1842. f The results of Cauchy were
extended by Poincare" in his prize memoir on the Problem of Three Bodies, t
and were proved again, following Cauchy 's Calcul des Limites, in LesMethodes
Nouvelles de la Mecanique Celeste, vol. I, pp. 58-63. The theorem will be
needed in this work in the form given by Poincare, viz. :
// thef t (Xi, . . . ,x n ,n; t) of equations (25) are analytic \\inxi,. . .,x
, m,
and t, and regular at x, = a t , n = 0, for all t ^.t^T, then p > can be taken so
small that the series (27) will converge for all to^t^T provided | m I< P-
To prove this theorem consider a comparison set of differential equations
I'^^.d/.r . , v , m; -i, . . ., n), (250
where the <p t are analytic in ffo, . . . ,y n , it, t, and regular at y,= |a 4 |=& 4 , /x=0
for all to ^ t^ T; and where, further, the coefficients of all powers of ^ 6,
and /x in the expansions of all the <p, are real, positive, and greater than the
moduli of the corresponding coefficients in the expansions of the /, for all the
values of t under consideration. Then positive constants M, p,r y ,. . .
,r
exist such that equations (25') can be written in the form
dy1 = Mjj.
"'"(>- 00-^) (i-^)'The conditions are evidently satisfied also by
dyt_ Mm_dt
"
(i_^ \ 1(2/i-fri) + (y,- h) + + (y- 6,) 1
'
(25")
where r is the smallest of r, ,. . .
,r n .
Suppose now the solutions of equations (25") are developed as powerseries in n of the form
fc2tfV. (27')Ju*0
There will be quadratures corresponding to (29), (30), .... Moreover,
by virtue of the hypotheses on the <p t ,
dt>
(i)
d^_dt
Tisaerand's Micanique Celeste, vol. Ill, chapter 6.
t See Cauchy's Collected Works, 1st series, vol. VII.
t Acta Mathcmatica, vol. XIII, pp. 5-266.
||The assumption that the /; are analytic in I is not necessary for the demonstration.
{Picard's Traiti d' Analyse, edition of 1905, vol. II, pp. 255-260.
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NON-LINEAR DIFFERENTIAL EQUATIONS. 13
for all <3 t $T. Therefore it follows that tf ^1 4? |
for alU ^ t T. Thenit is seen from the form of (31) that
dxfdt
>dt
for to <t^ T. From this it follows similarly that yf > \xf\. This processcan be continued indefinitely, giving by induction for the general term
I^5|* (
/| (37)
for t <it^T. Consequently, if the right members of (27') are convergentseries when
| p |
< p > for to^t^T, then likewise are the right membersof (27) convergent when
| p \< p > for the same range in t .
It is a simple matter to find the explicit expression for (27') by a direct
integration of (25"). Since the right members are the same for all i, we have
yi c l = y 2 -c 2 = = y n -c n ,
where c x ,. . ., c are constants of integration. By the initial conditions it
follows that c, Cj= b ( b s and y t b,= y t b } . Let this common value of
y t b t be y b. Then each equation of (25") becomes
dy MjxTt
"
(i_
a) (i_ n (y~ b )
)
'
( 25 "0
On integrating this equation and determining the constant of integration bythe condition that (y
b)=0at t = U, it is found that
0-$)The solution of this equation for (y b) is
.. r , r L 2n Afp ,,. .0--
J-^-)<-'- u> -
<38 >
Since (y b) = at t = U>, the negative sign must be taken before the radical.
It follows directly from equation (38) that whatever finite values n, M,r,p,and T-t may have, (y b) can be expanded as a convergent series in p for
to^t^T provided the condition
2nM\tx\
0-5)(T-kXl
is satisfied. This condition imposes the explicit limitation
I M I<
2nM(T-tp) ~1 (39)
r"*
p
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NON-LINEAR DIFFERENTIAL EQUATIONS. 15
14. Formal Solution of Differential Equations of Type II. The dif-
ferential equations
doc- = g t (x, ,. . .
,x n ;t) + nf i (x l ,
. . . ,x n , jtj t) (=
i, ...,), (40)
will be said to be of Type II if
(a) the g t {x u . . ., x;t) are independent of /x and not identically
zero;
(6) the g t (x lf . . ., x n ; t) and } t {xi, . . ., x n , ft', t) are analytic*in Xi, . . ., x, ix, and t;
(c) the g,(xi ,. . .
,x n ; t) and f t {x x ,
. . . x n , n ; t) are regular at
Xi = xf{t), m=0, for t^^t^T, where the xf are the solutions
of equations (40) for m = 0, and xf = at at i = t .
It follows from these conditions that the g t and f t can be expanded as powerseries in (x, zf) and /x, which converge if \x t xf [
< r ( > and| n \
< p > for
all t in the range t ^.t^T.
Equations (40) can be solved formally as power series in n having the
formx = 2z<V, (41)
1=0
where the as^ are functions of t, and where
x t (to)= a t . (42)
Upon substituting (41) in (40) and equating coefficients of correspond-
ing powers of n, it is found that
,(0)
dt
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16 PERIODIC ORBITS.
where c, ,. . .
, c. are the constants of integration which can be determined
in terms of the a t . Substituting these xT in (44) and integrating, we obtain
I? - 24? *,<*)+ J? (f), (48)
where the ^4"' are the constants of integration. After these solutions are
found, equations (45) can be integrated, and this process can be continued
to the k thstep, which gives
. xf = 2 ^;V <y (*) +/*?(<) (49)
The <p u (t) belonging to the complementary function are the same for each
step, but the Ff (t), which depend upon the right members of the differential
equation, are in general all different. The problem of finding the Ff\ the <p () ,
and the Ff depends upon the explicit form of the differential equations, andcan not be
givena
generaltreatment.
15. Determination of the Constants of Integration in Type II. Ateach step there are n constants of integration introduced which can be
determined in terms of the initial values of the x t . It follows from equa-tions (41), (42), (47), (48), . . .
,that
FT (c, , . . .,
c-; Q + 2 [
J 47 9u W+ F? &)V f a, . (50)
HenceF<
0)
( Cl ,. . .,c n ;t )=a t (i =l, . . . ,n),
|
(51)2 4? *,.)- -Ff (Q (* =i, . . . oc), I
Suppose the constants c ly . . .,c are uniquely determined in terms of a {
by the first set of equations of (51) . Then the F ( " become completely defined,
and from the second set of (51) the A 1-" are uniquely determined since the
determinant A =| <p (J (Q |
is the determinant of a fundamental set of solutions
at a regular point of the differential equations and is therefore not zero ( 18).
Then the Ff become entirely known and the Af are determined by a similar
set of l inear equations whose determinant is the same A. The whole processis unique and can be continued indefinitely.
16. Proof of the Convergence of the Solutions of Type II. Consider
the comparison set of differential equations
dv-g
= <Pt(y l) . . ., y; t) + M ft (y, ,
. . ., y n , M ; t), (40')
where the conditions corresponding to (a), (6), (c), and (42) of 14 are
satisfied, and where, in addition, the coefficients of the expansions of the <p {
and the ft as power series in (y t yT) and m are real, positive, and greaterthan the moduli of the corresponding coefficients in the expansions of the g,
and the /, for all t in the interval t ^ t g T. Suppose y t=
|a,
|
= 6, at t = t .
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NON-LINEAR DIFFERENTIAL EQUATIONS. 17
Equations (40') will be solved in the form
Vi = 2 tfV, (42')
where the yf are functions of t to be determined. It will be shown thatthe yf are real and positive, and that
|?>|a#lio*4i'*3f. (52)
The xf and yf are defined by
(53)
Since, by hypothesis, the integrands of the second set of equations are real,
positive, and greater thanthe
maximumvalues of the
moduliof the inte-
grands in the first set of equations in the interval fc <! < ^ T, it follows that
in the whole interval yf > \xf\.
The x'" and y are defined by
d4'= Slf ^ +/<> :*?.0;o;
^f 5z;
"?.Sfc^ + * <^" ' 2/
,
' 0; -
(54)
It follows from (48) and (51) that xf= at t = t .
Similarly yf-
at < = to . Equations (54) can be solved by Picard's approximation process.*
Let x2 and y t be the k'" approximations to x and yf. Then
*S -J?/* (*? ' * '^ (-i. ...;, ),
*s =.f,' [ 2 S6*B- + * ox* > <* ;
*)]*,
(55)
(56)
(57)
*7Vai d' Analyse, vol. II, edition of 1905, p. 340.
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18 PERIODIC ORBITS.
It follows from (55) and the relations between the /, and the ^ 4 that
Vu >\x ?l\ f r *o ^ t = T. Then, making use of the relations between the
coefficients of the expansions of the g, and the <p t ,it follows from (56) that
y ^ |xjy| for t ^ t ^ T; and from the method of forming the successive
approximations it is seen that, y{" > | x[l)
t | for t ^ t ^ T, for all k.
Now Picard has shown* that \im x"l= x for a sufficiently restricted
range of values of t. But equations (44) being linear, the range of values
for / is precisely that for which the differential equations are valid. f There-
fore we conclude that y >\ x[
l)
\for t ^ t ^ T. The corresponding relation
between yf and x can be proved in the same manner, and the process can
be continued step by step indefinitely. Consequently the inequalities (52)
are established. Hence, if the series (42') converge when| n \
< p', then the
series (42) also converge when| m |
< p' f or U ^ t ^ T.
Since
1<Pi\yi , > y , i),
it follows from the reference given in 11 that the conditions imposed upon(40') can be satisfied by equations of the formj
61"
O-SMg^*.]}
(58)
As a consequence of these equations (yt y)
)=(yj y< )
) + c J , where the
Cj are constants. Since y (= yf at t = t
,it follows that c,
= 0. Now let
z= 2 (</,-</;>) +M. (59)
Then, upon taking the sum of equations (58) with respect to i, we get
dz _ Mnz(l+z)
On integrating this equation and determining the constant of integration bythe condition that z -
n at t = t,
it is found that
log (1+m)2 Mn(t-Q
%(! + )- AM *
(61)
*Loc. cil. \Traitt d' Analyse, vol. Ill, edition of 1894, p. 91.
tSee Lei Milhodes Nowelles de la Micanique Ctlesle, vol. I, p. 60.
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NON-LINEAR DIFFERENTIAL EQUATIONS. 19
Solving this equation and determining the sign of the radical so that z = n att = t
, the expression for z becomes
,-- q+)' "^"(j+io 86(62)
2/x k-w(1+m)
2
where
# = Mn
4 MeK(r - w
(-5)
It follows from (62) that if \p\ < p, |m| < 1, and^j=y < 1, then z
can be expanded as a converging power series in m for t ^.t^T, and that in
this range for t the values of z are such that the expansion of the right mem-ber of (60) as a power series in z also converges. Consequently, the y, and x,
satisfying (40') and (40) respectively can also be expanded as convergingseries in /x for all t in the interval t ^. t ^ T.
The point to be noted in these results is that when the differential equa-tions are of the Types I or II, as defined above, and when the interval T t has
been chosen in advance and kept fixed, then the parameter /x, in which the
solutions are developed, can be taken so small in absolute value that the
series in which the solutions are expressed will all converge in the whole
interval U ^ / ^ T.
As in equations of Type I, there may be many parameters, Mi>Mi> >M*>
instead of a single one. The treatment can be reduced to the case of the single
one, just as in the preceding case.
The parameter can be generalized precisely as was explained in 13. It
is obvious that if there are many parameters they may all be generalized.
Since the generalization can be made in an infinite number of ways, a great
variety of possible expansions for these solutions is secured.
17. Case of Homogeneous Linear Equations. While the linear equa-
tions are included in those already treated, they deserve some special attention
for the reason that in their solutions the values of m are not restricted by so
many conditions. Consider the equations
^=id (1 (t)Xj (i=l n), (63)at i=i
where the are expansible as power series in n of the form
ao
6^ = S da n ,
t.o
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20 PERIODIC ORBITS.
which converge if \p\ < p for t*^. t ^ T. Suppose x t= a, at t = U- Then the
solutions can be developed as power series of the form
x< - 2 a? v\ (64)
precisely as in 14.
To find the realm of convergence in p of (64) , consider a comparison set of
differential equationsdy t
dt= 2 + (t) y> , (63')
>-i
where the ^n are expansible as power series in p of the form00
which converge provided | p \ < p for t ^ t^ T. Suppose also that \f/*) > |0*| for
to^.t^T. Develop the solutions of (63') in the form
</,= 2</<V. (64')Jr.O
It can be shown by the method used in proving the inequalities givenin (52) that if y t {Q -
\a t\
= b t ,then yf >\xf\ for U 5 t ^ T.
The conditions imposed on (63') are satisfied by the equations
dy, v Mi"
=
2 1 _M^'(63")
p
in which M is the maximum value of the \6 tj \
for to ^ t ^ 7\ It follows that
(y ( b,)=
(yj bj). Let the common value be (y b). Then (63") becomes
ifcfi. <- + -*- jvP P
The solution of this equation satisfying the initial conditions is
_M (64")
P
Hence y, and therefore ?/, and x t ,can be expanded as a power series in p. con-
verging for| m |
< P for to ^ < ^ 7\ That is, w/ien ^e differential equations are
linear the realm of convergence of the solutions in the parameter p is precisely the
same as that of the coefficients of the differential equations. Therefore, in those
simple cases in which the original equations are polynomials in p, the solu-
tions converge for all finite values of p.*
See Mtmoire sur let Groupes des Equations Liniaires, by Poincard, Acta Mathematica, vol. IV, p. 212.
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LINEAR DIFFERENTIAL EQUATIONS. 21
III. HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS WITHPERIODIC COEFFICIENTS.
18.The Determinant of a Fundamental Set of Solutions. Suppose
x' i= :ke tl {t)x l (-i, . . . ,*), (65)
where x\ is the derivative of x x with respect to t, is the set of linear homo-
geneous differential equations under consideration, and let
*n = *>(0> Z(2=
<Pa(t) ,. . . ,x tn
=<p in {t) ('-l, ...,), (66)
be a fundamental set of its solutions. The determinant of this set of solutions
maybe denoted
by A =\<Pij\. (67)
It will be shown that A can not vanish for any t for which the 6 n are all
regular. In the applications which follow, the 6 V are analytic in t and in
general regular for all finite values of t.
The result of differentiating A with respect to t is
A'=Sj^|,
where the index k denotes that in the k'" column the <p u are replaced by the
derivatives of the <p IL with respect to t. But it follows from (65) that
Hence A' can be written
A'=2
<p<t
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22 PERIODIC ORBITS.
where Alt is the minor of the element <p lk in A. But it is known from then
theory of determinants that 2 (-
1)*+<
?,* A* is zero when ;V, and equal to A
when j = i. Therefore*
A' = AS9,<-i
whence
*-*/*', (69)
where Ao is the value of A at t = 0. The initial conditions are taken so that
Ao^O. Thus A can vanish or become infinite only at the singularities of the
coefficients of the main diagonal of the differential equations.n
If 2 = the exponent vanishes and the determinant reduces to the
constant A . If the differential equations were originally of the second order,
having the form usually arising in celestial mechanicsn
x[ = 2 6 tJ Xj (i=
l, ...,),
they are equivalent to the system
x\ = y t , y' = 2 e tJ Xj (i=i, . . ., n),
which has the form of equations (65) . Since every 6 U of this set of equations
is zero, the determinant of any fundamental set of their solutions is a constant.
19. The Character of the Solutions of a Set of Linear HomogeneousDifferential Equations with Uniform Periodic Coefficients. Linear dif-
ferential equations with simply periodic coefficients were first treated byHillf in one of his celebrated memoirs on the lunar theory. About the
same time HermiteJ discovered the form of the solution of Lame's equation,
which has a doubly periodic coefficient. Starting from the results obtained byHermite, Picard|| showed that in general a fundamental set of solutions of a
linear differential equation of the n'horder with doubly periodic coefficients
of the first kind can be expressed by means of doubly periodic functions of the
second kind. In 1883, Floquet published a complete discussion of the char-
acter of the solutions of a linear differential equation of the n'"order which hassimply periodic coefficients. In this memoir Floquet gave not only the form
of the solutions in general, but he considered in detai l the forms of the solu-
tions when the fundamental equation has multiple roots. The forms of the
solutions being thus known, the efforts of later writers have been directed
Equation (69) was first developed by Jacobi in a somewhat different connection, Collected Works,vol. IV, p. 403.
t77ie Collected Works of G. W. Hill, vol. I, p. 243; Acta Mathematica, vol. VIII, pp. 136; also publishedat Cambridge, Mass. in 1877.
tComptes Rendus, 1877 et seq.
\\Comptes Rendus, 1879-80; Journal fur McUhematik, vol. 90 (1881).
lAnnales de I'Gcole Normale Superieure, 1883-1884.
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LINEAR DIFFERENTIAL EQUATIONS. 23
toward the discovery of practical methods for their actual construction,principally when the differential equation has the form
d 2x
-jjf+ (oo + a,cosJ + a 2 cos2 2 + -)x = 0. (70)
Different methods for constructing solutions of this equation have beenproposed by Lindemann,* Lindstedt,f Bruns,J Callandreau,|| Stieltjes, andHarzer.*|f
In what follows there will arise only equations with simply periodiccoefficients having the form
*,'= 2e tj Xj (i=l, . . . ,n), (71)
where the d tj are periodic functions of t with the period 2tr. It will beassumed that the d tJ are uniform analytic functions of t and are regularfor0^^27r. Let
Xa = <Pu (Oi z<2 = <Pt2 (t), . . . ,X in = <p in (t) (t = l, . . ., n),
be a fundamental set of solutions which satisfy the initial conditions
<p (0) = if ir*j and <p it (0) = 1. It is clear that n solutions can beconstructed with these n sets of initial conditions, and since their deter-
minant is unity at t = 0, they constitute a fundamental set of solutions.
Now make the transformation
*,-"*, (72)
where a is an undetermined constant. The differential equations become
y: + ay t = 2d 4J yj, (73)
any solution of which can be written in the form
y t -e-"* ZA,*, d'=i, . . . ,n), (74)j=i
where the A s are suitably chosen constants.
The question arises whether it is possible to determine a and the As in
such a manner that the y t shall be periodic in t with the period 2 t. From the
form of equations (73) it is clear that sufficient conditions for the periodicity
of the y t are that y ( (2w)=y i (0) (i=l, . . ., n). On imposing these condi-
tionsupon
(74), there results
- S A, [<p (J (2tt)- e
2^,, (0)] (i= 1, . . .
, n). (75)j=i
Either all the Aj are zero or the determinant must vanish. The former case
is trivial and we therefore impose the condition
\s(2*) -****, (0)|-0,
*Maihematische Annalen, vol. XXII, (1883), p. 117-123.
fAstronomische Nachrichlen, No. 2503 (1883). M4nwiresdel'AcadimiedeSt.Pilersbourg,yo\.XXI,'^oA.lAslronomische Nachrichten, No. 2533, 2553 (1884).
\\Astro7iomische Nachrichlen, No. 2547 (1884).
LAslronomische Nachrichlen, No. 2602 (1884).
^Aslronomische Nachrichten, Nos. 2850 and 2851 (1888).
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24 PERIODIC ORBITS.
where ^ (J (0) = if it*j and ^,,(0) = 1. Putting e 2ar =s and denoting tp u (2ir)
simply by <p ih this determinant becomes
<P2\ <P22~ S, <Pi3 )
<P3l } <P32 ) ^3.1 & f
<Pn3 <Ptm S
= 0. (76)
This is an equation of the n'" degree in s, the constant term of which can
not vanish since it is the determinant of a fundamental set of solutions.
It is known as the fundamental equation for the period 2w. Its roots can be
neither zero nor infinite, because the coefficient of s" is unity and the term
independent of s is A.
20. Solutions when the Roots of the Fundamental Equation are all
Distinct. Suppose the roots s s 2 ,. . . , s n of (76) are all distinct. Then
at least one of the first minors of (76) is distinct from zero when s is putequal tos t ,
and therefore the ratios of the A, are uniquely determined by (75).
For each s t a set of y t is determined by (74) involving one arbitrary constant.
Since this solution depends upon s 4 it will be designated by y ik ,and the
corresponding A t by A jk .
Since s e 2ar,the a is uniquely determined in terms of s except for the
additive constant v
Vl,where v is an integer. In every case the principal
value of a can be taken, for its other values simply remove periodic factors
from the y, . Consequently, for the n values of s there are n values a, andfrom equation (74) there are n solutions, one for each k from 1 to n,
i/ rt= e-
a*' i A jk <p u (t) (t-1, . . . i n), (77)
where the ratios of the A Jk are determined by (75). From (72), n solutions
of equations (71) are thus found, one for each k,
x, k - ea"'y ik (-l, ,,.,), (78)
where the y ik are periodic in t with the period 2ir.
These solutions (78) form a fundamental set, for, if they did not, there
would exist linear relations among the x lk of the form
2 Ck x lk (0 m1 = 1
(t'=
l, ...,), (79)
where not all the Ck = Q. Increasing t by 2t, it follows from the conditions
imposed upon the x, and y t that
2 Ck x ik (t + 2tt) = S Ck s k x lk (t) m 0,*-i i-i
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LINEAR DIFFERENTIAL EQUATIONS.
and similarly that
S Ct x ik {t + 4tt) = S Ck si x tk (t) m 0,
25
I
2 Ck x lk [t + 2(n-l)ir] = 2 C.s^Xn (t) = 0.
(80)
Since the Ct are not all zero it follows that the determinant of these equationsmust vanish; that is,
n>(o
1 ,1
Si , S22 2
Si , S2
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26 PERIODIC ORBITS.
zero. If these solutions are taken, the identities (82) can not be satisfied and
consequently equations (79) can not be satisfied. Therefore (78) constitute
a fundamental set of solutions.
21. Solutions when the Fundamental Equation has Multiple Roots.
Consider first the case where the fundamental equation has only two roots
equal. The notation can be chosen so that s 2 =Si. There are two cases
according as all, or not all, of the first minors of (76) vanish when s = Si.
Suppose first that all the first minors vanish for this value of s. Since s =Si is
only a double root, not all of the second minors can vanish . Hence two of the A t
can be taken arbitrarily and (75) can be solved for the remaining (n 2) of
them. Then equations (74) and (72) give the corresponding z t . Since the <p u
are linearly distinct two linearly distinct values of the y t can be obtained bytaking first one of the arbitrary A } equal to zero, and then the other equalto zero. Therefore in this case there are two linearly distinct solutions of
the formx n = e
a''y n , x t2 =e ait
y i2 (t-1, . . . ,n),
where the y n and y i2 are periodic in t with the period 2ir.
If, however, not all the first minors of (76) vanish for s = s 1} there is but a
single solution of this form belonging to the root s x of the fundamental equa-tion. Let x n = e
aily iX be this solution; it will be shown that the other one
belonging to this root has the form
' t
(y t 2+ty tl ) (i=
l, ...,n), (83)f 2
where the y i2 are periodic in t with the period 2x.
Before proceeding to the demonstration a lemma pertaining to a certain
type of transformation of a fundamental set of solutions will be proved.
Suppose the <p u constitute a fundamental set of solutions. Then define newfunctions \f/ ik by the relations
if/ (k= 2 A Jk tp ( j
i-k(i,k =
l, . . . ,n). (84)
The \f/ a also constitute a fundamental set of solutions provided no A kk = 0, for
the determinant of the \f/ ik is
n
| ft, |-
\-LtA lk <p lt \
= \A*\ |^ ( ,|,
where \A lk \and \<p tl \
are the determinants of the A Jk and <p ti respectively.The determinant
| <p (J \
is distinct from zero and
A Jk \
=
An > A 2 i
, A 22
,
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LINEAR DIFFERENTIAL EQUATIONS. 27
which is distinct from zero unless some AM is zero. A special case, which will
be used first, is that where all the elements except those in the first line andin the main diagonal are zero.
Now return to thepoint
under discussion.By hypothesis not
allthe
first minors of (76) vanish for s = s x . Let the notation be chosen so that oneof those which is distinct from zero is formed from the elements of the last
n 1 columns. Then A x must be distinct from zero in order not to get the
trivial case in which all the ^are zero. Now in place of <p u (i,j=l ,. . . ,ri)
as a fundamental set of solutions we can take, as a consequence of the lemma,
ea,<
Vn, <P<j (i=
l, . ,n;j =2, . . . ,n). (86)
Any solution can be expressed in the form
i. = fi 1 e 0l,j/ ( , + 2 5^ i , (t
=i, . . . ,n).
Now make the transformation x i2 = e a,t(y (2 + ty n ); whence
V = - tyn + B lVil e(a '- a '+ e" w 2 A j* .
Since by hypothesis the x a = eait
y n satisfy (71), it is found by substitution
that, if (83) are to constitute a solution, the y i2 must satisfy the equations
n
y' 2 + a^a - 2 dtjyjz- y (t-l, . . ., n). (87)
Since t enters only in the tJ and the y n , which are periodic with the period2ir, sufficient conditions that the y t2 shall be periodic with the period 2ir are
Vn (2tt)- y a (0) = - 2tt y tl (0) + 2B, [<p t , (2tt) e~^- Vt , (0)]
= 0.1=2
On transforming from the exponential to s, these equations give
-2tt t, y n (0) + 2 Bj [<p tj (2x)-
s, it,]= 0, (88)
where 6 U = if jVi and 5 1( =1.
The condition that equations (88) shall be consistent is
A = |y(0),p (2*) -*!, . . ., ^ ( (27r)-s 1 5 lB
|
= 0,
where A is the determinant formed from their coefficients. This equation
is satisfied, for if the fundamental equation is formed as usual from the
fundamental set (86), it is found that D=(s-s 1 )D 1= 0. Since D is inde-
pendent of the fundamental* set from which it is derived, and since D=
has the double root 8 - i, it follows that A (s,)= 0. Therefore (88) can be
solved uniquely for the B 2 ,. . .
,B n . These equations determine the y i2 ,
and through them the x i2 in the form given in (83).
Fuchs, Journal fur Malhemalik, vol. LXVI (1866), p. 133.
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28 PERIODIC ORBITS.
Now suppose s=Si is a triple root of the fundamental equation, but that
it is not a quadruple root. If all its minors of the first and second order
vanish for s = Si, three of the A s can be taken arbitrarily and three linearly
distinct solutions of the form
x u = ea ' f
y tl ,x l2 = e
a ''
y i2 , x i3 - e a,ty t3
can be determined, where the y n , y t2 , and y t3 are periodic in t with the period 2ir.
If all of the minors of the first order of the fundamental determinant
vanish, but not all of "those of the second order, then two of the Aj can be
taken arbitrarily, and two linearly distinct solutions of the form
Xn = c'
i/n , x {2= e 2/*2
will be obtained, where the y n and y t2 are again periodic.
In order to obtain a third solution associated with the root s, take as
a new fundamental set of solutions
l^'fn, ea*'y i2 , <p ti (i
=l, . . .
, n; j = 3, . . ., n),
so that any solution can be written in the form
x t= B, e
a >'
y tl + B 2 ea ''
y l2 + B, <p tJ (-l, . . . , n).
Now make the transformation
x, 3 = eait
[y (3 + t(y il + y t2 )];
whence
y t3= -^u-^< 2 + JB 1 e
(a - a ' ),
^/ )1 ^- ^ eca '- a ),
2/e+ea , S B 1<Pil .
tml
In a manner similar to that in the case just treated the periodicity conditions
on the y i3 lead to the equations
0=-2*- Sl t/ (1 (0) -27rs 1 t/ (2 (0)+2^[ v?u (27r) -Mil-J-3
As in the preceding case, it is found that for s=s x the B3 ,. . ., Bn are
uniquely determined and that the x, 3 have the form
x l3= e
a''[y i3 + t(y n + ?/)], (89)
where the y tl , y a , and y i3 are periodic in t with the period 2ir.
Suppose now that not all of the first minors of the fundamental determi-
nant vanish fors=s,. Then there will be one solution x,i = eai'y i i and another
x l2= e
a,<(y a + t y (l ). It will be shown that in this case the third solution
belonging to s, is of the form
x i3= e
a > l
[y i3 + ty a + \e yil \. (90)
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LINEAR DIFFERENTIAL EQUATIONS. 29
Take as a new fundamental set of solutions
e a,,2/.. , e
a >'
(y l2 + ty n ), *, (,>i t. . .
, n; j=3 ,. . .
, n). (90')
After defining the x, by
x t= B 1 e
a ' t
y ll + B 2 ea>'(y t2 +ty n ) + 2 B j<PlJ (;=i, . . .
, ),J = 3
make the transformation (90). Then the expressions for y i3 are
yi 3 =-ty t2 -%t2
y il + B l e(a >- a > )l
y n +B 2 e1 '- a
*(y i2 + ty ti )+e-a >' 2 Bj<p tJ .
If the x t3 constitute a solution of the original equations (71), the y i3 mustsatisfy the equations
n
y' 3 + a, y l3= 2 6 ti
yJ3-
yt2 ,
since the y n satisfy (73) and the y t2 satisfy (87). Hence sufficient conditionsthat the y l3 shall be periodic are that
y t3 (2ir)- y i3 (0) =
(t-1, . . ., n).
These conditions lead to the equations
0= -2TS l y l2 (0)- 2 7r2s l t/, l (0)+2 1 rfi 2 s l2 /(0)+ 2 5,k(2i) -* t M.
J=3
The terms y n (0) in the second column of the determinant of the coefficients
of these equations evidently may be suppressed. Let this determinant bedenoted by D2 . In order that these equations shall be consistent it is
necessary that D2= 0. This condition is satisfied; for if the fundamental
equation be formed from (90'), it is found that
D=(s-s l )i D2 .
But by hypothesis D admits (s=
s,) as a triple root. Therefore D2 (i) =and the equations are consistent. Since s = Si is a simple root of D2 , not
all of its first minors are zero. Therefore the B 3 ,. . .
, Bn are uniquely
determined, andthe x
i3have the form
(90).Suppose s=Si is a root of multiplicity I. There is then a group of solu-
tions, I in number, attached to this root. In general this group of solutions
will have the following form
x n = ea''y n 0'
=l, ,n),
*- ea''[y t2 + ty tl ),
x i2= e
a ' l
[y t3 + ty i2 + %?y u ], (91)
x= e^ly^ + ty,^ + +a^
tyi}.
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30 PERIODIC ORBITS.
If all the minors of the fundamental equation D = up to the order
k-l(k^.l), but not all of order k, vanish for s=s,, then there are A; solutions
of the first form, t. e. of the form
x ti= e a,
'y ti , x it =ea
''y t2 ,. . .
, t=e ai
'yik .
If now the fundamental set
ea ' (
y (i ,. . . , e a,<
y u , <p t , t+l ,. . . , ? ,
be taken and the equation in s formed, it is found that
D = (-,)*#- 0.
Since the roots of the fundamental equation are not changed by adoptingthe new fundamental set of solutions, Dt = has s =s l as a root of multiplicity
lk. Suppose all the minors of Dk of order g 1, but not all of order g,
vanish; then there are g solutions of the second form, viz.,
Zf.*+i= e a,i
[y t + +1 + t 2 y i} ~\ , ..., %+(= e
a ''k t+ ,+<S|/J.
If k+g<l, by a similar change of the fundamental set of solutions, it will
be found that
D= (s- Sl )t+ 'A + = 0.
Now Dk+e= admits s = Si as a root of multiplicity I (k+g) and there is
a certain number of solutions of the third type of (91), depending uponthe order of the minors of Dt+<l which do not all vanish for s=s t . Continuing,there is obtained finally I linearly independent solutions associated with s, ,
and in a similar way the solutions associated with the other roots of the
fundamental equation can be found.
22. The Characteristic Equation when the Coefficients of the Differ-
ential Equations are Expansible as Power Series in a Parameter /u. In
the preceding discussions no explicit reference was made to the parameters
upon which the tJ may depend. It will be assumed now that the 8 tJ are
expansible as power series in n whose coefficients separately are periodic in
t, and that the series converge for all finite values of t\i \n\ <p. It will be
assumed further that 6 u = a iJ ,where the a u are constants, for ^ = 0. Under
these conditions, which are often realized in practice and particularly in the
applications which follow, the discussion of the character of the solutions
can be made so as to lead to a convenient method for their practical con-
struction. The discussion will depend upon the principles of 19 and the
integration of the equations as power series in m-
Consider now the equations
n "
*,' = S exj = 2 [au + S flV]x, d =l, . . . , n), (92)
J ~ 1j=i
*"'
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LINEAR DIFFERENTIAL EQUATIONS. 31
where the a are constants, the ( * }
are periodic in t with the period 27r, and
S efjfj." converge for all real, finite values of t if \n\ <p. For n = equations
(92) admit xj0) = c,e
a "' as a solution, where the c ( are constants whose ratios
depend upon the coefficients of the differential equations, and a (0)is one of
the roots of the characteristic equation
Ona 2 i
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32 PERIODIC ORBITS.
Since the A) must not all be zero, the determinant of their coefficients must
vanish, whence
A =| [c^l-e^-^ + e-^l^ (2.) /] |
= 0. (98)
This equation has an infinite number of solutions, for if a = a, is a solution,
then also is a = a, + ** V 1, v any integer. The fundamental equation
corresponding to (76) is obtained by the transformation eiar = s. If the values
of s satisfying the fundamental equation are distinct, the correspondingvalues of a are distinct but not the converse, for if two values of a differ by an
imaginary integer the corresponding values of s are equal. Only those
values of a will be taken which reduce to the a (
,
0)for m = 0, the aj being
uniquely determined by (93).
Suppose now that two of the roots of the characteristic equation, say
o? and a(
i\ are equal. Then the solutions of (92) have in general the form
x t= Ajcn e
<U + 2 ^l/] + AS(c a +tc n ) e^' + 2 a/|
+ SA J [*V*l
+5<y]J=3
(99)
The exception to this general form is that tc n may be absent from the
second term for i 1,
. . ., n, and this possibility must be considered at
those places where it makes differences in the discussion.
After making the transformation Xt=eat
y,, the solutions for the y t are
yi = Ajc n e {a '
y -a)t + e-*< 2 x^^]+AS(c i2 + t cJ^-^'+r" 2 xg M*l
+ ^A i [c () e^)
-^+e-lx^^].(100)
The conditions for the periodicity of the y,, viz., y t (2ir) y,(0) =0, lead to
the determinant
A =|[c
(1 (l-e<a
-<>) + e-2 < ,
'2^H2 7 r)Mt
])
[c12 (l-e
2(a - a<,
',
^) + 2^c (I + 6-2 < )
'2 4>(27r)Mt
],
. .=
0,
\ (101)
where the elements which are not written are of the same form as those in (98) .
If, for m = 0, the characteristic equation has a root of higher order of
multiplicity, the fundamental equation is formed in a similar manner.
23. Solutions when of, of, . . .,a
( 0) are Distinct and their Differ-
ences are not Congruent to Zero mod. V^l. The part of (98) independentof n is
Ao ='u(l
_j(-r>
and the determinant\c (J \
isunity.
^=1
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LINEAR DIFFERENTIAL EQUATIONS. 33
If, in any particular case, (98) were an identity in p its ft solutions wouldbe simply = 0* In case it is not an identity, let
/ , "?+& (103)
and A becomes
A = Ao+m ftiw p) = (1- e^)
]J (l- e (
a "- a
)J (m)
+p/^(ft,p) = (j*k),
where F*(ft, p) is a ser ies in ft and p, converging for |ft| finite and |p| <p.Since, by hypothesis, no (af a
(
;
0)
) is an imaginary integer, the expan-sion of (104) as a power series in ft and p contains a term in ft of the first
degree and no term independent of both ft and p. Therefore (see 1 and2) it can be solved uniquely for ft as a power series in p of the form
ft = pP*(p). (105)
Substituting this value of ft in (103) and the resulting a in (97), n homo-geneous linear relations among A l ,
. . ., A n are obtained whose determi-
nant vanishes, but for p sufficiently small not all of its first minors vanish,since the roots of the determinant set equal to zero were all distinct for p = .
Therefore the ratios of the A t are uniquely determined as power series in p,
converging for |p| sufficiently small. When the ratios of the Aj have been
determined, the y* are determined as power series in p, and the coefficient of
each power of p separately is periodic in t. A solution is found similarlyfor each
af.The origin of the singularities which determine the radii of convergenceof the final solution series is known. If p is the smallest true radius of
convergence of the original solutions (95) as t varies from to 2r, then, in
general, the final solutions will converge only if| p |
< p. Consider the funda-
mental equation, A(s, p)=0, which is a polynomial in s of degree n and a
power series in p converging if \n \ <p. From the algebraic character of A it
follows that the only singularities introduced by solving for s in terms of pare branch-points, which are determined by the simultaneous equations
A(s,p) = 0,
^= 0. (106)
The variable $ can be eliminated from these equations by rational processes
and the eliminant will converge if |p| <p. Its zeros are branch-points for s
as defined by A(s, p) = 0.
The zeros of the eliminant which lie within| p |
= p can be found in any
particular numerical case by Picard's extension of Kronecker's method* pro-
vided the zeros are all simple. If there is a zero at p = p , then the solutions
for s as a power series in p converge only if |p| < |p |.If there is no po the
limit remains p.
Picard's Traite d' Analyse, vol. II, chap. 7.
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34 PERIODIC ORBITS.
Now consider ft as a function of n through its relation with s, viz.,
St = e'( ai
'>+ "*) = e
2ai" r
. e^ T
. If Sl has a branch-point for M= m, then ft also has
a branch-point at the same place since d/3/ds = 1/2 tts is distinct from zero
for all finite values of s. Therefore the series for ft converges only if \n\ < !mo| .
The root * is 8, = s?+ 2 sl
tV and a?' + ft = (l/27r)log[s<0) + 2 8? //].
If for any mi such that| m, I
< P we have|
s|
=|2 4 m! I ,
then ft has an
essential singularity at m = mi, and the series for it converges only if \n\ < |ah| .
The zeros determining these singularities can also be found in a special
numerical case by Picard's method. When \n\ satisfies the inequalities
imposed by these various possible singularities, the solutions are convergent
for all finite values of t.
24. Solutions when no twoaf
are equal but when af af is Congruentto Zero mod. V l. Suppose two roots of the characteristic equation for
H = 0, say a'"' and af, differ only by an imaginary integer, and that there
is no other such congruence among them. Then the equation corresponding
to (104) becomes
(1- e^Y II (l
- e'^*-8^
') +ft M^Cft , m) +m2Wi , m) = 0. (107)
The term of lowest degree in ft alone is 4tt2
P] . The term of lowest degree
in ix alone is at least of the second degree, and will in general be precisely
of the seconddegree.
This follows from the fact thatevery
term inevery
element of the first two columns of the determinant (98) contains in this
special case either ft or /x as a factor. In order to get the terms in it alone,
those involving ft are suppressed, and then the conclusion follows from the
fact that every term in the expansion of the determinant contains one
term from each of the first two columns. In a similar way if p of the a( }
are congruent to zero mod. V l, then the term of lowest degree in ft alone
is exactly of degree p, and in /x alone it is at least of degree p.
Consider the expansion of (107), which may be written in the form
tf + 7n&M + Yo2M2 + =0,
where y n , 702, are constants. The quadratic terms can be factored, giving
(ft 6iju)(ft &2M) + terms of higher degree =0.
If &! and 6 2 are distinct, as will in general be the case, the two solutions
of (107) are then (see 6),
fti = 6.M + M2
Pi(M), ft 2= 6 2M+ M
2
i2(M), (108)
where P, and P 2 are power series in fx. If bi = b 2 the solutions are powerseries in Vm or n, depending upon the terms of higher degree. If 7 02 is
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LINEAR DIFFERENTIAL EQUATIONS. 35
zero at least one of the solutions starts with a term of degree higher thanthe first in /* If the first term in m alone is ^ and if y u is zero, thenthe solution has the form
/3ii = c 1 m' a +-
, /3 I2 =- ClMv, + ....
But in general the solutions are of the type (108), and no other special caseswill be considered in detail; they can all be treated by the principles of
6 and 7. Thus, starting from the root of of the characteristic equation,two solutions are obtained. But it follows from the form of equations(98) and (104) that if the start were made from the root of ,
the same valuesfor ft would be found.
The other ft (k =3, . . .
, n) are found as in the preceding case, thesolutions from them are formed in the same way, and their realm of
convergence is limited by possible singularities of the same types.
25. Solutions when, for n = 0, the Characteristic Equation has a Mul-tiple Root. Suppose only two roots are equal, say of = of, and thatthere are none of the congruences treated above. Then, for m = 0, equation(101) becomes
a fi Aa ~ a T)'\ /1 2(a -
i
W)' r \ lo /i (a-tt
(
;V\Ao=Ca(l-e
v
>>), c I2 (l-ev ' '
) +2ttc,. . .,c tJ {\-eK '
),-
which easily reduces to
A.- (l -/-*>')' ]J '(i-/ 5K (109)
since the determinant \c tJ \is unity.
After the substitution a = af + ft is made in (109) and the result
expanded, it is found that the term of lowest degree in ft alone is 4 t 2
0[ .
If the determinant A for of = af is of the special form (98), the term of
lowest degree in fj. alone is at least of the second; but if in this case A is of
the general form (101), the term of lowest degree in n alone is in general of
the first. Except in the special cases the solutions of (101) in the vicinity
of the double root af are of the form
fc,-yjy)', ft^-^-M 1),
where P is a power series in ^, containing a term independent of m-
When the coefficient of fi is zero in the expansion of (101), the first
term in /x alone is of at least the second degree, and the problem is of the
type treated in the preceding article.
If, for n =0, p roots of the characteristic equation are equal, then for
these roots the expansion of (101) starts with $[ as the term of lowest degree
ft alone, and except in special cases the lowest degree of terms in m alone is
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36 PERIODIC ORBITS.
the first. Consequently in general for a?-F- = <C the solutions
of (101) are
/3 u = e' M'iVM') (j = l,...,p),
where e is any p lh root of unity.
Another case is that in which A = has a double root identically in m,
the conditions for which are
A(a, M)=0, |^(a,M)=0
for all| ju| sufficiently small. Suppose a 2 = c^ . If, for a = ai ,
all the first minors
of A are zero, the solutions of (97) for the ratios of the A, will carry two arbi-
traries, and the two solutions associated with a! will be obtained. If not all
the first minors of A vanish for a = o, , then in this way only one solution is
found. But it is known from thegeneral theory
of 21 that the second
solution has the form
x a = ea,
'(y a +ty tt ) (i= l,...,n).
On substituting these expressions in the differential equations and makinguse of the fact that e
aity ti are a solution, it is found that
V + *iV- 21 \a a + 2 C M*I Vi2= -Ua (*-li ...,*)
l- -i J
If the left members of these equations are set equal to zero, they become
precisely of the form of the equations satisfied by the y n . Consequentlyyt2=yn plus such particular integrals that the differential equations shall be
satisfied when the right members are retained. The method of finding the
particular integrals will be taken up in 29-31.
26. Construction of the Solutions when, for ju=
0, the Roots of the Char-
acteristic Equation are Distinct and their Differences are not Congruentto Zero mod. V 1.' The knowledge of the properties of the solutions and
their expansibility as power series in fx leads to convenient methods for
constructing them. Under the conditions that for n = the roots of the
characteristic equation are distinct and that the difference of no two of themis congruent to zero mod. V l, it has been shown that there are exactly n
distinct values of a expansible as converging power series in n, such that
x a = e akty ik {i
=l, . . . ,n), where the y* are purely periodic, constitute a
fundamental set of solutions. It will be assumed that for ft= no two o'"'
are equal and that the difference of no two of them is congruent to zero
mod. V 1, and it will be shown that the coefficients of the expansions of
the a* and y u are determined, except for a constant factor, by the conditions
that the differential equations shall be satisfied and that the y tk shall be
periodic in t with the period 2w.
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LINEAR DIFFERENTIAL EQUATIONS. 37
00
Suppose the value of y ik is y ik- ly^n
1
, where the series converge for
all |m| sufficiently small. It follows from the periodicity condition that
Zf* (2t) m' f 1 ^(0) M4
(*, *-l, . . . , ).
Since this relation is an identity, it follows that
i#(2*)-tff(0) (i,k =l, ... ,n).
Therefore each y ik separately is periodic with the period 2t.
Now the original differential equations (92) after making the transfor-
mation x i= e
aty i become
n
tf+fcr 2) [oo+itfv'V P-i,n).
(no)
For n the roots of the characteristic equation belonging to these equationsare of, of ,
. . ., of. Consider any one of them, as af . It has been shown
that for n ^ , but sufficiently small in absolute value, a t and the i/, t are
expansible in converging series of the form
a a * + <** M+ ; 2j a t n ,>>=o
y=yS? + y2V+ = s yV.(in)
On substituting (111) in (110), arranging as power series in m, and equatingcoefficients of corresponding powers in n ,
there results a series of sets of
equations from which a t and the y iK can be determined so that the y lk shall
be periodic with the period 2x. The determination is unique except for
an arbitrary constant factor of the y a - For simplicity of notation this
constant factor will be determined so that y(
k (0) = c u , provided c lk 9^Q,
and it can be restored in the final results by multiplying this particular
solution by an arbitrary constant.
Terms independent of n. The terms of the solution independent of /z are
definedby
the differentialequations
(j)'+afi#- 5/wff-o 0-i, ., n), (ii2)
the general solution of which is
n / (o) (o)\
jC- ltficjF'^ (-i, ...,n), (113)
where the iffi are the constants of integration. Since the tjf are periodic with
the period 2 r, and since, by hypothesis, af- of # mod. V^T, except when
j - k, every ij= if j*k. The initial value of yf is c
;therefore C = 1.
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38 PERIODIC ORBITS.
If c, were zero the initial condition would be imposed upon another y,not all of which can be zero at t = 0. The solution satisfying the conditions
laid down is then
|-A. (114)
Coefficients of n. The differential equations for the terms in the first
power of m are
The general solution for the terms homogeneous in y is
y=mic u e^-a' }t
, (116)
where the r?"* are the as yet undetermined constants of integration, and the c tJ
are the same as in(113).
Using the method of variation of parameters, we find
2 m'c ti eW- a^ l = -a?y+ 2 <%1% = (g(t), (117)
where the g\"(t) are periodic in t with the period 2ir. The determinant of
the coefficients of the (VJ*)' 1S
A=M' fci -e* Vj *'
which can not vanish for any finite value of i. Therefore the solutions of
equations (117) for (ij
(
'*)' are
C,2>'-.-W->%S, (118)
where the A," are periodic functions of t with the period 27r.
The solutions of (118) for j^k have the form
#--<'*-*">'P2+*S tfp4), ( 119 )
where the P (0 are periodic with the period 2t, and the B% are arbitrary
constants. For j = k equation (118) becomes
0tf)'-A8--tf + , (120)
where C is A" t
}
after the terms a^z/l" have been omitted from the k lh
column. It is a periodic function of t with the period 2x, and has in general
a term independent of t. It can be written in the form
f% = d? + Q?(t),
where d" is a constant and Q" (t) is a periodic function whose mean value
is zero. Then
()' = n -0 + W).
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LINEAR DIFFERENTIAL EQUATIONS. 39
It is clear that if ijg is to be periodic the right member of this equationmust not contain any constant terms. Therefore
<4"= C, (121)
and
nS-PSJ + flg, (122)
where Fg is periodic with the period 2tt and B is the constant of integration.
Upon substituting (119) and (122) in (116), the general solution withthe value of a* determined by (121) becomes
/<o) (o)\,
n
i- 2^ Clj e^ -*>>' + JCCvF*. (123)
In order that the y tk shall be periodic with the period 2w, all the Bjk must vanish
except Bu . From the condition that y u (0) =c n for all \n\ sufficiently small,it follows that y(0) = c lk and y[ (0) =
fj=
1, . . . oo ). From the con-dition that 2/i" = at t = it follows that
B2--7-2cP?(0).Therefore the solution satisfying all the conditions is
i/2 = 2 [* p * W- c
f c *2 ()1 ( 124 )
It remains to be shown that the integration of the coefficients of the
higher powers of n can be effected in a similar manner. Let it be supposedthat
ai", af ,. . .
, a*"" and the \g, y ,... , y%~l)
have been uniquelydetermined so that the yf (t) are periodic with the period 2t and that yf =
0,
1=1, . . . ,m 1. It will be shown that the y can be determined so as
to satisfy the same conditions.
From equations (96) it is found that
m 1
+ ^/L_
o*2/* + ^V tJ y jkJ.
p=i
(125)
Omittingthe terms included under the
signof summation with
respectto
p,these equations are identical in form with equations (115) except for the
superscripts (1) and (m). Obviously the integrations proceed with the index
(m) just as with the index (1), and the character of the process is no wise
altered by the inclusion of the terms under the sign of summation with
respect to p, for they are all periodic with the period 2x and do not changethe essential character of the g^\t) . Therefore a k
m) and the y can be uniquelydetermined so that the y shall satisfy the differential equations and be
periodic in t with the period 2w, and so that at the same time y (0) = 0.
The induction is complete and the process can be indefinitely continued. Thesolutions associated with the other a'j" are found in the same way.
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40 PERIODIC ORBITS.
27. Construction of the Solutions when the Difference of two Roots
of the Characteristic Equation is Congruent to Zero mod. V l. It will
be supposed now that of af is congruent to zero mod. V 1 and that this
relation is not satisfied by any other pair of af. The solutions associated
with ?*...., a( 0)
are computed by the method of 26 without modifica-
tion. It has been shown that in general a x ,a 2 and the y n , y i2 can be developed
as converging series in integral powers of ju- It will be assumed further that
the case under consideration is not an exceptional one.
The general solutions of (96) for the terms independent of ju is in this
case
vm _ v <o)c Mj ~ a
, )l
yn ^ Vji L tjV
Imposing the conditions that y shall be periodic with the period 2w and
that yfl (0) = c,these equations become, since af ai
0)is an imaginary integer,
y^=fl-r, 2 l
c
^)c n + V2 lcJ^-a ^ t
(t-1, ...,), (126)
where t} 2 l is so far arbitrary.
Coefficients of /* It follows from (96) that the coefficients of n must
satisfy the equations
The general solution of these equations when their right members are zero is
|C-Z/<*/4
' ' }(*= !,..., n). (128)
On considering the coefficients j as functions of t and imposing the
conditions that (127) shall be satisfied, it is found that
^vW c,/ -ttj y n -+-
z^(fiJ y n (t-1, . . .
, n).
On substituting the values of the pg1 from (126) and solving, there result
(*)'- -\
l)
(i- <) +<ag +# ),
(*)'- -rf* + *Ag) +D 2"l (t), (129)
where the A'^ and DJJ' are periodic functions of / with the period 2t/ a W_ (o)\,
depending upon the d and ey ' ' ;
. In the first two equations the unde-termined constants a'," and t$> enter only as they are exhibited explicitly.
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LINEAR DIFFERENTIAL EQUATIONS. 41
Equations (129) are to be integrated and the results substituted in (128).In order that the y
(
,[
}
shall be periodic the conditions must be imposed that
=
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42 PERIODIC ORBITS.
The general solution of these equations when the right members are neglected
is the same as (128) except that the superscripts are (2) instead of (1).
On varying the *, the equations corresponding to (129) are
(tfy=-*?il-V%C
f)+C
f<B%+B%A(t)+D(t),Oil/ Wl
KVn) -a-i Vn *i #21&)'- - - a^? + #M (0 +Dff(0,
(J -8, , n).
(134)
The undetermined constants af and Z?^" are exhibited explicitly in the first
two equations, and it is to be noted that Ai" and A",' are precisely the samefunctions of t as those which appeared in (129).
In order that these equations shall lead to periodic values of the y
the undetermined constants must satisfy the conditions
.< 2 > ( \ ,Cl2 \ _1_
Cl2 <" R (1 > _L k< R> J./)o, ^1
?j 21 ^+ aj J 21 -f- fc n # 21 +d u ,
n _ < 2 >. <D R< _1_ J>< R< _L W<2 >U= Oj T) 2l a, iJ 21 +0 21 iJ 2 i + 2 i,
0' = 3, . . . , n),B%
(135)
The first two equations are linear in a (
,
2) and B , and they determine these
quantities uniquely unless their determinant is zero. The determinant is
A=-i ,<> Cl2 j<"_i_ (
>
i -~ r l2i , On "T a-iC
c
,(1)
= -6 2 + a-+^[6<
A) + ^]
121 > 21""
1
On eliminating j? and a ( "by means of (130), it is found that
A=V5,where D is the discriminant of (131). Since by hypothesis D is not zero,
the determinant A is not zero. Hence af and -B 2"
are uniquely determined
by (135). Having determined JS", and af, equations (134) are integratedand the results are substituted in the equations corresponding to (128).
Then the conditions that y (0)= are imposed and the final solution at
this step becomes
tf (0) (o)\ I n| 1
(*-l, ,),(136)
where B is undetermined until the next step of the integration.
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LINEAR DIFFERENTIAL EQUATIONS. 43
The next step is similar to the preceding and all the equations are the
same except the superscripts are (3) and (2) in place of (2) and (1) respec-
tively. The determinant of the equations corresponding to (135) is preciselythe same. In fact all succeeding steps are the same, and the whole
processcan be repeated as many times as is desired. The solutions associated with
af, . . .,a ( 0)
are found as they were in 26.
For congruences of higher order similar methods can be used, and in
the cases which are exceptions to this mode of treatment the existence
discussion furnishes a sure guide for the construction of the solutions.
28. Construction of the Solutions when two Roots of the Characteristic
Equation are Equal. It will be supposed af = af and that all the
remaining af are mutually distinct and distinct from af. The solutions
depending upon af , . . ., af can be computed by the method of 26. It
has been shown that the two solutions proceeding from a^ are in general
expansible as power series in VJi. The detailed discussion will be madeonly for the general case, where
a 1 =af + a|V + af M+a 2 =af-aV + af M-
(137)
Terms independent oj ju. The terms independent of n are defined by
(^'+ftC-J l,4C0 (*-l,...,). (138)
The general solution of these equations is
y?? = v^c tl +v?!(c i2 +tc il )+ ifZc/*?-?)*'. (139)
In order that the t/f shall be periodic with the period 2ir and the initial value
c n of yfi shall be obtained, the V must satisfy the conditions
Vn=0 (j = 2,...,n).
The solution satisfying all the conditions is then
|C-*. (i= l,...,n). (140)
Coefficients of m} The coefficients of m* are defined by the equations
W+ ai iff- 2 0!#- -aft/f = -af c a (t-1, . . . , ). (141)
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44 PERIODIC ORBITS.
On neglecting the right members, the general solution of these equations is
|tf,
-*ff, + fl&,
fe.+fcn)+ ^Hjfe/*'"
(142)
The method of variation of parameters leads to the conditions
n / () o>n
Wiyc a +WlY(c a +UQ+ S Wl)'c t / a>_a
' ; ' = -a% (t-1, . . ., n).
On solving these equations for the (VAT* it is found that
(*S)' ?, G$)'0 (j =2, ...,n).
Consequently
$-*-?, $-3 (i-2, ...,). (143)
On substituting the values from (143) in (142), the result becomes
y" = (B\-a?t)c (l +B 2l (c {2 +tc (l )+ 2#X ( ' ' '
J=3
To satisfy the conditions for periodicity and to make y (0) = 0, the B n
must fulfill the relations
<?=
<*>, BSCu+BSfCa-O,< = fJ-8, ...,). (144)
Then the solutions satisfying all the conditions become
y =(-
c
t c " +Ca ) < > (145)
where the constant a"' remains as yet undetermined. It is to be observed that
not all the coefficients of a"' can vanish, for otherwise the determinant \c tJ \
itself would vanish.
Coefficients of p.. The coefficients of m are determined by the equations
(t/i?)'+- i.i*S- -fi/SFi/S+jSfiA 0-1. .> d46)
The solution of the homogeneous terms is of the same form as (142), and byvarying the constants of integration, it is found that
(OV n +(0'(c fi +fc)+ 2^ ,^-J-3
-af C(l- (-^C+ Cfi )(ar>)
s + 2 C<(147)
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LINEAR DIFFERENTIAL EQUATIONS.
The solutions of these equations for the (a(
jf)' are
(n?f)r -+e-<
a^%() (,'-8, . . . , ),
45
(148)
where the Af (t) and D(t) are known periodic functions of t. The first
equation gives rise to integrals of the type
a (ts[n itdt= x -^ c
.os i/4- ' sin U
)Jl cos JIM + s in J 1 ^ -
2 cos./ 1 -
The second equation gives rise to the corresponding integral
-aJZJtdt=p*jtWhen these results are substituted in equations (142) the terms of the
type 1 1 jt destroy each other. Hence at this step
1=3
J=3
(149)
where the P (
jl(t) are periodic functions of t, and &* and dfl are the constant
terms in Af,' and D . Equations (149) are the general solutions of equa-tions (147). In order to satisfy the conditions for periodicity and the
initial condition ^'(O) =0, the constants a"' and B' must fulfill the relations
#21 -o-i -r n a u>
= B%c n +B%c 12 +2P%(0),(150)
0-B* (J =3, ...,n).
The constant af still remains undetermined. The solutions now are
y%=(-c
fc n +c a )a?+^(t), (151)
where the *"' are known periodic functions of < having the period 2ir. After
making a choice as to a"', provided bfi^O, it is found that of is determined
uniquely by the periodicity conditions for t/Jf. The process can be con-
tinued indefinitely and the constants are determined uniquely. The other
solution associated with a is obtained by taking the other determination
of a"', and the solutions depending on af, . . ., af by the method of 26.
The chief types of cases have been treated, and the exceptions to them are
developed similarly accordingto the forms indicated in the existence
proofs.
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46 PERIODIC ORBITS.
IV. NON-HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS.
29. Case where the Right Members are Periodic with the Period
2w and the a, are Distinct. Take the set of differential equations
x' t-
i^x^g^t) (i=l, ...,), (152)
where the 6U and the g t {t) are periodic in t with the period 2tt. For the left
members set equal to zero the form of the solution of (152) is
x t= 2,A, (153)
where thejjj
are arbitrary constants, the a, are the characteristic exponentswhich are supposed distinct, and the y t}
are periodic in t with the period 2ir.
By the method of variation of parameters, it is found that
iv',ea%}
= g t (t). (154)
The determinant of the coefficients of therj'j
is the determinant of the funda-
mental set of solutions. Since the 6i}
are assumed to be regular for all finite
values of t, it follows from 18 that this determinant can not vanish for anyfinite value of t. This determinant is
s ajt
where A is the determinant of the y i}. If A, denotes that which A becomes
when thej
th column is
replaced bythe
gt (t), the solutions of
(154)for the
jj,are
',= |'e-a", (155)
;-/ :
and consequently
^e-a*dH-B,. (156)
The quotient A,/ A is a periodic function of t, continuous and finite in the
interval 0^^27r. Therefore it can be expanded into the Fourier series
^ = <# + [ojf cos mt + U sin mt] .
A m= l
If a2
j+m2
9^0 (i = l, . . . , n; m= \, . . ., oo), the integral becomes
J A a, ^ L a'+ra2 a'+m 2 J
v
so that t,-- wF f (0+B i ,
where theP,(0 are periodic with the period 2t, and the B, arc constants of
integration. On substituting these values of the t) }in (153), the general
solutions of (152) become
x (= J B,e*%+ jtPffllks
-l. ,n). (158)
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PARTICULAR SOLUTIONS OF LINEAR EQUATIONS. 47
Now suppose at = k V l, where k is an integer. Then the term
je~iVZ::rT
'[a k cos kt+b k sin &] dt
becomes, after the integration has been carried out,
^(a t -b k x^i)t+^r(a kV^i-b k) (cos2kt- V~^Tsm2kt).
Therefore the expression corresponding to (158) becomes in this case
x t= 5 B,e
a *y u + P 4 (0 + |(a,
-6, v^T) t e
a >'
y u , (159)
where the P,(t) are periodic with the period 2x.
Therefore, t/ /ie characteristic exponents are distinct and none of them is
congruent to zero mod. V l, and if the g t (t) are periodic with the period 2r,then the particular integrals are also periodic with the period 2w. But if some
of the characteristic exponents are congruent to zero mod. Vi, then the par-ticular integrals in general contain, in addition to periodic terms, the corre-
sponding parts of the complementary function multiplied by a constant times t.
30. Case where the Right Members are Periodic Terms Multiplied byan Exponential, and the a t are Distinct. Consider the case where the g t (t)
have the form
the f t (t) being periodic with the period 2t. When X = l V^T is a pure
imaginary,this form includes such cases as
g i (t)= 2[a k cos(k+l)t+b k sin(k+l)t}.
t
If the differential equations, which are now of the form
4-ie ti x,~<?'f t (t), (160)
are transformed by xt= e
u z( , they become
zi + ^-iy^m, (lei)
and have the same character as those treated in 29. If the character-istic exponents a
7of (160) are distinct, then the characteristic exponents
of (161) are a, X, and are also distinct. Applying the results of the pre-
ceding case, it is seen that if no a, X is congruent to zero mod. V^i, then
the solutions of (161) are
where the Qt (i) are periodic with the period 2ir. Therefore the solutions of
(160) are
x t= iB.e^'y.j+e^Q^t) (*-l, . . . , n), (162)
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48 PERIODIC ORBITS.
But if one of the a} , say a t ,
is congruent to X mod. v%-l, then the z y
have the form
and therefore the expressions for the x t become
x t= 2B, e> y {J +e*Q t (t) +c t t e
ay {t . (163)
These results may be stated as follows: // the g,(t) have the formg,(t)
= e^f^t), where f,(t) =f,(t + 2ir), and if none of the characteristic
exponents is congruent to X mod. V i, then the particular solution has the form
x^e^QS) (i=l,...,n),
where the Q,(<) are periodic with the period 2w;
but if one of the characteristic
exponents, a t ,is congruent to X mod. V^, then the particular solution has
the form
x^e^Q^+cJe^y* (i=l, . . . ,),
where the Q,(t) are periodic with the period 2ir.
31. Case where two Characteristic Exponents are Equal and the RightMembers are Periodic. Suppose o 2
=a, . Then the solutions of
x[ 2 6(JX
j=
(t'=
l, . . ., n),
in general have the form
*-% ea''y (1 +r, 2 e
a(y a +ty (l )+ 2 Vj e
a*y f} (i=l, . . .
, n), (164)
For the associated non-homogeneous equations
x'i -2 8
iJ xJ= g l (t) (165)
it is found by applying the method of the variation of parameters that
ea>'y tl v[+e
a "(y i2 +ty il )r,' + 2 e
a *y (J v'j
= g t (t) (t-1, )'.
J=3
On solving these equations for the rj' } ,the results are found to be
bifx = \g&) , (Va + ty a ), 2/ , ... , y t \e-
*vf
2=\ya , g t (t), y {3 ,. . .
, y,\e-
&v'j=\y<x , tom+tVa), V, '. . . , ytn\e-ajt
(j = 3, . . ., n),
where A is the determinant \y tj \
. The expansions of these determinants
have the form
^e-^P^D-e-HPAt), ^ = e-a "P 2 (0,
j
9',-e-<*J>(*) (i-3, . . ., n), j
where the Pj(t) (j=1, . . ., n) are periodic with the period 2t.
-a,(
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PARTICULAR SOLUTIONS OF LINEAR EQUATIONS. 49
Suppose no a, is congruent to zero mod. V^T. Then it is easilyfound that
Vl= e-^R, (0 - e~ a H R 2 (t) +, ,
r, 2
= e-a *R
2 (t)+B 2 , (167)
where R v (t), . . ., R (t) are periodic with the period 2ir. On substituting
these values in (164), the solution becomes
x i =B 1 ea"y n +B 2 e
a(y i2 +ty n ) + 2 5,e
a'y + 2 R
t (t) y irn
The 2 Rjy (J are periodic with the period 2 n. Hence, if two of the characteristic
exponents are equal but none of them is congruent to zero mod. V 1, then the
particular solution also is periodic with the period 2ir.
The case where one aj(j = 3,
. . ., n) is congruent to zero mod. v^ 17 !
is a combination of the present case with the second part of that treated in
29, and that where a 2=
a! is congruent to zero mod. %/< l does not differ
in any essentials from that where a 2=
a,= 0.
Now suppose a 2= a
1= 0. Then the equations which correspond to
(166) become
*J-P f (<) ~tP t (t), v' = P 2 (t), fae-^PM (i =3, . . .
, n).
The Pj(t) are periodic with the period 2-k and can be written in the form
Pj = aj + 2 [a kJ coskt+b kJ sin kt\.
Hence the t], are found, by integrating, to have the form
i I +2& 1 (0+<M--ia t f--iS,(0,
tl 2 =+R 2 (t)+a 2 t,
li-tT-RM+B, tf-3, . . . ,n),
where
R,{t)=R j {t + 2ir) (j=l,...,n).
These values substituted in (164) give for the complete solutions
n n
x t= B
i y n +B 2 (yn +ty n )+'L B ,eaj ty+[a 1 *+a 2 *
2
] y (l +a 2 ty (2 + 2 R t y u .
Hence, when the g t (t) are periodic with the period 2w, and when two of the char-
acteristic exponents are not only equal, but are zero, then the particular integral
involves not only t but, in general, also f outside of the trigonometric symbols.
Of course it may happen that al and a 2 are either or both zero.
Those particular cases have been treated which will be most useful in
in the applications which follow. Any others which may arise can be
discussed in a similar way.
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50 PERIODIC ORBITS.
V. EQUATIONS OF VARIATION AND THEIR CHARACTERISTICEXPONENTS.*
32. The Equations of Variations. The preceding considerations find
immediate application in dynamics in the study of small variations fromknown periodic solutions. Suppose there are given the equations
^=X t -i, . ...n), (168)
where the A', are functions of the xt , and that they are satisfied by
as,-*(0, (169)
the <p,(t) being periodic functions of t with the period 2w. These are called
the generating solutions.
Let the initial conditions be varied slightly and put
xt (0) = <Pi<P)+P t , (170)
where the ft are small arbitrary constants. The value of the x ( for any t will be
*-*(*)+{,(*), (171)
the, being functions of t which for at least a short interval of time will
remain small. On substituting (171) in (168) and expanding the right
members as power series in the_, ,
it is found that
% = 2 If^ *> +higher degree terms (t= 1, . . ., n), (172)
the xt being replaced by <p t (t) in the partial derivatives. Since the <p t (t) are
periodic in t, so also are all the coefficients of (172). The , are expansible as
power series in theft which, by the Cauchy-Poincare theorem, 16, convergefor any preassigned interval of time provided the |ft| are sufficiently small.
The differential equations for the linear terms in the ft are the linear
terms of (172), or
=f +!+ +f *-* ' < i73 >
These equations are known as the equations of variation.
Suppose the solution (169) contains an arbitrary constant c, that is,
one not contained in the differential equations (168). If c = c + y, the <p t (t)
arc expansible as power series in y of the form
*(0 =*?()+* y+hd
y2 + -i, ,..,)
Obviously
*'dc
J ^ 2dc
2 y ^ { l, . . ,n)
The subject of this section and other related questions have been treated by Poincar<5, Les MethodesNow>clles dz la Mecaniqus Celeste, vol. I, ehap. 4.
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EQUATIONS OF VARIATION AND CHARACTERISTIC EXPONENTS. 51
is a solution of equations (172), and consequently}
(0)
&^- (il,.../) (174)41
is a solution ofequations (173).
One such constant is always present when the Xtdo not contain t
explicitly and the <p t (t) are not mere constants, for then the origin of time is
arbitrary. Hence in this case
*-jg t <<-i,. ....1* awis a solution of (173). Another such constant usually present is the scale
factor which determines the size of the generating orbit. If there are p such
arbitrary constants in the generating solution, the equations of variation
have p solutions of the form (174).
If the equations (168) admit an integral which is independent of t,
F,(Xi ,. . .
, x.) =c,,
where c, is an arbitrary constant, the xtcan be replaced by <p t (t)-\- t
in the
integral and the integral can be expanded in powers of,
. The result is
dF lt ,dF lt , ,dF x.
,
. . , .
7= dx~^ dx~^2
' ' ' +^+ hl g nerde S ree terms.
The constant 7 is a power series in the /?, , and therefore the linear terms are
^*+*+;
+g*''(I76!
which is therefore an integral of equations (173). The coefficients dFx /dXj are
periodic functions of t, the xt having been replaced by <p t (t) after differentiation.
Si. Theorems on the Characteristic Exponents. The existence of
arbitrary constants in the generating solutions and the existence of integrals
of equations (168) have an intimate connection with the characteristic
exponents of the solutions. These solutions are in general of the form
fc-*"/(0 o;=i, ...,), (177)
the/, being periodic with the period 2ir. The solutions (175) have the form
(177), but since the <p t are periodic so also are their derivatives, and thecharacteristic exponent of this solution is zero. There is an exception only
if the <p tare constants, in which case the solution (175) disappears.
The solution obtained by differentiating with respect to the scale
constant, which will be denoted by a, will, in general, have the form
^'=^(0+^(0,
\f/ t and ^ being periodic. The characteristic exponent of this solution is
zero. If the generating solutions have p distinct arbitrary constants, the equa-
tions of variation will have at least p characteristic exponents equal to zero.
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52 PERIODIC ORBITS.
From the existence of the integral (176) it follows also that at least one
of the characteristic exponents is zero; for all solutions have the form
*.,= e"%(0 (i,j=l, . . . ,n),
and substituting them successively with respect to the index j in (176), we get
%-tj i '= e
~ ait ^ U=h...,n). (178)
The left members of these equations are periodic with the period 2ir, except,
perhaps, for coefficients which are polynomials in t. It follows, therefore,
that either the a,=0 mod V^l, or all the 7^ = 0. In this connection a
congruence has the same properties as an equality, and they need not be
distinguished from each other. If all the a } are distinct from zero, then
7 </>= =
1, . . ., n) and (178) becomes
2^/^ = (j-l,...,). (179)
Since the determinant| / |
5* 0, these equations can be satisfied only if
a*ff0 <-' >
Therefore, unless the integral (176) vanishes identically at least one of the
characteristic exponents is zero. Suppose that a x= and that 7
(
," ?* 0.
It is possible then to solve the equations corresponding to (178) uniquely
for the dFJdx t in terms of yf and the j tl.
Suppose now there is a second integral F 2 (x t , . . ., x)=c i . Then
2|g^= const.
On substituting in this equation successively the n fundamental solutions
for the ,it follows, since ^ = 0, that
Sf/ = 5<- 2|f;/o= ^^ (j
= 2, ...,n). (180)
If 0,5*0, and therefore 6^ = = 2, . . ., n), these equations can be
solved uniquely for dF^/dx, in terms of f tJ and 8[l)
. It results that, aside
from a constant factor,
dJ\^d_F\dx t
' dx t
'
and the second integral is identical with the first. But if F x and F 2 are
distinct, then there must be at least two characteristic exponents, say c^ and
a, , which are zero. Proceeding in this manner it follows that if the equa-tions of variation admit of p linearly distinct integrals not identically zero,
then there are p characteristic exponents equal to zero.
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EQUATIONS OF VARIATION AND CHARACTERISTIC EXPONENTS. 53
(-l, ...,*),
If the original differential equations have the form
d% = dVdt
2
dx,
which is the case usually in celestial mechanics, they may be reduced to
equations involving only first derivatives by writing
dx, _ dy, _ dVdt~ Vt ' ~dt~~dx {
If the generating solution is
and the equations of variation are formed by putting
* =*> (0 +6 , y t
=<p[{t) +Vt ,
there will result
(t-1, n).
d * -n~dt~ vi tt-1, n),
dm_ d*V d*V a 2 F(181)
eft dx, dx x
S1 '
dXi dXi2 ' '
dx, dx'
The main diagonal of the right members of these equations (considered as
a determinant matrix) contains only zero elements. Therefore, by 18, the
determinant of any fundamental set of solutions of these equations is a
constant. But the determinant of the fundamental set of solutions
i,-**&(0, y<j=eait
g tj (t) u=i, n).
has the form A = eofs ajt
P (t) . This must therefore be a constant, from whichit follows that the sum of the characteristic exponents is zero since P (t) has
the period 2ir.
Suppose #", nf and f , nf (i=l, . . ., ) are any two solutions of
Then
>
equations (181).
ddt
and also
dtf
dn? = d2 V (1)
a2 F (1)
d 2 Fdx,dx n
dt^ drfl
2)
dt dxidx
l
'
M2, + #L^ , +
dx,dx 2
+ a2 r
dxldx
at
From these equations it follows that
2/(* dF "' "dfJu
' A/A 6d<
6 dt)u -
The sum of these two equations is
dt a)
2 [(<l fc) drit
dt+tl
dt )-bnt^di)]=>
(182)
(183)
(184)
(185)
which can be written
,7^ (m fi m *?i J~ u -
at i=i
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54 PERIODIC ORBITS.
Consequently
2 (gV-SV) = const. (186)
The relation (186) between any two solutions leads to important conclu-
sions respecting the characteristic exponents. Suppose the {$" and ij (U) are
e = ea
f tl (t), v? = e^g (1 {t) 0-1 ,n;j = l ,2),
where / and g u are polynomials in t with periodic coefficients, and that theyconstitute a fundamental set of solutions. On substituting any two of these
solutions in (186) and dividing through by the exponential, there results
M/,^-/,,?,;)^/-"^. (187)(=1
It follows from the character of the left member of this equation that either
a.j+a t = 0, or 7 yt = 0. It will be shown, however, that y Jk. can not be zero
for every k.
Suppose j is kept fixed and give to k all the values from 1, . . .,
2n.
Suppose y Jk= (k =
1, . . .,
2 n). Then one equation of (187) is an
identity and the others are linear in the f tJand the g u . The determinant
of this set of linear equations is the determinant of the fundamental set
and is not zero. Hence they can be satisfied only by f tJ ^g tj =0. Butthis also is impossible since / and g u are a solution of the fundamental set.
Therefore not all the y jKcan be zero. Hence for some k
a,+a=
0, t^0. (188)But since a, is any one of the characteristic exponents, it follows that corre-
sponding to each characteristic exponent there is another one which differs
from it only in sign.
If two of the ajare equal but not equal to zero, then there are two others
which are also equal and which differ from the first two only in sign. In
order to show this suppose a,= a >+1
= a m . Then a m = a m+l , because from
(188) it follows that a,+a m = 0, y Ju i^0. If t,*= (k=l, . . .
, 2n, k*m),then (187) can be solved for / and g tJ uniquely in terms of f, k and g it
(k = l . . ., 2n). Now the corresponding equations for f tJ+1 and g Uj+l
will differ from (187) only in that j is replaced by j+1. They can besolved uniquely for / M+1 and g t , ]+l , but this solution will differ from the
solution for / and g tJ only by a constant factor. Since this is impossibleit follows that Yj+i.m+i^O, and consequently a J+l + a m+l = 0. In the samemanner it can be shown that if p of the a.
j are equal, then p other a.j are
also equal and differ from the first set only in sign.
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Chapter II.
ELLIPTIC MOTION.*
34. The Differential Equations of Motion. Consider two sphereswhose materials are arranged in homogeneous spherical layers concentricwith their centers. Then they attract each other as material points, their
orbits are plane curves, and the differential equations which the motion of
one relative to the other must satisfy are, in polar coordinates,
dt2 r
\dt)+
r2
~ u'
diVTtJ'"-{ }
In writing these equations the origin has been placed at one of the bodiesand the variables r and v are measured in the plane of motion.
Equations (1) are easily integrated, and the integrals show that the
relative motion is in a conic section for any initial conditions. If the initial
velocity is not too great the orbit is an ellipse, and the discussion will belimited to this case. While the ordinary integration of (1) shows that undercertain conditions the orbits are ellipses, it does not express the coordinates
explicitly in terms of the time. The explicit developments are obtained
through solving Kepler's equation, generally by Lagrange's method or bymeans of Bessel's functions. In treating elliptic motion as periodic motionthe expressions for r and v in terms of t will be derived directly from the
differential equations.On integrating the second equation of (1), and by means of this integral
eliminating dv/dt from the first, it is found that
2 dv d 2r c
2
F(m,+w 2 ) _ (t}.
Tdt
=C> dT~? + *~
Assume that the conditions for an elliptical orbit are satisfied, and let
a = themajor
semi-axis of the orbit;
e =the eccentricity of the orbit;
co = the mean angular velocity in the orbit;
T = the time the body passes its nearest apse;
t =w(t T) =the mean anomaly.
It is found from the integrals of (2) thatf
coV = A;2K+0, c
2 = A;i!
(m 1 -|-m 2 )a(l-e2
)=coV(l-e2). (3)
This chapter was written in 1900 and a brief account of it was published in the Astronomical Journal
Vol. XXV, May, 1907.
fMoulton's Introduction to Celestial Mechanics, pp. 173-8.55
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56 PERIODIC ORBITS.
On making use of (3) and using t as the independent variable, the
second equation of (2) becomes
_< + .e. (4)(It r r
Equation (4) is satisfied by the circular solution r = a. Let the radius in
the elliptic orbit be
r = a(l-pe), (5)
where p = 1, dp/dr =
,at t = 0. At the half period t = 7t, r = a(l+e). There-
fore p 1 at t = x. These are the extreme values of r in elliptical motion,and therefore + 1 ^p^ 1-
Upon substituting (5) in (4), the latter becomes
* + -2Z* =0 (6)
The second term in this equation can be expanded as a power series in e for
all the values of p if \e\ <1, as is explicitly assumed, giving
CO
f? + P = ! S (*'+!)[*'- (H-2)p>'-V, (7)
and the first equation of (2) becomes by the same substitutions
35. Form of the Solution. The solution of equation (7) will first be
considered. After it has been found, v is determined from (8) by a simple
quadrature.
Equation (7) belongs to the type treated in 14-16, and therefore
can be integrated as a power series in e, and \e\ can be taken so small that
the series will converge for ^ t ^ 2ir. Since the body moves so that the law
of areas is satisfied and completes a revolution in 2t, p is periodic with the
period 2ir. Consequently, if the series converges f or <! t ^ 2ir, it convergesfor all real values of t. It is, indeed, possible to find the precise limits for |e|
within which the series will converge for all values of t, and outside of which
they will diverge for some values of r. The problem was first solved byLaplace,* who found that the series converge for all r if e< 0.6627 . . .
,
which is far above the eccentricity of the orbit of any planet or satellite in
the solar system.
*Micanique Celeste, vol. V, Supplement; see also Tisserand's Mecanique CSleste, vol. I, chapter 16,and a demonstration by Hermitc, Conrs a la Fac. des Sci. de Paris, 3d edition (1886), p. 167.
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ELLIPTIC MOTION. 57
The solution of (7) can be written in the form
P=2 P ,(tK (9)J=0
where the Pj{t) are functions of r. According to 15 and the initial condi-
tions, the constants of integration which arise are to be determined by the
conditions
p (0)=
l, Pl (0)=0 (-l, ...oo)j ^(0)=0 -l,\..oo). (10)
As p is periodic with the period 2t, it follows that p(t+2it)=p(t); whence
iop,(r+27r)e
j =iop j (r)6
;. (11)
Since (11) is an identity in e it follows that Pj(t-\-2t) = Pj(t) . But this is
simply the definition of periodicity. Therefore each p t separately is periodic.
The body is at its nearest apse when r = 0, and the orbit is symmetricalwith respect to the line of apses. Therefore it follows that p is an evenfunction of r. Since p is periodic in r identically with respect to e, each
Pj (t) is expressible as a sum of cosines of integral multiples of t.
If the sign of e in (6) is changed, then the body is at its farthest apsewhen t = 0. Consequently changing the sign of e and increasing t by ir
does not change the value of r. Since r = a(l pe) ,it follows that
p/ry-^p/T+TrM-e)'-1
. (12)
Therefore when,/ is even, pj(r) involves cosines of only odd multiples of t;
and when j is odd, p } (t) involves cosines of only even multiples of t.
If we substitute (9) in (8), we get
where
=0 J=0
x=vi-e 2
2J(i+i) 2) c* *p':---p **gi *++< At+l
(13)
c, . = orr~n (*++ +**=*) ( 14 )t, . t 2 . . . . t, .
Suppose i,i,+ +i.j, + i is even; then there are two cases to be
considered, viz. (a) when i is even, and (6) when i is odd:
(a) When i is even an even number of t\ ,. . .
,t must be odd, and
the number of odd i\ multiplied by odd j x must be even. Therefore the
number of odd i x multiplied by even j\ must be even. All those factors p
in (13) for which i x is even involve only even multiples of t, and those for
which j x is even and i x odd involve only odd multiples of r. Since there
must be an even number of these terms involving only odd multiples of
r, their product involves only even multiplesof t.
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58 PERIODIC ORBITS.
(6) When i is odd it follows from (14) that an odd number of t, ,. . .
, i,
are odd, and from the hypothesis that i,Ji+ . . . +ij,+i is even it follows
that an odd number of i A j t ,. . .
, ij, are odd. A term i x j x can be odd
only if both t\ and j x are odd. When j x is odd the term involves only even
multiples of r whether raised to an odd or even power. Since the whole
number of odd i x is odd, and an odd number of them are multiplied byan odd j x ,
it follows that there is an even number of terms p'; ,where j x is
even and i x is odd. Therefore their product will be cosines of even mul-
tiples of t. That is, in the right member of (13) the coefficients of even
powers of e involve only even multiples of t.
It is easily proved in a similar way that the coefficients of odd powersof e in (13) are odd multiples of t.
Upon integrating (13), it is found that v is expansible as a power series
in e of the form
v = CT+Zv i e\ (15)
where vt
is a sum of sines of even multiples of t when i is even, and of odd
multiples of t when * is odd. The coefficient c is unity because, the ellipse
being fixed in space, v increases by precisely 2ir in a period.
36. Direct Construction of the Solution. Upon substituting equation
(9) in (7) and arranging in powers of e, we obtain
2 P ;e'+ 2 Pj e> = [l-3p>+[3 Po -6p oPl -6p^ 2
(16)+ [-6 Po p 2 +3p 1 (l-p 1 -6pD+2pS(3-5pD]e3
+ [-6p(p,+3p?+3p p 2 -2p 1 )+3p 2 (l-2p 1 )-5p2(8p 1 -2-3pDK . . ., j
where p* is the second derivative of p } with respect to t. Upon equatingcoefficients of like powers of e, the differential equations which define the
several coefficients become
(a) p o"'+Po = 0,
(6) tf+ft-l-apg,(c) p 2
r-f-p 2
= 3p (l-2 Pl -2p2
),
(d) p"+p 3= -6p p 2 -f 3p 1 (l-p 1 -6po)+2po(3-5po),
The only solution of (a) satisfying (10) is
p = cosr. (17)Then equation (6) becomes
i 13Pi+Pi= 2
~2
COS2t.
The solution of this equation satisfying (10) is
Pi=-f4c s2r. (18)
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ELLIPTIC MOTION. 59
In a similar manner equations (c), (d), . . . can be integrated in order,and their solutions are found to be
P 2=
g (-cost+cos3t), p 3 =-^(cos 2t-cos4t), (19)i
The general term of the solution is defined by an equation of the form
p'j+P^A^+A^ cos t+ +^coskt, (20)
where the A 1
^' are known constants. Since p sis periodic, A[
J)is zero for all
values of j, and since p s involves only even or odd multiples of r according as
j is odd or even, all the A^ with even subscripts are zero if j is odd, and all theA with odd subscripts are zero if j is even. On putting Af equal to zero,
the solution of (20) satisfying the initial conditions (10) is
[JL A'D -|*
AV)
-K n + 2 1 1 Jcos T ~ 2 x^- 1
cos Xt - (21)
On substituting (17), (18), (19), ... in (9) and (5), the final expressionfor r becomes
= a-U [cost] e +|[l cos 2r]e2
+! [cost cos3r]e3
+ |[cos2t-cos4t] e4 +
j-
(22)
On making the same substitutions in (8) and integrating, the explicit value
of v is found to be
v = t+ [2 sin r]e+ [f sin 2t] e2 + [- \
sin t+ g sin 3t] e3
(23)
37. Additional Properties of the Solution. It will be proved that no
Pj carries a higher multiple of t than j+l. It has been seen that it is true
for,7
=0, 1, 2,
3. It will be assumed that it is trueup
toj 1,
and then it
will be proved that it is true for the next step.
The general term in the right member of (7) is, apart from its numerical
coefficient, p1=t
'e'. After substituting the series (9) for p, any term of degree
j in e arising from this term has the form plpl'pl' p"V, where
\ + X, + X2 + +\ K= il, X,+2X 2 + +^+{1 =
7.
After eliminating * from equations, it is found that
X+2X, + 3X 2 + +(+l)X,=;l, (24)
where obviously k <j.
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60 PERIODIC ORBITS.
By hypothesis the highest multiples of t in p , pj ,. . .
, p K are respec-
tively 1, 2, . . ., k+1. Therefore the highest multiples in pi, p\', . . .
, p*' are
respectively X, 2X,, . . ., (k4-1)X*- Consequently the highest multiple in
the product p^ ... pi' is X+2X.+ +(/c-f-l)X K ,which is jl by (24).
Therefore the highest multiple in the expression for pMs j + 1 .
A similar discussion of (8), (13), and (15) shows that vtdoes not involve
multiples of t greater than ,;'.
38. Problem of the Rotating Ellipse. In certain cases where the motion
is not strictly elliptical, it is convenient to suppose the body moves in an
ellipse whose position and form are constantly changing. This conceptionis at the foundation of the theory of perturbations originated by Newton,and has been essential in the work of most writers on celestial mechanics.
One of thehistorically interesting
andimportant problems
has been the
theory of revolution of the line of apsides of the moon's orbit. WhenClairaut first made a computation of the rate of this revolution under the
supposition that it was due to perturbations of the moon's motion by the
sun, he obtained an amount about half as great as that furnished by obser-
vations.* Later work by himself and others has shown that the discrepancywas due to imperfections in his theory, but at first he sought to relieve the
difficulties by supposing that gravitation does not vary simply as the inverse
square of the distance, but that it also depends upon a term in the inverse
third power of the distance. We shall solve the problem of the motion for
this law of force as a further illustration of thepower
andsimplicity
of the
methods which are being used here. The steps in the solution are almost
exactly parallel to those used above, and it will be noted in the course of the
work that they would not be fundamentally different if the added term werenot such as to make the peculiar simplicity of this problem.
The differential equations of motion in this case are
* rT (
dv \ ifcVi+K) _ F (m. + m2 ) d( 2 dv\_ ,
df~ r\dt)
+ T "? M ' dA
rTt)~> (25)
where, for simplicity in the final formulas, the constant coefficient of the term
in the inverse third power of r is given the formA;2
(m 1 +m 2 )a/x. The p. is anarbitrary parameter.
The integral of the second equation of (25) is
r*f =c, (26)
by means of which the first equation reduces to
dV c2
Ar^+ra.,) _ A;2
(m,+m 2 ) , 97 s
__ d7^7+ ?
~? aM ' {27)
*See Tisserand's Micanique Ciksle, vol. Ill, chap. 4, and particularly articles 24 and 27.
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THE ROTATING ELLIPSE. 61
39. The Circular Solution. We shall first find a solution of (27) withan arbitrary constant of areas, c, for which r is constant. Let a and u bedefined by
k 2
(m l
+m 2 )=w2a 3
,c =
kVm 1+m 2 )a. (28)
Let r = a(l+p), where p is a constant. Then (27) becomes
1 1 1.
ft _L<v^2 /I I \3 M>a+py '
(i+pY (i+p)3
or, expanding as a power ser ies in p,
p-3p2
+6pz - =- M (l-3p+ ) (29)
By 1 and 2 this equation can be solved for p as a power series in n, con-
verging for|jut
| sufficiently small. In fact, the additional term has beenchosen in such a
waythat the
powerseries reduces to a
single term, butit
is evident that this condition is in no way essential to the process. It is
found at once that
P=-M- (30)
In this case the solutions of equations (25) and (26) are
r-(l-), .-^ypS-yfl-ri-S*!^-!,, (31)
involving the three constants of integration a, v,and T.
40. Existence of the Non-Circular Solutions. We shall now derive asolution of equations (25) corresponding to the elliptic solution in the ordi-
nary two-body problem. It will involve four constants of integration which
are arbitrary except for the restriction that the orbit shall not deviate too
widely from a circle, a condition which is imposed to secure convergence of
the series. The solution is thus seen to be the general solution.
Now let
c = fcV(mi+mi)a(l-e*), k2 {m^m 2 )
= wV,r = a(l- M-pe), w(t-T) = Vl + 5 r, \
(32)
where T is the time ofpassing
the nearestapse,
anda(l
/*e)
is the arbi-
trary initial value of r. Therefore the initial value of p is unity. Theconstants a and e are defined by the first and third equations at t = T, u bythe second, and 5 is a parameter to be determined later.* There are so far
three arbitrary constants of integration a, e, and T; the fourth is introduced
in integrating equation (26).
With these substitutions equation (27) becomes
ft+ ft+^-'j -O. (33)aV (1 n pe)
*Poincar6 introduces a parameter r somewhat analogous to this. Les Methodes Nouvelles de la Mecanique
Ciksle, vol. I, p. 61.
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62 PERIODIC ORBITS.
Equation (33) admits a periodic solution, as is known from the fact that the
orbit is a rotating ellipse. However, it will be shown directly by forming
the first integral of (27), viz.,
&i\c t -i+2 ^ m*+m ' ) '+ **}">> *>-9<r). (34)\dU r r r
Suppose the initial conditions are real; then <p(r )>0. For p = the equa-tion (p(r)=0 has two roots, viz., r
t= a(l e) ,
r 2= a(l+e) between which r
must vary. For p small it also has two roots, r[ and r 2 ,near r, and r 2 respec-
tively. Suppose r[<r 2 . Then <p{r) <0 if r<r[ or r>r' . Consequently it
follows from dr/dt =V<p(r) that if r is increasing at / = it will increase until
r = r 2 when the radical changes sign, after which it will decrease until the
radical changes sign again at r = r[. The period of a complete oscillation is
-*rdr
V^Jrj'
( 35 )
If r is periodic then p is periodic also.
The existence of the periodic solution can be established directly from
(33) and a proof made of the possibility of a construction similar to that used
in treating the elliptic motion. Equation (33) can be expanded in the form
p'' + (l + 5)p=(l +5)j[(l-3p>
+ 3p(l-2p2
)e2 + ]
+ [-3 P +3(l-4p2
)e+- -] M +[-6p+6e+ -]p
2
+- -J,
(36)
where the right member converges so long as |p+pej<l. By 14-16
this equation can be integrated so as to express p as a power series in 5, p,
and e of the form
p = P(d,n,e; r). (37)
The series will converge for all t in the interval 0^r^27r if \e\, jp|, \S\
are sufficiently small.
We now avail ourselves of the arbitrary parameter 5 to determine the
period. We will determine d so that the period shall be 2-k in r. Since
equation (36) does not involve t explicitly, sufficient conditions for perio-
dicity with the period 2t are
P (2ir)=p(0), p'(2tt)=p'(0). (38)
These equations are not independent, for (36) has an integral correspondingto (34), which is a relation between p' and p that is always satisfied.*
This integral has the form'
p/
'4-(l + 3)p2 = (l + 3)P(p,e >M) + C. (39)
Compare Lts MUhodes Nouvelks de la Mccanique Ctkstc, vol. I, p. 87.
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THE ROTATING ELLIPSE. 63
Suppose the particle is projected from an apse so that p' = at t = 0. ThenC is a power series in e, p, and the initial value of p, which is unity. Let the
general value of p be 1 +<r. Then the value of the integral at any time minusits
value att = will
be p'
2
plus a power series in a, e, and p, vanishingfor a = whatever e and p may be. The terms coming from the rightmember all involve e or p as a factor, but there is a term coming from p
2
which involves a alone to the first degree. Therefore (39) can be solved
uniquely for a as a power series in p'2
, e, p, vanishing with p' = 0. Henceif p' = at t = 2it, then will the first equation of (38) necessarily be satisfied,
and consequently it may be suppressed.This result can also be shown from a certain symmetry which is par-
ticularly simple in the present problem. Equation (36) can be written
in the form
Suppose p = l, p'= at t=0 and that these equations have the solution
P=/i(t), P'=/ 2 W.Now consider the differential equations obtained when (40) are trans-
formed by the substitution p=Pj , p'=-p[ ,t=-t
1. The equations in the
new variables are the same as in the old, and consequently if the initial
conditions are the same (p,=
l, p' = at t = 0), the solution is
Pi=fi(r 1 )=J(-T)=p, p!=/a(Ti)=/,(-T)= -p.
Therefore p is an even function of r and p' is an odd function of r.
Suppose p' = at r = r. Since it is an odd function it must also have
been zero at t = t. Since p is even in t it has the same value at r = t as
it has at r = ir. Consequently the system is the same at r = ir as it was at
t=tt, and the motion is periodic with the period 2w. Hence if p' =at t = 0, it is sufficient to satisfy the condition p = at t = -k in order to
secure a periodic solution of (33) with the period 2ir.
It will now be shown that the second equation of (38) can be solved
uniquely for 5 as a power series in p and e, vanishing with p = 0. It is found
by integrating (36) and imposing the initial conditions p ^ 1, p ^ that
P =Poo+Pi M+Poie+ " ' *
'
where
P 00= cos\/1 + 5t,
P.o=-
f Vl + t sin Vl + 5r,
i ,i
Poi=-i + icos2W + 8T.
(41)
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64 PERIODIC ORBITS.
On substituting these results in the second equation of (38) and expandingin powers of 5 also, we get
p'(2x)-p'(0) = 6
[-!*+(terms in 5, p, e)]
+m[ 3 7r + (terms in 5, p, e)l = 0.
(42)
There are no terms in e alone, for when p = 0, the orbit becomes a fixed
ellipse and 8 = satisfies the periodicity condition. Hence (42) can be
solved uniquely for 5 as a power series in p and e, vanishing with p =
.
When the value of d obtained from (42) is substituted in (37), p becomes
periodic in t with the period 2ir, and is expanded as a power series in p and e
which converges provided |pj and \e\ are sufficiently small. It can be written
p= 2 p f ,/iV, 8= 2 2 3,,/iV. (43)
From the reasoning of 35 it follows that each p tj separately is periodic.
The range of convergence of (43) is limited in two ways. In the first
place the inequalities 1
5\< 5
, | p |< p , |
e\< e must be satisfied in order that
(37) may converge for 0^t^2tt. Then the inequalities |p|<p, , |e|<e,must be satisfied in order that the solution of (42) shall converge and givefor
15
1a value less than 8 . When
| p |
and|
e\ satisfy both of these sets of
inequalities the convergence of (43) is assured for all t.
After the explicit development of equations (43) has been made the
results can be substituted in (26), when v will be determined by a quadra-ture. The final form of v is
v= S Si;mV (44)<-o ;=o
The constant parts of the vtj are independent of e since f or p = it was
found that v = r + periodic terms.
4 1 . Direct Construction of the Non-Circular Solution. In carrying out
the practical construction of the solution, we shall make use of the facts that
(a) p^l, p'^0 at r = 0, (6) p is expansible in the form (43), and (c) each
p tj separately is periodic with the period 2ir.
On substituting (43) in (36) and equating coefficients of equal powersof p and e, it is found that the several coefficients must satisfy
(A) Pm+Poo = 0,
(5) Po,+P<n = (l-3p2o),
(C) Pio+Pio= ( 5 u) + 3)Poo>
(D) Po2+Po2 = 3poo(l-2p 01 -2p^),
(E) Pn+P= -8 11 p 00 -5 1 oPoi + 51 o(l-3p2 )-6p 00 p 10 -3p ,+3(l 4p^,),
(F) P20+P20 = _&20POO ^10Pl0~35, o p oo 3p 10 6P00,
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THE ROTATING ELLIPSE. 65
The solution of (A) satisfying (a) is p 00= cos r. The solution of (B) is
given in equation (18). The solution of (C) is not periodic unless the coeffi-
cient of Poo is zero. Imposing also the condition (a), 5 10=
3, p 10= 0.
The termp
02 is
givenin
equation (19). Equation (E) becomes explicitlyPa + Pn = 3 /2 <5 n cosr 3/ 2 cos2r. Upon imposing conditions (a) and(c), the solution of this equation is found to be
*n =0, P n = -f+cos t+^cos2t. (45)
The explicit form of (F) now becomes p 2o+p 20= (- 8 20 +,S) cos r, whose
solution satisfying the conditions (a) and (c) is
8 20=
3, p 20= 0.
Hence the final expressions for p and 8 as power series in e and p. are
P = cost+ [-} + { cos2T]e+ [- l + cosr + l cos2r]pe
+ |[ cost +cos3r]e2-|- ,
5=-3/i+3 M2 +
(47)
(48)
1).
Then
where, from (32), t is to be replaced throughout by w(t T)/Vl + 8-
The differential equation defining the general term is
Po+Pi>= -8,jPoo+Fij(8K\ , Pkx) (k=1, . . ., z'-l; X = l, . . .
, j
Suppose all the p KX and 8 KX for which k<?' , X<,;' have been found,
this equation can be written in the form
pr ; +p u = ^f + (-5 <y +4 (
r)cosr+A 20, cos2r+ -Mi'* COS kt, (49)
there being no sine terms. Its solution satisfying conditions (a) and (c) is
f> A UJ)
Af> (A A \A -\ -++ ffrjww- rwi-ost - cos2t- cos KT.
l (50)
The 5(; and all the coefficients are uniquely determined. Therefore the
process can be continued without modification as far as may be desired.
After the transformations (32) equation (26) becomes
^= vT+^jl+2p+2pe +(-i+3p
2
)e2
+6p Me-f-3p2 +
J.
Upon substituting the value of p given in (48) and integrating, it is found
that
t;_ J ; o =[l+2p-f-3M
2 + ]T+[2sinT]e+[Jsin2T]e2 -
r-
(51) X >
where t is to be replaced by w(tT)/Vl + S. The four arbitrary constants
of integration are a, e, T, and v .
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66 PERIODIC ORBITS.
42. Properties of the Solution. It has been proved that each p tl
(except Poo) and p' Jvanish at t = 0, that each p it
is periodic in t with the
period 2rr, and that p is an even function in r. Hence each p tJ involves onlycosines of multiples of t. It has also been noted that 8 depends upon no
terms independent of p.
There is no term pm distinct from zero, for when e = the differential
equations are satisfied for the same initial values of the variables by p = 0.
Or, it is seen, from the development of the second term of (33), that the onlyterms which are independent of e involve p to the first degree alone. Conse-
quently the right members of the equations corresponding to (49), whichdefine p (0 ,
will be [ 5 w -|-/(p)] cos t alone. The periodicity condition makesit necessary to put 5 (0 =/(p) ,
and the initial conditions then make p <0= 0.
The expression for the right member of (51) has no term independentof p, except unity, in the non-periodic part, for when p = the ellipse is
fixed and this part reduces simply to r. On making use of all of these facts
and some simple artifices, the labor of actually constructing the series can
be very much reduced.
The radius r completes its period in t=2x, or = 2ir\/l-|-5/w. It follows
from (51) that the longitude of the radius has increased in this interval byVi + 5 [l+2p+3p
2 + ] 2t. Therefore the line of apsides has movedforward in this interval through the angle VT+S [l+2p+3p
2 + ] 2t 2 it.
Hence its average rate of angular motion in t is
d&
vT+M1+2 m+3m 2 + ] 2ir -2ir I
r,.9 -i
fKtn
w
where d> is the longitude of the nearest apse. The parameter p can bedetermined so as to secure any rate of revolution of the apsides not too great.
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Chapter III.
THE SPHERICAL PENDULUM.
I. SOLUTION OF THE Z-EQUATION.
43. The Differential Equations. The problem of the spherical pendu-lum falls in the class of those which can be treated by the methods of
periodic orbits. Its simplicity makes it particularly well suited to illustratingthese processes, and its value as an introduction to the subject is increased
by the fact that it is easy to verify the results experimentally. It is doubt-ful whether there is a problem which is superior in these respects.
Let us take a rectangular system of axes with the positive 2-axis
directed upward and with the origin at the fixed point of the pendulum.The pendulum is subject to gravity and the normal reaction, N, whichwe shall take with the positive sign when directed outward. If we representthe radius of gyration by I, the motion of the pendulum satisfies
x2
+y2
+z2 = l
2
,mx" =
Nj, my" =N^, mz'^N^-mg, (1)
where the accents indicate derivatives with respect to the time.
The last threeequations
of(1)
admit theintegral
m(x'2
+y'2
+z'2
) =mv2 = mg{-2z+c l ), (2)
where ct is the constant of integration.
The normal reaction exactly balances the centrifugal acceleration of the
pendulum due to its motion and the component of mg along the normal to
the surface; hence,, mv 2 mv 2
. zN=-- r -F N =-- r +mg J,
where FN is the normal component of mg. On making use of (2), we get
N= m i3z-
Ci) . ( 3 )
Hence equations (1) become
x2
+y2 +z
2 = t2
, y"=faz- Cl )y,
x"=(
$z-c^x, z"=f (3z-c t )z-g.
(4)
The last equation is independent of the others and is therefore solved first.
After it is solved the second gives x in terms of t, and then y can be found
from the first. 6 7
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68 PERIODIC ORBITS.
44. Transformation of the z-Equation. It will be convenient to trans-
form the last equation of (4). It admits the integral
z"=l(2z-c l )z
2
-g(2z-c 2 )=f(z), (5)
where c 2 is a constant of integration which is independent of c,. If wesubtract this equation from (2) and reduce the result by (2), we find
g(c,-c 2 )= (x*+y") (
1 -)-
jz'\ (6)
Now z2
^ f . In the case where the spherical pendulum reduces to the simple
pendulum z takes the value =I, and at the same time z' = . In this case
cl
c 3= 0. In the spherical pendulum z> I when z' = 0; consequently in
this case cl
c 2 >0. Hence in all cases of the physical problem C, c 2 5;0.
Now consider equation (5). If the initial conditions are real, zr
2 =f(z )
is zero or positive and
/(_) = -oo, /(--?(*-<*,) 0,
/(z,)=0 (-**+,f(+l)=-9(c l -c 2)m /(+oo) = +oo.
Therefore the equation /(z) = has always three real roots. Let thembe a, ,
a 2 ,and a 3 , where the notation is chosen so that a! ^ a 2 ^ a 3 . Then
equation (5) becomes
z' 2 " y (*
- a,) - a,) (z - a,) .
On comparing this equation with (5), it is seen that
2 (a 1 + a 2 +a 3 )=c l , a, a 2 +a 2 a 3 +a 3 a t=
f, 2a 1 a 2 o 3 =-c 2 f, (7)
It will now be shown that a 3 satisfies the inequalities I <! a 3 ^ 0. It
follows from the last of (7) that c 2 is negative if a 3 is positive. On puttingz' = and z = a 3 in (5), we get
c 2= 2a 3
+p(-2a3 +c 1 ).
By hypothesis the first term on the right is positive, and by (2) the secondcan not be negative. Therefore c 2 must be positive, which contradicts the
implication from the last of (7). Therefore a 3 ^0.Some special cases may be indicated :
(1) It follows from (5), (6), and (7) that a 3= -I implies that o,= +Z,
2a 2=
c,= c 2 , a, a 3
= 2l. The constant a 2 is not determined, and we shall
suppose it is less than +1. This case is that of the ordinary simple pendu-lum making finite oscillations. In the sub-case where a 2
= -I, we havec,= c a
= 21 and the solutions of (4) are x = y = 0, z= I.
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THE SPHERICAL PENDULUM. 69
V2) If a 2= +1, it follows from the same equations that a 3
= -I, 2a 1 ~C 1 *e L
The constant a x is not determined. If a, > I, we have the case of the simpleper dulum swinging round and round, and a,
- a 3 > 2 1 . In the sub-case wherea
x=
l,we have
c^c^+21, and the solutions of (4) are x=y=Q, z=
+l.(3) If a 3
=0, it follows that a 2 ^0. Therefore the second of (7) can
not be satisfied except by a 2=
0, ax= oo . Then, from the first equation we
get c,= oo. This is the case of revolution in the xy-p\a,ne with infinite
speed, and of course can not be realized physically. Excluding this caseand that of the simple pendulum, the constants a, ,
a 2 ,a 3 satisfy the
inequalities -Z<a 3 <0, l<a 2 <+l, ^> +1.Now make the transformation
z a 3 =(o 2 a 3 )u2
. (8)
Thenequation (5) becomes
Also let
u'2 = ^"i aa) -
(1- u 2
) (l - ^^ u 2
). (9)& I V a t a 3
/
where t is an arbitrary initial time and 8 is a constant as yet undefined.
The constant /x satisfies the inequalities 0^/x^l- Then (9) becomes
ii2
=(l+5)(l-u2
) (l-,xu2
)=F(u), (11)
where u is the first derivative of u with respect to the new independentvariable r. The first derivative of (11) is
w=(l+6)[-U+M)w+2 Mw3
]. (12)
45. First Demonstration that the Solution of (12) is Periodic, and that
u and the Period are Expansible as Power Series in ju. It will first be
shown that, for any initial conditions belonging to the physical problem,
except when ix=
\, the solution of (12) is periodic. By the fundamental
existence theorem of the solutions of differential equations* the solutions of
(12) are regular in t for all finite values of r and u. For real initial condi-
tions the coefficients of u and its derivatives expanded as power series in t
are real, and by analytic continuation they remain real for all finite real
values of r provided u does not become infinite. Now consider the curve
F = F(u). Suppose u = u at t = and that u is positive. Then u is
increasing at a rate which is proportional to the square root of F(u ), and it
continues to increase until u = 1 . It can not increase beyond + 1 for then uwould become a pure imaginary, and it has just been shown that it alwaysremains real. It can not remain constantly equal to 1 unless n = \, for
otherwise u 1 does not satisfy (12) . Therefore, unless n = 1, u will increase
*Picard's Traiti d' Analyse, vol. II, chap. 11, 111.
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70 PERIODIC ORBITS.
to 1 and then decrease to 1; then, in a similar way, it changes at u = 1
from a decreasing to an increasing function. That is, at u= 1 the func-
tion F(u) changes sign and u varies periodically between + 1 and 1 .
This result follows, of course, from the fact that in the present problemu is the sine amplitude of t, one of whose properties is that of havinga real period, but the argument given above applies to much more general
cases, and the result can be read from the diagram for F = F(u). It maybe mentioned in passing that the imaginary period of the elliptic function is
associated in a similar -way with the portions of the curve between + 1 and
+ 1/Vjl, and between 1/ V/I and 1 .
The period of a complete oscillation is found from (11) to be
p= 2 r+ l du"+ du
J-i V(l-u 2
)(l-uu2
)'(13)vT+sJ-i V(i-0(i- M<)
which is finite unless n = l. We shall exclude this exceptional case. It is
well known that
p =vfeE 1 +(i)"*+G3)W +G:l:::S
tl))>+ ] <>
In t the period is
That is, the period is expansible as a power series in /x, and in the presentsimple case the series converges provided | /x |
< 1 .
The constant 8 has so far remained undetermined. If we let
v^r-[i+(i)V+(a)V+ ],
the period in t will be simply 2w. On solving this equation, we find thatthe required value of 5 is
5=|m+^m
2
+, (16)
which is a power ser ies in /x.
Now consider equation (12). By 14-16, this equation can be inte-
grated as a power series in n, and |m| can be taken so small that the series
will converge for all t in the interval O^t^t, chosen arbitrarily in advance.If t, is greater than P and the constant of integration is chosen so that
(11) is the first integral of (12), that is, so that the period of the motion is
P, then it follows from the periodicity of the solution that the series will
converge for all finite values of r. That is, both u and T arc expansibleas power series in n.
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THE SPHERICAL PENDULUM. 71
46. Second Demonstration that the Solution of Equation (12) is Peri-
odic. -While the proof of 45 is sufficient for the construction of the solution,
it will be instructive to give another demonstration of its periodicity. In
many problemsthe former methods can not be
applied.By 14-16, equation (12) can be integrated as a power series in the two
parameters m and 8 of the form
l-2 2w(t)5V, (17)( = 1=0
where the u tJ are functions of r to be determined by the conditions that
(17) shall satisfy (12) and the initial conditions, and where |5| and \n\ can
be taken so small that the ser ies wil l converge for any interval O^t^t^chosen arbitrarily in advance. We may take the initial values w(0)=0,
u(0)=a,and from (11) we see that in order to
getthe same solution as
before we must put a= Vl-\-8 at the end. The subsequent steps of this
demonstration would not be essentially modified if we took general initial
conditions. From these initial conditions we get
= u tJ (0) V> a = tt(0) v,(=0 1=0 <=0 1=0
from which it follows that
u(0)=0 (i,i=0, oo), MO0 (0)=
a, <(0)=0 (i+j>0). (18)
On substituting (17) in (12) and equating coefficients of corresponding
powers of 5 and n, we get
Woo + ^oo = 0, W1 o + W10 =-W 00 , (19)
The solution of the first of these equations satisfying (18) is
u 00= a sin r. (20)
Upon substituting this result in the right member of the second equation
of (19), integrating, and imposing the conditions (18), we find
w,o=
2sirXT +
2rC0ST - ( 21 )
Hence we have
u = a sin r + % [- sin t + r cos t] 5 + higher powers of 5 and m- (22)
Since the right member of (12) does not contain r explicitly, sufficient
conditions that u shall be periodic with the period 2x in r are
u (2 tt)- u (0)
= 0, u (2 x)-
(0)= 0. (23)
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72 PERIODIC ORBITS.
It will now be shown that the second one of these equations is necessarily
satisfied when the first is fulfilled. Let w = 0+v, u =a-\-'v, where v and v
vanish at t = 0. Then (11) becomes
{a + vf= (1 +5)
(l-
a
)(l-nv
2).
On making use of the fact that l + 5 = a, we have
2av+v a+a(l+n)v
i
-anv' = 0.
There are two solutions of this equation for v, but the one which vanishes
at t = must be used. It has the form
v=vp(v), (24)
where p(v) is a power series in v. Since u(0)=v(0) =0, it follows from
the first equation of (23) that w(27r)=0. Then, from (24) we havev (2w) =u(2tt) m(0)=0. That is, by virtue of the existence of the
integral (11), the second equation of (23) is a consequence of the first.
Now let us consider the solution of the first equation of (23). Uponsubstituting u from (22), we get
= Tab + terms of higher degree in 8 and /j.. (25)
It follows from the theorems of 1-3 that this equation can be solved for 5
uniquely in the form
= mP,(m), (26)
where p,(/x) is a power series in /x, which converges if \n\ is sufficiently small.On substituting this result in (17), we have
U=lu i {r) l x1
, (27)
which converges for all 0^t^2it for \/x\ sufficiently small. It is sufficient
that| fx |
and1
51 satisfy the conditions necessary to insure the convergence of
(17) and the solution of (25). These conditions can both be satisfied byvalues of m different from zero because the expression for 8, given in equa-tion (26), carries m as a factor.
Hence, the periodicity conditions having been satisfied, we have provedthat the solution is periodic. It has been found to be expansible as a powerseries in it, and the period, which in t is
r= 27rZV2(l + 5)
V<jf(ai-a 3 )
is also expansible uniquely as a power series in n. It is clear that this modeof demonstration applies to a wide class of equations, for the explicit values
of only the first terms of the right member of the differential equation, the
general properties of its convergence, and the existence of a first integral
have been used.
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THE SPHERICAL PENDULUM. 73
47. Third Proof that the Solution of Equation (12) is Periodic Thereis a certain symmetry property of the solutions which can be used to
simplify the demonstration that the motion is periodic. It will be shownthat the motion is
symmetricalin
t with respect to the value w=
0, which,by (8), corresponds to z = a 3 ,
or the lowest point reached by the pendulum.Suppose u = 0, ii = a at t = and that the solution of (12) for these
initial conditions is
w =/i(t), /,(0)=0, *-/.(*), / 2 (0)=a. (28)
Now make the transformation
u= v, u = v, t=<t. (29)
Then (12) becomes
v = (l + 8)[-(l+iJL)v+2 l xv 3 l (30)
Hence if v = 0, v = a at a = 0, the solution of (30) is
v-M*), /.(0)=0, #-/,(), M0) = a, (31)
where /, and / 2 are the same functions of <r that /, and f 2 ,of (28), are of t.
On substituting (28) and (31) in (29), we get
w = / 1 (r)=-/ 1 (-r), *-/,(r)-+/.(-r).
Therefore u is an odd function of t when u = at t = 0, and hence it
follows that sufficient conditions that the solution of (12) shall be periodic
with the period 2ir are
w(0)=0, !*(*)= 0. (32)
The solution (22) was obtained with the initial condition w(0) = 0. Hencethe second equation of (32) becomes
= - a8+ higher powers of 6 and /x- (33)
The solution of this equation for 5 and the further discussion are preciselylike the treatment of (25), and lead to the same results.
48. Direct Construction of the Solution. It has been proved that 8
can be expanded as a power series of the form
S = 5 iM + 5 2 m*+ (34)
such that, when (12) is integrated as a power series in m of the form
u = u +u 1 h+u 2 ij.
2 + (35)
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74 PERIODIC ORBITS.
with the initial conditions u(0)=0, ii(0)=a, u will be periodic with the
period 2t. In fact, the value of 5 is given in (16), but we shall make use
only of its expansibility in this form; in more complicated problems its
explicit value would not, in general, be known. Since u is periodic with the
period 2ir, we have
u(2t+t)-u(t)= 2 [t*,(2T + T)-u,(*)]/i' = 0,
and this equation holds for all \n\ sufficiently small. Therefore
uj (2t+t)-u ] (t)=0, (j
=0, l, . . . oo). (36)
Hence each us separately is periodic with the period 2w.
Instead of determining the solution by the initial conditions u(0) = 0,
u (0)-
0, we may use u (0)=
0, u(w/2)= 1.
Or, by (35) ,
2 u,(0V=0, 2 u,(W-i.Therefore we have
o(0) = 0, *(g)-l.
,(0)=0, w,(f)=0,=
1, 2, . . . oo).
(37)
Now we determine u and h by the conditions that (12) shall be satisfied
identically in /x and t, and that the conditions (36) and (37) shall be ful-
filled. By direct substitution and equating of coefficients, we find
^0 + ^0=0,
w 1 +w 1 =-(1 + 5 1 )w +2m*,
w2 +W2=-(5i + 5 2 )w -(l-f-5 I )w 2 +6WoW 1 +25 1 o
(38)
The solution of the first equation of (38) satisfying (37) is u = sin t.
Then the second equation of (38) becomes
^i+w^-Q + SjsinT-l- | sinr-i sin3r. (39)
In order that the solution of this equation shall be periodic we must set the
coefficient of sinr equal to zero. Then the solution satisfying (37) is
,- j, !,-- [anr+anSrl. (40)
Upon substituting the results already obtained in the third equationof (38), we find
(1\ 27 1 ^- + 8 2 ) siiit+ sinr- - sin3r- sin5r. (41)
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THE SPHERICAL PENDULUM. 75
Upon setting the coefficient of sinT equal to zero, as before, and integrating
subject to the conditions (37), we find
5 2=
,u 2
=[7 sinr + 8 sin3r+ sin5r]. (42)
The induction to the general term can now be made. We assume that
u,
. . .,u
i_ l ; 5j ,
. . ., (_! have been determined and that it has been
found that us
is a sum of sines of odd multiplies of r, of which the highestis 2j + l. The differential equation for the coefficient of \x is
ut +u t
= -S t u +F t (8 K ,MX ) (k, X = 0, : . .
, t-l), (43)
where the F t are linear in the 8 K and of the third degree in the U\ . The
general term of Ft
is
wherera = or 1,
"i+" 2 +"3=lor3, (44)
ritK-\-v l\ 1 -\-v 2 \ a -\-v 3 '\ 3
= i or i1 (v l +v 2 +v 3= l or 3).
It follows from the second of these equations that there are an odd numberof odd Vj . Consequently T t is a sum of sines of odd multiplies of r. The
highest multiple of r is
tf (= *
1 (2A 1 + l) +^ 2 (2X 2 +1) + *> 3 (2X 3 +1). .
On reducing this expression by the third of equation (44), it is found that
Nf= 2mic+2i+l, the greatest value of which is, by (44),
Nt= 2i+l. (45)
Hence (43) may be written
u+u i= [-8 t +A?]smT+A 3
i) sm3T+ +4h an(2t+l)r, (46)
where the A l
2 \ +l are known constants.
In order that the solution of (46) shall be periodic, the condition
*-A?'
( 47 )
must be satisfied, which uniquely determines 8 t . Then the solution of
equation (46) satisfying the conditions (37) is
,-? sin r+a? sin 3r+ + &., sin(2i + l)r,
4<Sa ^' =_
4X(x+T)(X l '---' 1h
a,= Zl(- l) a^~ Z 4X(X+1)
(48)
X = l A-i
The solution at this step has the same form as that which was assumed for
u a ,. . .
, rt ( _, , and the induction is therefore complete.
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76 PERIODIC ORBITS.
On collecting results, we have for the first terms of the solution
M= [sinr]+ ^r [sinT+sin3r]M+ 5L [7sinT+8sin3T+sin5T]/i2
+-10 L . , ...jp ,
256
(49)
(50)
5 = 0+|M+gM2 +
And substituting these results in (8), we get for the final result
2 = a 3 +|(a 1 -Oj)[l-COS2T]M + ^(a 1 -a 3 ) [1-COs4t]m2
+ S5(o 1 -o 3 ) [16+3 cos2t- 16 cos4t-3cos6t]m3 + J
the expression for T agreeing with that found in (15).
49. Construction of the Solution from the Integral. In the direct
construction of the solution we have made no explicit use of the existence of
the integral (11). We shall show that it can be used to check the computa-
tions, or to furnish the solution itself.
Equation (11) can be written in the form
<p(u, u, /x)=0.
Since it, u, and h can be expanded as converging power series in fi, we have
> = >o+^iM+*>2M2 + ' * =- (51)
Since this equation is an identity in n, it follows that
<Pi=
<P t (u ,u ,5 2 ,. . .
,u
t ,u t , 6,) =0 (*-0, ...), (52)
where <p t is a polynomial in u ,. . . , 6, . It follows from (11) that <p,
is linear in the 5X , of the second degree in ii\, and ie\,, and of the second or
fourth degree in u\ and wx,. Therefore <p tis a sum of cosines of even
multiples of t. It is seen without difficulty that the highest multiple of
t in <pi is 2i+2. Hence we have
<p i= B?+B?co&2T+ +fl*? +1 cos(2i+2)T = 0. (53)
Since this equation holds for all values of r, it follows that
B&=0(f=l, . . . oo; X=0, . . . ,t+l). (54)
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THE SPHERICAL PENDULUM. 77
The B^lare functions of the a$ +1 and 5( ,
and equations (54) constitute
a searching check upon the computation of these quantities. If by somenumerical accident an error were not indicated at any particular step, it
would be revealed by the failure to satisfy (54) at some later step.But the c4x+i can be computed from (54) , as will now be shown. Suppose
u a ,. . .
,m
< _ 1 ;5 2 ,
. . .., _, have been computed. Then we find from
the explicit expression (11) that
<Pi= 2u o u t +2u u- 5,(1 -ul)+^ t (u x ,
u x ,5 2X ) (X =
, t-1),
the \J/ ( being known functions. Hence, using the notation of the first equa-tion of (48), we get
^ = 2ar>+0-i5 4+C =0,
5 2 = 0+4a^-i5 1 +C< = 0,
5 2 x= 2(X-l)a 2 x _ 1 +2(X+l)aS +1 +C 2 x
= (X-2 i),
B2 +2= 2ia% +l +0+C<X =
0,
(55)
where the C2 x (X =0, . . .
, i+1) are known constants. On solving these
equations beginning with the last, we find
(o _ C 2<+2 (o _ X-f- 1 (o C 2 xtt 2<+l n- >
a 2X-l~ A I a 2X 2C\ 1 )* ' ' " ' ' ''
0,-oa 3 -\-ZL 2 ,o 1 --d J -L ,
(56)
which uniquely determine 8 { and the a 2 x +1 .
Let us apply these equations to the computation of the first terms of u.
Suppose M = sinr and take i = 1. Then we find from (11) that
<p t= 2u u
x +2u u x-
5,(1-
u\) +u\- u\ ;
whence
< - 2ai" - 15l + 1
= 0, B? = 4a;"-
1 1,= 0, B? = 2a -
f= 0.
Therefore(1)
_1 *
i. n(1)
a 3 _ 16' ~ 2 ttl 16 '
agreeing with the results already found. The process can be continued as
far as may be desired.
Two different methods of computing the solutions have been developed.
They are both reduced by the general discussion to the mere routine of
handling trigonometric terms. When they are both used each serves as a
check on the other. There is little advantage with either over the other, so
far as their convenience is concerned; the integral has a slight advantage in
that when using it the computations are made with cosines, and the disad-
vantage of involving u to the fourth degree instead of only to the third.
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78 PERIODIC ORBITS.
II. DIGRESSION ON HILL'S EQUATION.
50. The x-Equation. The value of z was given explicitly in (50), and
since it is periodic the second equation of (4), after transforming to theindependent variable t, has the form
x+[a2 +0 lM +0 2 M
2 + ]x =0, (57)
where a is a constant independent of /x, and 0, , 2 ,. . . are periodic functions
of t, having in this case the period it. Since the period can always be made2-k by a linear transformation on the independent variable, we shall supposefor the sake of uniformity that the period is 2-k.
This problem belongs to the class which was treated in Chapter I, andcan be transformed to the form considered there. But it is now in the form
used first by Hill, and later by Bruns, Stieltjes, Harzer, Callandreau, etc.,and because of its historical interest and for the sake of comparison with
this earlier work, it will be treated directly in the form (57) .
51. The Characteristic Equation. Suppose that with the initial con-
ditions x(0) =1, x(Q) =0 the solution of (57) is
x = <p( T ), *-*(t), v(0) =l, #0)-0,
and that with the initial conditions x(0) =0, x(0) =1, the solution of (57) is
*-*(t), i*(r), *(0)=0, *(0)= 1.
The determinant
<p(.t), v?(t)
<Kt), f (r)
being equal to unity at r = 0, these solutions form a fundamental set. In
fact, A is independent of t for an equation of the form (57), as was shown in
18. It follows from the initial conditions that its value is unity. Hencethe general solution of (57) is
x^^tO+c^t), x = c1 <p(t)+c 2 \},(t). (58)
where c, and c 2 are arbitrary constants.
Now let us make the transformation
x = e", (59)
where a is an undetermined constant. Then equation (57) becomes
+2a+a 2
+[a2 +0 1 M+0 2 M
2 + J 1-0. (60)
The general solution of this equation is, by (58) and (59),
5 = e~ aT[cMr) +C,*(t)], k = e" ar
[c^(r) +c2*(t)}
- ae- aT[Cl <pM +c,*(t)]. (61)
A =
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hill's differential equation. 79
We now raise the question whether it is possible to determine a in sucha way that shall be periodic with the period 2tt. It follows from the formof (60) that sufficient conditions for the periodicity of with the period 2ir are
(27r)-(0)=0, !(2)-i(0)-0.
On substituting from (61), we get, after making use of the initial values of <p,
ip, \f/, and fa
[e-2a
>(27r) -l] Cl +e-
2aV(27r)c 2
=0, 1
e-2aV(27r)c 1 + [e-
2aV(2x)-l]c 2
= 0.
'
J
(62)
In order that these equations may have a solution other than the trivial onec,= c 2
=0, the determinant of the coefficients of c, and c 2 must vanish; or,
D = e-4 <p{2ir)-e
2a\ *(2x)
<K2tt), +(2ir)-e2a * (63)
Since A is equal to unity, equation (63) is a reciprocal equation, and becomes
Z) = ( e ^)^_^(2 7r )+^(2x)]e2a ' r +l =
0, (64)
of which the roots are e 2a,x and e-2a
'". (This is not the a, of 44.)
When the value of e2a ' T which satisfies (64) is substituted in (62) the
ratio of c, to c 2 is determined. Then equations (61) give(1) and (I)
periodic with the period 2x. We get a second solution<2)
and<2)
by usingthe other root e~
2a'\ The (1) and (2>
will each carry an arbitrary factor.
We shall determine these factors so that U)(0)
= <2)
(0) =1, and multiply
the solutions by arbitrary constants at the end.
Consider equation (64). If |p(27r)+^(27r)| <2, at is a pure imaginary;
if <p(2w)+y/(2ir)>2, a, is real; and if <p(2ir)+^(2ir)< 2, at
is complex. In
the first case x remains finite for all real values of r; in the second case x
becomes infinite as r becomes infinite through real values; and in the third,
x = oo for r = oo except for special initial conditions. It is found from (57)
that <p(2ir)=
\j/(2ir)= cos a w for /j,
= 0. Therefore the part of a, which is inde-
pendent of fi is the pure imaginary aVi. Suppose a is not an integer;then a, is a pure imaginary for all real values of n whose modulus is sufficiently
small. If a is an integer, the value of c^ for real values of n whose modulusis small may be purely imaginary, real, or complex according to the values
of <p(2w) and ^(2ir).
Some of the more important properties of(1) and <2)
will be derived.
There are two particular solutions of (57) of the form x = e a,T such that a,
is a constant reducing to a V l for n = 0, and such that is periodic with the
period 2ir,viz. e a,T (1) and e -a,T <2>
. The coefficients of (57) by hypothesis are
all real, and the 6, are sums of cosines of multiples of t. Therefore, if the
sign of V l be changed in a solution the result will be a solution. Suppose
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80 PERIODIC ORBITS.
a and n have such values that o, is a pure imaginary. Then it follows that( "
(\^TY) = 0>( V l). Similarly, since (57) is unchanged by changing
the sign of t, it follows that(1)
(r)= (2)
(-
r) . And finally, since (57) is
unchanged by changing the sign of both V l and t, it follows that
r(v^=T,T)=r(-v^T,-r), |(v^t,r)-{*(-V=T,-r). Therefore
in the expressions forll> and <2) the coefficients of the cosine terms are
real and of the sine terms pure imaginaries, and (1) and <2)differ only in
the sign of V l. Hence, writing them as Fourier series, we have
(1) = 2 [a jcos jr +Vlbj sin jr] ,
(2, = 2 [a jcos jt V lbj sinjr],
where the a,j and the bj are real constants. It follows from this that it is
sufficient tocompute
(,).
Any solution of (57) can be expressed linearly in terms ea,T (1> and
e-a,r(2). It follows from the initial values of <p, <p,(1)
, and (2) and the
equations above that
Sincell) and C2)
are periodic with the period 2 x, and since their initial values
are unity, we have
I( e
2a.x +e -2alT) m cosh2o 1
7T = (27r).
But by (64), e2a''+e-
2a,T = ^(2T)+^(27r). Therefore v (2t) =^(2r).When a t
=/3 V l is a pure imaginary, as it is in many physical problems,
we havecos27r =
<p(27r). (65)
This equation has the same form as that developed by Hill in his memoir on
the motion of the Lunar Perigee, Acta Mathemetica, vol. VIII, pp. 1-36, andCollected Works, vol. I, pp. 243-270. It is also derived differently in
Tisserand's Mecanique Celeste, vol. Ill, chap. 1.
Equation (64) or (65) furnishes a means of computing the transcendental
a, because, under the hypotheses on (57), <p can always be found, for exampleas a power series in n, with any desired degree of accuracy. Though this
constitutes a complete solution of the problem and there are no difficulties
in carrying it out except those of the lengthy computations, we shall find
it convenient to make use of more of the properties of equation (57), andto find both ai and , otherwise.
When m= the general solution of (57) is known to be
z = a,eoV= > r +a 2 e- av/:r ' T
,x = aV^l [a l e aV
~iT -a 2 e- a '
vr=:iT] l (66)
where a, anda
2 are arbitrary constants.
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HILL S DIFFERENTIAL EQUATION. 81
It follows from the form of (57) that x can be expanded as a power series
in n which will converge for 0<t^2tt if \n\ is sufficiently small. Hence
Therefore equation (63) becomes
cos2a7r-e 2a,r + 2 ^(2*V, -sin2air + &,(2it)mx
(67)
D = e"w OS
-osin2ax +2 <p x (27r)M\cos2a7r-e2a,r
4- 2 ^ x (2ir)/ix
= 0. (68)
This equation expresses the condition that (57) shall have a periodic solution
of the form (59), where is periodic with the period 2w. If it is satisfied
by a = a it is also satisfied by a = a +v Vl, where v is any integer. Thesedifferent values of a do not, however, lead to distinct values of x. Weshall use only those values which reduce to a V l for n = 0.
52. The Form of the Solution. For n = the principal solutions of (68)
are o,0) = +a V l and ci2
0) = a V l. There are three cases depending
uponthe value of a:
Case I. a 5*0 and 2 a not an integer.
Case II. a t^O and 2a an integer.
Case III. a = and therefore a = of = 0.
Case /. This may be regarded as being the general case, and is that
actually found and discussed by Hill and later writers on the same subject.
It follows from the form of (68) that
D = P(a, M), (69)
where P is a power series in a and n which vanishes f or /x= 0, a = a V l.
It is also easily found for /x= and a = * a V l that
<>P = 4 T ^azT sin2a7r e* 2 "^^*,da
which is distinct from zero under the conditions of Case I. Therefore it follows
from the theory of implicit functions that (68) can be solved for a in the form
a1 =+aV^T+< ,
M+ aiV+ ' * *i
a 2 =aV l +C4" ju + <42>
M2 + ' ' '
,
(70)
where the series converge for |/x| sufficiently small. Since the equation for a
is a reciprocal equation in e2
", it follows that a[J) = - 0^0 =
1, 2, ... ; , ).
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82 PERIODIC ORBITS.
If we substitute either of the series (70) in (62) ,we shall have the ratio
of c, and c 2 expressed as a power series in m- If this result and equations
(67) are substituted in (61), will be expressed as a power series in n,
converging for \n\ sufficiently small, and carrying one arbitrary constant
factor. We shall omit the superfix and adopt the notation
? = o+?,M + ^M2+ (71)
Since the periodicity conditions have been satisfied, is periodic and it
follows from its expansibility that each, separately is periodic. Hence
it follows from this property and the initial condition (0) = 1 that
&(2t+t)-,(t)-0 (4*0,1, ... 00), I
(0)=
1, &(0) =(t= l,2, . . . oo). j
(72)
It will be shown when the solutions are constructed that these properties
uniquely define their coefficients.
Case II. In this case we find from (68) f or /x= 0, a = a V^T, that
fa~=0
' 10=^(2^)^(2^-^(2^)^(2^,
if? =^ ^ = (-1)^2^(2^(2.)].
Hence if we let a = aV 1+/3, the expression for D has the form
47r2
J3
2 +c n /JM+c 02 M2 + ' =0, (73)
where in general c n and c 02 are distinct from zero. In order not to multiplycases indefinitely we shall suppose that c n and c 02 are distinct from zero andthat the discriminant of the quadratic terms of (73), viz. 5 = c
2
ll 16n 2c
l
is distinct from zero. Suppose the quadratic terms factor into
K02 1
4tt2
(/3-M 03-M,where now fe,?^6 2 since 5?^0.* Then by the theory of implicit functions
equation (73) is solvable for /3 as converging power series of the form
,3 1= 6 lM +#V+ , /3 2
= 6 2m+&V+ (74)
Hence we get two solutions for /3, and consequently for a, as power series in
M starting from the root a= +aV l for /x= 0. There are two similar
ones obtained by starting from a = a V l for n = 0, but they do not lead
to distinct solutions since they differ from the former values by purely
imaginary integers. Then, by means of (62) and (61), we obtain the final
solutions as before.
There are other sub-cases, for example b 1= b 2 , all of which can be
treated by the theory of implicit functions, but they will be omitted.
*Since = 0, and 0= 01 are the roots of (63), it follows that in this case &, = &,, and that they are
therefore distinct unless 6i = 6,=0.
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86 PERIODIC ORBITS.
equations, and we shall assume here that they are distinct. This was the
case treated in 52, Case II, and when the two values of a[l)
are equal the
solutions may be developable in a different form. After o"' has been
found, b can be obtained at once from either of equations (88). There is
a difficulty only if S 21 = 5 22 = 0, when one solution for b becomes infinite;
but in this case we impose a different initial condition on .
After satisfying equation (88), the integrals of (87) have the form
bF-PM+B?, 61 > = Q 1 (r)+ 2
where P, (r) and Q, (t) are periodic functions of r, and P"' and P"' are unde-
termined constants of integration. These results substituted in equation
(80) give, after applying the condition ,(0) =0,
&-Jtf+P I <r)+Q 1 (r)ri
^%^() +^[P, (0) +0,()J Y
U
^(90)
where P"' is so far undetermined.
It is necessary to carry the integration one step further in order to
prove that the general term satisfying the periodicity condition and the
initial condition can be found. The differential equation for the coefficient
of ju
2is
l + 2a V=ll= -ai^av^T&r+e-*^] 1 m)-2aV^la[
1) B (
;) -d
1 B[1) +F 2 (r), j
where ai2) and B[
l) are undetermined constants, and where F 2 (t) is an entirely
known periodic function of r.
In the case under consideration the equations corresponding to (88) are
-2aV=lba? )
-(2aV=la[i) +d 1 )B[
l) +d 2=
(),
j- of + 2a V=l 8 2 , B|!) + 5 2
=0,
where d 2 and 8 2 are known constants depending on P 2 . The unknowns
af and P'," enter (92) linearly, and, by means of (88), the determinant of
their coefficients becomes 4a v^-T a', , which, by hypothesis, is distinct
from zero. Therefore a'," and B[l) are uniquely determined by these equa-
tions. When equations (92) are satisfied, the solution of (91) satisfying the
initial condition is
i,= B<?+P a (r)+Q a (T)e-*'*=
iT
-{QM+ ^T ^ +4]}e-2oV ^ T
) (93)
which has the same form as (90). Therefore the next step can be taken in
the same manner. Thus it is seen that the process is unique after the choice
of the sign of a"' is made, and in this way two solutions which satisfy the
periodicity and initial conditions are obtained.
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hill's differential equation. 87
55. Direct Construction of the Solutions in Case III. In this case the
solutions were proved to have, in general, the form
= o+iVM+fcM+ , a = + a(1V+a <2V+ (94)
Substituting these equations in (60), we have for the term independent of VJ1
u=o,of which the solution satisfying the conditions (72) is
fc-1. (95)
The differential equation which the coefficient of Vm must satisfy is
and the solution of this equation which satisfies the conditions (72) is
i=
0,a
(l) = an undetermined constant.(96)
The differential equation which defines the coefficient of /x is then
2 +2a<2 > = -2a m ^-(O'g,-^- -()-*, .
When the left member of this equation is set equal to zero, its general solution
is found to be6,-&r+* (97)
where 6f and o 2
2)are the constants of integration. On making use of the
variation of parameters and imposing the condition that the differential
equation shall be satisfied, we find
6i2, = -[(a a,
)'+0,], of = +[(O 2
+0>] r. (98)
In order that, when the first of (98) is integrated and the result substituted
in (97), there shall be no term proportional to r2
,the condition
(")+ d,-0 (99)
must be imposed, where d xis the constant part of 6
l. This equation deter-
mines two values of a(1> which differ only in sign, and they are reals or pure
imaginaries when the coefficients of 6^ are real.
Since there are no more arbitraries available in (98), no more conditions
can be satisfied. The first equation of (98) gives rise to integrals of the type
n C sin It. J*. -t ai C0S 4*ajj cosJ TdT ~ *
7sinJr -
J
The second of (98) gives rise to the corresponding and associated integrals
i C sin j T a i T cos I &] sin+ ajJ
r cos J Tdr= +"J
sin J T + J cosJ T -
Hence, imposing the condition (99), integrating (98), and substituting the
results in (97), we have^ 2= F 2 (r)-P 2 (0), (100)
where P 2 (t) is periodic and a (2)is as yet undetermined.
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88 PERIODIC ORBITS.
In determining the coefficient of v', the equations for \j? and &f corre-
sponding to (98) are found to be
ir--.[2.a,>+^(T)], b?= +[2a<V
2
>+* 3 (T)] r,
where <pjj) is a periodic function of t. If d 3 is the constant part of <p 3 , wemust impose the condition
2o a, o(2) 4-d 3
= 0,
which uniquely determines a (2)if a a)
is distinct from zero, as it is in general.
If a(1) = the expansion -may be of another form, for this is an exceptional case
in the existence discussion, and it is necessary to go to higher terms of the
differential equation to determine the character of the solution. But
limiting ourselves here to the case where a (I>is distinct from zero, the solution
is carried out as in the preceding step. All the succeeding steps are the
same except for the indices and the numerical values of the coefficients.
III. SOLUTION OF THE X AND Y-EQUATIONS FOR THESPHERICAL PENDULUM.
56. Application to the Spherical Pendulum. On transforming from t to
t as the independent variable in the second equation of (4), and making use
of (50), we get
x+ia'+d, M+0 2 m2 + ]x = 0,
2 2(2o,+o,) 3r ,, > , ( 101 )
a = v ' ' 3Z, 0,= - -
[0, + (0,-0,) cos 2t], ....Cli 3 Oil 3 >
Obviously a will not in general be an integer. It will be shown that the
only integral value it can have in the problem of the spherical pendulum is
unity. Suppose a equals the integer n. In this case the second of (101) gives
(4-n2
)o,= -(24-n2
)o 3 .
It was shown in 43 that in the problem of the spherical pendulum a, is
positive and a 3 is negative. Therefore n- must be unity or zero. In the
former case we have a,= a 3 , which, because of the inequalities satisfied by
a, and a 3 ,can be true only if a,
= -f /, a 3= I. This is the special case of
thesimple pendulum.
If w = we have4a,
= 2a3 , which,
because of the
inequalities to which a, and a 3 are subject, can not be satisfied. Therefore a
is not an integer and the equations can be integrated by the methods of 53.
Upon omitting the subscript on the a in e aT, so as not to confuse it with
a, , ou , o 3 defined in 43, it is found by actual computation that
_ +3V2 aiV^I?0 *
>
4V(o,-o,) (2 a, + a 3 )
,. (1) (a, a 3 ) ,. V/ 2V /
(a 1a 3 )(2a, + a 3) ,
- .= ",/ \ (C0S2t 1) + ^ , \ ' - V^-l Sin2r,4(a, + a 3 )
v
4(0,4-0,)
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THE SPHERICAL PENDULUM. 89
The other solution is found from this one simply by changing the sign of
V 1. Hence the general solution is
a:*=4eaT
|(1)
(r)+Be- ^ C2)
(r) ,
1 V^^ 4V(a 1 -a 3 )(2a 1 +a 3 )^ J
V_i *'= fv ^V^at + a, , ___3v
/2 a, ,
1 Vai -a 3 4V(a,-.
(102)_ vWUi-aaH^ + as) v^ sm 2r~L + . . .
4(ai+a 3 ) J
+ V2V(a t
-a,) ( gai+aj v-ZT S in2r~L +4(,ai+a 3 ) J
It can be shown from the properties of 0, , 2 ,. . . and the method of
constructing the solutions, that the coefficients of mj in(1) and (2)
are cosines
and sines of even multiples of r, the highest multiple being 2j.
It follows from the form of equations (102) that, for real initial condi-
tions, A and B must be conjugate complex quantities, 2A=A 1 V l A 2 ,
and 2B = A1 + V^1 A 2 . Hence the solution takes the form
x =A 1 [x 1cos\t x 2 sinXT]+^ 2 [ ;c i sinXr + x 2 cosXr] ,
3^ = 1'
, \ (cos2t-1) m + (cosines) m2 + *
,4^ + aJ
^VK-aH^a^a,)^^^)^^^^ . . .
4(a, + a 3 )
(103)
where Aland A 2 are arbitrary constants.
Since the second and third equations of (4) have the same form, the
solution of the latter can differ from that of the former only in the constants
of integration. Therefore
?/= B
1 [a; 1 cosXT x 2 sinXr]+fi 2 [Xi sinXr-f x 2 cosXr].
The constants Al ,
A 2 ,B l ,
and B 2 are subject to the conditions that
equations (2), the first equation of (4), and the relation
xx+yy+zz=0
shall be satisfied. This leaves one arbitrary which may be used to dispose
of the orientation of the xy-axes at t = 0. Let the axes be chosen so that
x = at r = 0. Then,since z also vanishes at t = 0, we have from this equation
and the values of x and y above
x = ^l1 [x 1
cosXT-a;2 sinXr], V
= B 2[x
1 sinXr+z, cosXt]. (104)
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90 PERIODIC ORBITS.
Making use of (104), it is found from (2) and the first of (4), that at t = 0,
A'-r-n* R2 _ 212
(l+8)(2a t +c 1 )
Therefore, the solution is completely determined when the positive directions
on the x and y-axes are chosen. The well-known properties of the motion
can easily be derived from equations (104).
The variables x and y always oscillate around their initial values since
they are made up of terms which are the product of two periodic functions
that are always finite. Since the period of x l and x 2 is ir, the solutions are
periodic and the curves described by the spherical pendulum are re-entrant
provided X is a rational number. Let the period of xt and x 2 be P 1
= r,
and that of sinXr and cosXt be P 2
=2ir/\. Then,when P x and P a are
commensurable,
Pg = 2 = 2jPi X V'
p and q being relatively prime integers. Hence the least common multipleof the two periods is
P = pP 2= 2qP 1
= ^ =2qir. (105)A
In the period P the variables z, x x , and x 2 make 2q complete oscillations,and sinXr and cosXt make p complete oscillations. In the independentvariable r the period of z, z, , and x 2 is independent of n, but P 2 is
a continuous function of ix. In the original independent variable t both
periods are continuous functions of t. But in either variable the period Pis a discontinuous function of n, being finite only when the ratio of P, to
P 2 is rational. It is seen from the solution expressed in terms of t, in
which Pj is constant with respect to fi, that this ratio fills a portion of
the linear continuum, and therefore that onby exceptionally is it rational.
57. Application to the Simple Pendulum. Since the problem of the
simple pendulum is a special case under that of the spherical pendulum, it
can be treated by the same methods. Of course, it is not advisable to doso in practice, for x and z must satisfy the relation
x 2 + z2 =l 2
, (106)
from which x can be found when z has been determined.
Before discussing the properties of x we must find the expression for z
in this case. Since in the simple pendulum z always passes through the
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THE SPHERICAL PENDULUM. 93
The conditions that the solution of the second of these equations shall be
periodic are
-2V^fai 1,
6'-|6-|v'^T=
o, -a;1 '- &>-
fV=I0,(114)
of which the solutions are
a < =,
6<> = -Iv^T. (115)
Hence, we see by direct computation from the differential equations that in
this problem a is independent of n up to \x at least .
58. Application of the Integral Relations. We now return to the con-
sideration of the problem of the spherical pendulum. Since z and x havebeen determined the value of y can be found from the first equation of (4).
But it will be noticed that in this work no explicit use has been made of theintegral (2). Now x, y, and z must satisfy the differential equations, givenin the last three equations of (4), and the integral relations
ttl-
0-3 i (HO)
xx+yy+zz=0, xy-yx = c 3 ,
where the second equation is determined from (2) by changing the inde-
pendent variable from t to r; the third equation expresses the fact that the
motion must be along the surface of the sphere; and the fourth equation is
obtained from the second and third equations of (4), and expresses the factthat the projection of the areal velocity on the xy-plane is constant.
The solutions of the second and third equations of (4) have been shown
to have the form
x = aleaT
1 +a 2 e-aT
Z 2 , y = b, eaT ^+b 2 e~
a ^ 2 , (117)
where a,, a 2 , &,, and& 2 are arbitrary constants, ,and 2 are power series in ju
and are periodic with the period t, and a is a pure imaginary which is also a
power series in n, but is not an integer. Moreover, ^ and 2 are conjugate
complex quantities.
If we make use of (117), the first equation of (116) becomes
{a\+b\) e*aT+(a
2
2 +b2
2 ) e-,aT
+2( ai a 2 +b xb a )te> = V-*- (U8)
Now it has been shown that z2
is expansible as a power series in ju and that
it is periodic with the periodic w.
Before proceeding further we shall prove a lemma. Suppose there is
given
F(t)-i ajeaj*<pj(t)=0.
Suppose the <pj(t) are not identically zero, that they are periodic with the period
2tt, and that no two of the aj are equal or differ by an imaginary integer.
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96 PERIODIC ORBITS.
where the B^ ,. . .
, B 2 / are known constants. Since equations (127) are
identities in t, we have
AJf-SJ? (*-, . ,ijj-0, ... x). (129)
On substituting (122) in the second of (116), we get from this integral
W+ [**!+ life2 " +[<+%] [a^ 2 -^]
2e-
2
"-2(a 1a 2 +6 1
6 2 ) [a2
tf,
+ a(^ 1 ^-$ 1 b)-$ i y=-i 2 + 4^
(1 + 6)[-g + a. + a. + a,].
ai a3
When we reduce this equation by (120) and (121), we obtain
4a;[a(f,f i -g 1 ,) + f,^l = aa
(f-28
)-i8 + 4m + 8)
[-g+a 1 + a,+a,]. (130)ai a3
Now, on substituting (122) and the series for z in this equation, we get an
expression of the formH +H 1 +H 2 S+ =K +K lfJ L+K 2 n
2 +From the fact that this expression is an identity in n, it follows that
H i= K) (j=0, ... x). (131)
The Kj are known except for the expansions of a. The constant K involves
(a(0)
)2
,and the K
j {j =l, . . . oo) involve the a
U)linearly.
On referring to (122), we see that the Htand the K, have the form
#,= (T+C 2"cos2r+ +C 2?cos2 <7Y, 1
K, =D+D cos2r+ +0*? cos2ir. j
Since (131) are identities in r, it follows that
C%-D& (t= 0, . . . ,j;j = 0, ... x). (133)
It will now be shown that equations (123), (129), and (133) determine
uniquely the a 2 \y
, &",' , a[J
\ a U)(j>0), in the order of increasing values
of j when z and 5 are known. To do this it is necessary to develop the
explicit forms of (129) and (133) by reference to equations (121) and (130).
It is necessary to eliminate a 2 and I2 from their right members by equations
(124) and the first of (10). When./ = 0, we get from (121) and (130)
4(a[0)
)2 =-2a 3 (a 1 + a 3 ),
4 ci3 (2 ai + as) (ai + as)0=-(a
(0,
)2
[2a 3 (a 1 + a 3 )]-(134)
ai d3
The first of these equations determines a except as to sign. The sign of aj0)
depends upon which is taken as the positive end of the -axis. The second
equation gives
(a(0)
)
5 = - a2 = - 2(2ai+(fa)
, (135)ai cu
agreeing with the result in (101).
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THE SPHERICAL PENDULUM.
When ./=1, we find from (121) and (130) that
^ ,, = 8(ar)2 < )
+8ar<'=-(a^-aD-2a 3 C = JB:i)
,
ii" = 8(aj )*a+0 = 0-2a,4 1) = ^,)
,
C = - - 4 a(0 >
a 3 (a, + a,) a( " -
(a<>)2 W~S) "2 (*
<0>
)2 a 3 C?'
+4K-2(a 1 + a 3 )
2
] + 4a3(.2ai+a3)
[C-(a 1 + a 3 )3 1 ] =/)<",
97
oi a 3
^ 1)
=-16(af>)V0)
V^Tb<') = -2(a^a 3 41) + 4a3(2a ' + a3)
4') =
)<",ai a3
(136)
to which we must add the first equation of (123) for j= 1. The unknownsin these five equations are a"\ a?\ af\ am
,and b
(
2
l)
,which enter linearly.
The second equation determines a'; then a is found from the first of (123) ;
then the first of (136) defines a"', while au>
and b1
" are given by the last twoequations of (136).
We shall apply (136) in computing the first terms of the solutions.
We find from (49) and (50) that
V 1
2' (137)
Upon substituting in (136) and solving these equations and the first of
(123) for a'", aj", a[l)
,a a)
,and 6 in order, we get
,(i)
_(oi eu)
a ~~~~
4U.+0,)'
.(i) .+3 V2ai V-l
4 V(oi 03)(2ai+os)
< =
bw =2
(ai a 3 ) V^+a,4 V-2a 3
V2(dr-cto)"(2"ar+fa")
4(o, + a 3 )
(138)
agreeing exactly with the results obtained in (102).
For a general value of j the equations corresponding to (136) arc
"0 1iC = 8(a)*a<'> +8a a;'
, +Z" >!
A$=8(a>ya$+0 +25? --B
C=C^= -4a
(0)a
3 ( ai +a
3 ) a^+BS",C<? = - 16 >)
2 a (0) V^T t 6J? + <?= *>
= a<') +a" , + +a :
(-l,
(t-1,
21
, (139)
The unknowns in these equations, after a\ a[k
\ a[k)
,b (fc-0, . . .
, j-1)have been determined, are a ( J)
, a[}
\ oj?, a U), h% (t-1, . . . J). The
A%, CJ?, and D (t-1, . . . ,/) are known quantities depending uponthe coefficients having smaller numbers for the superscripts. The j+2 equa-
tions of the first two and the last lines define uniquely the j+2 quantities
and a^t (i = 0, ,j). The equation of the third line determines
aU)
, and the equations of the fourth line, the coefficients b^ (t-1,. .
. ,j).
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Chapter IV.
PERIODIC ORBITS ABOUT AN OBLATE SPHEROID.
By William Duncan MacMillan.
59. Introduction. The orbit of a particle about an oblate spheroid is
not, in general, closed geometrically. The motion of the particle is
not,therefore, in general, periodic from a geometric point of view. But if weconsider the orbit as described by the particle in a revolving meridian planewhich passes constantly through the particle, several classes of closed orbits
can be found in which the motion is periodic. The failure of these orbits
to close in space arises from the incommensurability of the period of rotation
of the line of nodes with the period of motion in the revolving plane. Whenthese periods happen to be commensurable the orbits are closed in space andthe motion is therefore periodic, though the period may be very great.
Indeed, it seems that much of the difficulty in giving mathematical expres-sions to the orbits about an oblate spheroid rests upon the incommensura-
bility of periods. The difficulty arising from the node can be overcome in
the manner just described, but elsewhere it is more troublesome.
Orbits closed in the revolving plane are considered most convenientlyin two general classes: I, Those which re-enter after one revolution; II, those
which re-enter after mam r revolutions. The existence of both classes is
established in this chapter and convenient methods for constructing the
solutions are given. Orbits which re-enter after the first revolution are
naturally the simpler and will be considered in the first part of the chapter.
Those lying in the equatorial plane of the spheroid become straight lines in
the revolving plane, and within the realm of convergence of the series
employed all orbits in the equatorial plane are periodic. When the motion
is not in the equatorial plane there exists one, and only one, orbit for
assigned values of the inclination and the mean distance. These orbits reduce
to circles with the vanishing of the oblateness of the spheroid.
In considering orbits which re-enter only after many revolutions the
differential equations are found to be very complex, and one would despair
of ever finding any of these orbits by direct computation. However, a
proof of their existence and a method for the constructions of the solutions
are given by the aid of theorems on the character of the solutions of non-
homogeneous linear differential equations with periodic coefficients.
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104 PERIODIC ORBITS.
Equation (10) can be integrated as a power series in 5 and e with the
initial values
p=-l, p' = 0.
ByPoincare's extension of Cauchy's theorem, 14-16, this solution con-
verges for values of 8 and e sufficiently small, and for all values of T in the
interval ^ T <: T, where T is finite, but otherwise arbitrary.
The condition for periodicity is simply
p' = 0atT = 2\ (11)
If we choose T = t, an inspection of equation (10) shows that for e = the
solution for p is periodic with the period 2ir, whatever may be the value of 8.
Consequently equation (11) must carry e as a factor. After integrating
equation (10), we find that the condition (11) is, explicitly,
-+a x \Trbe- [- + -a, + a*+
ga 2 J
7re2 + higher degree terms. (12)
Upon dividing out the factor e, there remains an equation in which the
linear terms in 8 and e are present, and this equation can be solved for 5 as
a power series in e. We find
5 =[-l+20
2
p2
+(f^-02
) M<+ -]e+ (13)
If this value of 8 be substituted in equation (10), it will then admit periodicsolutions for
p havingthe
period2ir for all values of e
sufficientlysmall.
Furthermore the solution as a power series in e is unique.
64. Existence of Periodic Orbits which are Inclined to the EquatorialPlane. For p
2 = the differential equations (6) admit the circular solution
c2
=l, r =l, v = t, <2= 0. (14)
In order to investigate the existence of orbits not lying in the equatorial
plane, but having the period 2t for p2
?^0, let us put
r=l+p, q = + c, c! = l + , (15)
and take the initial conditions
p = a, p' = 0, (7 = 0, a' =j3iJL.
The conditions for periodicity are then
p =<r = Q at T = 7T.
We have three arbitrary constants at our disposal, a, 0, and , and twoconditions to be satisfied. We will therefore let /3 remain arbitrary anddetermine a and e so as to satisfy the two conditions.
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ORBITS ABOUT AN OBLATE SPHEROID. 105
After making the substitutions (15) and expanding, equations (6) become
(a) p" + p = -3pe + 3p2
+!cr2
-0?M2 + 6p
2
e-6p3
-6p<r2
+4p0, ju
2
-f- higher degree terms, | (16)
(6) a" + a = 3 p a 6 p <x + ? a3
a 6\ p2 + higher degree terms.
In order to integrate these equations let us put
p= 2 p iM e'ay, a= 2 d*V. (17)
The put and or ot can be found by successive integrations, the constants of
integration being determined so as to satisfy the initial conditions. In the
seriesthus obtained put
t = t.The two conditions for periodicity give
the two equations
(a) p (it)= = a x t + a 2 e a + a 3 e
3 + a t ea + a 5 e a + a 6 a3 + a 7 e p
2
+ a 8 aM2 +a 9 p
4 +(6) a (ir) =0 =
/3 /u [6[ 6 + 6 2 e
2 + &3 a 2 +6 4 ea-|-6 5/ u2 + ],
(18)
Equation (18a) involves only the even powers of p, while (186) involves
only the odd powers. After dividing (186) by /3p, we can solve it fore as a
power series in a and p2of the form
= C 1 a 2 + c 2 M2 + c 3 a 3 + C 4 aj Lr + C 5 p 4+ (19)
On substituting (19) in (18a), we obtain a series of the form
(a)= d l ap
2 +d i a3 +d 3 p
i +d i a2
p2 +d i a i + ' '' ' '
( 20 )
If in this equation we make the substitution
(6) -(*-$)*we obtain
(c) o=p 4
[/ iT +/ 2 p2
+/,7m2
+/472
p2 + I;
which can be solved uniquely for y as a power series in p2
. This solution
substituted in (206) gives a as a power series in p.2
. This value of a sub-
stituted in (19) gives e as a power series in p.2
. We thus have a solution
a = p2
P, (p2
) ,e = p
2 P 2 (p2
) ,- arbitrary,
where P 4 and P 2 are power series in p2
. Newton's parallelogram shows that
equation (20a) has two additional solutions, but as they are imaginary we
we shall notdevelop
them.
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106 PERIODIC ORBITS.
65. Existence of Orbits in a Meridian Plane. If in equations (6) we putthe area constant c equal to zero, the motion of the particle is in a meridian
plane; that is, the plane has ceased to revolve, and the orbit in this plane is
the true orbit. After changing to polar coordinates by the substitution
r = p cos<p,
the differential equations are
q = psm<p,
i -| + |cos2^>+ -cos4<?v"-vWY+ p t - --**-
p<
4-
e\ M2 +
p?"+2pV =- sin2<p 7 sin 4^2i 4
p'^M2+
(21)
(22)
For p2 = 0, equations (21) have the periodic solution
p = l, <p = r;
that is, a circle. For p2
7^0, we will put
p = l+p, <p= T + a,
with the initial values
p = a, p' = 0, <r = 0, <r' = j3,
where a and /3 are two new arbitraries. By 14-16, p, p', a, and <r' are
expansible as power series in a, /3, and p2 with r entering the coefficients.
The conditions for periodicity are that at t = x
p' = <r = 0.
If we integrate equations (21) and then put r = x, we obtain from the
periodicity conditions two equations of the form
(a) a- (T) = = a 1 a+a 2J8+a 3 a2 +a 4 a/3+a 6 /3
2 +a 6 p2 + a
7p4+
(6) p'(tt)=0= +& 3 a 2 +& 4 a/3+& 5 /32 +0.p 2 +6 7 p
4 +The first of equations (22) can be solved for a as a power series in /3
and p2
. This expression for a substituted in (6) gives rise to an equationof the form
(c) O = c 1 0p 2 +c 2 /33
+c,/32
p2 +c 4 p
4 +
This equation has the same form as (20) and can be solved in the sameway, giving a solution for /S as a power series in p
2
, vanishing with p2
. This
expression for fi substituted in the equation for a gives a unique value for
a as a power series in p2
, vanishing with p2
. Therefore periodic orbits exist
for pVO, which are analytic continuations of circular orbits for p = 0.
We have thus proved the existence of the following three classes of
periodic orbits which have the period 27r:
I. Orbits lying in the equatorial plane whose generating orbit is a circle.
II. Orbits inclined to the equatorial plane whose generating orbit is a circle.
III. Orbits in a meridian plane whose generating orbit is a circle.
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108 PERIODIC ORBITS.
On substituting the values of p and p, in (25c), we find
fp+P*= +[m-*i(3-3; m2
-) + (3-602
M2
)]
+ [61 (-3+60
2V+(20 4 +120 2
V )
+ (-3+1202M
2
+(-y04 +90 2
)/x< .)]cost
+[5
1 (3-302
M2
) + (3-180^* ) ]cos2T
+[~3-40
2/x
2
-]cos3T.
In order that the solution of this equation shall be periodic the coefficient
of cos T must be zero. This is the condition that determines 5, , and con-
sequently5 1 =-1+2^ m
2
+(|^-^)m4 +
which agrees with (13) of the existence proof. With this value of 5j equa-tion (28) becomes
f^+P 2=
[5 2 +302
M2
-] + [-902M
2
]cos2T+[3-402
p2
. .
.] Cos3t.
The solution of this equation which satisfies the initial conditions is
P 2= 6 2 (l-cosT) + [30
2
p2 + . .
.]+[|_|^ M2 + . .
.]cosT
I (29)+ [302p
2 + ]cos2T+[-f + |02
/x2+ -]cos3T. !
The constant 6 2 is as yet entirely arbitrary. It is determined by the
periodicity condition on p 3 in the same manner that 6 S was determined by the
periodicity condition on p 2 . Without giving the details of the computation,its value is found to be
S 2 =-60 2
M2 +
This method of integration can be carried as far as is desired. In order to
show this, let us suppose that p ,. . .
, p_i have been computed and that
all the constants are known except 8 n - t . From (25 d) we have
ff+Pn=5-3p o 5 n _ 1 +[3-302
/x2
.]**_+/.(* . ., p- 2 ), (30)
where /(p ,. . .
, p_ 2 ) is a polynomial in the p } and contains only knownterms. It is easy to see that p_i depends upon b n _ x in the following way,
p_i = 5_,(l cosT)+ known terms.
Equation (30) may therefore be written
|^n +P=5 n +[3-3
2M
2 ]-,+ [-3 + 6 #/** :]_, cost
+[3-30
2p
2
+] cos2T+ known terms.
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ORBITS ABOUT AN OBLATE SPHEROID. 109
In order that the solution of this equation shall be periodic the coefficient of
cost must be zero. This condition determines 5_,. The equation canthen be integrated, and the constants of integration will be determined by
the conditions that, at T=
0,
Everything is then determined with the exception of b, and we have
p=(l cosT)5+known terms.
On substituting the values of 8 1 and 8 2 in the solution as far as it has
been computed, we find
p = -cost,
Pi=[|+^iM2
+(^-4^)M
4 -
-]+[-^m
2
+(-^+3^)m4
-JcosT
+ [" ? + I*V+ ({*?+) "4 "
-]cos2T,
p 2 =[-3^ M2+
]+ [f-|^M
2+-]cosT +
[302
M2 -
-]cos2T
+ [-| + ^ 2M
2
---]cos3T,
5 1 =-1+2^ m2
+(|^-^)m4 -
,5 2= O-60 2
M2 +
From these expressions the series for r becomes
(a) r=l-ecosT +[[{
+ ^ 2M
2 +(^-40 2V-f '
]
+ [-02m
2
+(-|04 + 30 2
)m4 + -]cosT
+[-i+i5^+(i<+<)^ -]cos2T
}e2 USD
+{[-30
2
p2 + -] + [+|-|^M
2
"JCOST
+ [3^ M2
-]cos2T + [-f +}02
m2
-]cos3Tje3 +
On substituting this value of r in the equation (86), transforming to the
independent variable T, and integrating, we find
(b) v-v=+{[i+eis+(\e\+2(%)s+ ]
+ [^M2
+(-f0H-f02
) MH- -]e2 +
-}t
+[[2+22
p2
+(0t+4C>M4 +
-JsinTJ-c
+[[22
m2
+(j0}-6<)m4+ -]sinT
+ [f + f^+(~ *+!)**+'
-]sin2Tp
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110 PERIODIC ORBITS.
Equations (31a) and (316) are the periodic solutions sought. If we return
to the symbols defined in the original differential equations (1) by means of
equations (5), with the additional notation
n a/I-0V-30^m4 '
= ">
we have the following expressions for the polar coordinates:
/? =a|l-ecos
J '< + [|-|cos2j/f]e2
+[|cos^<-|cos3j'f]e3 +
+a^[(|-^
cos ^+i cos2, '0 e21(32)
+(-^-l cos ^+B cos2 " i +i) cos3j '0 e3 + ]*+ }>
v v <> = vt+2 sinj^-e+Tsh^pf .e 2 +
+ (fsin^+|sin2^) e2 + :
]m2
+^[- -]m4 +
(33)
Equations (32) and (33) contain four arbitrary constants,* a, e, v, and J -
Since the differential equations of motion in the equatorial plane are of the
fourth order, these series, within the realm of their convergence, represent
the general solution. The expression for the radius vector, R, is always
periodic with the period 2-k/v. At the expirationof this
periodv has
increased by the quantity
2 *||*(5+S*+ )+-
-1=2.9 (34)
in excess of 2x; that is, the line of apsides has rotated forward by this
amount. If 9 is commensurable with unity the orbit is eventually closed
geometrically. If 9 = I /J ,where / and J are relatively prime integers,
then v = 2(I + J)ir at t = 2Jw/v, and the particle is at its initial position
with its initial components of velocity. The particle has completed I+Jrevolutions, and the line of apsides has completed / revolutions. Themean sidereal period is
P =VWTW (35)
Equation (34) for the rotation of the line of apsides has an interesting
application in the case of Jupiter's fifth satellite. On the hypothesis that
Jupiter is a homogeneous spheroid whose equatorial diameter is 90,190miles and whose polar diameter is 84,570 miles, that the mean distance of
the satellite is 112,500 miles, that the eccentricity of its orbit is .006, and that
*The constant a is also contained implicitly in v through the constant n, and t can obviously be
replaced by(t t t ) since I does not occur
explicitlyin the differential
equations(1).
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112 PERIODIC ORBITS.
Coefficient of p. The coefficient of p is defined by q'+q^O, and the
solution of this equation satisfying the initial conditions is
ft-pTsinr. (39)
Coefficient of p2
. The coefficient of p2
,from (37), is defined by
P:.+P 2=f?:+^-^=(f/3
2 +c 2 -^)-f^cos2r.
The solution of this equation which satisfies the assigned initial conditions is
p 2 =(i / 32 +c 2 -0j) + a 2 COST + j0
2 cos2r.
The constant c 2 is determined by the periodicity condition on q 3 ,where it
is found that it must have the value c 2= 20, /3
2
; and c^, which is deter-
mined by the periodicity condition on p 4 ,is found to be zero. If we
anticipate these determinations, we have
p 2 =(02 -/3 2
)+i/J2 cos2r. (40)
Coefficient of n3
. The coefficient of p3
,from (38), is defined by
g'-Hb-.(a*+f-3*i) =(3 0M-3ci-6 0j)jSsinT+ f a 2 /3sin2r.
In order that the solution shall be periodic it is necessary that the coefficient
of sinr be zero. Therefore c 2 =202
/32
. On substituting this value and
integrating, we find
g,=/3, sin r 703 /3 sin2T.
From the initial conditions we must have ^(O) = 0, and therefore
&=fa 2 .
But it will be shown in the next step that a,= 0, and consequently that
q^O. (41)
Coefficient of p4
. It follows from (37) that the coefficient of p4
is
defined by
p;+P <=
3p2
+3(/ l?3 -6p 292
-ff /l
,
+(402
-3c 2 )p 2 +f02
9
2
+c 4. (42)
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(44)
114 PERIODIC ORBITS.
On collecting the various coefficients, we have the following series
(a) r=l + [(8l-\F)+\Fcos2T]S+[(-3e*-? 6*F-l- P)
+ (-^2
+^/34
) cos2t -^ cos4t] m4
+
(6) 5 = [^sinT]M+[0]Ms
+[-f^^sinr+^^sin3r] M5 +
(c) -*; o= [l-^M
2
-(f04+!O 2y+ ]r+[-^
2
sin2r] M2
+ [&P i -^ i
)sin2T + k (ii sin4r] M
4 + ,
(d) C2 = 1 + [202
-/?] m2
+[-404 -^ 2
/3
2
]m4 +
In this solution the constants* a, j8, v 0) and t are arbitrary. As is shownby equation (44c) the nodes regress, the measure of regression being
27r[0
2
M2
+(f04 +i0 2
/3
2y+ ]
The generating orbit of these solutions is a circle in the equatorial
plane. A circle having any assigned inclination might have been used, e. g.,
r = Vl s2 sin 2
T, g = ssinT, v = tan -1[Vl s
2tanr], (45)
where s is the sine of the inclination. The solution thus obtained wouldhave been identical with
(44). If we should
expand (45)as
powerseries in
s and put s = /3/z, we should find that the terms thus obtained are identical
with the terms independent of 0, in the solution which has been worked out.
It might therefore be of assistance in the physical interpretation of the
constants to put /3/x= s in the series (44).
68. Construction of Periodic Solutions in a Meridian Plane. When the
constant c is zero the motion is in a meridian plane. The equations of
motion (21) are
, -| + |cos2^+7Cos4^
p-p(< Py+j- 2
= 4 2
pi
4 _ 0*
M
2
+-sin2<p -sin4<?
p<p"+2 P '(<py -;-i- - -el M
2 +
(46)
We have proved in 65 the existence of periodic solutions of these equationsas power series in m
2
, which, for m2 =
0, reduce to the circle p =1, <p
= r.
Let us therefore put
p = l+p 2 M2 +p 4 M
4 + ,> = r + VJ 2 M
2 + ?4M4 + ' ' '
*Thc constant o is containedimplicitly through
rand 0];
seeequations (4).
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116 PERIODIC ORBITS.
The differential equations for p t and <p A are just the same as for p 2 and <p 2
except in the right members. The process of integration isj ust the same. In
the right members only even multiples of t occur except in terms carrying the
undetermined constant c 3 as a factor. In the equation corresponding to (48d)
there will be a term in cost carrying c 3 as a factor. But the integral from this
term will be non-periodic unless c 3= 0. Upon putting c 3
=0, the integration
proceeds just as before and the constants are determined in the same manner.
The same steps are repeated in the coefficients of m\ and so on for all higher
powers. Therefore no odd multiples of r occur in the solution. We have,
therefore,
p = l + [-i + |cos2r+^cos4r]^ M2 + ,
^ = r + [-^sin2r-| 5 sin4r]^M2 +
(49)
Since the series involve only even multiples of t, the orbits are symmetricalwith respect to both the r-axis and the g-axis.
This completes the formal construction of the solutions of which the
existence was proved in 63, 64, and 65.
II. ORBITS RE-ENTRANT AFTER MANY REVOLUTIONS.
69. The Differential Equations. The orbits which we have previouslyconsidered have had the common property of involving only the period 2x.
Since thisperiod
is
independentof the oblateness of the
spheroid, the deriva-tion of these orbits has been relatively simple. We shall proceed now to
investigate a class of orbits which involves, beside the period 2w, another
period 2t/\, where X is a function of the oblateness of the spheroid, the
inclination of the orbit to the equator, and the mean distance of the particle.
We will start out from the solution which involved an arbitrary inclination
(67). Into this solution four arbitrary constants were introduced, viz.,
inclination, mean distance, longitude of node, and the epoch. Two morearbitraries are necessary for a complete solution, viz., constants correspond-
ing to the eccentricity and to the longitude of perihelion. In what follows
weshall
introduce the constant corresponding to eccentricity.We have found for the differential equations a certain solution, givenin (44), which we may write
r = <p(j8, m; t), ? = lK/3, m; t), c2 = c
2
,
which is symmetric with respect to the equatorial plane. That is to say, at
t = the particle is in the equatorial plane and its motion is perpendicularto the radius vector. Its initial distance is (p (0).
Suppose now we change the initial distance slightly and also the initial
velocity so that
r(0) =*(0)+a, g(0) =0, r'(0)=0, q\0)=f'(0)+y,
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ORBITS ABOUT AN OBLATE SPHEROID. 117
and give an increment to the constant of areas so that c? = c2+e. Can
we determine these three constants a, y, and e, as functions of /3 and p, in
such a manner that the series for r and q shall be periodic? The solutions can
be expressed in the form
r=<p(fi,n;T)+p, q=f(P, m; t)+<t, c2 =c 2
+e,
where p and a are the necessary additions to <p and \p. If now we substitute
these expressions in the differential equations (6), all the terms independentof p, <x, and e will drop out, and there will remain the following differential
equations for p and a :
(a) p" +
|l+ [(-4^+f^)-f^cos2r] M
2
+[(20^+6^/32
+|^)
+ (^^-|^)cos2t + | j8 4
cos4t] m4 + jp+jf- 3/3 suitJm
+ [(-302
0+/3 3
)sinT-/33
sin3r]p3 +
j*
={3+ [(-16^+6^
2
)-12^cos2r]p2
+[(9O^+f0^2
+|^)+ (^^
2
-|/34
)cos2r+^/34
COs4r]p4+
jp2
+{[-12/3 sin r]p
+ [(-9O02
/3+f /33
)sinr-^/33
sin3r]p3+ U
+{f+[(f ^-f/3
2
)+f
/32
cos2r]p
2 +]<r
2 +
+[{
1 +[(- 3 ^+!^2
)-!^ cos2r ]^ 2+ [( 15 ^-i e^ 2+ l^)
+ (f^/32
-|^)cos2r + f /3<cos4r]p<+ ]
+{-3 + [(l202
-3/32
)+3/32
cos2r]p2 +
-}p+ ]e,
(6)(r"+jl
+ [-|/32
+|^COs2r] M2
+[-4e2
/3
2
+7e2
^cos2T]p4 +
-}<r
+|[-3/3sinr]p+[(-30
2
)3+|/3
3
)sinr-|/3
3
sin3r]M
3+-}p
1[-6/3sinr]p+[(-210
2
/3+f /33
)sinr-f /33
sin3r]p34-
jp2
3 + [(3^-f /32
)+f /32
cos2r]p2+
-}p<7
[f /3 sin r]p+ ]**+
In the first of these equations the coefficients of all the terms containing
odd powers of a involve only sines of odd multiples of t and odd powers of
p;all other coefficients involve only even powers of p and cosines of even
(50)
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(58)
(59)
120 PERIODIC ORBITS.
Coefficients of tf. The coefficients of /z2 are defined by
* ,2+'j=-2tX s ?o-2^o-ai^i> #+&= 2iM'J-M'o flitt J (57)
or, expanding by (53) and (56),
ft + ft =f P y?~ f 7? cos2r + f Y Sin2r + [4^af + 2*A,af] sinr
+ [402< ) -2iX 2 a 2
o>
]cosr + 3/32a 2
O) sin3r + 3,82
ai0>
cos3r,
^+ft-|/8a-|/J-< cos2r + f/3ar>sin2r + [^7 3
0> +2iX 2 Ti)
]sinr
+2i\y? cost.
In order to satisfy the periodicity conditions we must have
2x\a + 402
af =O, 2iX 2 7i
0, + f /32
7 20) =
0,j
4^2
ai0> -2t'X 2 af
> = 0, +2lX 2 7 20> = 0.
The last two of these equations are satisfied by taking
yr-Yf-0.On solving the first two, we find
\=2f lt a-ia 2
0) = 0.
Equations (59) can also be satisfied by
X2. a'
' = a2
0) =7 f
=0, t1
0>
arbitrary.
This leads to the development of the solution in which the characteristic
exponent is zero, which will be discussed, beginning with equations (80), byusing the integral relations.
It was known at the outset that the two values of X are equal but of
opposite sign. We will choose the one with the positive sign. The solution
for the negative X can be derived from it. The condition a ta 2
0) =still leaves us with an arbitrary constant. Since the equations are linear,
this constant will enter the solution linearly, and may therefore be taken
equal to unity, inasmuch as the solution after development is multiplied
by an arbitrary constant. We will take then a[0) = l, which therefore
makes a 20) = i. Consequently
< = cosr i sinr, &> = 0. (60)
On integrating equations (58) with these values of oj0) and a 2
0)
,we get
ft=|/3721, + a[
2> COSr + afsinr+i^7 2
,,
COs2r-}^,
'sin2T
+3/32
cos3r-3,82
tsin3r,
ft = | j8a<"+7(
1
2)
cost+7 2
2)
sinr+ liSa'1 '
cos2r-| pa[l)sin2r.
2 i- - 2 . /i . 12 i
2^ 2 *~
2
(61)
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ORBITS ABOUT AN OBLATE SPHEROID. 123
Similarly, in so far as <p J+2 depends upon constants as yet undetermined, itis found from equations (52) that
'
<p" +2 + <p J+2= -2i\ v
'
J -2i\ J+a <p' +[(40
2 -/3
2
)
+ f/3
2
cos2r]^+[3/3
sin t]^= +[-2\ +2 +4:6
2M-ia<) +A J+2 ]cost
| (66)
+ [2i\ j+2 + 4tf(i a<J) + a<") +B J+2 ]
sinr
+3/32
a^cos3r + 3/32
afsin3r,where A j+2 and J5 J+2 are the known terms in the coefficients of cost andsinr respectively. From the periodicity condition we must have
-2\ i+i +16\{ a}-ia?)+A j+2=
Q, 2i\ J+2 +4e2
1 (ia[+a 2 )+B J+2= 0. (67)
The solution of these equations is
X ;+2= i
[A j+2 + iB J+2 ], >-t4 = - i4 **~fl j
B*(68)
As has already been pointed out, we can choose a 2" =
0, and we have then
u) _ A J+2 iB j+2 . .
i 80 2 (b9 '
In order to show that X ;+2 and a'" are real, it will be sufficient to showthat A i+2 is real and that BJ+2 is a pure imaginary. This is readily provedby induction, for, up to j = A inclusive, we have
am j+i'
j-\ i~i
<Pj= S mK coSKT+i 2 n K sinnT, ^j= i 2 f KC0SKT + 2 gf K sinKT,
where mK ,wK , f K , and gr K are all real. From the form of the differential
equations it follows at once that the same forms hold for j = 5, then 6, andso on. That is, A J+2 is real while B 1+2 is purely imaginary.
Furthermore, it is to be noticed that A j+2 and B J+2 do not contain anyterms in /3 independent of d\ ,
and consequently the 8\ ,which appears in the
denominator of a[n
,will divide out. This is proved as follows : If d\ be
put equal to zero in the differential equations, then equations (52) become
the equations of variation of a circular orbit in the ordinary two-bodyproblem, the plane of the circle being inclined to the plane of reference byan angle whose sine is 0n = s. Equations (6) can then be written
^_ Q-s*)(l-e2
) _ r R __ q =Q (7Q)
where the constant c2
is given the form (1 s2
)(l e2
). For these equationswe have the solution
r _ (1-e2
) Vl -s 2sin
2( g-7T) (1 -e 2
)ssin(fl- ft) ,_ nl+ec6s(0-0 o )
ql+ecos(0-0 o )
K J
where
6-e =(T-T )+2esm(T-T )+
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124 PERIODIC ORBITS.
If now we form the equations of variation by varying r, q, and e, that is
by putting
r = r +p, q = q<>+<r, e = e +e,
where r , q t , and e are the values in (71), we find
dR ,dR ,dR dQ ,dQ , dQ no ,
dr dq de dr dq de
Three solutions of equations (72) are given by
(1)
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(76)
ORBITS ABOUT AN OBLATE SPHEROID. 125
The solution which we have obtained may be written
Pm = e
tXT[<pm+i<p], a w = e
lXT Wl)
+i+ i2)
], (75)whero
^(1> =
cosT+[|/32
cosT-|^cos3r] M2
+[a 4 cosr+(-f^/32 -/3 4
)cos3r
+ ^ 18
4
cos5t] m4 + ,
r w --Bbt+|>8in3T]MH[(J^i8+^)8in3r-^8in5r>+r = [-^sin2r> + [(-^^-^3)sin2r] M
3 + ,
'
+ (-^^2
-f /33
)cos2r+ -]M
3 +
By putting e'Xr
=cosXt+{ sinXr, the solution takes the form
^^[^"cosXr- sinXrl+it^'cosXr+^'sinXr], 1
^'^[^"cosXr- 'sinXrl+tt^cosXr+^sinXr].j
(7?)
We have thus one solution of the differential equations. A second solution
can be derived from it by merely changing the sign of i, or
P^^t^'cosXT-^sinXrl-tt^'cosXT+^sinXr], 1
^^[^cosXr-^'sinXrJ-if^cosXT+^sinXr].
By adding and subtracting these two solutions, we have finally
p = A[pw+ p}+B[p m -p], a = A[a
m+ a]+B[a w-*},
A and B being arbitrary constants.
As above developed, there is a certain arbitrariness in these solutions,
owing to the manner in which the constants of integration were determined.
They may be reduced to a normal form by multiplying each solution by the
proper series in p,2 with constant coefficients. By this process we can make
p(
"(0)+p(2
>(0)=
1, a\0) -a <2)
(0) -#*. (79)
Since [p(1)
p(2)
] and [a w+a] are sine series, they vanish for t = 0.
The third and fourth solutions of the equations of variation are
p*-Tt ** H> Pi~ D
~teT' ai - D ~dT' (80)
the r and q being defined by equations (44). In performing the differentia-
tion in this last solution it will be remembered that r and 0, are functions
of a. The third solution also can be normalized by giving the arbitrary
constant such a form that
P 3 (0)=
0, r,(0)=C/V
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126 PERIODIC ORBITS.
As already stated, the fourth solution is non-periodic and has the form
*- j>[
tS? + *J '*- D
[*
d
+A> (81)
where ^ 4 and \f/ t are periodic functions of r with the period 2ir. As in the
previous solutions, this can be normalized so that at t =
The functions <p 4 and ^ 4 are also easily found by substituting (81) in the
equations of variation and solving for these variables (which must be periodic) .
Upon carrying out the foregoing operations, we find the followingfundamental set of solutions:
p=+yl|cos(l-X)r+ [-^2
cos(l-X)r+ f /3
2
COs(l+X)r
-|^cos(3-X)r] M2
+[(-|a 4 +f^/32 + |/3
4
)cos(l-X)r
+(! a <-i04
)cos ( 1+x ) T + (-!^
2
-!/54
) cos ( 3 - x ) r
-i !/?cos(3 + X)t+ i ^cos(5-X)t] m
4 + j+#[ sin(l -X) r + [(-I^-|^) sin(l
- X)r- /32
sin(l + X)r
-^ 2
sin(3-X)r] M2 +
+ c{[-^2
sin2r] M2
+[(^2
/32
-^)sin2r + ^sin4r]M4 +
+>{{ [| /32
+| /32 cos 2t] m
2 + [(f e\p -|/J<)
+ (-|e2
/32 + |^)cos2r-i^cos4r] M
4 +
+rf[J^sin2r] M
4+[(^^+^6
)sin2T-|/3,i
sin4r]M6 +
a= +Al [ j8 sinXr -|/3sin(2
-X) r] M+ [f
2
/3sinXr
-^02
/3sin(2-X)r] M3 +
+ #[[f /3cosXr + | /3cos(2-X)r]M+[-|-02
/3cosXr
+ 2
/3cos(2-X)r] M3 +
+<7J[/3cost]m+[0]m
3 +}
+Z>{{[{^sinr]M+ [(-^
2
^ + |/33
) sinr] M3 +
}
+r{[-f/3
3
cosr]M3
+[(-S
2
/33
+l^)co
^]^+"
'}}
(82)
(83)
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128 PERIODIC ORBITS.
where the (p,) and (<r t ) are the complementary functions, and the |, and j?,
are the particular integrals of which the w, are the periodic parts, andwhere a and b are constants which depend upon the differential equations.
Let us suppose that g(r) is an even function of r and that/(T) is an oddfunction of r, and let us seek the character of the solutions which satisfy
the initial conditions
*(0)-*(0)-0.
On changing t into t in equations (86), we get
J 7 -
^;Pi(-t)= -p 2 (-t), ^p 2 (-t)= -2 p 1 (-T)+0 3 tn(-T)-0(T),
^<j- 2 (-t)=-o- 2 (-t), ^o- 2 (-t)=+ 3 p 1 (-t)-0 4
(t1 (-t)+/(t).
(88)
(89)
From equations (86) and (88) we obtain, by eliminating g (t) and/(r), the
differential equations
^;[p 1W-p 1 (-t))= [p 2 (t)+p 2 (-t)],
[P2(T)+P i (-T)]=TAp 1 (T)- Pl (-T)]+e' [<Tl(T)+<r l (-T)],
fo[<Tl(T)+<Ti( T)]= W2 (T)-<T 2 (-T)],
| : [<r 2 (r)-(T 2 (-r)] = [9;[p 1 (r)-p 1 (-p)] + i:[p 1 (r)+p 1 (-T)].
These equations are the same as the original homogeneous set (84). Hencetheir general solutions have the same form, viz.,
p i (T)-p 1 (-T)=Aa i (T) + Ba 1 (T) + Ca 3 (T)-\-D[a i (r) + T7 (T)],
P i (T)+p 2 (-r)=Aj 1 (r) + Bj 2 (r) + Cj i (r)+D[%(r) + r%(T)],
^(r)+^(-r) = Ay, (r) + By 2 i^ + Cy, (t)+DF,(t)+t7(t)],
<T 2 (T)-<T 2 (-T)=AT (T) + BT (T) + CT (T)+D[J (T)+T5 3 (T)].
(90)
Upon putting t = 0, we find from the first and the fourth of these
equations that
= AZ2 (0)+Da i (0) }= 4l 2 (0)+Z)5 4 (0).
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ORBITS ABOUT AN OBLATE SPHEROID. 129
Either A = D =0, or the determinant a 2 (0)5 4 (0)-a 4 (0)5" 2 (0)=0. But it is
readily verified that this determinant is not zero. Therefore A=D = 0.
By virtue of the hypotheses made on the initial values, it follows from the
second and thirdequations
that
= Bft(0)+CA(0), 0=*By t (0)+CyM,
and hence B = C = 0; consequently
Pi(t)-Pi(-t)=0, P 2 (T)+P 2 (-r)=0,
<t 1 (t)+<t 1 (-t)=0, <r 2 (r)-<r 2 (-r)=0.
Since these equations are identities in r, we have the following theorem :
(91)
Theorem I. If g(r) is an even function of r andf{r) is an odd function
of r, and if p 2 (0) =<r 1 (0) =0, then p^r) and o- 2 (t) are even functions of r, and
P 2 (t) and (t^t) are odd functions of t.
In the same way it can be shown that if g(r) is odd and /(t) is even and
if p,(0) = <r 2 (0) = 0, then p x and <r 2 are odd, and p 2 and a 1 are even.
Let us suppose now that g(r) contains only even multiples of r, and
that /(r) contains only odd multiples of r. The general form of the solution
will be the same as (87), and, , 2 , ij, ,
and r? 2 satisfy the differential equa-
tions
;t!,(t)-&(t) 3 fUr)=e 2 M+e, m(r)+g(r) f
a,T ar
nM = %(r), v*(r) =1 fe(r) + <U(t) +/(r).
(92)
Let us denote ^(j+t) by (t) and ^(t+tt) by i&(t). Then by changing t
into r+7r in (92), we have
&t) = &t), (T) = +e 2 a[(T)-o 3 v' (T)+ g(r),
|-ii(r)-^(r), f uiW = - i(r)^ *(t) -/to-
From equations (92) and (93) it follows that
(93)
(94)
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(95)
(96)
130 PERIODIC ORBITS.
The solutions of these equations, which have the same form as (84), are
^-^ = 4a 2 (r)+J5^(T) + C^(T)+Z)[r 4 (T)+T^(T)],
^-^=M(T)+5^ (r) + C|:(r)+Z)[^(T)+r|:(r)],
^+^ = ^7 1 (r)+^(r)+C^(r)+Z>[^(T)+r^(r)],
r, a + v' = AT (t) +J5^(r) + CT (r) +D[T (t) +t! (t)].
On forming these expressions directly from (87), we get
& fLftM i(H-*r) (a7r+2 &7I' !!
) a3_&7r ^ Ta3 "'" a4 ^
l 8 -^ = "2(r)- 2 (r+7r)-(a7r+|67r2
)^-67r(r5;+I),
1i+?[ = W3(T)+w 3 (T+7r) (a7r+-6ir2
)7 4 -67r(T7 4 +7 3 ),
l2+'72 = ''4('")+w 4 (T+7r) (a7r+|&7r2
)5 3 67r(r5 3 + 5 4 ).
A comparison of equations (95) and (96) shows that
A=B =0, C=-(a7r+!&7r
2
), D=-bT,
CO^T) W1 (t + 7t)=0, O> 3 (t)+C0 3 (t + 7|-)=0,
oj 2 (t) o) 2 (t+7t) = 0, w4 (t)+co 4 (t+tt)=0.
Therefore a^ (t) and a> 2 (r) contain only even multiples of t, while w3 (t) andoj 4 (t) contain only odd multiples of r, and by carrying this result into equa-tion (87), we have
i= iM+aTa 3 +&[-T
2 a 3 +T o 4 ], %= <a t (j)+aTy t +b[ji> 7 4 +t 7J,
These results may be expressed in
(97)
Theorem II. If g(r) contains only even multiples of r andf(r) contains
only odd multiples of t, then a and 2 contain only even multiples of r, and771 and r; 2 contain only odd multiples of r.
If in addition to the above hypotheses we suppose that g{r) is an evenfunction of r and/(r) is an odd function of r, then & and % are even functionsand 2 and % are odd functions; therefore 6 = 0. But if g(r) is an oddfunction and/(r) is an even function of t, then t and ri 2 are odd functions and, and t^ are even functions, and in this case a 0.
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132 PERIODIC ORBITS.
72. Integration of the Differential Equations. It will be convenient
hereafter to use as notation for the fundamental set of solutions, (82) and
(83), of the equations of variation
p - A a, (t) +B a, (t) +C o, (t) +D [o 4 (r) +r a, (t)],
(r = ^T l (r)+5 7 ,(T) + C ,
T 1 (r)+Z)[7s(r)+r7(r)],
where the a and 7-f unctions are characterized as follows :
Oj(t) involves only terms of the form cos [(2n+l)X] t,
(101)
7iWo,(t)
7M.(t)
74 (t)
o 4 (t)
7 3 (t)
(
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ORBITS ABOUT AN OBLATE SPHEROID. 137
By 30 terms involving multiples of Xr give rise only to periodic termsin the solution. By 31 those parts of Rm and Sm which are independentof X.give rise to terms in the solution which have the form
P=VM+c[Ta,{T), ff = P 2 (r)+c;T7 4 (r),
where p,(t) and p 2 (r) are periodic with the period 2?r. Consequently thefunctions x 1 (r) and y x (t) have the form
x 1 (r)=P 1 (r)+c 1 ra 3 (T), y,{r)=P 1 (r)+c i Ty i (j), (116)
where jP^t) and P t (r) are periodic with the period 2o\
Coefficients of a.8. The differential equations for these terms are
Pno + 9 2 Pno + 3 <Tno= Rm , cr no + 4 O'no + #3 Puc m8mi (11 7)
where/t llo
= 2 JAj OO 4 olo [0 2OO a 2 + llo a 2 7 1 +0 O2o7i]
+ [A AZ+AZ AZ\ [26 M02 (ra 3 + eO +0 no j Tl (ra^+a,) + a 2 (r7 4 +7 3 ) f
+20020 7 1 (t7 4 +7 3 )]
+2^;2
oJ,^)
o [02o O (ra, + o 4 )2 +0 n o(ra 3 + a 4 )(T7 4 + 73)+0o2o(r74 + 73)
2
];
and aS 110 is obtained from Rno by replacing dm by 6 iJt .
The functions Rno and Sm differ from R200 and aS 200 only in the con-stants A, Jk . The initial conditions impose the same conditional equations.
Consequently the solutions differ only in the constants A iik , so that we can
express them at once without computation in the form
Pno = 2 A lw A 0l0 x^t) + [A lM A 0l0 -{- A 100 A 010 ]x 2 (t) +2 A 100 A 010 x 3 (t) ,I
am= 2AZ AZVtM+lAZ AZ+AZ AZWr)+2AZAZy 3 (T), J
where the x,(t) and ^(t) are the same functions of r as in (115).
Coefficients of 82
. By symmetry with the coefficient of a 2
,it is seen that
P^A^x^+AZAZx^+A^x^r),|
^o = ^o%2
^(r)+^;i
1
>o^ )
o2/ 2 (r)+^ 2
2/3(r). J
Coefficients of e2
. Since the coefficients of the first powers of a and 5 are
homogeneous of the first degree in the A's, the coefficients of a 2
, a5, and 82
are homogeneous of the second degree in the A's. The coefficients of the
first power of e are not homogeneous in the A's; hence the coefficients of
the second power are not homogeneous. But if the functions a 6 and 7 6
were zero the coefficient of the first power of e would be homogeneous, andtherefore the second also. By symmetry, therefore, we can at once write
down the terms involving the ^4's to the second degree. To these must be
added terms in the first degree in the A's, and one term independent of
the A's.
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13S PERIODIC ORBITS.
The differential equations for these terms are
P(X)2 + 6 t pm+ 6 3 (Tm= Root , Co02 + O* a 00l + ^PcKH = "to ) (120)
where"'002 "200P001 1 "noPooi^ooi 1 "o2oooi "T "101P001 >
O(|02=
"2O0P0OI I "lloPoOl^OOl T" "(BO^OOl
The terms involved in P^ are shown in the following table, where the
coefficients of the constants given in the first line are the products of the
functions in their respective columns and the functions of the same line
in the last column. Thus, one of the coefficients of A is Oj 7 X UO ,and
this coefficient comes from the expansion of p M1 am .
Origin ofterm
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142 PERIODIC ORBITS.
On changing the x, into y~, and u into v, we get ^" . Hence
~i _ i \v aw a > 4> j)"i rEi_^.neioJ ^p1 L 10 o 01 l0 ^oou l w v J
J(2)
f J(1)
J(2)
_ J(
4<D "1
ri _ V* ~\I
J(2'2
["6 _ 2/_6~]I
_ -^100^100 -^001 ^100 " 001 J Lw yj' 100 Lw j, J
J
But T- - -1 and T- 6 -^~\ vanish since
x^^u, Vi^c^v, x t= c 4 u, y t =c t v,
as is readily seen from (116) and (123). On referring to (122), it is also seen
that does not vanish, but is equal to c,- -"
. Hence
\ 4 (2) \A m A A m A a) ]f^ i7~I_ , _ _ ^ioq l/Moo -fl-ooi -^100 -^ooiJ \Xj __ /4 I
C20 C20 J (2)3
-fl-ooi U v A(136)
Without repeating the detai ls of the computation, we find similarly
[02-02j= -
AWT A (1) A (2) _ A (2) A (1) 1 I A (2)\ A U> A (2) _ A (2) A <
010llMOO-' i 001 L 100^ x 00lJA (2)3
-<iooi
J (2) r ^ (1) J (2) _ A (2) ^ (1)-^OlO I/^OIO -^0 01 -^010 -^OOl.
A (2)3iloni
[a _ y*] .
Lw v J
(137)
On substituting in (133) the values obtained for the coefficients, we find
that the second degree terms in a and 5 are factorable, giving
n i^- [x *
y*][ A n-\-A<2)
X4- .~| T^ 4a>
Am
Am
Am
in^12731
J l-^ioo a ^-^oio "t J M /ljoo^iooi Ji 100 Jl 00l ja
+ (^010 ^001~~
^oio ^.ooiy ^~f~ I-
(138)
There are therefore two solutions for 5 as power series in a.
On substituting the two solutions of (138) for 5 in (130), we find the
two corresponding values of e. We thus obtain the two solutions
A (0 A (2) _ A (2) A (1)
moo -"-po i"- loo -^o oi
A (1) A (2) _ A (2) A (1)
-^oio -^ooi -^oio -^ooi
a +
7l(0)-qTa(0)
~a 6 (0)-aa 4 (0)a + %(0)
a 6 (0)a+
A (1) A (2> _ A (2) A (1)
-"10 -^-010 -<*100 -" -010
A ) A ) _ A (2) A (a +
1
5=-
e =
a 5 (0)-aa 4 (0)
,4(2)
a +
-floi
1(2) 1(2)
-"Moo _ _ "^OlO S_l_A (2)
aJT2) I
"001 -"001
o,(0)'
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144 PERIODIC ORBITS.
Coefficients of & . The coefficients of e2
satisfy the equations
Pi + 0J P2 + ^ 2-
^001 J + 0H>1 l Pi + 0*00 Pi+^UO Pi ""l + ^OJO "l= ^2 I
<j\ +6 t (T 2 + d 3 P 2=
0200 Pi+0110 Pi O"i + 0O2O 0"l= ^2
(146)
Every term of R2 and S 2 contains either A a>,
e s ,or Cj as a factor. Arranged
in this manner, we have
R2= +4 (,,J
[ 6^ a2
2 +d m a, 7l +Bm y\ ]
+A me, [IZ^Os o,+5 110 (7 l a 6 +Os7 6 ) +2^7! Ye+^ioi o]
+ J [ 02oo <*6 + ^iio a 6 7 6 + 0020 7e + 0ioi <* 6 ]
+ *2[0OOlL
S 2 =+A (m
[0^+07,0^71+^0 71]
+vl(1>
e 1 [ 20 2oo a, a 6 +0 llo (7i ae + a, 7 6 ) +20o2 O -y, 7 6 ]
+ 1 [ 0200 tte + ^llO a 6 76 + 0020 7o ]
In order to understand the character of the solution of equations (146), wemust examine the character of R2 and S 2 . The coefficient of A lm in both R2
and *S 2 is homogeneous of the second degree in a, and y t. Its expansion there-
fore involves terms carrying 2\r and terms independent of Xt. By 30, the
solution for the terms in 2Xr is periodic. The terms independent of Xt are
cosines of even multiples of t in R2 , and sines of odd multiples of t in S 2 .
These terms have the same character as those in the coefficients of el and e 2 ,
and will be considered under the discussion of those terms.The coefficients of Aw
e r in both R2 and S 2 are homogeneous of the
first degree in a 2 and 7, ,all terms of which carry the first multiple of At.
By Theorem IV, 71, the expression for p 2 will carry the term tcl^t) , andfor <r 2 ,
the term T7 2 (t) . Non-periodic terms of this character do not arise
elsewhere in the solution. Hence, in order to avoid them, we must take
either Aw or 4 = 0. If we choose Aw = 0, then, by (145), x is determined
and has the value e1= l/a^, thus agreeing with the first solution (139) of
the existence proof. But if we choose e^O, so that by (145) Am = l/<^,we are in agreement with the second solution (140) of the existence proof.
We will commence by developing the first solution, in which
A> =0, %-I.
FIRST SOLUTION.
Since A a) = 0, all terms in R2 and S 2 which carry Xt, or any multiple of
Xt, vanish. There remain
R2= e
2
t [0 2OO <h + no a 6 7 + 0^ 7e + 0ioi a e] + 2 0ioi >
C 1 47")
S 2
=e\ [0joo al + no
a7 + O2O 7e] J
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ORBITS ABOUT AN OBLATE SPHEROID. 145
We have also
Pl = ^l), *-*&*.. Cl= i- (148)
Ctc OLs Cl<)
It follows from (147) that R2 contains only cosines of even multiples oft, and S 2 contains only sines of odd multiples of r. Since p 2 is an evenfunction of t and <r 2 is an odd function of r, the solution is
p 2= yl
(2)a 2 (r)+Z)
C2
'[Ta3(r) + a 4 (T)] + k(T)+a 2 ra 3 (r)]+e 2 [a 6 (r)+ara 8
(r)],|
^ = A (2,
Tl (r)+i)(2,
[TT 4 (r)+T3(r)] + [r2(r)+a 2 rT4(T)] + 2 [T 5 (T)+aTT 1 (r)].j
(
In this solution the terms are grouped according to their origin. Thefirst two terms are the complementary function. The third arises fromthe terms carrying e
2as a factor. The fourth arises from the terms having e 2
as a factor, a 2 is a constant depending upon the coefficients of e2
in the
differential equations, and ^(t) and y 5 (t) are the same functions as in
the coefficient of the first power of e, t] 2 (r) and f 2 (t) are periodic functions
of r with the period 2-k, and so constituted that rj 2 (r) contains only cosines
of even multiples of r, and f 2 (r) contains only sines of odd multiples of r.
In order that p 2 and o- 2 shall be periodic we must have
which makesDm = a 2 ae 2 ,
p 2= A i2)
a, (t) +e 2 a 6 (t) +% (t)- a 2 a 4 (t), 1
<r 2= ^ (2)
T 1 (T)+6 2 7 6 (r)+r 2 (r)-a 2 Ta(r).(150)
In order that we may satisfy the initial conditions, we must have p 2= at
t = 0, which determines ^ by the equation
_ a i a i -y 2 -A a)a,
2 at
It is obvious that A 2 ,which so far is arbitrary, must be zero, for in the
coefficient of e3
it will give rise to terms involving the first multiple of Xr.
All such terms will carry A <2)as a factor; hence to avoid non-periodic
terms of this character, we choose Aw = 0. Anticipating this step, we have
p 2= Ct^^a 6 (r)+ )72 (T)-a 2 a 1 (T),
a*(151)
r i -M^ T,Cr)+r.<r)-flb'*(T) l
a 6
so that p 2 contains only cosines of even multiples of r, and <r 2 contains only
sines of odd multiples of t.
It only remains to show that this process of integration can be carried
on indefinitely. On assuming that up to and including p,_, and <7,_i every
p } and <jj is periodic with the period 2x, and that the p, contain only cosines
of even multiplesof r and the
07 onlysines of odd multiples of r, except that
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148 PERIODIC ORBITS.
on the coefficient of 2 in the differential equations. In order that this
solution shall be periodic, it is necessary that
which reduces p 2 and <r, to
p 2= ^ (2)
a,(T)+^(X, T)+77 2 (T)-a 2 a 4 (T)+e 2 a 8 (T),J
(r s= ^ (2,
Tl (r)+t/' 2 (X, T)+r 2 (r)-a 2T3 (r)+e 27 6 (T). J
To satisfy the initial conditions we must have p 2 (0)=0. Hence,since Oj(0) =
1, A is defined by
A (2 ' - -ft (0)-
r? 2 (0) +a 2 a 4 (0)- e 2 a 6 (0),
where 2 is a constant which is determined by the periodicity condition on
the coefficient of e
;3
Coefficients of e3
. The coefficients of the third degree terms are defined by
P3 + O2P3 + 03^3 = #3 1 <h + Q&z + #3 Ps= ^s > ( 1 59)
where
R} = ^OOl 3 + 0101 2Pi + 202OO PlP2 + 0ilo[ Cr 2Pi + 'lP2] +2 0^0 <T X (T t + 3OO Pj
+ 0210 P^i + 0120 ""lPi+0030 "l >
^3~ +2 2OO PlP 2 + 0110 IV2P1 + ^lPa] + 2 0020 O
-
! (Tj + 0300 Pi
+ 0210 Pi 0"l +0120 0"l Pi +0030 "l
In classifying the terms which belong to the expansion of R 3 and S 3 ,
we bear in mind that
1. The 6 ilt in R3 involve only cosines of even multiples of r, except those
which are coefficients of odd powers of a (i. e., where j is odd), andthese involve only sines of odd jnultiples. The opposite is the case in
the 8 tjk of S 3 . If j is even, the B ilk involve only sines of odd multiplesof t. If j is odd, the d iJt involve only cosines of even multiples.
2. The terms independent of X involve only cosines of even multiples of t
in the expressions for p x and p 2 ,and only sines of odd multiples of t in
the expressions for a x and cr 2 .
It is seen, then, that in those terms of R 3 which are independent of X
only cosines of even multiples of t enter; and in those terms of S 3 which
are independent of X only sines of odd multiples enter. In the processof integration, therefore, two types of non-periodic terms arise. First,
those coming from the terms which involve the first multiple of Xr, and
secondly, those coming from the terms which are independent of X. It is
important, therefore, to separate the various terms into three classes, (a) terms
independent of X, (6) terms involving first multiple of Xt only, (c) terms
involving multiples of Xt higher than the first.
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(5)
154 PERIODIC ORBITS.
Suppose the coordinates of (a), (6), or (c) are =, i?=0, f = 0, the value
of depending upon which point is in question. It will not be necessaryto distinguish among them except in numerical computation. Now give the
infinitesimalbody
a small displacement from one of these points, and a
small velocity with respect to the finite masses such that
= +*', ^ = 0+ y', r = 0+ z',
^l-no-^ dn=()4-dl ^_ n i dz'
dt^
4t' dtu-t
"
dt dt^
dt'
The differential equations (1) are transformed by these relations into
dV _ 2 dy' _ dU _ , p , , , 2 ~
(6)
^ = V($o+x'+m)2
+?/'2
+2", r 2= vU-l-r-z'+M)
2
+2/'2+z",
where Pi , P 2 , and P 3 are power series in x', ?/'2
, and z'2
.
77. Regions of Convergence of the Series P] ,P 2 ,
P 3. It follows from
the form of U in equations (6) that P l} P t ,and P 3 converge for the common
region of convergence of the expansions of \/r x and 1/V 2 . We are consider-
ing only real values of x', y', and z', and consequently the conditions for
the convergence of the expansions of 1/r, and l/r 2 as power series in x', y',
and z' are respectively
_, 2x' a;/2+t/
/2 + 2 /2
^|
,
t2x
f
g"+y*+g* < , ,
^^o-l+M^fe-l + M)2 +
(7)
The surfaces which bound the regions of convergence of the expansions are
obtained by replacing these inequalities by equalities. For the convergenceof the expansion of 1/r, ,
the equations of the bounding surfaces are
x'2
+i/'2 + 2
' 2 +2a +M)x'+(| +M)2 = 0,
J
The first is the equation of the point occupied by the finite body 1 m-
The second is the equation of a sphere whose center is at 1 m and whoseradius is V2( +m)
2- The convergence of the expansion of 1/r, holds
for thespace
between thepoint
and thissphere.
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OSCILLATING SATELLITES FIRST METHOD. 155
The equations of corresponding surfaces for the expansion of l/r 2 arc
x'2+ 2 /'
2 + 2' 2 +2fe-l+ M) a; '+fe-l+M) 2 = 0,
]
(9)
z'2
+y'2
+2'2
+2(& ) -l+ M)z'-( -l+ M)2
= 0. J
These are respectively the equations of the point occupied by the massn and of a sphere whose center is at n and whose radius is V2( 1+m)
2-
The convergence of the expansion of l/r 2 holds for all points within this
sphere except the center.
The distances from 1 n and n to the point (a) are respectively +/xand 1+m- The radii of the spheres which have been defined in (8) and
(9) are V2 times these distances. Since V~2 ( +m) 1> v^ ( 1+m), the
sphere around 1 n as a center is entirely outside of the one around n as a
center. Consequently, the series P x ,P 2 ,
and P 3 converge in the case of the
transformation to the point (a) for all points within the sphere whose center is
at m and whose radius is V2 ( +m)> except the point ju. itself.
The distances from 1 -^ and n to the point (b) are +n and ->/( l+/i)2
.
The radii of the two spheres are V2 times these distances, and hence theyboth include the point (6) in their interiors. In this case the two spheresintersect unless m is small, when one will be entirely within the other.
The distances from 1 n and // to the point (c) are n and 1 p.
Since V2 (1 m) 1 > V2 ( ju), the sphere around n as a center
includes in its interior the one around 1-ju as a center. The latter includes
(c) in its interior, and everywhere within it, except at 1 n, the series P lf
P t , and P s converge.
78. Introduction of the Parameters e and 8 . Let us now make the
transformations
x' =xe', y'
=yt', z' = ze' (eVO), t-t Q =(l + 8)r, (10)
where e' and 8 are constant, but at present undetermined, parameters.
Then equations (6) become
g -2(1+5) f=
{\ +8YP
1 {x, y\ *2
)
T=(l + 8T[X 1+X 2 e
f +
^+2{\ + 8)f ={l + 8yyP 2 (x, y\ z2
)
(1+5)2
[F 1+F 2 e' +
^?- (1 + 8YZ =(l + 8yzP 3 (x, y\ 2
2
)
+X n (e')n - i +
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156 PERIODIC ORBITS.
We shall now generalize the parameter e' (see 13) by replacing it
everywhere by e, where e may have the value zero or any value in its
neighborhood. When e ^ e', the differential equations belong to a purelymathematical problem; but when e = e' they belong to the physical problem.Since the value of e' has not been specified except that it is distinct from
zero, the generalization may appear trivial, but the same method can be
used where the parameter corresponding to e' does not have this arbitrary
character, and where the device is of the highest importance. We havetherefore to consider the differential equations
** _ 2(1 + 5) g=(l + 5)2
[X 1+X 2 + +X n -'+ ],
^ + 2(1 + 5)^= (l + 8y[Y i + Y2 e+ +F B- 1 + ], [ (12)
** a+tyz = a+5m+z 2 6+ . . . +z n e-i + ]
79. Jacobi's Integral. Equations (1) admit the integral
(i)'+$)"+(f )-**-.where C is the constant of integration. This integral was first given byJacobi in Corn-pies Rendus de VAcademie des Sciences de Paris, vol. Ill, p. 59.
For equations (12) there is the corresponding integral
where U is now a power series in e.
80. The Symmetry Theorem. Let us consider the solution of equations
(12) and suppose that at t = we have
_ dy _ dz _ _ _ dx _ nx-c,, d~~
Cl ' Tt~ C" V ~ Z ~Tt~ '
that is, that the infinitesimal body crosses the x-axis perpendicularly att = 0. The solution will have the form
*=/i(t), y=f 2 (r), -/t(r),
dx f i i n dy f f , -, dz ft ( s
57 =/ ' (T) ' ^ =/ * (r) '
dr =h{T) -
Now transform equations (12) by the substitution
x = +x', y=-y', z=-z', t=-t',d& = _^ dy = j-d]/ dz . dz'
dr
"
dr'
'
dr
= '^dr'
'
dr^
dr'
'
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158 PERIODIC ORBITS.
82. General Solutions of Equations (12) for 8 = e = 0. On referring
to equations (1) and the succeeding transformations, we find that equations
(12), for 5 = =0, become explicitly*
K+3$-r,- (X-A)y,
'*-Z --At.
(14)
where
A-r
< +r(o,3-
[( ^ +/x)2]j+
[( ^_ 1+)li)2], (10)
The third equation of (14) is independent of the first two, and its
general solution is
z = c l cosVAT+c 2 sinVAT, (16)
where c t and c 2 are the constants of integration.
The first two equations of (14) are linear and homogeneous, and theyhave constant coefficients. To find their solution, let
x = Ke Xr, y = Le* T
, (17)
where K and L are constants. The conditions that these expressions shall
identically satisfy (14) are
[\2
-a+2A)]K-2\L = 0, 2\K+[\2
-(l-A)~\L = 0. (18)
In order that these equations may have a solution for K and L other than
the trivial one K = L = 0, we must impose the condition
X2 -(1+2A), -2XAs
+2x, y-(i-A)= \ i +(2-A)\*+(l-A)(l+2A) = 0. (19)
We shall now discuss the roots of this biquadratic equation in X. Its
discriminant is
D = (2-AY-4(l-A)(l+2A) = (9A-8) A. (20)
We shall show that I A is negative for each of the points (a), (6), and (c)
for all values of juf, and therefore that D is positive. From (15) we have
1-A =1--jmt -p = (1
/*)[_! ^wJ+/i|_l ^bjsJ-(21)
At the points (a), (6), and (c) we have
dx dn dx dn dx
See also Introduction to Celestial Mechanics, and Charlier's Mechanik des Himmels, vol. II, pp. 117-137.
fFirat proved for all fi by H. C. Plummer, Monthly Notices of Royal Astronomical Society, vol. LXII (1901).
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OSCILLATING SATELLITES FIRST METHOD. 165
The solution of this equation gives us the periodic orbits in question. Wehave the two arbitrary parameters 7 and S, and we shall show first that
we can not give 5 an arbitrary value and solve the equation for 7 as a
power seriesin
e, vanishing withe.
Suppose that 5 is neither zero nor an integer. Then equation (44) is not
satisfied by 7 = e = 0, and the solution can not be made. Now suppose 8 is
zero or an integer. Then the left member of (44) vanishes, and the equationis divisible by e. It is, in fact, divisible by e
2. It is seen from (34) that x
and y do not enter in the last equation except in terms involving e as a
factor, and since the a, defined in (43) enter only through x and y, the
part of eP coming from the first four equations is divisible by e2
. It is
seen also that the part of the right member of the last equation of (34)
which is independent of x and y is multiplied by e2
. Therefore the right
memberof
(44)is divisible
bye2
.
Weshall
now provethat after e
2
hasbeen divided out there is left a term which is independent of 7 and e, andwhich is distinct from zero.
Terms in e2 in the right member of (44) are introduced both through
the a, defined in (43), and directly in the integration of the last equation of
(34). The terms obtained in the former way involve B as a factor and
depend upon a and p; the terms entering in the latter way carry C as a
factor. Hence, if the coefficient of e2
is to be identically zero, the parts
involving B and C as factors separately must be zero. We shall verify
that the part involving C as a factor is distinct from zero.
Thecoefficient of e
2 in(44),
so far as it isindependent
ofB and 7,
is
defined by the equation
-8TOaF sin3V7(1+ ^ )r -
(45)
The solution of this equation satisfying the conditions z 2= 0, z[
=0, at 7 = 0, is
(46)
z -= 64ffa+- sinVT(1+i)r "
wMfW'TCOsVJa+>)T
+ 6lJtW' Sin3V:T(1+ ' )T '
Consequently the last equation of (42) becomes
*(!9-*<>=[-8ATO+---]'
=- (47)
Hence, after division by e2
,there is a term independent of both 7 and e,
and the solution for 7 as a power series in e, vanishing with e, does not exist.
That is,periodic
solutions of the type in question do not exist.
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OSCILLATING SATELLITES FIRST METHOD. 171
The general term in Rn has the form of (60), where the subscripts and
exponents satisfy the relations:
(a") fi' + + /v is an even integer because the right member of
the third equation of (12) is a function of y2
.
(b") juf + ' +M" is an odd integer because the right member of
this equation involves only odd powers of z.
(c"), (d"), (e"), and (f") are the same as (c), (d), (e), and (f) respectively.
Suppose Mi+ +M* is an even integer. Then (a"), (0"), (7"), (6"),
and (e") are the same as (a), (18), (7), (5), and (e) respectively. Therefore in
this case the left member of (f") is an even integer. It is shown similarly that
the same result is true when ju t + +/x K is an odd integer. Therefore, Ris identically zero if n is odd, and in general is not identically zero if n is even.
The discussion now naturally divides into two cases, viz., where n is
even, and where n is odd. We shall treat them separately.
Case I. We shall prove that if n is even, R is a sum of sines of odd
multiples of VAt, the highest multiple being w+ 1. Consider the generalterm (60). The z Xi
. are all cosines of even multiples of VAt; therefore
the product x1*
1 x1*'
is a sum of cosines of even multiples. Because
of property 3 and the properties of the products of cosines of multiples of
an argument, it follows that the highest multiple which occurs is
M1 (X 1 + 1)+ +H K (K+l)=Hi\+ +MK X*+M 1 + +Mk. (62)
Similarly, from properties 4 and (a") ,it follows that wf* y"'' is a
sum of cosines of even multiples of VAt, the highest multiple being
Mi (>W+1)+ * ' +/vOC+l)=Mi^+ +Mk'^'+Mi+ +/v- (63)
From properties 5 and (b"), it follows that^'',
zM"" is a sum of sines
of odd multiples of VAt, the highest multiple being
*f(xr+i)+ +*&te+i)-Mfxf+ +mI"C+/x;'+ +/v<. (64)
On taking the product of these three sets of terms, we find that R is a sumof sines of odd multiples of VAt, the highest multiple being
N = n 1 \+ +mAc+mX+ ' +Mk'Ak'+m!X'+ +Mk'X'" 1
+ M!+ +M+Mi+ +/V+m('+ +/V'-J
By (f"), which is of the same form as (f), we have
N = n+l-(p l q l +p 2 q 2 ).
For those terms in which q^q^ 0, we have, as the largest value of N,
N = n+1. (66)
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OSCILLATING SATELLITES FIRST METHOD. 177
If we substitute the solutions (49), or rather the equivalent series for x, y, z,
and their derivatives, and the series for 5 in (77), and then arrange as a
power series in e, we have
F = F (x ,. . .
,<jfy +Fl (x ,x lf . . . ,)*+
~\~^n\X ,. . .
, x, . . ., , ,
o 2 ,. . .
, 6)e -f-= constant.
(78)
(79)
Since this is an identity in e, each coefficient is a constant. We shall nowwork out the form of the general term of (78). It is seen from (77) that
the function F n will contain terms of the types
^,&W fyfy=i, %>%=>, and (60).dr rfr dr dr dr dr
It follows from theproperties
ofx, y, and
zthat the terms of the first three
types are zero unless n is even.
Now consider the term of type (60). All the properties of its exponentsand subscripts are the same as when it belonged to P except that in the
relation (f ) the 1 in the left member must be replaced by 2. Hencewe see that this term is also identically zero unless n is even.
Since F involves only even powers of the y, and z, ,it is a sum of cosines
of even multiples of Vat. It follows from the relations (a), . . ., (f)
(the last one modified as indicated) that the highest multiple is n+2. Hencewe may write
F n = D' w +D'ln)
cos2VAT+ - +D'cos2JVAt j
+ +D^' 2 cos(n+2)VTr = constant,
Since F is identically a constant, we have
D' ln) = constant, D' 2ia) = =2)^=0. (80)
The quantities D'\ . . ., D& depend upon a (
;\ . . ., a?; 0?, ....,?;
yf ,. . .
, yf. Suppose all the at , p Jf 7j up to <-
2)
, p;~2)
, yf-" have
been computed and are known to be accurate. Equations (80) can then
be used in two ways, as we shall show. First, they test the accuracy of
the computation of the a^~v
, ft"~l
\ y{*~ l
\ for these quantities must have
such values that the equations shall be satisfied. And secondly, we can
compute the e^"
-", ftj~
l) from equations of the type of (69), and then find
the y and 5 directly from (80) without referring to the differential equations
of the type of (67).
The first use is obvious and we need to consider further only the second.
We are working under the hypothesis that n is even. Therefore the afand the /3
(
;
B) are identically zero. It is seen from (77) that the only terms
which can introduce the y" and 5 are
2Tr It + 2Az z + 2As z l- ( 81 )
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OSCILLATING SATELLITES FIRST METHOD. 181
This equation is satisfied, at t = 2ir/a, by v t= f = f =
0; and we find fromthe explicit form of F and the transformation (33) that, for these values of
the variables and a r= = a
4= 5 = e = 0,
~=ia 1 [n2 a 2 -1-2 A].
But from equations (19) and (28) we have
a 2 +l + 2A 2an =o*+l-A
Therefore nV 1 2 A = a 2 +n 2
{A 1), which is always positive. Conse-
quently, for r = 2t/(t, (88) can be solved uniquely for i\ in terms of a f , y,
5, e, v 2 ,v 3 ,
v t , , f', and this solution vanishes for v 2 = v 3 = v 4 = f = f' = 0.
Hence, if we impose the condition that v 2= w3
= i> 4= f = f
' = at r = 27r/<r,
the equation vl= at r = 2x/<r will be satisfied. Therefore the first equation
is redundant, and it will be suppressed.It will be shown that the orbits of Class B lie in the xy-plane. It
follows from the form of the last equation of (34) and the initial values of z
and z' that P 5 and P e contain 7 as a factor. Therefore the last two equationsof (88) contain 7 as a factor. The explicit form of the next to the last one is
P.Qf) -P 6 (0)=^sin^^ + 7i3
(a,,7,5,e)=0. (89)
When VA/a is not an integer the only solution of this equation, vanishingwith the parameters in terms of which the solution is made, is 7 = 0. Thecase where VA/a is commensurable will be considered in connection with
the orbits of Class C. Therefore z = 0, and the orbits are plane curves.
Necessary and sufficient conditions for the existence of the periodicsolutions of Class B reduce to (87), where i = 2, 3, 4. The explicit forms
of these equations are
0=(a 1 + a 2 )[e-2T(I+J,v=r
-l]+6Q 2 (a 1 ,. . .
,a 4 , 8, e),
0= a 3 [e+2T ^ a+5)
-lj+e^K, . . . ,a 4 , 5, e),
0= a 4 [e_2,r
" <1+5>
-l]+eQ 4 (a,, . . .,a 4 , 5, e).
We have three equations to satisfy and five arbitrary parameters,besides e, at our disposal. The parameter a x enters only in the combination
ai + a,, and since at
is as yet subject only to the condition that it shall not
vanish, we may let it absorb a x. We may determine t
, which enters in the
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OSCILLATING SATELLITES FIRST METHOD. 183
93. Direct Construction of the Solutions for Class B. It has beenshown that in the periodic orbits of Class B the coordinates are uniquely
developable in series of the form
u t= 2 w4
uV (t=
l, . . . ,4), 5= 2 SjeJ
, (93)
where the u\n are periodic functions of r with the period 2t/<t, and where
the 5 j are constants.
We have seen that without loss of generality we can, put, at r = 0,
u 1= u i
= a 1 =^, u 3= u A . (94)
It follows from (33) and (93) that we have also
x= 2 *,', y = 2 y t el
. (95)i=0 i=0
Upon substituting (93) in (34), arranging as power series in e, and equatingcoefficients of corresponding powers of e, we obtain a series of sets of
differential equations from which the u (
f
n can be determined.
The terms independent of e are defined by
(IT UT
a,T ar
The solution of these equations which satisfies the periodicity conditions
and the initial conditions is
uT=^eaV^ T
, u?=e-^ T, uf =
Q, <-0. (96)
From these results and (33) we get
x = a cos (xt, y = na sin or. (97)
The terms of the first degree in e are defined by
__ aV-lu, - + aV-ld lUl -r-4(wi<r
_np) ^f 2 (mp+n<T)
d< . ^r u > _ _ aV r8 M(o, _ 3mB[-2xl+yl] _ 3Bx y
du? _ > m _ 3nB[-2xl + yl] ,3Bx y
>
dr3
4(mo--wp) 2(mp+n<r)
^C x - 4- 3nfl[- 2^+y?] i%Bx y ,
drtP 4 -r
4( mr-
np )
" r
2(mp+n<
r)
(98)
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184 PERIODIC ORBITS.
The conditions that the solutions of these equations shall be periodic with
the period 2ir/a are that the right members shall be periodic with this
period, that the right member of the first equation shall not contain the
term e aV ^ lT,and that the right member of the second equation shall not
contain the term e -<r%/=rTr. The first condition is satisfied, and on referring
to (97) we see that the second and third conditions can be satisfied only
by 5,= . Therefore we put 5, equal to zero.
Upon substituting from (97), equations (98) become in full
du _ / -to m , 3 mBa'ltf - 2) - (n
2+ 2) co s 2 or] . 3 wBa asin 2 <tt
dr1
8(m<r np) V^l 4(mp-\-n<r)
(99)
du^ ,/ -
a) = _ 3mJ?a 2[(n
2-2)-(tt
2+2)cos2o-T] ,
3 nBa 2sin 2 or
drU*
8(m7 np) V^T 4(mp+n<r)
du _ a, _ 3 ni?a2
[ (n2 -
2) -(w2 + 2) cos 2 (7t] 3nJ?a 2
sin2<rr
dr 3
8(m<r np) 4(rap+n<r)
du, (,) = ,
3wBa 2[( n
2
-2)-(n2+2) cos2q-T] _ 3 nBa 2
sin 2 o r
dr*
8(m<r np) 4(mp+n<r)
The solution of the first equation of (99) has the form
w (.) = c(.)
e <rv=ir _|_ gffl _|_ a ) cog 2 _ ^AZ7 &< s i n 2 (rr, ( 100)
where cj" is the arbitrary constant of integration. Upon substituting this
expression in the first of (99) and equating coefficients of like functions
of t, we get
, 3mBa 2
(n2-2)+
8(ni(rnp)<T'
m = , mBa\n 2
+2) nBa?(
11
8(ni(r np)<r 2(mp+n<r)(r'
bm = _ mBa 2
{n2
+2) . nBa 2
124(ra<r np)a 4 (rap+no-)<r
(101)
It follows from the form of (99) that the solution of the second equationcan be obtained from that of the first by changing the sign of V l.
Therefore
a$-a$, ag-ag, 6--iff. (102)
The solution of the third equation of (99) has the form
u^-cfV+aff+aS? cos2rT+6^ sin2<7r,
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OSCILLATING SATELLITES FIRST METHOD. 185
where, because of the periodicity condition, c"5 = 0. We find by substitution
in the differential equations that
aa)
= +<= +
u 32 r*^42
3nBa2
(n
2
-2)8(manp)pSnBa i
(n2
+2) P + ZnBtfa8 (ma-np) (4 <x
2 +p 2
) 2 (mp+n<r) (4 o 2 +p 2
)
'
7>>- _^<u-_L 3nBa 2
(n2+ 2)o- ,U32 "42 T a /_. ITTTl 2
_l 2V "T
3nBa 2
4(mo--np)(4o-2 +p 2
) ^4 (rap + no-)(4o-2 +p 2
)
(103)
Since x = a and ?/= at t = for all values of e, we have x x (0) = y x (0) = 0.
Upon substituting uj, . . .,n"' in (33) and applying these conditions, we find
Mi r w .
Then the expressions for Xj and ^ become, by (33) ,
x,= 2 [<> +aj] - 2(a>+a+a+] cos err +2 [ag + ag] cos 2rr,
y l =+2n[< ) +<+< 2
)
+a< 2)
] sin<rT+2[n6g+m&g] sin2<rr,
6\= 0.
(104)
(105)
In order to see how the construction goes in general, the process mustbe continued one step further. The differential equations which define the
-next terms are
dulOB
-oV=\u? = + cr/=7< l + 3mB[-2x xl + y y l ] _ 3B\x& i xMi {
2 (ma np) V 1 2 (rap + no-)
^_ +<rv^Twf = - rV^=T<V
, raC[2xg-3x j/ g] , 3(7,4x2
j/ -j/^|
(rao- np) V 1 4 (rap + no-)
3m^[-2x x 1 + y y 1 ] _ 35)3^ +3^,}2 (rao- np) V 1 2 (rap + no-)
ra(7[2x^-3x t/g]
+3C]4x
2
y -i/g }
?
(rao- np) V 1 4 (rap + no-)
j P^a n t ...~ _ ^ ~t... I \
2(ma np) 2(mp+na)
_ nC{2x3 -3x a yl ] _ 3C{*x
2
y -y3
\>
ma np i(mp+na)
du ? j-.i - l . 3nB[-2x x 1 + ?/ y l ] , 3g]z y 1 +g,y2 (rao- np) 2 (rap + no-)
+ nC[2xg-3x y2
] 3C|4x2
y -^[
manp 4 (rap + no-)
(106)
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(107)
186 PERIODIC ORBITS.
In order that the solutions of these equations shall be periodic with the
period 2x/<r, the right members must be periodic with this period, the right
member of the first equation must not contain the term e ay/ ^ iT, and the
right member of the second equation must not contain the term e~ ay/ ^ riT.
The first condition is satisfied as the equations stand. The expressione <rv^ir en ters through w{
0)
,x x lt y y lt x y lt and Xj?/ . The sum of its
coefficients must be put equal to zero; this condition determines o 2 by the
equation
8 m _ 3 mB [4> + off) +2((4U+<) +n(nb[ +mb%)]
2
2(m<Tnp)p
_ 3B[-2n(a{+a)
)+n(a{ Iu+<4 u
) + (n6gf+w6g)]2(mp+na)<r
3ma 2C(2-n i) _ 3na'C(4-3n
2
)
8{ma np)
a 32(rap + no)
<r
This disposes of all the arbitraries appearing in the equations, and the
third condition remains to be satisfied. Upon comparing the first and second
equations of (106), we see that the signs of the [ ] in the second are oppositethe signs of the corresponding terms in the first, and that corresponding
{ \have the same sign in the two equations. It is observed that the [ ]
are sums of cosines of multiples of or, while the\ \
are sums of sines
of the same arguments. Since 8 2 enters in the second equation with the
sign opposite to that in the first, it follows, as a consequence of the prop-erties of [ ] and
\ \ ,that the condition on the second equation is satisfied
by the same value of 6 2 as that which satisfies the condition on the first.
We now proceed to the general term. Suppose x l ,. . .
, x v ^;y l ,
. . ., ?/_! have been computed, and that they have been found to
have the following properties:
1. The Xj are sums of cosines of multiples of gt.
2. The y } are sums of sines of multiples of or.
3. The highest multiple of <jt in x } and y, isj + l.
4. The [ ] is an even function of y.
5. The \ \is an odd function of y.
The equations defining thecoefficients of e
are
dr l
2{m<T-np)V-l 2(rap+n<r)
*+<rv ^r ur = -*v=\ <>s- -
dr 2 2 "2(m<r-np) V-l 2(rap+n<r)
dr 3
2(ma np) 2 (rap + n<r)
d< + u w = + n[J + U ,
dr *2(ni(T np) 2 (rap + no-)
where the 5, ,. . .
, o_, are included in [ ] and ] },
;108)
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OSCILLATING SATELLITES FIRST METHOD. 187
It follows from properties 1, 2, 4, and 5 that [ ] is a sum of cosines of
multiples of <tt, and that { J is a sum of sines of multiples of or. The
geaeral term of [ ] is
*-< <- <; (109)
The exponents and subscripts of this expression satisfy the conditions:
(a) Mi+ +/v is an even integer because of 4.
(b) q is or 1, since 5 enters (34) linearly.
(c) mA+ +hkK+h'iK+ +HK>K>+pq+Mi+ +M* + Mi+ +/V 1=".
The product x[ ocfc is a sum of cosines of multiples of or, by 1.
There is an even number of odd /*,,..., /v, by (a). Those factors w\ for
which ii sare odd are sums of sines of multiples of or. The product of an
even number of such factors is a sum of cosines of ar. It follows that
T v is a sum of cosines of multiples of or.
The highest multiple of or in T v is
N = n l (\+l)+ +n K (\ K +l)+n' (\[+l)+ +n'AK'+l).
It follows from (c) that
N = v + l-pq = v + l when q = 0. (110)
Therefore [ ] is a sum of cosines of multiples of ar, the highest multiple
being v+1.It can be shown in a similar way that
] ^is a sum of sines of multiples
of or, the highest multiple being v-\-l.
Equations (108) can be written, therefore, in the form
du?_ ay/ZZiu? = + aa v 1 oV=\ Te'
J,, 00
[v+i
-i ( "+l 1
e-aV^ T
8 v
du
mv l
-v+\
[v+l
-i [v+l T
2 Arcosiarl -jS ^"siniVr k
dr
du?
*- - P u{?= +n\X Al
v)cosi<TT
L(=o J
+pt*f- -n[s ^cosiVtt]
+
+
rs^rsin^rj
[ "s^'sin tVrj
(111)
where the Af and <" are all known real constants.
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188 PERIODIC ORBITS.
In order that the solution of equations (111) shall be periodic, we must
impose the conditions that the coefficient of e aV=lT in the first equation, andof e -,rV=TT in the second equation, shall be zero. It is easily seen that the
two conditions are identical, and they uniquely determine 5 by the equation
ab VI a ( "> _ A R<">(112)
The periodic solutions of (111) are of the form
v+l _ v+1< - < + < e ffV=TT + c? e~ aV^ T+ 2 < cos jar - V - 1 2 &# sinjVr,
<> = < + 2 < cos>r+ 2 b% sin jar,
< = <+ 2 <> cos ja T + 2 fc^sinjVr,
^=2
.7=1 J=l
(113)
where e?5 and CJ" are arbitrary constants of integration.
Upon substituting (113) in (111) and equating coefficients of corre-
sponding functions of r, we find
io i^20 A o Ia
a%=+a%=-[mA?-B?\
u(> _ _ ijW _ _ (jAj -\-Bj )
U ~ u~a(f-l)
a 30 "t" a 40 ~~ I "^0 >
P
M
(i=2, . .
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OSCILLATING SATELLITES FIRST METHOD. 189
The arbitraries ef and cf are determined by the conditions that x v=
and t/= at r = 0. Upon applying these conditions, the final results are
x -2[a>2*j---2 Zl [<+<] cos<rr +2 's [<+<>] cosjVr,
i/, =2n 2 [<+a] sincrr+2 "s [n6J?+m&$ tan jar.
(116)
These expressions have the properties 1 and 2. Since x, y ,
x t , y x also
have these properties, the induction is complete and x depends only uponcosines of multiples of or, and y upon sines of the same argument. Theorbits are therefore symmetrical with respect to the z-axis.
94.Additional Properties of the Orbits of Class B. It is observed
that x and y a are homogeneous of the first degree in a, and that x, , y t ,and
5 2 are homogeneous of the second degree in a. It is easily proved byinduction, making use of the general term (109), that x v and y v are homo-
geneous of degree v+1 in a, and that 8 p+l is homogeneous of degree v+1 in a.
Consequently, it follows from (10) that the actual coordinates, x' and y' ,
carry in each term of their expansions e' and a to the same degree. That
is, e' and a are equivalent to a single parameter, and we may put one of
them, say a, equal to unity without any loss of generality. Then the
explicit expressions for the coordinates become
x' = [cosrT]e'+2[(aS,) +oJ) - (<C+a+<+a) cosar
+ 2
>
+a32)
)cos2< n .]6/2 + . . .
;
y'=
[- n sin ar] e' + 2 [ + n + a + ag + ag) sin <tt
+ (w6<1
2
) +m6< 2
>
) sin2<rr]'2 +
(117)
Since n is a positive constant, it follows that in all cases the motion in
these orbits is retrograde. For small values of e' the orbits are approxi-
mately elliptical in form, the axes of the ellipses coinciding with the
x' and i/'-axes.
The integral can be applied as before to check the computation, for
it has the form
F(x, y, j^> ^)=F +F l e+F 2 e2 + =constant.
Since this equation is an identity in e, each Fj separately is a constant. It
is seen that F, is a sum of cosines of multiples of ar, the highest multiple
being j + 2. Since the equation is an identity in r, the coefficient of each
cosjVt(j =1, . . .
, v-\-2) separately is zero. These coefficients involve
the a\" and b^ linearly, and their vanishing constitutes a check on all
the a\i\ bl^ f rom the beginning of the computation to the step under
consideration.
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190 PERIODIC ORBITS.
95. Numerical Example of Orbits of Class B. In Darwin's memoir,cited at the beginning of this chapter, there are a few examples of orbits of
this class in the vicinity of the equilibrium points (a) and (6). In all his
computations Darwin took one finite mass ten times that of the other.
To be able to compare the results of this analysis with his orbits, we shall
apply the formulas for the same ratio of the masses. This was the ratio
used in the computation of 90, and the first part of the table given there
can be used here.
Upon making use of the preceding computations and (36), (101), (103),
(104), (107), and (117), we get the following table of results:
Coefficient.
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OSCILLATING SATELLITES FIRST METHOD. 193
In the solution of the third and fourth equations it is unnecessary to
compute the terms which carry a 3 and a4 as factors. Omitting these terms,
the divisors for the third equation are respectively p, 2<ri p, 2aip,
and the terms coming from ( nEc2
/2A) cos2 VAt are
nEpc2
cos2V At + nEc7=? sin2V^T.2(A+ P
2
)Awu " v "' '
(4A+p2 )VA
For the fourth equation the corresponding quantities are
nEpc2
n.r-r. nEc 2
p, 2<ri+p, 2<ri+p, cos 2 VJt2(4A+p 2
)Awa " v "'
(4A+p2 )VA
The solution of the last equation of (124) is
== sin2V At.
z,= + L|" cos xTat+L sin \TXt +SB
did
+ zBckci
2(<r-2crVT) V~A
SBcaii
2(p2 -2ipVA)VA
. zBa 2 ci
,(-<t+VX)(t
(-p+VAt)T
2(<t2 + 2<tVa)VA
'3 Bead
(,("+ V7)tT
(p+VAI)T
2(/+2?pVT) VT
+
2(<t2 + 2(tVA)VA
3 Bead2(p
2 +2ipVA)\/A
(_<r_V^4)[T_|_
2(a2 -2a\ rA) V~A
3 Bead2(p
2
-2ipV~J)V~A,(p-VAt)T
=j^e(_p_\ At)T
(126)
The constants of integration K[v
, , K[x>
are determined by the
conditions that u[l)
, u[l)
shall vanish at t = 0, and the constants
Lf and L< by the conditions ,(())=
z((0) =0.
It is necessary to compute all non-periodic terms of u u u 2 ,and z which
are of the second degree in e and which are independent of a, ,. . .
,a
,
and 5. The right members of the differential equations involve
[}w = lB[-2x x
l +y y l +z zl }+2C[2xl-3x y
2
Q -3x zl],
| !
<2, = 3B-;x o2/l +a- : t/ !- +ic\-44ih+ti+v&.
The quantities xA and y {are defined by
x^u^ + iV + uy+u'", y I -m(, U) -O+m-i ).
(127)
In order to get all the non-periodic parts of the solutions at this step, the
terms of the differential equations which are non-periodic, that is, which
carry p in the exponential, must be retained; in the first and second
equations the terms in e"T and e~
atTrespectively must be retained; and in
the z-equation the terms in cos VA t and sin VJt must be retained, for
these periodic terms give rise to terms in the solution which are multiplied
by r, and which therefore are not periodic.
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194 PERIODIC ORBITS.
2ir7) 2 trHThe conditions for a periodic solution with the period
*- =rj
= T are
- u t (T)- u t (0)
= [if (T) - uT (0)] + K\T) -mJ(0)] e
+K)
(D-wf)
(0)]
J
+ (i
= i,4),
= z(T)- z (0) =
[z (D - z (0)] + [z, (T) -z, (0)] e
+ [z 2 (!T)-z 2 (0)]e'+
, (128)
By means of the steps explained on pages 191 to 193, we find explicitly
u?\T) -u(0) = (a 1 +a 1 )[e+J " a+8,J -
1], u?(T)-v?>(0) = a 3 [e+2xp
l-l] ,
u?{T) - uf (0) = (a 2 +a 2 )[e-w(l+i>< -
1], uf{T) - M (0) =a 4 [<f2
"*-l] , (129)
2,(T)- 2.(0)=vz(
c
1 + 8)sin2T(l + g);
u a)( D - wH)
(0) = i1 2mEi{2 - mni )-F(ni + m)]a l a 3
[ 2mEi{ 2 -f- mm) F(ni m)] a^ ~\ r />+ 2t pt il+
2<n-p jLe
*J
,f
[ 2mJgt'(2 + ram')+/'T
(m' ra)]^1 P
, [ 2mEi{ 2 mni) F( m' + ra)]a,a 4 \ r -**? _ , ~i+2ai + p jL
6 lJ'
mW/'Pi <o/ay- f [-2m-gt(2-mm)-^(m + w)]aia 3
w, ^; ws <u;-j 2<n' + p
, [ 2mEi(2 + mni) + F (ni m)]^ |r +2*pt _ j"l
+{[ 2mEi(2 + /Mm') F(ni m)]^^
2ai p
, [ + 2mEi{2 mni ) F(ni + m)] a^a t \ r - 2 *pt _ j"|
u? (T7
) u'n(0) = j
[-^( 2 +^)-
wA>>+
2nE(2-rit
)a l a t
[nE{2 + tf)-nFi\a\ 2nEc 2
lr.+ 2 ^7_il+2<rt+p ( 4 ^ + p
s
)pjLe X
J'
Ma> ^ _ wa)/)= f [ng(2+n^-wft]^ + 2nE(2-n*)a l a 1
[-nE(2+ n 1)- nFi]< 2nE& \ r -**v _ , "i
+ 2^~p (4^ + p')pJ I6
'-I'
mfV\ ,(tt\- GBca 3 r+2'p^ ,i 6flca 4 r -*> . -i.
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OSCILLATING SATELLITES FIRST METHOD. 195
, 2nE(2-n 2
)a 1 a 2. [nE (2 + n 2
)- nFi]a\
p 2ai-\-p2nE& ]fqi_ <h ] r**p^ il
(iA+p*)p}\p 2<ji-p\\-e
*J
_ imEi \ [nE{2 + n 2)- nFi]a\ , 2nE(2 -n 2 )^,
1 2ai + p p
2ai-p (4A + p2
)p\\p~i~2ai + p)
X \e*"* -ll +2m^([-^ (2 + n2) - wF^l J
[ 2crt p
_|_ 2w
J
g(2-n2
)a 1 a !!,
[ttff(2 +n2
) -nA']a2
p 2ai + p
_ 2n.Ec 2
1 fai , en 1 r **>$ n
~(4A + p2 )pJlp"
h2 (rl-pjL
e\J
.,, 2m *nE \^ lE{2 + ^ ~ nFi ^< + 2nJg ( 2
~ n *) q i q
I 2<7t + p p
, [-tt.E(2 + n 2
)-nFt]a2
_ 2?lEc 2
] [at _ a,]
2(7i-p ~1JA + pTp\\p 2ai + p)
x IV 2*^ ll mF \ [~nE(2+n i
)-nFi}alL J
\2<n
p, 2nE(2-n 2
)a l a i , [n.E(2 + n 2) -nFtja
2
p 2<rt + p
2n.Ec 2
1 fat a, 1|y^4 il
"(4^ + P2
)pJIp 2*i-pJLe "'J
F J [n.E(2 + w2)- nFt]a
2
_,_ 2nff (2- n 2 )^^
L 2<n + p p
, [-?i^(2+n2
)-nFt]a2
_ 2n.Ec 21 fai ,
a aj
2<n'-p JjA+TTpj {p^~ 2ai + p]
x [Y2
^ _ i] _ nF A\-nE{2 + n*)-nFi]a\L -JL 2<n p
, 2nE(2-n 2
)a l a i , [n.E(2+n2)- nFi]al _ 2 nEc 2
]
p 2cn + p (4^+p2
)pJ
fa, ,a 2 ] r^ _ ^ ^ r [nff(2 + n 2)-nFt]a
2
lp 2ai-p\\- J-T 1 2<rt+p
, 2nE(2-n 2)a l
a 2 , [-n.E(2-f tt2
)-ni^]a;p 2<ri p
_ 2nE& IK _ a 8 Ir.-^^.i-i(4^+p s
)pI1p 2<n+pJ L M
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196 PERIODIC ORBITS.
where
L = 4 mETi f[2m(2+n a
)-4nF] + [-2n PE(2 + n 2) + 4n<rF]
1 3a 4<72 +p 2
_ 4i?(2-w 2)(mp + tt(r)
], 2 mn i ETi I [~4m^(2+n 2
) + 2nF]
, 2mn[2(r^(2 + n 2) + pF]
] _ , F y f [2m(2+n 2) + 2wF]
4tr2 +p 2
J 1 3<r
_ 2m[2o-ff(2+w2) + pF] , 2n[pE(2+n
i
)-<rF} }4<r
2
H-p2 +
4 ff2 + p
2
j
+nFTi [ 4(2-n 2)(mp + na) l imECTi
\ <rp ] B
+
(2 - w2)
"J"1"
~52nFCTi ZtfFCTi
B 2B
M 4mE 2 Ti(m P + n<r) SmBETi nEFTi(mp + na)
apA'
(4A-a 2
)A apA
mECTi nFCTiAB 4AB
There are corresponding equations for u^(T) u(0) and z 2 (T) z 2 (0).
The third and fourth equations of (128) can be solved for a 3 and a 4 as
power series in e, a, ,a 2 ,
and 8, vanishing with e. The explicit results are
(130)
\ [+nE(2+n i
) + nFi]al _ 2w(2-rt 2)a 1 q i
3
1 2ai p p
-{
, [-nE(2+n 2
)+nFi]al ,2nEc 2
1
2<n+p" h
(4^+p2
)pJ
"i"
-7?ff(2+?t2
)+flFt]a2
_ 2nE(2-n 1
)a l a %
2ol+p p
. [+ nE(2+n i
)+nFi]al , 2nc 2
]
2<n-p" h
(4^+p2
)pJ6i "
(131)
When these expressions for a 3 and a 4 are substituted in the first, second, andfifth equations of (128), we obtain
= (a, + a 1
)[e+2,r ' ,5
'-l]+(a 1 + a1
)[(a1 + a 1 )(a 2 + a 2 )L 1 + c 2 iVf
1]e2+ ,
= (a 2 + a 2)[e-
2 "' 5,
-l]-(a 2 + a 2)[(a
14-a 1 )(o 2 + a 2 )L I +c 2 M1
]2 +
n c ainoJ^t ZBT\E<?[ mp + na, 1/ -m<r
, np \~| [=vT(T+5y
Sln27r(?5-IXIXL-TT + 2(4^^7 2 +
4Z+p- )J
65(ai4-ai)(ai+a 2 )cAA -a
'
+ ^{2(4-n2
)(a 1 + a 1 )(a 2 + a 2 )c-f|}e2 +
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OSCILLATING SATELLITES FIRST METHOD. 197
After removing the factor c from the last equation, solving for 5, and
substituting the result in the first two equations, we have
where
= (a i + a 1 )[(a 1 + a1 )(a 2 + a i )L+c 2M] e
2 +0= (a, + a,)[(a 1 + a
1 )(a, + a 1 )L + cI
M] e2 +
L= Jji 9B 2
, 3C(4-n 2)
2*jri
"(iA-a 2 )VA
~t"
2VA= j
M= i_i_
^BE [ mp+na ,
l^/ ma , np \~|
2wJW^
2A* L (rp+
2 V 4 A - ff2 "*"
4 A + p2 /Jirpi 2.A* L <rp SV^-^ 4i + p! /J 16-4'
(133)
(134)
Equations (133) can not be solved for atand a 2 as power series in e,
vanishing with e, unless
a l [a 1 a 2L+c2
M] = 0, a 2 [a 1 a i L+c2
M]=
0. (135)
One solution of these equations is 0,-0,-0, and with this determination of
aY and a 2 , equations (133) are uniquely solvable for a
xand a 2 as power series
in e, vanishing with e. In this case the generating solution reduces to the
form of that of Class A. But the orbits of Class A heretofore treated were
those for which V~A and a are incommensurable. This restriction was not
necessary in order to prove that orbits exist which re-enter after one revo-
lution, but it was not certain that there are not others re-entering only after
many revolutions. The uniqueness of the solution of (133), for a i= a 2
=0,
proves that all of the orbits of Class A, of the analytic type under consideration
here, re-enter after a single revolution.
At the beginning of the present discussion the assumption was madethat c is distinct from zero, and this permitted the suppression of the last
equation of (122). If we had assumed that al
is distinct from zero, wecould have suppressed the first equation of (122); and solving in a different
order, we should finally have arrived at two equations corresponding to (133)
containing (c + y) as a factor. The equations would have been found solvable
after imposing the condition c = 0, and we should have arrived at the con-
clusion that all orbits of Class B re-enter after one revolution.
Equations (135) also have the solution
a l a t L+<*M = 0. (136)
This equation defines c when ax and a 2 have been given arbitrary values.
If the orbits are to be real, 0, and o 2 must be conjugate complex quantities.
Under these circumstances their product is positive, and L and Mmust be
opposite in sign in order that c shall be real.
After the condition (136) has been applied, equations (133) become
0=(rt l + a,)[a la 2 +rt 2 a 1 -f a^lL + e [a, ,
a 2 , e]+ ,1 .
0=(a 2 +a 2 )[a 1a 2 +a 2 a 1 + a
1a 2 lL+e[a 1 ,
a 2 , e]+ j
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198 PERIODIC ORBITS.
Since the terms of these equations which are independent of t are identical,
except for the non-vanishing factors a^a^ and flj+Oj ,it follows that if
one is solved for a, and the result substituted in the other, the latter becomes
divisible by e. After dividing out e, there is a term independent of Oj and
6 which must be equal to zero in order that the equation may be solved
for a, as a power series in e, vanishing with t. This term involves the
coefficients of e3 in the original solutions (122), since e
3 has been divided
out. Likewise, terms enter from lower powers of e through the elimination
of a 3 ,a 4 ,
and 5. It is not possible to construct these terms without an
unreasonable amount of work. But we see from the way in which they
originate that they are homogeneous of the fourth degree in a 2 and a 2 .
Unless one or the other of these constants is absent, their ratio is determined
by this constant term set equal to zero. If one is absent, the only solution
is the other set equal to zero, which throws us back on Class A, which
has been already completely treated.
Suppose both constants are present and that their ratio is determined.
Since they must be conjugate in order that the orbit may be real, the solu-
tion for the ratio has the form
cti a+b V=l q 2 -6 2 + 2abV=la,
"a-b V-l
'
a 2 +6 2
It is clear that only for special values of the coefficients, which might never
be possible in the problem, could the solution for the ratio have this form.
The complexity of the problem is such that no further attempt will be made
here to determine whether there exist solutions of Class C which are dis-
tinct from those of Class A and Class B.
If the attempt is made to construct the periodic solution of which
(32) are the terms independent of e, no difficulty will be encountered until
the terms in e2 are reached. Then it will be found that equations (135)
must be satisfied in order that the solutions at this step shall be periodic.
That is, step by step, the construction agrees with the existence, thoughthe computation is somewhat less laborious.
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(11)
202 PERIODIC ORBITS.
100. Existence of Periodic Solutions. Since the right members of
equation (7) do not contain t explicitly, sufficient conditions that (10) shall
be a periodic solution with the period T are
=ii,(T)- 11,(0)
=a 1 [e
+V/riT
-l] + p 1 (T)-p I (0),
O-tt.CD-u.CO) = a 2 |V<VriT
-l] +ft(T)-p 1 (0),
= u 3 (T)-u 3 (0)=
*ie+
>>>'t
-l] + p,(T)-p,(0),
= u 4 (T)-w 4 (0) = a 4 [e-".T
-l] + p 4 (T)-p 4 (0),
0= 2 (T)- Z (0)=^=sinVT T +p t (T)-p s (0),
0=2'(T)-2'(0)=7[cosVT T-l]+p 6 (T)-p 6 (0).
The last two equations of ( 1 1 ) are satisfied by y = 0. Suppose y j* 0. Thenit follows from the form of the integral (8) that unless \/A T = (2w + 1)t/2,where n is an integer, the last equation is a consequence of the first five.
We shall suppose T does not have one of these special values, and we shall
suppress the last equation since it is a redundant condition. The first
five equations are to be solved for a, ,. . .
,a 4 ,
and y in terms of X, andwe can use only those solutions which vanish with X . These equations are
satisfied by a,= = a 4
= y = 0. In order that this may be not the onlysolution vanishing with X, the determinant of the coefficients of the linear
terms in a,, . . .,
a 4 ,and 7 must be zero. This condition is explicitly
[/a*T _
1] [-V=iT _
1] [>T _
j] [e->. T _
j]sin vj oT . 0. (12)
This equation has the solutions
Tl = VT ' T2= ^f (* an integer). (13)
Consider first the solution T = T x . For this value of T the determinant
of the linear terms in a, ,. . .
,a 4 of the first four equations of (11) is distinct
from zero unless v<jJ2 VX is an integer. This condition can not be fulfilled
for all v unless <r is an integral multiple of 2 V~A . Now since w = <r V 1
satisfies
w<+(2-AV+(l+2,4 )(l-.4 )=0 (
this condition can not be fulfilled unless A is negative, but from its
definition in (3), A is always positive. Therefore there are values of v for
which vaJ2 \/A~ is not an integer, and one of these values of v is necessarily
unity. It follows that the first four equations of (11) can be solved for
a,, . . .,a 4 as power series in y and X. Since u lf . . .
, w4 ,and z vanish
with a,= = a 4
= 7 =0, these solutions vanish with 7, and since the
first four equations of (7) are functions of z2
, they carry 72 as a factor.
On substituting the solutions of the first four equations of (11) in the fifth,
it becomes a power series in 7 and X alone, and is divisible by 7.
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OSCILLATING SATELLITES SECOND METHOD. 203
In order to prove the possibility of the solution of the fifth equationfor 7 in terms of X, and to determine the character of the solution, we mustwork out the first terms of the series. Terms in X alone can not be intro-
duced from the solutions of the first four equations of (10) unless the fifth
equation is divisible by 72
,for the former carry y
2 as a factor. We shall
show first that the fifth equation of (11) has a term in 7X.
It is seen from (7) and (3) that the coefficient of 7X in the expressionfor z is defined by
> . A . A,- A'Li ""!! " *M VA sin \ rA n t. (14)
The solution of this equation satisfying the conditions that z(0) = and
< 1 (0)=0is
2,,^-^sinv^+^cosVi;*. (15)
It follows from the non-periodic term of this equation that the fifth equationof (11) has a term in 7X, and therefore that the solutions exist. To get their
character we must find the terms of lowest degree in 7 alone.
The coefficients of y2 are defined by
d< "- a * - 4- oI I*
0> VZ7 1
dtl
du?-0)
dt+ <r <' 0) =-
2 ( Wo (To W0P0)
m [ ]'
2 -'v^T
I I (2, 0)
j L
J.. a. o)aU 3 _ n ,,<2,0>_Po
M3dt
dul^dt
+ Pyr= +
2 ( Wo <r W0P0)
2(w <r WoPo)
n [ ]
++
2(w p +WoO'o)
I I (2. 0)
2(WoPo+WoO"o)
J I (2. 0)
2(WoPo+W ffo)
I I (2, 0)
* f
2 ( Wo (To noPo) 2(w p +WoO'o)
whereZj,o 1 -^o^j.o ~r"3 -Oo^i.o^i,
(16)
[ ]
<2 ' 0, =-fJ3 [-2z?, +2/?. + 4o],
i i (2, 0) O _,
.
1 f<5 -Do^i.oi/i, (17)
wj2,0 *
carrying 72 as a factor.e shall need only the terms u"-
m,
Hence in the first four equations we may omit x h0 and y h0 . From equa-
tions (G) and (10) we have
^ 0= 2A ~~2A cos2Vir>l
>
^1,0 ^i,o=
1 2
+
+
2V^ a/y r
e (-r.+vTjV=it _ e (_<r,-vT) v^I< "1
a : ,7
+
2V7; v^
a*y
2VX V
(+pQ+-^i vCi)i _ (+ p,-vt; vri)(
-P,+Va, ^=i)t p( _p,_VT, VZ1)I
(18)
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OSCILLATING SATELLITES SECOND METHOD. 205
we find for the required terms
35 0%0 z1 ,=35
^[2(ai5/ ) +; 2
o'
u, )-K ' 0,
+ar ' 0,
)]sinV7;<n D Q 0)
+ AiS *?*- y(f*** + e-^ u
)sin VT tVA
+ -^^- (ep ' + r. )
sin VT^,(22)
(23)
VA
Therefore the part of the solution of (21) not having the period T^ is
-7 o / a \ rr sm(<x +VA )t + -,
".
^ sin(o--\/X)J
n D ^.(2.0)
off n , 0) .
- ffiT 7\ / . sin(p fl V=T-V^*.(po+2 V^ V-l) V^ po
vo 0/
We can now write the conditions for the existence of periodic solutions.
Upon using the results just obtained, we find for the values of (11)
tTT.vCj R 2 r- _^rovv^l
[ -']- 2K-:)( V,['" ^- i ]+
o-*|V3 -i]- 2 (m J4:% , +P ;)J^ "'] +
-.[" -']-i( ^.-^(W ,;) J e **_-']
+
ft R T y7T<r V3I
+ [(-l)'-l]^'+
(ii l i
^ V=T [(-l)'-*
-l]
-TTi ftr^r(-l)'e^-ll+7Ti%Vr(-ire"^r.-ll(4^ + Po)p Lv
J (iA a + p2
) Po LK
J
6M-7_ Tr iv/^r ili
6J?o!r'7~(4ii
-**)>. v^T L
( i} e ~ XJ+
(4^- <r* 7=t
(-''""'O-feysM'^- 1]
^(4A+pJ)p Lv '
J
(24)
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20S PERIODIC ORBITS.
We have one equation for the determination of the parameters a, and
a,,. Consequently we may impose one condition upon them. It will be
convenient to take a, and a 2 so that
x' = 0at t = 0, (32)
a condition which is satisfied in every closed orbit for which the right mem-bers of (2) converge. From equations (6) it follows that this condition is
o- v^T (a 1-a 2 )+p (a 3 a 4 )=0. (33)
We shall regard this equation as a relation between a, and a 2 which is to be
used in connection with (31) for the determination of these parameters.In order to complete the proof of the existence of the solutions it is
necessary to discuss the second and third degree terms of (31). It follows
from (7) and (3) that the terms of the second degree are determined by
dt
(2)
-oj,uT=-m a [ ]
w i-\ r,
du?dt + Pa uf=+n,[ r +I I (2)
(34)
where
2(m <r n p )
I I (2) .
1 f'
2(mopo+no<ro)(35)
It follows from the forms of the right members of these four equationsthat their solutions will contain Poisson terms,* whose coefficients involve
Xa, ,. . .
,Xa 4 respectively as factors. The coefficients of all the other
terms arc of the second degree in a, ,. . .
, a, and linear in B,and those
which are not periodic involve a 3 or a 4 at least to the first degree. Conse-
quently, when we solve the third and fourth equations of (11) for a 3 and a 4 ,
the results will start with terms of the second degree in a, , cu, ,and X. When
these results are substituted in the second equation of (11), it will contain
a term in X ctj and terms of the third degree as the lowest in a, and au alone.
If we now eliminate ax by means of (33), we have an equation whose terms
of lowest degree are OjX and a| . We shall verify first that the coefficient
of a 2 X is not zero.
*In Celestial Mechanics terms which are of the form of / multiplied by cosine or sine terms are called
Poisson terms, from the results in Poisson's theorem on the invariability of the major axes of the
planetary orbits.
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OSCILLATING SATELLITES SECOND METHOD. 209
It follows from (34) and (35) that the Poisson terms in f are
(2> = _ m A'\a i te-<r ' v =' u ^V^Ti^aj/c-''^'
2
(mo (To n<>po) V^l 2(m po+no <To)
Hence, so far as these terms are concerned, we find
-, /2tt\ _ w (2)/qx = _ irA'Xat f 2m, n "I
>2\<r / O-oV 1 (.moffo WoPo W po+Wo(7oJ
which, by equations (25) 2 reduces to
&) - tew = 4
T5yffftf?(1+ /^ )] Xa < ( 36 )
Vffo/ (1+2^1 ) K+ p ) cr V 1v
Therefore the coefficient of Xa 2 in (31) is not zero unless A' = 0. But A'?*0
except for the center of libration (6) when the finite masses are equal, andin this case a different generalization of the parameter p can be made to
keep it distinct from zero. Consequently, since all the equations are iden-
tically satisfied by a,= =o 4=
0, the equation obtained by eliminating
o, between (31) and (33) is divisible by a,, after which there is a term
in X alone whose coefficient is distinct from zero. Therefore the equationcan be solved for a, in terms of X, vanishing with X, and the periodic
solutions exist.
The form of the solution depends upon the degree of the term of the
lowest degree in a 2 alone in the final equation after at ,
a 3 ,and a 4 are elimi-
nated. It is easy to show that the coefficient of ai) in this equation is not
identically zero. It has been shown that the terms arising from the solu-
tions of (34) involve B linearly. There are also terms of the third degreein o,, . . .
,o 4 arising from the terms of the third order. The terms of
w, of the third order are defined by
dih i v <rZT %P* = mot J v' 1 ]__[ , (37)
dt2
2(rao<r noPo) 2(wop +no<ro)
where
[ f ) = 3B [-2x i x 2 +y l y i \+2C [2xl-3x l yl],
The Poisson terms in the solution of (37) involving C as a factor are
= _ 3(7, m (2- n ) a^ t e-v^i' + 3(? n (4
- 3 njj) a.cjfe-'.^"2
"
(m ao noPo) V 1 4(ra po+w cro) V 1
Hence we find for these terms, after some reductions,
f(g)-<W- w+j& si ^'iSS**+- 3
'-H'(38
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210 PERIODIC ORBITS.
It follows from equations (5) that-j \
of (38) does not identically vanish.
Hence the coefficient of a, a^ in the expression for u 2 (2ir/a 3 ) w, (0) consists
of terms multiplied by B plus non-vanishing terms multiplied by C .
Now suppose a,, a 3 ,and a 4 are eliminated from the second equation
of (11) by means of (33) and the third and fourth equations of (11). It
follows from the properties of these equations that in the result the term
independent of X and of lowest degree is of the third degree in a 2 . Its
coefficient consists of two parts, one of which is (38) and contains C as a
factor, while the other contains B a as a factor. If the coefficient of B a is
identically zero, the coefficient of a 3
2 is distinct from zero, because, as we have
seen, the part involving C is distinct from zero. Even if the coefficient of
B is not zero, it is easy to show that the sum of the two parts of the
coefficient of a\ can not be identically zero for each of the three libration
points (a), (6), and (c).
Consider the points (a) and (6). The quantities A ,<r
, p , m , n ,
and C are the same function of /x for both points, but B is different because
of the change of sign in its second term [eq. (3)]. Consequently, the sum of
the terms in B and C can not be identically zero in /j, for both the pointsof libration (a) and (6). Hence in this case the second, third, and fourth
equations of (11) and (33) are solvable for o, ,. . .
,o 4 as power series
in X*, vanishing with X. Therefore the periodic solutions with the period
2ir/<r e are expansible as power series in X*.
In a manner similar to that used to prove that (29) are series which
represent orbits of Class A, it can be shown that the orbits now under con-
sideration belong to Class B.
101. Direct Construction of the Solutions for Class A. As in the
method of Chapter V, the coordinates in the orbits of Class A are most con-
veniently obtained from the x, y, and z-equations. Consequently we start
from equations (2) and (3). Since the solution is periodic for all |XJ
|suf-
ficiently small, each term of the expansion separately is periodic with the
period 2ir; and since 2 = at t = 0, each term in the expansion of z separatelyvanishes at < = 0.
The coefficients of X* are defined by
x!-2y' -(l+2A )x 1=
0, y:+2x' -(l-A )y l= 0, <-M*-0. (39)
The solutions of these equations satisfying the periodicity and initial con-
ditions are
*i = Vi=
0, *i=-^T sinv/ ^o t, ( 40 )
where c, is so far undetermined.
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OSCILLATING SATELLITES SECOND METHOD. 211
The coefficients of X are defined by
<-2| / :-(l + 2i ) a ; 2 =fB [-2x;+|/;+4
yl+2x i -(\- A ) y 2 = 3 B.x.y,,
Z% T'ljZj = o JJ Xl
Zl
.
(41)
Upon making use of (40), integrating, and applying the periodicity andinitial conditions, we have
x >
4(l + 2,4 )A +4(1-7A +1SAI)A
C0S ^ VA^
z2= + ^j-sinVAj,
(42)
where c 2 is so far arbitrary.
It will be necessary to carry the computation two steps further in order
to show how the general term is found. The coefficients of X 3 are defined by
x:-2yi-(l+2A )x 3= SB z
lz 2= ^^[l- cos 2VT t],
yl+2x' -(l- A B)y 3= 0,
z"3 +A z 3 = -A'z l +ZB&z l +\ C zl
A & / j-*_i_ 3 t'oCi
-^anVT,t+j Ain v^ t+ 1
Cj^ [
3 sin VAJ- sin 3 VT 1] (43)
_ 27ff(l-3.A +14A;)ci sinv/x<8(1 + 2A.) (1 -74,+ l&AJ) Aj
9^f3^rf&(1-1A %+AX)ISAI
Consider first the solution of the third equation. In order that it shall be
periodicthe coefficient of
sinV^i must be zero,or
_ JL c 4-5. & - wBl{l-zA +uAS)c\ =Q (U)VA lT 8 ^i 8(1+2A.)(1-7A,4-18AJ)A!'
This equation, which is identical with (26) of the existence proof, has the
solutions
2V2A A'Ci ' Ci_ n~ 8flj(i-3ii.+i4i ir (45)
3\ C(l + 2^)(l-7Z 4-lW
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OSCILLATING SATELLITES SECOND METHOD. 213
It can be shown without difficulty that the solutions have the following
properties :
1. Therc2i+1
,
y 2j+l,
z2J
areidentically
zero =1, 2, . . . oo).
2. The Xj, y lr z s involve c y homogeneously to the degree,/.
3. The x 2j are sums of cosines of even multiples of \^A t, the highest
multiple being 2j.
4. The y 2j are sums of sines of even multiples of y/~A a t, the highest
multiple being 2j.
5. The z 2j+1 are sums of sines of odd multiples of V~A~ t, the highest
multiple being 2j-\-l.
6. Changing the sine of c t is equivalent to changing the sign of X }
,
which is equivalent to increasing t by ic/VT^. Therefore the
two values of c x (or X*) belong to the same physical orbit, the
origin of time being different by half a period in the two cases.
7. The orbits are symmetrical with respect to the z-axis, the :n/-plane,
and the xz-plane.
It is not necessary to go into the proofs of these properties, which are the
same, so far as the comparison can be made, as those found in 87.
102. Direct Construction of the Solutions for Class B. For these
orbits it is advantageous to use the first four equations of (7), the last onebeing identically zero. We have proved that the solutions are expansibleas power series in X }
,that the coefficients of each power of X* are periodic
with the period 2ir/<r ,and that x' = at t = for all X.
The coefficients of X* are defined by the differential equations
dUl - <r v^T < =0, % - PX - 0.
(50)dt
v *" "'dt
dt+<7 V 1Ui U
'
dt
The periodic solutions of these equations are seen to be
<-a l ^ v=I,J uf = a 2 e-"^ u
, u = u < = 0. (51)
From equations (6) and the initial value of x' it is found that
a 1= a 2
=| a (l)
,x t= a (1) cos <r Q t, j/,
= n a a)sin <r a t, (52)
where the coefficient a 0)is so far undetermined.
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214 PERIODIC ORBITS.
The coefficients of X are defined by
du% (2) i
dt
du?'
I I (2)
(Woffo /loPo) V 1 (WoPo+Woffo)
+ ff ft*. Mj reft
2(mo <7- nPo) V 1 (moPo+Woffo)
l I (2)
I l"
dufdt
duf
Po<=" nj ]
(2)
+l I (2)
I f
dt^ p * Ul ^
(nhao-noPo)
(m <r n p )
'
(mo Po+ Wo oo)
+J t
J L(niopo+noao)
where
(53)
[ r=fB [-2x?+^]= + |5 (a(l)
)s [-(2-n )-(2+n )cos2(r <],
I I ( _ T> =-|fl (a
(1)
)2 w sin2<r *.
(54)
The periodic solutions of these equations satisfying a;' = are
where
tt a (B e^+a+a 00B2*,*-t&ffsin2r,<,
- a (8)e-
<r
"+af) +a ) cos2c +i6'
2
2
)
sin2(r <,
uf-M*
+<+a<2)
cos2*r *+ 6 sin 2 (r, J,
+ off + 32 cos 2 <r 1 - 6 sin 2 <r *,
a (2)is so far undetermined,
3m fi (2-n )(aa>
)2
<2) =
fl(2, =
"30
<= +
//2) = -
"l2
8(moOo nopo)<T
3n B (2-nl)(ay
8(m.o<ronopo)p<i
m B (2+nl) (aa)Y
8(Wo(To ttoPoVo
m Bo(2+^ )(a")
)2
4(m <ro noPo)o"
B(a a))
2
+
U32 +n fl (2+tt2
)p (a(1)
)2
8( ffo noPo) (ial + pl)
6<2) = 3 Jgo(2+r ?2
)r (a<1)
)2
4(moffo-noPo)(4<rJ+p
2)
(55)
2{mopo-\-noao)<ro
v B (aa)y
4(m po+^o 0o) ffo
3n <r (a'")2
2 ( mo po + n <To ) ( 4 <rl + pi)
3n /? Po (a'")2
4(mopo+no-o)(4(7-o+p5)
(56)
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216 PERIODIC ORBITS.
The right members of the differential equations for the terms of the
next order are
[ T =A'x t +\ B [-2xl+y\-4x 1 x 3 +2y 1 y s } + C [6xlx i -6x 1 y l y 2 -Sx i y\] ,
\\M=~A'y t +lB {x,y i +x 1 y 3 +x i y 1 \ + ^C >-8x 1 x 2 y l -4x\y i +3y
i
l y i \.
(60)
Before integrating the first equation the coefficient of e a,v=Tt in its rightmember must be put equal to zero. It is found from an examination of
the terms of (60) that a <2)is involved linearly but not homogeneously.
Moreover, a <2)is the only unknown quantity in this coefficient. Therefore
a w is uniquely determined by setting the coefficient of e'"^ 11equal to zero.
The condition that the solution of the second equation shall be periodicis identical with that imposed by the first. The particular integrals of the
third and fourthequations
areperiodic.At the i th
step the right members of the differential equations involve
[ ]
w = A'x i^+^B [-2x l x t_ 1 +y l y t ^] + C [6xlx i ^-Qx 1 y l y i _ 2 -3x t _ i y\}+ ,
| \
w^ = -^'^- 2 +f5 |x 1 y < _ 1 +a; t_
1 y 1 !+|C ;-8.T 1^x ( _ 2 -4^^_ 2 +3^^_ 2 ;+- ,
(61)
where the parts not explicitly written are independent of a ( '~ 2). In order
that the solution of the first equation shall be periodic it is necessary that
the coefficient of e' w=rt in its right member be put equal to zero. Thiscoefficient carries a <i_2>
linearly and in general non-homogeneously, and the
known factor by which a (<_2) is multiplied is precisely the same as that of
a a)in the equation by which the latter was determined. Therefore the arbi-
trary a <1_2)is uniquely determined by setting the coefficient of e <7,v/=T<
equalto zero, for it carries no other unknown. When this condition is satisfied
the coefficient of e~ v:=u in the second equation is zero, and the entire solu-
tion at this step is periodic. Therefore after the sign of o (1) has been chosen
the process is unique, and it can be continued indefinitely.
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Chapter VII.
OSCILLATING SATELLITES WHEN THE FINITEMASSES DESCRIBE ELLIPTICAL ORBITS.
103. The Differential Equations of Motion. Suppose the finite bodies
describe ellipses whose eccentricity is e. Let 1 /x and p (p^0.5) representtheir masses, and then determine the linear and time units so that their meandistance apart and the gravitational constant shall be unity. With these
units their mean angular motion is unity.
Now refer the system to a set of rectangular axes with the origin at
the center of gravity, and let the direction of the axes be so chosen that the
i7-plane is the plane of motion of the finite bodies. Suppose the 7?-axes
rotate with the constant angular rate unity around the f-axis in the direction
of motion of the finite masses, and suppose 1 p and p are on the -axis
when they are at the apses of their orbits. Then the differential equationsof motion for the infinitesimal body are
where
df dt*
ri rl
cFn , 9 d _ _ (1 pXq m) niy m)dt
2"+
dtn
r\ ri'
<k = a-M)r hjdt* ri ri
'
(1)
r^Va-S^+CTj-T^+r, r 2= V(-| 2 )
2
+U-77 2 )2 +r,
and where , , 2 , r) l , and ?j 2 are determined by the fact that the finite bodies
move in ellipses.
If we let p, and v t be the polar coordinates of 1 /x referred to fixed
axes having their origin at the center of mass of the system, and p 2 and t> 2
the corresponding coordinates of p, we have
& = -p 1 cos( 1 -0, & - +acos(i>,-0,
7],= -pjSinOi-0, 77 2 =+p 2 sin(> 2 -0,
Pi^+M jl ecost+ (1 cos 2<)+[
Pt =(l-n)fl-eco&t+j(l-cos2t)+ },
v l= v i =+t + 2esmt+ - e
2 sin2+ ,
where the initial value of t has been so determined that the bodies are at
their nearest apses at1 = 0.
m
(2)
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218 PERIODIC ORBITS.
104. The Elliptical Solution. In order to get the Lagrangian elliptical
solution for the infinitesimal body we consider first the two-body problem.The equations of motion for the infinitesimal body subject to the attraction
of a mass mare,
when referred to therotating axes,
r = V?+v 2 +?-
(3)
We shall consider the solution in the ^-plane. If the eccentricity of the
orbit is e and if the mean motion with respect to the fixed axes is unity,
then the solution with the same determination of the origin of time and
apses as in (2) is
= rcos(# t), r)= rsm(v t), f = 0,
r = k l mi il-e cost-i-^ (l-cos2i)+
}>
v = 2esint+ 4e 2 sin 2t+4
(4)
It will now be shown that equations (1) will be satisfied if the infini-
tesimal body moves so that the ratios of its coordinates to the correspondingcoordinates of the finite masses have certain constant values. Let the
coordinates in these special solutions be represented by , rj ,and 0; then
=f 5=5 M| ^-~t+m-' *- i+m'
(o)
Upon making use of (2) and (5), it is found that
.-*,= +l _ ti{ l_ My *>-%-+ i_ m( i + M)'
t -t Mo = _ MVo?s i (i i its' Vo Vi
i-/i(i+M)" " 2 i- M (i+M)
1 ^i-m(i+m)
r2
^r- M (r+^/)
(6)
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 219
Then equations (1) become
? "_2/_ - _ [(i-mW*- m][i- m (i+M)]2
So tVo Co- "
~^p-f>
>" = _ [(i-m)M 3
+m]Ci-m(i+M)] 3
fio M3
'
rl'
(7)
The first two of these equations are of the same form as (3), and their solu-
tions corresponding to (4) are
=r cos (-0, ij, = r sin(i;-0, f = 0,
r.- [a-KW-dff-e a+Ml'
!1_ eco8(+
v = t + 2esmt+^e 2sin2t+
4
I
(8)
From these equations and (2), we find
& = vo = _ [(i-mW 2
-/j]Mi-m(i + M)P .
and from (6),
& = 38 = _ [1-M(1+M )]
&""
r?i
"
M(l+M)
On equating these two expressions for the ratio &/, = ^j/^ ,and rationalizing,
we have
(l- fx)M6 +3(l-n)M t +3(l-)M 3 -3 txM2 -3 fxM-n = 0. (9)
It is easily verified that starting from the expressions for the ratio / 2
the same quintic equation is obtained. Therefore, for those values of Msatisfying (9), equations (2) and (8) are a particular solution of the three-
body problem where one mass is infinitesimal. As is well known, there
are three real solutions, one for each ordering of the three masses. As in
Chapter V, we shall call them (a), (&), and (c) in the order of decreasingvalues of their x-coordinates.
Equation (9) is Lagrange's quintic in case one mass is infinitesimal and
the units are chosen so that the masses of the finite bodies are 1-ju and yu.
For example, if in equation (60), page 216, of Introduction to Celestial
Mechanics, we put w,=
l-ju, m2 =0, and m3 =n, we get equation (9).
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220 PERIODIC ORBITS.
105. Equations for the Oscillations. We shall study the oscillations in
the vicinity of the Lagrangian solutions. For this purpose we make the
transformation
= +*, v = v+y, f = 0+2 (10)
in equations (1), and expand as power series in x, y, and z. After this
transformation and expansion, we let
M= Mo+ X (11)
in those places where n appears explicitly. This is not the only way in
which the original n can be divided into the new n and n f> +\, and sometimes
others are advisable. The coefficients of the various powers of x, y, and z
are expansible as power series in e, the terms independent of e being constants,
as is seen from (2) and (8). We find from (6) and (8) that
G,-&-rf[l-**- |(l-cos20 +]>
$,-&-r[i-eeo8*-|(l-eos2*) +]>
*-%?[ +2esmt+^sm2t +]>U,-ih-f[ +2esmt+^sin2t +'].
r,- Vtti-eO'+Oj.-*)' -
f[l- coe* +
]>
r, =V(o-y 2
+(%-rj i )2 = ^
0)
[l-ecosi +]>- _ i r (i-M)M
2 - M-|
J
,_ _ Mj r (i-M)M 2 - M
-|
*
.r' ~M j Li-m(i+M)J ?2 Li- M(i+M)J
Consequently, after making use of these expansions and the transformations
(10) and (11), equations (1) become
x"-2y'-h+2A+()Aecost-3Aei
(l-5cos2t)+ -\x
-\6Aesint+^Ael sm2t+ -\y = X,
y"+2x'- J6,4esin*+hA(?sm2t+ -\x
-jl-^-3^1ecos<+|^c2
(3-7cos20+ ]y=Y,
z" +l+A + 3Aecost+^Aei
(l+3cos2t)+ -\z = Z,
(13)
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 221
where
j I
1 Mo i Moyi i" (0)3 -r cms '
'l '2
+1
- 6|_f8a psajc
sin <+ 4 yX
4- 3 [^-^]- 12 [^r =^] eco ^ +-}*
2
+{l[Tr-^] +6 [1
7r=rrO eco ^ + }v2
+
+
F= + \-6| r
(oy3-776T3jesini+
-l
+{[ rp- r4r3]+3[
ri-
r
i]e cosi+
}
+{4^=?]+ i2[
1
?--^] ecos< +}
+ {+3[j ^m ^]esini+ }*'+;
Z= + \[py
3-
-(6T3J +3[pr-
pa]e cos t+
JzX
+{6[
1^ = ^] esini+ }*+
(14)
the signs in the [ ] being + +,+ , , according as the point (a),
(b), or (c) is under consideration. All terms up to the second order inclusive
in x, y, z, and X are written.
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222 PERIODIC ORBITS.
106. The Symmetry Theorem. It follows from (2), (8), and (1) that
the right members of (12) have the following properties:
(a) The X and Y involve only even powers of z, and Z involves only
odd powers ofz.
(b) In X the coefficients of all terms involving even powers of y are
sums of cosines of integral multiples of t, and the coefficients of
all terms involving odd powers of y are sums of sines of integral
multiples of t.
(c) In Y the coefficients of all terms involving even powers of y are
sums of sines of integral multiples of t, and the coefficients of all
terms involving odd powers of y are sums of cosines of integral
multiples of t.
(d) In Z the coefficients of all terms involving even powers of y are
sums of cosines of integral multiples of t, and the coefficients ofall terms involving odd powers of y are sums of sines of integral
multiples of t.
Suppose the initial conditions arc
x = a, x' = 0, y = 0, if'-ft z = 0, z' = y. (15)
Then the solutions of (12) are
x=f(a, 0, 0, 0, 0, y; t), x'=f'(a, 0, 0, (3, 0, y) t),
y = g(a, 0, 0, 0, 0, T ; 0i -/(, 0, 0, 0, 0, y; t),
z=h(a, 0, 0, 0, 0, T ; 0, z' = h'(a, 0, 0, 0, 0, y; t).
(16)
Now make the transformation
x=+x, x'=-x', y=-y, y'=+y', *--*, *'=+*', t=-t. (17)
It follows from the properties (a) ,. . .
, (d) that the form of equations
(13) is not changed by this transformation. Consequently the solutions
with the initial conditions
x = a,
x' = 0, y = 0, *P, 1-0, z' = y,
are identical with (16), and we have, making use of (17),
x(t)=x(t)=x(-t) = +x(-t), x'(t) = x'(t)=x'{-t)=-x\-t),'
y(t)=y(t) = y(-t) = -y(-t), y'(t) = y>(t)=y'(-t) = + y '(-t), (18)
z(t)=z(t)=z(-t)=-z(-t), z'{t)=z'{J)=z'(-t) = +z'{-t). .
Therefore, with the initial conditions (15), x, y', and z' arc even functions of
t, while x', y, and z are odd functions of t. That is, if the infinitesimal
body crosses the x-axis perpendicularly when the finite bodies are at an
apse, its motion is symmetrical with respect to the z-axis.
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 223
107. Integration of Equations (13). The Terms of the First Degree.
Suppose the initial conditions are
3(0) -<H, x'(0) = a 2 , y(0) = a 3 , y'(0) = a A , (0) = a s> z'(0) = a e. (19)
We shall now integrate equations (13) as power series in a, ,a 2 , a, ,
a 4 ,
a 5 ,a 6 , and A. Since there are no terms in the differential equations inde-
pendent of x, y, and z and their derivatives, there will be no terms in the
solutions independent of c^ ,. . .
,a 6
.
The terms of the first degree in a x ,. . .
,a 6 are defined by the differ-
ential equations
xi-2y[-[l+2A+6Aecoat+ ~\x l -[6Aesint+ -^ =0,
y'{+2x[-[6Aesmt+ ]x x-[l-A-3Aecost+ ]&-0, [ (20)
z"+[A+3Aecost + ^Ae2
(l+3cos2t)+ ]%-0,
subject to the initial conditions (19). The coefficients are power series in
e, periodic with the period 2w, and reduce to constants for e = 0. The first
two equations are independent of the third, and conversely.
For e = equations (20) become simply
x';-2y[-[l+2A}x 1 =0 > y';+2x[-[l-A]y 1= 0, zl+Az^Q. (21)
From the results obtained in 23-25 it follows that the properties of the
solutions of (20) depend upon the character of the roots of the characteristic
equation of (21). This equation for the first two of (21) is
si +(2-A)s i
+(l-A)(l+2A)=0. (22)
Two roots of this equation can be equal only if A has one of the values-
1/2, 0, 8/9, or 1 . The first two are excluded by the fact that A is necessarily
positive. When n is near n in value, as it will always be taken here, A is
greater than unity and, consequently, the last two values are excluded.
Equation (22) has two pairs of roots equal in numerical value but
opposite in sign, and for A > 1 two of them are pure imaginaries and twoare real. Let us represent them by <r V l , p > where a a and p are
real. We now raise the question whether two of the roots of (22) can differ
by an imaginary integer. In order that this may be so we must have
s = cr V^T= -j\T^l,
where j is an integer. Since this value of s must satisfy (22), we find
^-4(2-A)j 2 +16(l-A)(l+ 2A) = 0;
whence164=j 2 +4v/9j 4
-56i2 + 144. (23)
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224 PERIODIC ORBITS.
The question is whether there are integral values of j giving admissible
values for A by (23). To insure the convergence of the final series it will be
necessary to take m nearly equal to n, and we shall suppose at once that
this condition is satisfied. It was shown in 82 that when Mo = M the value
of A exceeds unity for each of the points (a), (6), and (c) for all values of m-
From the expansions given in equations (42), p. 206, of Introduction to
Celestial Mechanics, it is seen that for very small values of n the values
of A are approximately 4, 4, and 1 for the points of libration (a), (6),
and (c) respectively. In the numerical example of 90, for which n = \/\\,it was found that A is 2.25, 6,51. and 1.08 for the points of libration (a),
(6), and (c) respectively. In the extreme case of /x= 1/2 we easily find from
the formulas of 76 and 82 that the values of A are 1.56, 8, and 1.56 for
the points of libration (a), (b), and (c) respectively. While n varies from
to 1/2 the values of A vary from 4 to 1.56, 4 to 8, and 1 to 1.56 for the
points (a), (6), and (c) respectively. From this we see what values of A are
possible in the actual problem, and we shall be able to determine whether jtakes integral values for any of them.
When the negative sign is taken before the radical in (23), the function
16 A has its greatest value of zero for j2 = 4. Consequently we get no
admissible values of A using this sign before the radical.
When the positive sign is taken before the radical in (23), j and A are
found to have the following relations:
If j = 1,
2, 3 ,
4,
5,
6 ,. . . ,
then A =0.92, 1.00, 2.01, 3.73, 5.94, 8.70,
and A is larger than 8.7 for all j larger than 6. The values of A for jbetween 2 and 6 are admissible for either the point (a) or the point (6),
but not for the point (c). Consequently since A, and therefore a, is a
continuous function of m and n ,there exist pairs of values of n and n for
which the difference of the imaginary roots of (22) is an imaginary integer,
but they are exceptional.
The characteristic equation for the third equation of (21) is simply
s2
+A = 0.
Hence the solution of the third equation of (20) does not take an exceptional
form unless 2 VA is an integer, or
fA =Zj- (j an integer).
It follows from the discussion above that the admissible values of A satis-
fying this relation are 2.25, 4.00, and 6.25. These values belong only to
the point (a) or the point (6), and in no case can the congruence be satisfied
for the point (c).
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(24)
OSCILLATING SATELLITES, ELLIPTICAL CASE. 225
According to the results obtained in 23, the solutions and character-
istic exponents of (20) are always expansible as power series in e except whenA has the special values noted above; and according to the results obtainedin 24, the same result is true, in general, even if the roots of the character-
istic equations differ by imaginary integers. However, in the latter case
the construction of the solutions is quite different.
It was proved in 33 that in equations of the type under consideration
here the characteristic exponents occur in pairs which are equal numericallybut opposite in sign. Therefore the solutions of (20) are of the form
2,=
Oj e aV ^' u t + a 2 e-ff ^ / "=T w2 + a 3 e
pt u 3 + a 4 e~pt u A ,
x[ = a x e,"^11
[a \ r ^\u 1 + u[] + a 2 e^^ 1 ''
[-
<s V=7 u 2 + u' 2 ]
+ a 3 d* [ Pu 3 + u' s ] +a 4 e-*[ - pu t + w4 ] ,
Vi a, eav/ ^ T '
v x + a 2 e-'^1 '
i\ + a 3 e"' v 3 + a 4 e~pt
v 4 ,
y[ =a, **=* [a V~^l i\ + v[] + a 2 e
-^~ u[-
<x V~=J v 2 + v' 2 ]
+a 3 epl
[pv 3 +v' ]+a i e-pt[-pv i + v[},
z x= c 1 e^
vz=rY 'w 1 +c 2 e~0,v ^^ , wi ,
z[= c x e aV= T, [wx^Tw l +w' \ +c 2 e~
Uv '=* t
[-o>\ r ^iw i +w' ],
where
aj ,. . .
,a 4 ,
cx ,
c,,
are arbitrary constants of integration ,
a = (r o +0-ie + a- 2 e2 + ,
tt4= uf
) + wi
<1) e+tl <
(
V-|- ,
p = p.+p.e+fte 8 -!- ,tr vl
0)
+v?e+v?)
e i + ,
=VZ+ 1 e+tt s e*+ , a> 4 -jT+? ) +wf >^+ ,
uf, vf, w are constants.
The initial values of the v, and w t can be taken equal to unity without
loss of generality, and will be so chosen. Moreover, the u i} v i} and wt are
periodic with the period 2w, and since this property holds for all e for which
the series converge, each u\'\ v\n
,and w\
nseparately is periodic with the
period 2t. The coefficients of these series can be found by the methods
set forth in 26. The a t and c, are uniquely expressible in terms of the
initial values of x, x', y, y' , z, and z', the a, of equations (19), becausethe solutions (24) constitute a fundamental set, by hypothesis, and the
determinant of the coefficients of the a t and c, is therefore distinct from
zero. We may use either the a ( and c, or the a t as arbitraries.
The characteristic exponents <r, p, and o> are real for e sufficiently
small, as we shall now show. The arises from the third equation of (21),
which is of the same form as that treated in 50, where it was shown that
the characteristic exponents are pure imaginaries in this case. ThetrV 1 and p are roots of an equation of the form
A(a, e)=0, (26)
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 227
Since these relations are identities in t, we have
ujt) = -u 2 (-l), ^l 3 (t) = -U t (-t), *(0 *(-!), V3 (t)=V t (-t), W^^Wii-t).
Therefore, if the u s , v, , and Wj are arranged as Fourier series, they satisfy
the relations
u y= +2 [A j cos jt+Bj am jt], Pj
= +2 [E 3 cos jt+F, am jt],
m2= 2 [A j coa jt Bj am jt], iu = +2 [E, cos jt F s am jt],
u 3= +2 [Cj coa jt+Dj sin jt], v 3
= +2 [G } cosjt+Hj sin jt],
u t= 2 [Cj coajt Dj ainjt], v A
= +2 [G t coajt-Hj ainjt],
Wt = + 2 [Kj cos jt + Lj sin jt] ,wl =+'L[K ] cos jt L s sin jt] .
(28)
Since the Uj , v s , and to, are defined by linear equations they are indepen-dent of the initial conditions. But the equations a^-a*, a 3 =-a it c
l =-c i
hold only for the initial conditions (5).
108. The terms of the Second Degree. The terms of the second degreein the a t and X are found from equations (13) and (14) to be defined by
x 2
,
-2y' -[l+2A+6Aecoat+ -]x 2-
[6Aesin*+ ]&-XyZ+2x' -[6Aeainl+ ]
x 2 -[l-A-3Aecoat+ -]y 2= Y2 , \ (29)
+ [A+ 34ecos<+ ]z 2
- Z},
where
Xi= + [-2A'- GA'ecoat-] ]x,X+[- 6A'eamt-\ ]r/,X
+ [-SB- 12Becoat-]
~]x\ + [ -24J5esinH ],y,
+ [+|5 + 6ecos*+]j/J + [fl+6flecosf+- -}z\ +
F 2 = +[- 64'esinH ]^X+[ A'+ 3A'ecoat-\ ]y,X
+ [-12e sinH ]zj + [3 + 12ecosH ~\x 1 y l
+ [+ 9fie sinH ]*S +[ + 3esinH ]zjH ,
Z 2=
+[,4'+3A'ecosH ]z,X + [3fi+125ecos<H ]x,z,
+ [ 6Z?esin*+ ~\y 1 z l +
A ' - l 1 U - + 1 Mo + Moft- (0)3
~Z(073 ' + (0)4 Zffift
'
'1'
'1'
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228 PERIODIC ORBITS.
It is necessary for further work to determine some of the properties of
the solutions of equations (29). These properties depend upon the form
and properties of their right members, which are given in equations (30).
The general character of the coefficients in these equations easily follows
from the properties (a), . . ., (d) of 106. On referring to the results which
were developed in Chapter I, it is found that :
[1] The solutions of (29) consist in the first place of the complementary
functions, and they are identical in form with (24). The arbitraries
a, and c, which "appear are uniquely determined by the conditions
x, (0) = x 2 (0) = y t (0)= y 2 (0) = z 2 (0) = z 2 (0)
=0, where x 2 , . . . , z' 2 are
the complete solutions of (29) [15].
[2] There are terms arising from those parts of the right members which
contain X as a factor. These right members consist of sums of terms
which are periodic with the period 2-k multiplied by one of the funda-mental exponentials e +ffv/ ^ T(
, e~ aV:= ~~"
,e +pt
,e~ pt
,e uvr ^'
, and e~ uVzri1.
Hence it follows from the results of 30 that the corresponding parts
of the solutions consist of sums of periodic terms whose period is 2ir
multiplied by these same exponentials, plus t times the corresponding
part of the complementary function. We shall be particularly interested
in those terms which contain t as a factor. They are linear in the a,
and c,, and for e = the parts having the period 2-k reduce to con-
stants. The expressions for x 2 and y 2 do not depend upon cx and Cj ,
and the expression for z 2 is independent of a t ,. . .
,a 4 .
[3] Next consider the parts of the right members of (29) which are inde-
pendent of X . They consist of sums of periodic terms having the period2ir multiplied by the squares and second-degree products of the funda-
mental exponentials. Therefore, except in the special case where
a or co is an integer, the exponents of the exponentials in the right
members are not congruent to any of the characteristic exponentsmod. V 1 ;
hence it follows by 30 that corresponding parts of the
solutions consist of sums of periodic terms, period 2w, multiplied bythese same exponentials. In particular, there are no terms containingt as a factor. These terms are homogeneous of the second degree in
the a, and the c, .
109. The Terms of the Third Degree. The terms of the third degreein the a t ,
c ( ,and X are defined by equations whose left members are identical
in form with the left members of (29). The right members contain terms
which are
(a) linear in X and of the second degree in xy , y x ,
and , ;
(b) linear in X and of the first degree in x 2 , y 2 ,and z 2 ;
(c) of the third degree in x lf y lf and 2,; and
(d) of the first degree in x lf y lf z, and in x 2 , y 2 ,and z 2 .
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 229
The solutions have the following properties:
[4] There are the complementary functions identical in form with the
expressions (24) . The constants which appear in them are determined
by the conditions that x 3 (0)=x 3 (0)=y 3 (0)=y 3 (0)=z 3 (0)=z' (0)=0.
[5] The part of the solution coming from the terms (a) consists of sumsof periodic terms, period 2ir, multiplied by the squares and second-
degree products of the fundamental exponentials.
[6] The terms (b) give rise to terms of two different - classes because x 2 ,
y 2 , and z 2 consist of terms of two different types. There are terms
which contain X as a factor, and a part of these are sums of periodic
terms, period 2ir, multiplied by t times the fundamental exponentials;the remaining part lacks the factor t. These terms are homogeneous
and linear in the a, and the c, . The corresponding parts of the solu-
tions are sums of periodic terms, period 2x, multiplied partly by t2
,
partly by t, and partly by f. They are all homogeneous and linear
in the a, and the c, ,the x and y- terms being independent of the c< ,
andthe z-terms of the a, .
The terms of the solutions coming from the other part of (b) are
sums of periodic terms, period 2ir, multiplied by squares and second-
degree products of the fundamental exponentials. They are homo-
geneous of the second degree in the a t and the c ( .
[7] The terms of the type (c) are homogeneous of the third degree in the
a, and the c t . They consist of terms of two classes, the first of which
are sums of periodic terms, period 2t, multiplied by the fundamental
exponentials to the first degree; and the second of which are sums of
periodic terms, period 2x, multiplied by cubes and non-canceling third-
degree products of the fundamental exponentials. The corresponding
parts of the solutions consist respectively of t times sums of periodic
terms, period 2x, multiplied by the fundamental exponentials to the
first degree, and sums of periodic terms, period 2ir, multiplied bycubes and non-canceling third-degree products of the fundamental
exponentials.
[8] The part of the solution coming from the terms (d) consists of terms
of two kinds, the first depending upon those parts of x 2 , y 2 ,and z 2
which contain X as a factor, and the second depending upon those parts
of x 2 , y 2 ,and z 2 which are independent of X. The parts of the solu-
tions corresponding to the first are homogeneous of the second degree
in the a ( and the c { ,and they are sums of periodic terms, period 27r,
multiplied by squares and second-degree products of the fundamental
exponentials, and some of these products contain t as a factor while
others do not. The other parts of the solutions coming from the terms
(d)have the
propertiesof those
comingfrom (c).
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230 PERIODIC ORBITS.
110. General Properties of the Solutions. It will be necessary to use
the following general properties of the solutions:
[9] Since the right members of the first two equations of (13) involve only
even powers of z, it follows that x and y are even functions of c t and c,
taken together.
[10] Since the right member of the third equation of (13) is an odd function
of z, it follows that z is an odd function of c, and c 2 taken together, and
that z identically vanishes for c 1=
Ci= 0.
[11] Since the right members of the first two equations of (13) vanish
identically for x = y = z = 0, but not f or x = y = 0, it follows that x and yvanish identically for a v
= = a 4= c t
= c 2 =0, but not for
a,= = a 4= 0, e^O, c 2 5^0.
[12] Since the equations reduce to those having constant terms for e = 0,
it follows that the sums of the periodic terms, period 2-k, reduce to
constants for e = 0.
111. Conditions for the Existence of Symmetrical Periodic Orbits. Thedifferential equations are periodic in t with the period 2t. Consequentlythe period of the periodic solutions, if they exist, will be T = 2nw, where n
is an integer. When the initial conditions are such that the orbit of the
infinitesimal body is symmetrical, as defined in 106, then sufficient con-
ditions for the existence of the periodic solutions are
a!+a 2= 0, a 3 +a 4
= 0, c,+c 8= 0,
z'(f ) -x'(0) =0, 2/(f) -2/(0) =0, z(f ) -z(0) =0.(31)
These equations are power series in a lt . . .,
a t ,c l} Cj, and X, and
vanish identically with a,= = a 4
= c x=
Cj= 0. In order that they may
have any solution for a, ,. . . ,a iy c lt and C;, , vanishing with X, aside from
this one, the determinant of the coefficients of the linear terms in a, ,. . .
, a 4 ,
Cj ,and c 2 must vanish. It follows from (28) that
uM = -uM, i(f) .(f)i <(0) = + u' 2 (0), *((f)- + 4(f),
",(0) = -,(0), u 3 (!)
= -tt, (|) ,u' 3 (0) = + u'M, u' 3 (|)
= + u\ (|) ,
t\(0) = + i> 2 (0), e,(f)-+*(f), v[(0) = -v! (0), ^(|)=-fKf),|(32)
vAo)=+v i (o), v t (f)-+4 (|), *(o)*-:(o), *(f)--;(f),
wM-+*(<>), K7l
(|)
=
+^(|),u<(0) = -
2 (o),
,;(!)--/(!),
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 231
and therefore the determinant of the coefficients of the linear terms of (31) is
found from (24) to be
* A = A1 A2 , (33)
A,=
i:
,
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232
where
PERIODIC ORBITS.
C^aV^lV^+v'^) , C-/*(l)+*(f)
The determinants become, after some reductions,
A-4^1 ) ^3
X^
*( <!)
A=-2[ wV^T^(f)-^(f)]^(f).
(37)
Since D^ and Z) 2 are determinants of fundamental sets of solutions of
linear differential equations for the regular point t = T/2, they are distinct
from zero. Therefore their second factors are not zero, and equations
(36) can be satisfied only by
[e * -e r-v^I* 2
]=0 or
[e>\/_i- - u vC
2
]=0. (38)
If either of these equations is satisfied, A is zero.
In order that one of equations (38) shall be satisfied it is necessarythat either
aT = 2N l ir, or wT = 2N 2 tt (N l , N2 integers) .
Since T = 2nir, where n is an integer, these conditions become
nor co =
n
(39)
(40)
Hence the conditions for the existence of the symmetrical periodic solutions
in question can be satisfied only when a or u is rational. These quanti-
ties, given in (25), are power series in e and they depend upon /x and /*<> andthe way n is generalized when the transformation n = n a +\ is made. Since
er
and w are continuous functions of n, /x ,
and e, the rationality of atleast one of them at a time can be assured. It should be noted further that
<r and co depend upon /*, ju ,and the mode of generalization of n, but that
they are independent of e.
In any case |X| can be taken so small that the series will converge, but
the periodic solution does not belong to the physical problem except whenX = M~ Mo- Suppose the series diverge for this value of X. Theoreticallythe values of the coordinates for this value of X can be obtained by analytic
continuation with respect to X as the argument from the periodic solution
which exists for a smaller value of X. There is an exception only if the
function has a natural boundary, or if X=
/x Mo is a singular point.
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 233
112. The Existence of Three-Dimensional Symmetrical Periodic
Orbits. Suppose w is the rational number u = N/n, where N and n are
relatively prime integers, and take T=2nir. Suppose aT is not an integral
multiple of 2t. Then A x ^0 and the first four equations of (31) can besolved for a lf . . .
,a 4 uniquely as power series in c lt c 2 , and X, vanishing
identically for C!= c 2
=0, by property [11]. The a x ,
. . .,
a 4 are evenfunctions of c x and c 2 taken together, by property [9]. When these results
are substituted in the last two equations of (31), they become power series
in Cj ,c 2 ,
and X. These series are of odd degree in c, and c 2 taken together,
by property [10], and therefore vanish identically for cx= c 2
= 0. Thesubstitution of the values f or a
x ,. . .
,a 4 does not change the linear terms,
for the first four equations were even in c : and c 2 alone.
Let c 2 be eliminated by means of the fifth equation of (31). Thenc, is a factor of the result, which has the form
= c 1 [a 01 X+a 20 c+ .
]. (41)
The solution c t= is trivial and we are interested only in those obtained by
setting the other factor equal to zero. The second factor set equal to zero
is satisfied by c x= X = 0. If o 01 is distinct from zero, solutions for c x as power
series in fractional powers of X certainly exist. If a y ,is the first a JQ which
does not vanish, then the solutions are expansible as power series in Xv \In particular, if a 20 is distinct from zero the solutions are expansible as
power series in X*. If the number of solutions is odd, only one is real; andif
even, onlytwo are
real,and these are real
onlyfor
positiveor
negativevalues of X according as a 01 and a,, are unlike or like in sign.
It is clear a priori that the number of solutions will be even, for there
is nothing of a dynamical nature by which to distinguish the two sides of
the xy-pl&nc. Consequently, if any initial projection gives rise to a periodic
orbit, a symmetrically opposite one with respect to the xy-plsme will also
produce a periodic orbit.
It remains to show that o 01 is distinct from zero. To prove this the
terms of the second degree in the a t , c,, and X must be considered (108).It follows from the form of (29) and (30) that a 01 depends only upon the
2-equation,for it is not
changed bythe substitution of the solutions of the
first four equations of (31) for a x ,. . .
,a 4 in the last two. Hence ai
depends only upon the solution of
z"+[A+3Aecost+ -]z 2= [A'+3A'ecost+ -} Zl , (42)
where ^_ ,_
It follows from the properties of wlt w2 ,u and the general theory of 29
and 30 that the coefficient a 01 is a power series in e. For e = it was given
in equation (26), of Chapter VI, where it was shown to be {-VfAaA'. This
being distinct from zero, e can be taken so small that the series for o ul is
distinct from zero.
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234 PERIODIC ORBITS.
Similarly, a^ is made up of a constant part distinct from zero plus a
converging series in e, and is therefore distinct from zero for e sufficiently
small. Suppose e = in (41) and let the value of c x obtained from solvingthe
resulting equationbe
cf;then let c x
=c{
0)
(l + y).It is
easilyfound
from (41) that dy/de is a power series in e and y which is distinct fromzero for y = e = provided X has such a value that cj
0>?^0. Hence the
solution of (41) can be written in the form
Cl= \*p(\), (43)
where p^X') is a power series in X* whose coefficients are power series in e.
On substituting this result in the solutions of the first four equations of (31)
for the a, as power series in c, and X, we have a, ,. . . , a 4 expressed as power
series in X 1. But since a t ,
. . ., a t contain only even powers of c x , they have
X instead of X* as a factor after Cj is eliminatedby
(43). Theexpressionsfor the coordinates become, since x and y are functions of c\,
x = \P^;t), |f-XP,(X*;0, 2 = X*P 3 (X*;0, (44)
where P l ,P i , and P 3 are power series in X*.
Since the problem is dynamically symmetrical with respect to the
xr/-plane, a solution symmetrically opposite with respect to the xy-p\a,ne
exists for all |X| sufficiently small. Therefore z must be an odd series in
X*, and x and y even series in X*. Hence equations (44) become
X-X^OjO, 2/= X&(X;0, z = XQ3 (X;0, (45)
where Q,, Q2 , and Q3 are power series in X.
We suppose that e>0 and therefore that the finite bodies are at their
perifoci* at 2=0. The infinitesimal body crosses the z-axis perpendicularlyat t=0 and at t-T/2. It follows from the symmetry of the motion that
it can cross the z-axis perpendicularly only at the end and middle of the
true period. Therefore if n and N are relatively prime, T = 2nir = 2Nir/u)is the true period.
The two cases I, n even, and II, n odd, merit a little further discussion.
In the first N is odd because n and N are relatively prime; in the second it
may be either odd or even.
113. Case I, n even, N odd. The infinitesimal body crosses the z-axis
perpendicularly at t = and also at t = T/2 = mr =2nV, where n' is an
integer. Since at both of these epochs the finite bodies are at their peri-
foci, the infinitesimal body crosses the z-axis perpendicularly only when the
finite bodies are at their perifoci. It follows from this that the infini-
tesimal body crosses the z-axis perpendicularly at the same point but in
the opposite direction with respect to the zy-plane at 2=0 and t = T/2. To
'Perifoci will be used to denote the points at which the finite bodies are nearest each other, andapofoci those at which they are most remote from each other. These points correspond (o perihelia
and aphelia in planetary motion.
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 235
prove it suppose the two points were different. Then we should have four
solutions corresponding to X* at each of the points, and it is known that
thei* are but two. The values of z'(0) and z'{T/2) are opposite in sign
because otherwise the period of the motion would be T/2.It can now be shown that in this case the orbits obtained by taking
the two signs before X* are geometrically the same one. Consider the
orbit defined by the positive sign before X*. At the time t - T/2 the
infinitesimal body crosses the x-axis perpendicularly at the point at whichit crossed at =
0, but in the opposite ^-direction. We may consider T/2as an origin of time for defining orbits which cross the z-axis perpendicu-
larly. It has been shown that there is but one with the given ^-direction
of motion, and at t = T/2+ T/2 = T the infinitesimal body will again cross the
x-axis perpendicularly with the opposite z-direction, viz., with that which
it had at t = 0. Since this orbit was unique, it follows that the two orbitswhich correspond to the double sign before \ h are geometrically the same,but that in one the infinitesimal body is half a period ahead of its positionin the other one. That is, changing the sign of X in the solution is
equivalent to adding T/2 to t.
When the solutions are actually constructed and are reduced to the
trigonometric form, they involve sines and cosines of (jo:-\-k)t, where j
and k are integers. Since x, y', and z' are even functions of t, they will
involve only cosines, and since x', y, and z are odd functions of t, theywill involve only sines. Since x and y are even series in X*, it follows from
the foregoing properties that in them
[(;+*)] s [0*+*)(+r/2)] .
These identities are satisfied if, and only if,
cos (jw+k) T/2 = cos (j'co+A;) 2rn' = 1 (n' an integer) ;
therefore
(jw+k)n'=j +kn' = p (p an integer).
Since N is odd it follows from this relation that j is necessarily even.
Similarly,in the case of the series for z the relation
sin [0+AO*]- sin [U*+k)(t+T/2)]
must be fulfilled. It follows from this identity in t that
(ju>+k)n' = j ^ + kn'=2-
(p an integer).
This relation can be satisfied only if j is an odd integer.
There are solutions in this case which have not been obtained by the
analysis as given. The orbits which have been discussed intersect the x-axis
perpendicularly when the finite bodies are at their perifoci, and obliquely,
if at all, when they are at their apofoci. Similarly, supposing t = when the
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236 PERIODIC ORBITS.
finite bodies are at their apofoci, it can be proved that there are orbits
with the same period in which the infinitesimal body crosses the x-axis per-
pendicularly when the finite bodies are at their apofoci, and obliquely, if
at all, when they are at their perifoci.
1 14. Case II, n odd. In the present case the infinitesimal body crosses
the x-axis perpendicularly when the finite bodies are at their perifoci andalso when they are at their apofoci, because T/2 is an odd multiple of -k.
If N is even, the solution for + X is geometrically distinct from that
for X*. To prove this, we note that |X| can be taken so small that the
sign of the series for z' is determined by its first term. If e is small, z' has
its sign determined by the constant parts of wt and w2 . Now the first
parts of these first terms, viz., e aVZ ^' and e~ ws/zzlt, have the period 2ir/u.
SinceT=2Ntt/u,
and N is an eveninteger,
it follows that thesign
of z' is
the same at t = and t = T/2. Now suppose X to increase to the value
belonging to the physical problem. If the sign of z' (T/2) changes it mustdo so by passing through zero. But in this case, since z(T/2) is also zero,
z would be identically zero. Therefore z' (0) would also be zero and would
change sign for the same value of X, and z' would still have the same sign
at t = and t=T/2. Consequently, the z-component of velocity at t=T/2is not equal to the negative of that at t = 0. Hence, when n is odd and Nis even, the orbits for +X* and X* are geometrically distinct, because a
single orbit can cross the x-axis perpendicularly but twice.
Ifn and N
are bothodd,
the infinitesimalbody
crosses the x-axisper-
pendicularly, as before, both when the finite bodies are at their perifoci
and also when they are at their apofoci. But though in this case the sign
of z' at t =0 is opposite to that at t = T/2, the orbits for +X 1 and -X } are dis-
tinct; for otherwise the motion of the infinitesimal body would be precisely
the same while the finite bodies were moving from perifoci to apofoci as
while they were moving from apofoci to perifoci. This is impossible
because t enters the differential equations differently in the two cases.
If we set up the problem taking t = when the finite masses are at their
apofoci, we shall find similarly two solutions; but they will be identical
with these, for in these the infinitesimal body crosses the x-axis perpen-dicularly when the finite bodies are at their apofoci.
The numbers n and N have so far been taken relatively prime, andT=2w7r. We shall now inquire whether there are other solutions with the
period T' = kT, where k is an integer. The determinant Aj vanishes for this
value of t. Solutions having this period certainly exist, for they include as
special cases those with the period T. Proceeding as in finding equation
(41), the corresponding steps are taken, and it is found that in this case a 01
and a 20 differ from their former values only by multiples of 2r. Therefore the
number of solutions with the period T' is the same as with the period T. Hence
there are no new solutions whose periods are multiples of those considered.
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 237
115. Convergence. The existence of the symmetrical three-dimensional
periodic orbits has been proved except for a discussion of the convergenceof the series which have been employed, a matter which must now be taken
up. The nature of the difficulty will first be pointed out. Equations (13)were integrated as power series in the a t ,
c ( ,and X. It was shown in 14-16
that for any preassigned T the moduli of these parameters can be taken so
small that the series converge for all <! ^ T. The limits on the moduli of
the a,, c,, and X are functions of n, /z a ,and e. But T can not be taken
arbitrarily in advance, for it is a discontinuous function of w, which is in
turn a function of n, ju , and e. It is not evident a priori that values of
fi, n ,and e, satisfying the relations n m =
X, o=/(m, Mo> e ), exist such that
all the series which are employed are convergent.The final parameters of the solutions are p, /z , e, and the mode of
generalizing /z is arbitrary. Suppose the ratio of the finite masses is given,that is, that n is a fixed number. Suppose also that the mode of general-
izing fi has been determined. It will be shown that values of ju and e exist
such that the series all converge for X = /x /vIn equations (25) it was shown that
co = \/Z+w 1c+w 2 e2 + j
where VA and the co, are functions of /z . Moreover, a detailed examina-
tion of the functional relation shows they are continuous functions of /* .
Suppose for any /* such that |m Mol=i>0 the series for w converges for
\e\ ^i7 u where i\ x depends on e^ It is easy to show from the nature of the
dependence of the series for w upon n and ju that q l > 0. Take any particu-
lar /xi" such that \n /4|4 and suppose that, while e runs over the range
to rj u the value of to (n , e) runs over the range co 04", 0) = VA to w (/4", ij,)-
For brevity, let w <0> and wH>represent the smallest and largest values of w.
An examination shows that u, ,w2 ,
. . .are not all identically zero, and it
follows from this that|u ( " - a>
(0)
|> 0. Let mJ take all real values such
that |/x Mo"|^ eiand find the corresponding values of w <0) and co
(1). Let
w be the greatest co<0)
,and a>,
(1) be the least co(1)
. The value of Cj can be
taken so small that w,a)
-w<0)
>0, and hence from the continuity of a as afunction of n it follows that u takes all values satisfying the inequalities
r ^w ^wj". Take any rational value of w depending on p, fi ,and e which
satisfies these inequalities, say
No=n
(46)
where N and n are relatively prime integers. Then determine T= !F by
the equation
7>2 7r=^- (47)
Wo
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238 PERIODIC ORBITS.
Now consider the series (31), in which we put T= T . Since they vanish
identically for a, = c, =0, the discussion in 14-16 shows that, for any values
of X and e for which the differential equations are regular, r 4 > and p>0can be so determined that the series converge for all ^ T ^ T provided
|o,|<r,, |c,|<p ( . The r, and p< are functions of p , X, and e. We mayeliminate X by the relation p = p -f X, and we shall take |pp | = e
i and
<! e ^ 77, ,where , and 77, are both distinct from zero and have such values
that the differential equations are regular for |p p | ^i , e<jr?i . Let rj0) and p,
<0)
be the least values of r< and p t as p and e take all values satisfying the ine-
qualities |m Pol =u ^ e = ^i- It has been shown that when the solutions
of (31) exist they have the form
a t= \p t (X
J
, e) c,= -c 2= X*p 6 (X
}
, e) , (48)
where the p t are power series in X!
and e. These series will converge andgive \a t \<rf\ |c,|<p,
(0)
provided 0<|X|^ 2 ^e 1 ,for all e<ij t
. That is,
since the series vanish identically for X = 0, the limits on the a t and c ( can be
controlled by X alone. Choose any X satisfying the inequality |X|^j and
determine p by the relation p p = X. It was shown above that for this
p there exists an e < ^ such that w (p , e)= w = N /n . Hence for this p
and e all the series employed are convergent; that is, the existence of certain
solutions of the type in question is proved.The question might be asked whether the solutions exist if both p and e
are given in advance. It is not easy to make the answer in general, but
it is clear that the mode of generalization of p into p and p + X opens awide range of possibilities. This is so unless, indeed, the realm of validity
of the results is independent of this process. Suppose p, e, and the initial
conditions are given such that the motion is periodic. The coordinates
may be represented by
x = Fx {n,t), y = F i ( t x,t), t-F t (p,f).
Consider the expansions of the functions F, as power series in X, where
/i=
ju 4-X in at least part of the places in which p occurs. It is clear that the
realm of convergence of the series depends upon the manner of this trans-
formation. For example, if F l= F t
= F 3= sin/(l+X)(2+p), and if we put
l+M=l+p +X, 2+p = 2+p, then the functions are expansible as powerseries in X which converge if |X| <l-fp ,
where X and p are subject to the
condition h -\-\ =^l. But if we put l+p = l+p, 2+p = 2+p +X, then
the series in X are convergent if |X| <2+p where X and p are subject to the
condition n + \=n. For example, if p = l/3 in the first case the scries
converge if |X|<2/3, and in the second case if |p|<7/6. Now it is clear
from the variety of ways in which X can be introduced that convergence of
the series can be secured in many, if not all, cases when p and e are givenin advance.
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 239
1 16. The Existence of Two-Dimensional Symmetrical Periodic Orbits.
The last two equations of (31) are satisfied identically by cy= c 2
=Q, in which
case the orbits become plane curves. We have to consider the solution of
the first four equations for a x , . . . , a 4 in terms of X.
The determinant A, equation (33), now becomes simply A = At
. It
follows from (34) that the condition A1= can be satisfied only by <r=N/n,where N and n are integers which we shall suppose are relatively prime.
We shall solve the first three equations of (31) for a 2 ,a 3 ,
and a 4 as
power series in a x and X. This solution exists and is unique provided the
determinant of the coefficients of the linear terms in a 2 ,a 3 ,
and a 4 is
distinct from zero. It follows from (34) that this determinant is
T T -
D=-A 3 [e^-e-p
-?],
which is distinct from zero unless A 3 is zero.If A 3 were zero we should use the first, second, and fourth equations of
(31). The determinant of the coefficients of the linear terms of a 2 , a,,
and a 4 in these equations is found from (34) to be
d *(j)|V*-.Tf],which is distinct from zero unless v 3 (T/2) is zero. But A 3 and v 3 (T/2) can
not both vanish, for then Dt of (37) would be zero, whereas it is distinct
from zero. Therefore a 2 ,a 3 ,
and a 4 can always be eliminated by meansof three of equations (31), leaving a single equation of the form
= a 1 [ft 1 X+/3 10 a 1 +/3 11 a 1 X+& aJ+ ] (49)
This equation carries a t as a factor because equations (31) are identically
satisfied by a, = = a 4= 0.
Now consider equation (49) . The trivial solution a x= will be rejected,
and we shall attempt to solve for a t as power series in X I/J,where j is an
integer. If j8 01 is distinct from zero such a solution certainly exists, and j
is determined by the first & which is distinct from zero. The coefficient /3 01
is a power series in e whose term independent of e was found in Chapter VI,
equation (26), to be distinct from zero. Consequently for e sufficiently
small /3 01 is distinct from zero, and the solutions exist.
Now consider /3 10 . It was shown in Chapter VI, 100, that the part
of this coefficient independent of e is zero. It will now be shown that it is
identically zero. The solutions of the first three equations of (31) for
a i} a 3) and a 4 contain no terms of the first degree in a x alone, for, if we
eliminate a 2 and a 3 by means of the first two equations, we have
A 3 [e"T e~ PY
~\a t= 0+terms of second and higher degrees. (50)
It follows from property [3] of 108 that the terms in the solutions of the second
degree in the a, do not contain t as a factor. Hence, the only terms of the
second degree in the a which are not periodic with the period T are those
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240 PERIODIC ORBITS.
which contain a\ or a] as a factor. Hence the solution (50) for a, has a
term of at least the third degree in a t as the term of lowest degree in a,
alone. Consequently, when the first three equations of (31) are solved for
a 2 ,a 3 ,
and a 4 and substituted in the last one, the result contains no term
of the second degree in a x . That is, /3 10 is identically zero.
It is not necessary to consider j3 u if /3 20 is distinct from zero. In ChapterVI, 100, it was shown that the part of /3 20 which is independent of e is
distinct from zero. It is also distinct from zero for e sufficiently small.
Therefore the solutions are expansible as power series in X* of the form
x = \ i P(V;t), y = \*Q(\i ; t).
One value of the double sign belongs to the orbit when the infinitesimal
body crosses the axis in one direction, and the other when it crosses it in
the other direction.
There are two cases to be considered, according as n is even or odd in
the expression a = N/n.
1 1 7. Case I, n even. The infinitesimal body crosses the x-axis perpen-
dicularly at t = and at t=T/2 = mr = 2n'ir, where n' is an integer. Since
2nV is an integral multiple of the period of revolution of the finite bodies,and since the infinitesimal body crosses the x-axis perpendicularly only at
the beginning and middle of the period, it follows that it crosses the x-axis
perpendicularly only when the finite bodies are at their perifoci. It follows,
as in the case of the three dimensionalorbits,
that the orbitbelonging
to X*
is not geometrically distinct from that belonging to +X !
;in particular, that
one orbit can be obtained from the other by increasing t by T/2.Consider the terms which are even in \K They are not altered by
changing the sign of X*. Consequently in these terms
sin
cos [(>+*)] a +Z[(jcr+k)(t + T/2)]
from which it follows that j is an even integer. In the terms of odd degreein X 5 we have
sin
cos [U*+k)t]ft -
Z[(j<r+k)(t+T/2)],
from which it follows that j is an odd integer.
If we should set up the problem starting the infinitesimal body per-
pendicularly to the x-axis when the finite bodies are at their apofoci, wcshould find similarly two geometrically identical orbits in which the infini-
tesimal body crosses the x-axis obliquely when the finite bodies are at their
perifoci. That is, when n is even there are two classes of geometricallydistinct orbits of given period which intersect the x-axis perpendicularly; in
one, the periodic orbits intersect it thus only when the finite bodies are at
their perifoci, and in the otheronly
whenthey
are at theirapofoci.
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 241
118. Case II, n odd. If in the case where n is odd the infinitesimal bodycrosses the x-axis perpendicularly at t = and the finite bodies are at their
perifoci, then it also crosses the .r-axis perpendicularly at t = T/2 when the
finite bodies are at their apofoci. If N is even, the infinitesimal bodycrosses the x-axis in the same direction at t = and t = T/2. Hence the
orbits for +X* and X are in this case geometrically distinct.
If N is odd, the orbits for -f-X* and X* are also distinct, because other-
wise the motion of the infinitesimal body would be the same while the finite
bodies are moving from perifoci to apofoci as while they are moving from
apofoci to perifoci. This is impossible because t enters the differential
equations differently in the two cases.
Similarly, starting when the finite bodies are at their apofoci, two geo-
metrically distinct orbits are obtained, in both of which the infinitesimal
body crosses the x-axis perpendicularly when the finite bodies are at their
apofoci, and also when they are at their perifoci. These orbits, therefore,
are identical with those obtained starting when the finite bodies were at their
perifoci.
CONSTRUCTION OF THREE-DIMENSIONAL PERIODIC SOLUTIONS.
119. Defining Properties of the Solutions. It has been shown that
the periodic solutions have the form
x = 2 x 2) \J
, y= 2 y 2J X', z= 2 ^,X , (51)1=1 1=\ l=i
where the x j} y S) and z t are all periodic with the period 2ir/w. It has been
shown that symmetrical orbits exist, two for each value of X, and their
coefficients are uniquely determined by the periodicity conditions and
z (0)= 0. It follows from this last relation and from the fact that the series
for x, y, and z converge for all|X
| sufficiently small, that each Zj(0) sepa-
rately is zero.
120. Coefficient of X . It follows from (13) that this term is defined by
z" + [A + 3Aecost+^Aei
(l+3cos2t)+ -Jz^O. (52)
This equation is of the type of that treated in 53, and its general solution
is of the form
z,= ciV^'u^e; t) +c (
2
l)e- a ^= ll wi (e; t), (53)
where
w = v / A+a) 1e+w s e2-|- ,
w l= l+w?
) e+w (
?e2 + ,
while tit, differs from wx only in the sign of V^i, and where each w[J)
is
separately periodic with the period 2ir. Since (53) is unchanged by changing
the signs of botht
and V l,it follows that in the
expressionsfor w
yand
w2
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242 PERIODIC ORBITS.
the coefficients of the cosine terms are real, and those of the sine terms are
purely imaginary. Since w\(e; 0) = wi (e; 0) = 1 for all values of e, it follows
from the condition ,(0) =0 that c[l)
+c^ = 0.
It will now be shown that co is a series in even powers of e. The coeffi-
cient of 2, in (52) is derived from the expansion of r^3 and r 2
- \ Now the
expressions for ^ and r 2 are unchanged if in them e is replaced by e andt by t-\-ir. Consequently, if eP^^^'w^e; t) is a solution of (52), then also
e u( -> v/ = 7((+ ' r)M) 1 ( e; t+w) is a solution. Since any solution can be expressed
linearly in terms of the two solutions in (53), and since the solutions nowunder consideration hold for all e sufficiently small, and, for e = 0, differ only
by the factor e^ ,v ^ ltT,
it follows that for any e they differ only by a constant
factor. Therefore
-*^=>, (e; t)- Ce u( -" v^ Tc,+,r, w; l (-e; t+v) = 0.
This relation is satisfied identically in t and e. Since w1 is periodic with
the period 2t, it follows also that
efaWV=ltr e' WVT=t wl {e; t)-e
wi -' )V=Ti 'C^- t)VzrT,,+ir) w1 (-e; t+v) = 0.
Hence we have two homogeneous linear equations in efaV>v ^ Tl wl {e; t) and
(; e t-)N^To+x) w?^_ e.
t-j-ir), and as these quantities are not identically zero,
the determinant
* '
I *I _ w(-e)\f=T2K -i(i)v^iJ
must vanish. Since by definition w must reduce to VA for e = 0, we have
w ( e)=a;(e); that is, co is a function of e2
.
Upon carrying out the computation by the method of 53, we find
[ (54), f_ 9A(l-15A + 18A* + 8A s
) _ 9 A' ,, 184 V^T+l
"
8(l-iD(l-4A) (l-4A)iC0St+
(1-iAYSmt
, 9A(l-34-2A !
) 9A ! (1-54)^T. J ,, ....+8(l-A)(l-4A)
C 82t8(l-A)(l-4A)
Sm2t\
e +
It will be noticed that the coefficients of cos,;7 and sin/7 have eJas a factor
so far as they are written. This is a general property of the differential
equations, and an examination of the process of integration, as explainedin 53, shows it is also a general property of the solutions.
If we had solved the differential equations for x v and y, , we should havefound that these quantities are identically zero because of the periodicityconditions to which the solutions are subject.
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 243
121. Coefficients of X. It follows from (13) and (14) that these terms
are defined by
(55)X2-2y' -[l+2A+6Aecost+ -]a; 2 -[6,4esin<+ -]?/ 2 =r 2 ,
y2+2x' [6 Ac sin t+ ]a; 2 [l A SAecost+ -]?/ 2 =Zj,
where
Xi =[^B+6Becost+ ]**,
F 2=
[3Besin*+ ]**,
r> _ + 1 Mo + Mo+ r (0)l (0)4
'
'l '2
the signs in 5 being the first, second, or third pair according as the point
(a), (6), or (c) is in question. By means of (53) and (54), we find
z\= +(c-)^
w-jl-[ 14^(l-cosO +
1
1
^^rTsin<] e+}
+2cl C'{l-[ 1-^ z (l-cosi)]e+ }
,
,(I)N2 -2o,v t f, r (\A n ., 12A' A
j r - ,1 I
+ 6Bcc !{i
+[ 1^A + f5|4co S(
](;+
j
+3B(c 2 '>)' e(2
+Li3~4A +T3IA cosf -I3H v^Tsm^-f j,
F,= 35(0 2e2w,/r7
'{[sin]e+ j+6c',)
c 2
1)
j[sin<]e+ ]
+3flc 2
1)V2
^'{[sin*]e+---i
The character of the solutions of equations of the type to which (55)
belong was discussed in 29. It was there shown that they consist of the
complementary function plus terms of the same character as X, and F 2 .
It follows from the hypothesis that the imaginary characteristic exponentcrV l, arising in the solution of (55), and wV l are incommensurable,that the constants of integration must all be put equal to zero. The par-ticular integrals can be found most conveniently by assuming their form
with undetermined coefficients, and then defining them by the conditions
that theequations
shall be identically satisfied.
(56)
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244 PERIODIC ORBITS.
We shall need the following properties of the solutions of equations
(55). They are homogeneous of the second degree in c"'e u>/ ^ rT ' andc
<i>e-u> %/=7 The terms in the solutions multiplied by (c^')
2differ from those
multiplied by (cjl)
)
2
only in the sign of V l, because this is a property of the
right members of the differential equations. If throughout equations (55)
we change y 2 into -y t , V^T into v/^T, and t into -t, the equations are
unchanged. Therefore, in the expression for x t the coefficients of the cosine
terms are real and the coefficients of the sine terms are purely imaginary.The opposite is true for y 2 . The terms in x, having c^Cj" as a factor involve
only cosines and are independent of the exponentials ef v ^7 'li and e~ wVZrr', while
those in y t having c"'^" as a factor involve only sines and are also indepen-dent of the exponentials e wv/trT ' and e~ wv/ ^ T
'. It follows from these properties
and (? = - d? ,that x 2 (0)
= y 2 (0) = 0.
Certain divisors are introduced in the integration of (55). When the
right members of these equations are omitted, the solutions are of the form
(24). On using the method of the variation of parameters, we find
a ;= M) (t-i, ...,4),
where F ( (t) has the form of X2 and F 2 ,and where A is the determinant
of the fundamental set of solutions (24). It follows from the principles
of 18 applied to this case that A is constant. The expressions F,(t)
contain terms of the types given in (56). Consequently the a, contain
terms of the types
Aa t= (ciyje^^'lAj cosjt+ sT^TBj sinjt]dt
+ (.c)i
Je-uVzrTt
[A J cosjt- V~=TBjsmjt~]dt+c?)
(*"fC J cosjt dt;
or, performing the indicated integrations,
A a, = +@^ ***= [2 a V^i cos jt+j sinjt]
- ^1-4^ [jV=Tcos./*+28in;*]
+ ^jj^ e"*"^ 7 '
[-2a> y/^\cosjt+jsinjt\
J
+ ^ ;
ff.4 t [jyT^T cosjt+2w sinjt] + SlS. C, sin jt.
Therefore, the divisor j2 4 2
appears in terms involving t in the form
e *2a/=T/cos^ an( j ^he di v i sor j j n those involving t in sin^.
It was seen that in the expression for zl the coefficients of cosjt and
sin jt carry e' as a factor. Consequently it is true also for z\, and therefore
for the A,, B,, and Ct .
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 245
122. Coefficient of X'\ It is found from (13) , (14), and (51) that
z"+\_A + 3Aecost-] ]z 3= Z,,
Z 3=+{pa
-pw -\ k +|
3 H \x 2 z,
+{l[1^ +
if ]+ * '
}^+{6^sin<+
j&Zi
(57)
The coefficients of z,, a^Zj, and z\ are series involving only cosines, andthe coefficient of y 2 z x is a series involving only sines.
Now consider the solutions of (57). The general solution of the left
member set equal to zero is
z 3= Cx e
uV=T, w1 + C, i~" v=t %,
where C 1 and C2 are the constants of integration. By the method of the
variation of parameters, the conditions on C, and C2 that (57) shall be
satisfied when its right member is included are
e"^ 1 ' wx C[ + grwsn w^ = o,
e"^ 71 '
[ v^MWj+wJlCl+e-"^ 1 '
[- V=iw t +w' ]C' = Z 3 .
Upon solving (58) for C( and Cj , we get
A C( = - wt Z 3 e""^ 77', A C 2
= +w 1 Z 3 e uV=~ t
,
(58)
(59)
where
A =w l , w2
UVI Wl -\-w[ , WV 1^2 + ^2
It follows from the results of 18 that in this case A is constant, and since
wx and w2 are power series in e the determinant A is also a power series in e.
In order that the solutions of (57) shall be periodic with the period Tit is necessary and sufficient that the right members of (59) contain no
constant terms. We must therefore pick out the terms in w2 Z 3 and
wx Z 3 which are constants multiplied by e?"^ 11 and e'"^ 1 '
respectively, and
set their coefficients equal to zero.
Consider the first term of Z 3 . Upon substituting the value of zxfrom
(53), we see that the constant part of the coefficient of e wv/=T '
coming from
WjZ, is the constant part of the product
-c^Wittdpa
-
pn+3fiecos<+ -
<J-
(60)
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246 PERIODIC ORBITS.
The corresponding part of the coefficient of e~ uV~' coming from +w t Z, is
the constant part of
+cj ;,>,J pai
-pr
+35ecosf+-J-
(61)
It follows from the properties of w1 and wt that their product involves onlycosines and that the coefficients of their product are all real. Hence the
constant parts of (60) and (61) are power series in e which, aside from the
coefficients c[l) and Cj
1
', differ only in sign, and the parts independent of e
are respectively
Ci y r(o>3
r (o)ij,t^ 2 \r
m)ir mi )
Considernow
the termscoming
from thatpart
ofZ s
which contains
x t z x as a factor. The constant parts of the coefficients of e"""^* and e~ Uy/=~u
in the products
-w^z^SB+UBecost ]
and
+w l x 1 z 1 [ZB+\2Becost ]
respectively are required. It follows from the properties of w, and to, that
u>,[3.B + 12Bcos*+ ]
and
w2 [3B+12Becost+ ]
differ only in the sign of V l, which is a factor of all the sine terms,
while the coefficients of the cosine terms are all real. These productsdo not involve the exponentials e wV^ it and e~' vr=Jt
, In the product x 2 z x the
coefficients of e""^' and e~ wVZ=^ 1 are multiplied by (O'cJ" and c"' (c"')2
respec-
tively. It follows from the properties of x 2 and zx that in this product
the coefficients of (c^Y 4" e uV ^' and cf (cj )**-"^' differ only in the sign of
V l,which is a factor of all the sine
terms,while the coefficients of the
cosine terms are all real.
The typical terms of the products tv i ['3B+ 12/?ecosJ ] and that
part of the product x 2 z xwhich contain e uV^ i ' as a factor are respectively
AjQO&jt V lBiSWJt, (ci)
)t
c?)
[a J cosjt+ V lbjBUijt].
The corresponding terms from +w\[3B + \2Becost + ] and the part
of x % z x containing e~ us r=ru as a factor are respectively
+AjCOsjt-
-f^lBjsinjt, c^ic^fiajcos jt
-V^Tbjsinjt].
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 247
The constant parts of the products of these terms are respectively
Since these properties hold each term individually, it follows that the con-
stant parts of the coefficients of e"^ 1 ' and e~ Va= ~u in w2 x 2 Zy[%B+ ] and
-\-w l x i z 1 [ZB-\- '
] are respectively of the form
-(<f)*4D [Ci+C+ ], +c[ (cmC a +C 1 e+ ]. (62)
The corresponding discussion shows that the same result is true for the
third and fourth terms of Z 3 . Therefore, in order that the solutions of
(57) shall be periodic, we must impose the conditions
O-JTcf+IKcfW, = Kc?+Lc[l)
(c(
i y, (63)
where K and L are constants and power ser ies in e. It follows from the
corresponding work in Chapter VI that both K and L have terms independentof e which are distinct from zero. The solutions of (63) are
c ; -e -0, K+Lc?c? = 0. (64)
The former leads to the trivial solution x=y=z=0 and need not be con-
sidered. Since cj"= c"
} the latter gives
c=^| = [X,+X 1 + ], (65)
and ej and c( " are determined except as to sign. The orbits which correspond
to the two values of c{ "
are geometrically identical or distinct according as
T is an odd or even multiple of 2w (see 113-114).
After the conditions (63) are satisfied, the solutions of (57) are
z 3= cJV^'ttJ, (e; t) +<f e-f* w2 (e ; t) +P?\t) +P?(0, (66)
where Pf is linear and homogeneous in c{l>e WN/^ rT ' and ce~ uV=l
', and where Pfis homogeneous of the third degree in c
{
"e uV=It and c^e~ uVr=lt . In the right
member of (57) the coefficients of all cosine terms are real, and the coeffi-
cients of all sine terms are purely imaginary; and moreover those which are
multiplied by cj", (O 3
, (c"')2
^" differ from those respectively which are
multiplied by c, (O 3
,c"' (c"')
2
only in the sign of V^T. It follows that
P{3) and Pf have these properties also. The wl and w2 are given in (53)
and (54).
The P[3) and P 3
3)are entirely known functions, while cf and c"' are
subject to the relation
2,(0)=c5
3)
+<f +PJ
3,
(0)+Pf(0) =0.
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248 PERIODIC ORBITS.
It follows from the properties of the expressions Pf (t) and P (t) and
c|l)
=-c;u that Pr(0)=P;
3)
(0)=0. Hence ef and <f are subject to the
condition
*(0)-<f+<f0.(67)
Therefore but one undetermined constant remains in (66).
The divisors introduced at this step can be found as they were in the
preceding step. They are the determinant of the fundamental set of
solutions of the z-equation, f ui2
, andj2
9oj2
, the j2
co2
appearing with
terms which involve e* Wv/=T '
jt, and the j2 9 2 with those which involve
e *w=Ti coejt The coefficients of*jt contain e' as a factor.
123. Coefficients of \\ The differential equations at this step are
(68)a-2yl-[l+2A+6AecosH ]z 4
-[64dsin* -\ ]y i
= Xi ,
y"+2x[ [6Aesint+ -]x 4 [lA 3Aecost + ~\y i =Y i ,
where XA and F 4 are of even degree in e a,v/ ^ T( and e~ wV=Tl considered together.It follows, as in the case of x 2 and y 2 , that in the right member of the first
equation the coefficients of all cosine terms are real, and those of all sine
terms are purely imaginary ;the opposite is true in the right member of the
second equation. The coefficients of cfej8 and (c"')
4differ from those of
c" cf and (<')* respectively only in the sign of V l. The coefficients of
(0*(0* are real, independent of e?"1^ 1 ' and tT^ r=u
iand involve only cosines.
The solutions of (68) have the same properties. It follows from these
properties and c[1) =
(%*, that x[ (0) = y t (0) = 0.
The divisors introduced in the solution are the determinant of the
fundamental set of solutions of the x and ^-equations, f 16w 2
, j2
4co2
,
and j in connection with the terms involving e* 4WN/TrT'a?
3jt, e* "^ 11 *
jt,
and ljt respectively. The coefficients of these terms contain e'as a factor.
124. Coefficient of X' A. It is necessary to consider this step in orderto be able to make the general discussion. The differential equation defin-
ing z 5 is
g+[A+SAmo6t+ ]z 6= Z 5 , (69)
where it is found from (14) and properties (a) and (d) of 106 that Z & is
made up of real cosine series multiplied into ,, x 2 z3 ,
a1^, x\z { , y\z ly
z\z 3 , z\, and of real sine series multiplied into y 2 z 3 and */ 4 z,. It follows
from the properties of z n z 3 ,x 2 , y it x t , and y t that Z 5 is of odd degree
in e" v ~' and e- "^^' taken together; that the coefficients of the cosine terms
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 249
are real, and those of the sine terms purely imaginary; that cf and cf enter
linearly; that the coefficients of cf\ (e?>)*f , ffc'V, (e?*)*^)1
differ from
the. coefficients of cf, (O'cf, c'?c?c? and (<f)2 (0 3
respectively only in
the sign of V l; and that the terms which involve cosjY and sinji havee' as a factor in their coefficients.
Now consider the solution of (69). The discussion proceeds as in the
case of (57), and it is found that, in order that the solution shall be periodic,
the constant terms in wi Z i e~ uV^ T ' and +w x Z h e uVtrT < must be zero. It
follows from the properties of Z 6 enumerated above that these conditions
are of the form
[A.+A^^^+AAc^^+A^yic^^O, J
(70)
where Alt . . .,AA are power series in e. Upon reducing by means of the
last equation of (64), we get
[A,- ^] <f + 4l <f + Af <f =
0,
[A ~ ^] c
" )+ ^iF c" )+ ^ c
"' =0 -
(71)
Since c=c ( " and c= cf\ these equations are equivalent and define
c[
3)
provideda A t K A*K l
Ax L Uis distinct from zero. In the case e = of Chapter VI the corresponding
coefficient was distinct from zero. Since the coefficient in the present case
is a power series in e and reduces to that of the former case for e = 0, it
follows that it is distinct from zero for e sufficiently small.
When equations (71) are satisfied, the solution of (69) has the form
z b= c? (T^'Wt+c? e-^-^Wt+PrW+P^W+PFit), (72)
where Pf(i) is linear and homogeneous in c">
e Uv/ ^ T ' and cf>
e _Wv^ r ' taken
together, and P(t) and P(t) are homogeneous of the third and fifth
degrees respectively in ew^~' and e~ uVrrTl taken together.
It follows from the facts noted above that Z 6 is unchanged if c}M
,
c'
f -f V^T are interchanged with c, 4, v^7, and that the coeffi-
cients of the cosine terms are real; hence the solution (72) has the same
properties. Consequently, since Cj"= -
c{", cf = -c,0>
,it follows that
P?(Q) = P? (0)= P^
6,
(0)= 0; and since z 6 (0) =
0, that
if+cF-O. (73)
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250 PERIODIC ORBITS.
125. The General Step for the x and ^-Equations. Suppose z, ,x t ,
y lt . . ., z^-i, _,, y t -2 have been computed and that they have the
following properties:
(a) Thex
2Jand
y 2}are even functions of e"^ 77 ' and e -0 "^*' taken
together,and the z 2J+x are odd functions of e?"^' and e _Wv '^ T ' taken together.
(b) In the x 2l and z 2J+i the constant parts of the coefficients of all
cosine terms are real, and those of all sine terms are purely imaginary.
(c) In the y 2 , the constant parts of the coefficients of all cosine terms
are purely imaginary, and those of all sine terms are real.
(d) In the x 2j and y 21 the highest powers of efaV^ Tt and e _Uv/=T ' are 2j,
and in z 2i+1 they are 2j+l.
(e) The coefficients of (0*(<f)* (?)H<f )*(<f )*' (C)*' in the
expressions for x 2i , y 2) , and z 2 y+i differ from the coefficients of
(0''(<f )''*
(<?)*(? )*(? )* (cD l - only in the sign of V^7.(f) The constants of integration arising at the step 2j + 1, viz.,
cf+1> and c\
j+l(j=l, . . . , v2), must satisfy the relations
c w+ mM<2J+ g ^*fU =M(2y+i) c(i) = _ Mw+0 ^
where MttJ+l) is a power series in e, in order that the solution at
the step 2,7+3 shall be periodic. The constants cf~l) and c 2
v ~ u
remain arbitrary at the step 2 v.
(g) The divisors introduced at the step 2j are the determinant of
the fundamental set of solutions of the x and ^/-equations and
k, A;2 -4 2
, &2 -16V, . . ., k*-jW in the coefficients of c
a ?*kt;
and at the step 2j+l they are the determinant of the funda-
mental set of solutions of the z- equation and k, k 2u? A;
29co
2
,
...,k*-(2j+l)W.
The differential equations which define x 2v and y 2 , are
xl-2y 2 -[l+2A+6Aecost ]*- [6Aeeint
y t[+2x 2v [QAesint '-~\x w [1 A ZAecost
It follows from the properties (a), (b), (c), and (d) of 106 that X2V
and Y2V are even functions of Zj, . . . , z 2v -i taken together; that in X2y
the coefficients of all terms which are of even degree in y 2) y if . . ., y 2v _ t
taken together are sums of cosines of integral multiples of t; that in Xly
the coefficients of all terms which are of odd degree in y 2 , y it . . . , y lp _ t
taken together are sums of sines of integral multiples of t ; that the last two
properties are reversed in the case of F,; and that if in Xiv or Ytv the
general term has as a factor
then
2\+ +2j\ 2] +2fi t + i-2Kii u +v l + {2l-\)v v _ l ^.2 v.
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 251
It follows from these properties and (a), . . ., (g) that Xiv and Y2v are
even functions of e w%/=T' and e~ wV=I ' taken together; that in Xiv the constant
parts of the coefficients of all cosine terms are real, and those of all sine
terms are purely imaginary; that in F 2 the last property is reversed; thatin X2v and Yiv the highest powers of e Uv/ = r ' and e~ wV=T( are 2v; and that the
coefficients of (0 A(0'' : (cftH^TW)* (<?)*- differ from the
coefficients of (f)*(D* (?)''(<f ^(eh* ' (0'"' only in the signof </=T.
It follows from the form of (74) and these properties that the periodicsolutions of equations (74) (i. e., the particular integrals) have the
properties (a), . . ., (g) so far as they pertain to x 2j and y 2] ; and then
from c,a, = -ci", that a,(0) = y(0) = 0.
126. The General Step for the z-Equation. The differential equationdefining z 2 ,+i is
zl +l + [A+3Aecost+ -~\z 2v+i= Z 2^ l . (75)
It follows from the properties (a), . . ., (g) of 106 that Z 2y+l is of
odd degree in z 1} z 3 ,. . .
, c*_i taken together, and that it contains z 2v -\
linearly. In Z 2y+] the coefficients of terms which are of even degree in
Vtt ' t V* taken together are cosines of integral multiples of t, and the
coefficients of those terms which are odd in the same quantities are sines
of integral multiples of t.
It follows from these properties and (a), . . ., (g) of 125 that Z 2v+l
is an odd function of e wV=*' and e~ wV=ri ' taken together; that the highest
power of e wV=Tl and e _uV=T 'is 2j/+1; that cf~" and c 2
"-" enter linearly;
that the constant parts of the coefficients of all cosine terms are real,
and that those of all sine terms are purely imaginary; and that the coeffi-
cients of (cl1
')'1
(c[")'< (O' 1
(cf)1 - differ from the coefficients of
(O* <%Y CO*' ' ' (cD4 *
only in the sign of v^T.Now consider the solution of (75). The conditions that it shall be
periodic with the period T are that in w2Z 2v+l and +tc,Z H. 1 the constant
parts of the coefficients of e Wv^ =I/ and e~ aV=Itrespectively shall be equal to
zero. The z 2) .-i enters Z 2v+i linearly and has the same coefficient that z 3 has
in Z %. Hence, from the relations of the preceding paragraph and equations
(70), we have
[^ 1 +^ 2
' )
]cri, +^3(0 2
cr-i, +p 2 , +1 (c 1
( ,o=o,j
where P 2v+ i is a polynomial of odd degree in c"' and c 2
1} taken together. It
is supposed that C?'"1" 1 ' and c 2
2 ' +l)(j=
l, . . .,
v 2) have been eliminated
at the successive steps by the equations corresponding to (71). If the
general term in P 2 , +i is (0'(c 2 'T, then ; and k satisfy the relation
i-*+l. (77)
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252 PERIODIC ORBITS.
On reducing (76) by means of (64), making use of (77) and (f) of 125,
and the fact that cjl) = c[, it is seen that equations (76) are equivalent,
and that cf~x) = cf~
v>is uniquely determined as a power series in e. Since
in Z 2 +l the coefficient of (cj / (cj )'' (O* 1
(cf )*- differs from that
of (0'' Wic?)* (cD*-only in the sign of v^T, it follows that
the solution has the same property, and from this that z iv+l (0) = 0.
For small values of v there will be no other terms than those considered
above in wt Z +i and -{-WyZiv+i, which are constants multiplied by the
exponentials e wV=Xt and e~ >v ^ it
respectively. When there are no other terms
and when equations (76) are satisfied, the solution of (75) is periodic and z 2v+ i
has all the properties of z 2t -\ specified in 125.
But since u=N/n, where n and N integers, there is a value of v for which
other terms of the type in question can arise. It follows from the properties
of Z 2v+l that wt Z 2v+l and +w 1 Z 2v+1 contain respectively the terms
-C M(cfT +l e + < iv+1 ^ v ^'[a a +a 1 cost+ +a k coskt+
V l&jsintf V ibisinkt ],
+C (p
\c?y +l e-'- +1 >" v =~u[a +a 1 co8t+ + a* cos Art +
+ V^T6 1 sinJ+ +V~^Tb t smkt+ ].
(78)
Now e 9* v>ay/=i> = e uV=~u[cos2vwt + \ r= lsm2vwt]. Consequently terms
of the type in question will arise if 2vu = k, k being an integer. Uponsubstituting the value of w, this relation becomes 2vn =
kN,which can
be satisfied when 2v becomes a multiple of N. Suppose the integer n is
odd. In this case when 2v = N, the smallest A; satisfying the relation,
viz. A; = n, is obtained. The term in which this occurs is multiplied byy*+Vt e == x JV+,/ '
e". But if n is even and iV odd, the relation is satisfied first
for increasing values of v when 2v = 2N, and then A; = 2n. The term in which
this relation occurs is multiplied by X2 "" 1"' 7 '
e2 " = \ iN+ ''' e
2". After terms of this
type once appear they in general occur similarly at all subsequent steps of
the integration.
The coefficients of e "^' and e~ aV ^ Tt obtained from (78) are respec-
tively -Ciy+iic^y^and
+C w+l (c2
)
y+l
, where C2V+X
is a constantmultiplied
by e" or e2"
according as iV is even or odd. Therefore, when these terms
arise we have in place of equations (76) .
[4 l 4-^ c r,
cr]crl, +A 3 (o ^cr l '+^ +1 (^^c;
lv^ +I (^^ +l =o. f
Consequently cf- " and c^
2 " - "are determined in this case as well as in that
in which the terms multiplied by C, +1 do not arise. This completes the
proofof the
possibilityof
constructingthe solutions.
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 253
APPLICATION OF THE INTEGRAL.
127. Form of theIntegral. Equations (1) can be written in the form
(80)
dt2
dt d' dt2
df'
After the transformations (10) and (11), these equations have the form
*--V-g. /W-f. ^-f, (81)
where now t/ is a power series in x, y, and z2
and contains no terms lowerthan the second degree in x, y, and z. It contains terms independent of
X and others, for the particular transformation (11), multiplied by X to the
first degree only. The coefficients in the series for U are power series in
e whose coefficients, in turn, are periodic in t with the period 2ir, and which
reduce to constants for t = 0.
The first terms of U are seen from equations (10) and (11) to be
U = + U+2A+Aeco$t+ ]x2
+[~6;lesin*+ ~\xy
+ ^[l-A-3Aecost+ ] y2
- ^A+3Aecost+ ]zs
+|_-(o)3
+ pn~ 3 ^3
-r mjecost+ J
x 2X
-[6(^3
-r
(T3)e
sin Z+Jxy\ \ (82)
+g [ r
<6T-
r m + 3(j.m
--(or )ecosi+ '. \
"
J2/
2X
+g [^3
-rm + 3
( r<,3
-rToy )ecos<+ Jz
2 X
+ EG5P + i*) + f ft** + >.(+ y+The integral of equations (81), analogous to the Jacobi integral in the
case where U does not involve t explicitly, is
x'2
+7/'2+z'
2 =2C/-2f(^)^+C, (83)
where (dU/dt) is the partial derivative of U with respect to t so far as t occurs
explicitly, and not as it enters through x, y, and z. This partial derivative
is zero fore
equalto
zero,and therefore it contains e as a factor.
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254 PERIODIC ORBITS.
128. The Integral in Case of the Periodic Solutions. In the periodic
solutions which have been considered, x, x\ y, and y' are expansible as
power series in X, while z and z' are expansible in odd powers of X*. It
follows from property (a) of 106 that U is a power series in z2
. Therefore
when the expansions of all these variables are substituted in (83), the result
is a power series in integral powers of X of the form
F = F 1 \+F 2 \*+ +F,\>+ =C, (84)
where, of course, F and the F } involve the integral sign which arises from
the right members of (83), and where C is the constant of integration.
Since (84) converges and is satisfied for all |X| sufficiently small, it
follows that
F.-C, (-l f ... ao), (85)
the C being constants. Since the series for x, y, and z have the forms
x = \[x 2 +x i \+ ' +x 2J \'~l + ' '
],
y^[y t +y^+ -fv v _1 + L
z = ~Ki
[z 1 +z 3 \+ +z 2J+1 \ i + ],
it follows that F v involves x 2 , x' 2 , y 2 , y 2 ,. . .
,x 2v . 2 , a4- 2 , y 2 -i, y' - 2 ,
z l ,z[, . . ., Vrts 3 2"-i- Equation (85) therefore has the form
*t>{%2jy%2j! y 2 )i y 2 ii z 2j+i> z 2i+n v i
J ~dt~( X2 J> y^n z 2j+d t)at = C v , (86)
where P v and Qy are polynomials in the indicated arguments with t enteringthe coefficients in sines and cosines. The subscript j runs from to v1.The derivatives enter (86) only in the form x' 2] ^ 2 x' 2v . 2) , y 2] - 2 y' v - 2l ,
andz 2i-\ z 2-t) Suppose the general term of P v or Q is
x i . . . x . y> . . . y
' . Z I . . . g2/1, 2M, "2ll[ *%flf 2M"+1 2^'+l
The exponents and subscripts satisfy the relation
+ [K+ ' ' +K] = 2v.
The partial derivative of Qn with respect to t is taken only so far as t
enters explicitly in the coefficients of x 2j ,. . .
,z ij+l ,
but the integral mustbe computed for t entering both explicitly and also implicitly throughthe x tJ , . . . ,
z 2 j+i
It was shown in 125 that the x 2J ,the y 2J , and the z 2J+l have the form
x 2j= 2 x^V"^', y 2J
= 2 iff*****, z 2J+1 = 2 ag^e****^', (87)
tj tjt=-j-i
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 255
where the x%? , the yff\ and the z^+i1 '
are power series in e whose coefficients
are periodic with the period 2w. In x and z+il)
the coefficients of the cosine
terms are real, and those of the sine terms are purely imaginary; the opposite
is true in y. It follows from these properties and those of F v enumeratedon page 254 that P, and dQ/dt can be written in the form
+P.- 2 P'V"^
at i=-v(88)
where P and R are periodic with the period 2t . Since in U the coefficients
of odd powers of y are multiplied by sine series, it follows from the properties of
the Zj2
/', the y^f, and the z^;1 '
that in P the coefficients of the cosine terms
are real, and those of the sine terms are purely imaginary; the opposite is
true in R?\
Theintegrals coming
fromJ -~^
dt are of thetypes
V=lj f^'cos jtdt = -yjgjg^ coajt + j ^}p rinjf,
r ' sin jtdt =/'
Therefore we have
i tto cos.?* H a .^ smj*.f-iWci ?-4k?u*
(89)
2 S?V*=* (90)
where the"'
are periodic with the period 2r. Moreover, the coefficients of
the cosine terms are real, and those of the sine terms are purely imaginary.
It follows from these properties that (85) can be written in the form
(*)/,2twv^=T/ c, (91)
where the F are periodic with the period 27r. The coefficients of the cosine
terms are real, and those of the sine terms are purely imaginary. If we let
4-*=**m 9t (92)
and make use of the fact that the ff are periodic with the period 2ir, we
get from (91)
2 F,tt
VtuVrrT
'<rt=CV , 2 Ff^^'al
=C , ,
2F?e"^t__l tm-r *=-
These equations and (91) can be satisfied only if either
FT-C, FT-0 {k=-v, -i,+i,..or
^fff-C,. (93)
+ v), (94)
1,
.
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256 PERIODIC ORBITS.
This determinant is the well-known product of the differences of the a+,
taken in all possible pairs, and is distinct from zero unless a relation of the
form <r t=
<rj is satisfied. Since <r,=
<r\, a t= a[, this relation can be satisfied
only if ffJ~J
=l; or, because of (92), only if
Since u = N/n, this equation can be fulfilled only if {ij)/n is an integer.
But then two or more of the exponentials of (91) are equal in value, and
the number of terms under the summation sign is reduced by combiningsimilar ones. With this understanding as to the reduction of (91), A can
not vanish and equations (94) must be fulfilled. It is clear that when these
equations are satisfied all relations of the form of (93) are satisfied.
The F"' are explicit power series in e, and they also involve e implicitly
in w, which is a power series in this same parameter. Now o> enters in twoways. It is introduced as a factor of certain terms either to the first or
second degree by the derivatives which occur in (83), and to the first degree
by the integral, as is shown by (89). The integral also introduces it in the
denominators in the form j2 4kW. We shall substitute for w its series
in e wherever it enters in the first way. This will not change the character
of the convergence. But where co enters in the second way we shall regarde as an independent parameter and leave it implicitly in w. Then equa-tions (94) can be written in the form
F?' = 2FeJ = C= S CV,
F? = 2 FZ e'=0 {k=-v .-i.+i,..., +).
Since these equations are identically satisfied in e, we have
F?,=C V , F%~0. (95)
The F") are sines and cosines of integral multiples of t. On making use
of the properties established in 126, we have
F= S [a,cosp+\^&,sinp*] =C,.,,
J>-0
F = i [a?J , cos pt + V^T 6?) , sin pt]= 0.
Since these equations are identities in t, we have finally
a ^.o= C./> ai),
=by] iP
= (p=l, ,j),
(96)
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 257
129. Determination of the Coefficients of z 2J+l when e=0. Equations(96) are relations among the coefficients of the solutions and may be used for
checking the computations. The control is very effective because at each
step all the preceding coefficients are in general involved. But equations(96) can also be used, step by step, for the determination of the coefficients
of the expansion for z when the coefficients of the expansions for x and yare determined, alternately with those for z, from the first two equations of
(13). Before taking up the general problem we shall treat the case of e = 0.
When this condition is satisfied the integral becomes
x'2
+y'2+z'
2 =[x
2 +y 2+A (2x2 -y 2 -z 2
) +3Bz 2
+( r 4-,-^)(-x=+2/
2 +z 2
)X+fC o2 <+ -]+C,
r>1 ~M i **<>
'1 's
In this case w = Va and equations (87) become
x 2 ,= 2 o^e"^"', y 21= 2 ^fe
2*^^ 1', z S)+1
= 2 Y,(
Saa e +OVIV = i
',(98)
where the <42
/', /?*', and y+l) are constants. We shall show how to
compute the y%+? for successive values of j.
Terms for j = 0. In this case, since x =y =0, equation (97) becomes,
as a consequence of the last of (98),
- A [(tJ-'O2 e-*^^' - 2 7f 7< + (y<)
1 e^v^T< ]=
'-M(HT*Y e -^^-'+2 7 r 1)
75" + (t^)2e2 ^^'] + C, .
Since this equation is an identity in t, we get 4^7,-l>
yf = Cl. Since C, is
unknown, this equation imposes no relation on 7j_1) and yf. But since
z l (0) = 0, it follows that t{-1> = yf, and there remains the single unde-
termined constant 7"'.
Terms in x and y for j=l. It follows from (13) and (14) that x 2 and
y 2 are defined by the equations
x 1
y
-2y' -[l+2A]x 2 =lBzl=lB(yme- tv * v= T '+e*^-><-2],
V:+2x' -[l- A]y 2 =0.
The solutions of these equations, which have the period 2ir/\/A, are
r _ 3B(2+ 3A) {y?)\ -^ ^-u,W(y?f 35(1 +3.1) (y?)
2
.^- ^ t
2~
2(l-7^ + 18A 2
)
^1 + 2A
-
2TT=7;L + 18A 2
)
y*- +-(r-7A+isA
i
)
[e e J-
(99)
(100)
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258 PERIODIC ORBITS.
Terms in z for j=l. It follows from (97) that the terms for j = 1 are
2z[z,
3 +2Az l z > =-[x?+y'1
} + (l+2A)x\+(l-A)yl(101)
Since z 3 has the form
2,= 7f 3) e~ wXv=s, +yi~^ F^^+iQ e^ v^ 7 '+7 ) e 1^^', (102)
equation (101) gives rise to the relations
(103)
There is another equation which is useless for present purposes because it
involves the unknown constant C2 . Since z 3 (0) = 0, we have also
y^+yl^+yr+y?^. (104)
It follows from equations (103) and (104) that
Y;-3, = -7,
0), Yi-
l> =-7 3
a). (105)
In order that yl~3)
shall be the same as determined by both equations of
(103), we must impose the condition
27(1 -ZA + UA*)B%yll)
y _ 9 .t0y , (1 1 \
( Wy _ , lom2(1 + 24) (1-7A+18A) 2
C > +Krf
i
ifV (Yl } _U ' (1Ub;
which determines y except as to sign. Then yl
3
~ s) and yf are uniquelydetermined in terms of 7"', but 7"' remains so far arbitrary.
The problem for e = was treated in Chapter VI. If we compare
equations (100) of the present work with equations (42) of page 211, wefind that 7"' and Cj are related by the equation
(1)\4c\ = 4A(y?)
Upon making use of this relation, it is seen that equations (106) of this
chapter and (44) of Chapter VI are identical.
Terms in z and y for j = 2. It follows from (13) and (14) that we have
in this case
x!-2y'<-[l+2A]x i= SBz 1 z i +P i , 2/ ;
,
+2^-[l-^]t/ 4= Q4 , (107)
where P t and QA are entirely known periodic functions of t having the period
2ir/V~A. We wish the details of the solutions of these equations only so
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 259
far as they depend upon the undetermined constant TjI)= fl~
l)- So far
as these terms are involved, the right member of the first equation is
3By?y?[e-" iv=I ' + <P* v=Tf]. Hence the solutions of (107) are
Xi ~1-7A + 18A 2 [e +C J+ T+2A + ^ 4 '
!/- + i-7^ + 18A2 [e 6 i+Qt '
(108)
where P K and Q4 are known periodic functions of t.
Terms in z forj = 2. It follows from (97) that these terms are defined by
2z 1 zl+2Az lz i =-2x 2 x[-2y 2 y' +2(l+2A)x 2 x i +2(l-A)y 2 y i
+6Rt 2 ^ 3 +2 (^m - ^3)21*3 + 3 C z?z 3 +fl 6 ,
(109)
where R b is a known periodic function of t. The expression for z 6 has the
form
(110)
On substituting this expression in (109) and equating the coefficients of the
several powers of e^ >/=r', beginning with e~ 6 *a N/=T
', we get
8 4 t!1^ -6 ^ known function of y= -8Ay? yf,
4v47;,,
[7l-3,
+37:-5,
i=3C ( T r)V ,
+functionof T i
l, = -4^ 7 r[7r+37r],
8^7[7r3) =
[
"1
1
-?i1
++l8^
)
-IZCofo^+functionof y?= SAy^y?.
(HI)
There is also an equation, coming from the terms independent of e vlvZT *',
which involves the unknown C, and need not be written. The first equation
uniquely defines yl~b
\ which equals y. In order that the second andthird equations shall be consistent, we must impose the relation
[ T^i-fisA 8 +^o] (7i
1,
)3
7f = known function of T (112)
Therefore 7 is uniquely determined, since its coefficient in this equation is
positive; and then the second or third of (111) defines 7s-3
', which is the
negative of y.Now, on imposing the condition that z 5 (0) =0, we get yl~ 1)=
7"', and
7^ remains undetermined.
All succeeding steps are precisely similar to the one which has just
been explained. The parts of the equations which contain undetermined
coefficients differ from those of (111) and (112) only in the subscripts.
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260 PERIODIC ORBITS.
1 30. Case when e^O. General Equations for z^ . I t wil l now be shown
that when the coefficients of the series for x and y are determined from equa-tions (13) and (14), the coefficients of the expansion for z can be determined
from the integral (83) . We shall need the partial derivative of U with respect
to t so far as this variable occurs explicitly. We find from (82) that
(j^=-[3Aesmt+lz 2
+[6Aecos+ Jxy+ ^AesinH- ]y
2
+|"|AesinJ+
-lz2 +
[3(^0,3-
^m) esmt+Jx
2 \
-[6(pi>j--<wi)ecos+ JxyX |J (^ojs-pajesinf-f- -jy2X
-[^( rm-^ 3)esmt+ .]^X-[f(^+ r^)esin<+ ]*+
Since x = y t= 0, we find from (82), (83), and (113) that the integral is
2;
2
=-[A+3Aecos*+ -]z2
-/[3Aesin*+ ^zldt+C,. (114)
In the notation of equations (87), the expression for z 2 has the form
(113)
where
Z =jz(-1)
o-uV^ltjigto gwv'-i*
t-0 *-0
m- s Wr^cosp<+-/=T/3i:^sinp<],p-0
& - S [Ccospi+V^TiCsinpt].p=0
(115)
On substituting these expressions in equation (114),it is
found that
,-2uv=it+ [-ui(z[-y+(z'
< - 1) y- 2a>V=iz[-l)
z[l - l)
]e-
+ [-* (O l+(2i'
H);
2 +2o J V^l2l1)
{ ]+"=*+ [2a;
2
^-8
* +2z[( ~ l)
z[a) - 2 wv^T (z{-
u<" - * ") ]
=
-[A+3AcosH-'
][(zi-1,
)2e-
2 ' oVirT'+(z;
i)
)2e2Uv '~ 1
+2z;-1,
z[1)
]
-([ZAeaint + ][(zr
i)
)2 e-
2wV=T'+(z;
i)
)2e2a,v ^ ,
+2z{-1)
zi1>
]^-|-C 1.
(116)
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 261
Before the integration the series for z[~l) and zf must be substituted
from (115). Consider the coefficient of e~^ s/=lt under the integral sign.
It is a sum of cosines and sines of integral multiples of t. Suppose V iAj
and Bj are the coefficients of cos jt and sin jt respectively. It follows from(89) that in the integral we have in place of these terms
j^ cosjt + Vf=$ sin jt, f[^ cosjt-If^mit (H7)
respectively. The corresponding formulas for the coefficient of e 2Wv/;rr '
are obtained from these simply by changing the sign of . The terms
independent of the exponentials which involve the cosine and sine of jt
are divided by -\-j and j respectively. Consequently, in all cases it is
easy to write down the explicit equations for the identities.
In the notation of (91) the coefficients of er*v=*J
e +i " v=Tl,and e in
(116) are respectively fj-0
, F?, and Ff\ and we have
F <- =F < =0) F?L<! lt
Since these functions are power series in e, we have in accordance with the
notation of (95)
V-*5-A *Z-C*> (us)
And these functions in turn have the form
Ft? = 2 [a^cos^+v^Tfc;-1 '
sinpZ]=0,J>=0
F% - s [CfCosP'+v^TCsinpflsO,J>=0
J
FZ - 2 [a h ,coapt+>r=Tb,,f&np(\ = Ch ,.J>=0
Since these equations are all identities in t, we have
(0) C a ia) =
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262
whence
PERIODIC ORBITS.
(z[-l)
y = (z^y+2z^z^e +[2aig> +()]*+z\-z'r'= +zWr l)
e
+[&&*'H&&*y+tf-y = 0+0 +(z 1 .
<
r")2
>2 +
=(0 2 +2**S +[ 2*5 *3 +(0 2
>2 +
= +iS^f< +[ *S*2? +zz[f\e*+= +0 +?)*>
,+_ (-1) _(1) , /-(- -<1> , _(U ,(-1) \ p
o +c;> +[ $?*$+*&*&]*+= o +*gi&-* +[ JW^+fSifi*]*+= co* +2co 1 e +[ ws
+2a> w,]e!! +
(O 2
(Cy
z[-l)
z[w =0+0
2[l,
2 1
,< - 1)
(120)
(121)
Equating to zero the coefficients of the various exponentials of (116), we get
(4-a>s
)(zl-l)
)2 + (z.'
(_l))
2 - 2o>V=Tz{-l)
z'1
'- i) =-(z[-y[SAecoat+ ]
-T[3^esini+ JGO**
(^-w!
)(02
+(2ia,
)2
+2coV"^l2;i,2 1
' a>=-(z;
i,
)2
[3^cos<+ ]
-f[3^6sin<+ ](z?ydt
2(A-w')zi-z?)
+2z[i - l)
z[m -2o,V^[zl-
l)
z[m-z?
)
z[i - i)
)^
-22l-,)
2j,>
[3^ecos<+ ]-2Jz[-l)
z?)
[3Aesmt+ ] +C lt
where the coefficients under the integral sign / must be transformed byequations (117) instead of forming the ordinary integrals, and where C t is an
undetermined constant. These equations are power series in e, and setting
their coefficients equal to zero we have equations (119).
131. Coefficients of e. On referring to (120), we find for these terms
-^(z^r= -A(z^y, -*(&- -aw, 2afc::,) C= -t^c+e,.,.
Since z^O) =0, we must add to these equations z[~" + z = 0. It follows
from these equations that
<*\= A, z^=-zZ, 4A(zY = CU0 , (122)
and 2["=
oj" 0i0remains as
yetundetermined since C
Mis an unknown
constant.
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 263
(123)
132. Coefficients of e. On referring to (120), (117), and (115), we getat this step
-2u>lz
( -
[a^.+o^cosH- V^T/^sin*]-2co
^ C^1
')2
-2a> V=Tz? [ -akfiSffiH- v^t/STui cos*]= -%(/-?)* Aco&t
- 2z l - l) A [a^+a^cos^ V^ip^aint] +^0^[cost +2co V=Tsin(|,
-2wlz^[a^ U0 +a^, 1 cost+ V~^T^ hl smt] -2u oUl (C)'
+2 v^Tz}u
[ -oj;)
liI sin<+ V^T/^cosZ] = -3(z) 2 4cosi
-2zi!U[a;:)
1 .+aiu
1 . 1cos<+^^^^, 1 sin<] + ^^[cosi-2co ^^Tsini],
+4 WoWl 2<: zi:) + 2g{*? [a<:i.o+a,
ai.i cosZ+ V=T0SL, sin*]
+*R [ &+& cos<+ v^Tttsin*]}
-2, v^lte [-
,sin*
+ V^TjfiB., cos*] -jft [- sin*+ V=T-fl?,
cos*]}=
-ta^agU cos*-2a{z{:u[oiJi.,+0.! cos*+ V^TigL, sin*]
+^0 [ a^'o+o^ cos*+ v^lfltSsin*]}
+ 6*? >,A eM#+C M.
Since these equations are identities in *, we find, after making use of equa-tions (122), that
-2AzWo}- - 2 VA Ul (z^y= -2Az[-"a^,
-2AzXU^+2VAz^fc=-3A(zXy-2Az^a<-+ ^jjfffl*
-2Az[^^+2VAz^a^=-2Az[^^+ **^p' ;
-2A3a. - 2^1^(0*- -24z<C.,
-4va^(^+2ii<; B<,+2il^aa(--24^ u a^-2ilSa{^4C u ,
+2A^^+2ii2W^+2^U ^-2v^^T.V*+A(^'-2Az^aZ,-2Az^a^-6A(z^y,
=-2Az^tfl-2AzX.
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264 PERIODIC ORBITS.
From the first two sets of these equations we get
i= U, a
i.i,i= ~ a
i,i,i=
i _ 4 a > Pi.1,1 =+Pi,i.i = -j^^ s z i,o-
- ^Vila 1 , 1 , 1 .( v liS4J
The first of the last three equations imposes no condition upon the unknowncoefficients since it involves the undetermined Cu ,
and the second and
third equations of the last set become identities. The coefficients ajjjjj and
aj" ,which are still undetermined, are not involved in these equations.
The fact that o>, is zero was known in advance, for it was proved in
120 that w is a series in even powers of e. It has also been shown that
2,"l) and zj", aside from constant factors, differ only in the sign of V 1.
Since z(0) = these constant factors differ only in sign, from which it follows
that a{~^ differs from a{", only in sign, while (t 9 and #" are equal for all
j and p. Applying the condition z(0) = to the terms under consideration at
present, we have z^l>
(0) + z," (0) = 0. On making use of (124), this equationleads to the result
<--C> (125)
and o"J alone remains undetermined at this step. Of course, it should be
noted that a",,, and 0"}., are expressed linearly in terms of the undetermined
constant z{"= a^ ,
whose value will be fixed when we treat the coefficient
of X2 in the integral.
133. Coefficients of e1 and e*. From the consideration of this step we
can infer the character of the process in general. Because of the relations
between aj^,", j8^ and a{", #" it is sufficient to equate to zero the
coefficient of e 2&,v/ ^ T ' in (116). Since we are interested only in the possi-
bility of determining the unknown coefficients, it will be sufficient to write
out the equations explicitly only so far as they involve these unknowns.
Upon equating to zero the terms independent of t and the coefficients of
cost, eos2t, sint, and sin2Z in order in the coefficient of e2
, we find
-2VJ(z!!i)2co 2
= r(z;)2
, -iVItfttfL-gf^, -4VJzM,=<^,, 1
(126)
-2VJO&-/&, -4VZ 02,-/5,, J
where /(I>
is known and where g?^ ,. . .
, /{j i2 are homogeneous functions of
the second degree in z"i =a['o, and a"J, ,
and are linear in aj" alone. The first
equation uniquely determines co 2 ;the remainder determine 0^ ,
. . .,a" 2il
uniquely when z" and a"| become known, as they do when the coefficient
of X2is considered. The coefficient a{| 20 is so far entirely arbitrary.
The coefficients of e* lead to similar equations. The first has wt in
place of co 2 ,and the left members of the remainder involve #".,, . . .
,
P^'i.t, a"i,, . . ., a"^, the numerical coefficient of /3['i and a"i being
2p -s/ZzJ'J . The right members are homogeneous second-degree functions
of a|fl
o, , , o"-i.o- The aj'i remains arbitrary.
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 265
134. General Equations for v= 1. The coefficients of z 3 are determinedfrom F 2
= 0. From (82) , (83), and (113) we find explicitly that
2z[z' +2[A+3Aecost+ ~]z l z 3 =-(x 2Y-(y 2y+[l+2A+6Aecost+ .]*j
+2[6^4.esinf+ ]x 2 t/ 2 + [6^1esini+ -~\y\
+[ 7 is-^r3
+3(^i-^)
S )ecos<+,
]l
$+[f(fp50,
+jjb)
+ f (j^r+ $m)ecoat +
]zl+2j*[3^esin<+. .
.]x2
2 dt
-2 \[6Aecost+ -~\x 2 y t dt-j [34esin+ ^\y\dt
-2J [34esinJ+ ]z x z 3 dt+ []^(^-^)e&mt+ \\dt
+3J
r
[(^ +^)esin<+ .]*}*+. .+(?,.
The 2 and i/ 2 are determined from equations (55), and it is seen from
these equations that they are homogeneous of the second degree in the
coefficients of z x . Other properties of the solutions are given in 121,
among which is that they are of even degree in e wV=il and e~ uVZZIt.
The expression for z 3 has the form
2 3= 2^
,)e- w= T
'+2<-l) e-"^= T
'+41)ewV= I, +2 s
,)e 3 "^= I<
. (128)
The coefficients of z(
3
~ 3) and zl~l)
differ from those of zf and zf respectively,
aside from constant factors, only in the sign of V l, and these constant
factors differ only in sign. Hence it is sufficient to determine the coeffi-
cients of zf and zf, which have the form
tf = S * e\ zf = X [C, cospi+^TTft^ sinpi],
zf= 2
ig e\ ig= 2 [a
l
3
l,cospt+V=i ff^smpt].t=0 J> =
(129)
135. Terms Independent of e. Equations (128) and (129) are to be
substituted in (127) and the coefficients of f*^ and e**^ set equal to
zero. These terms are power series in e whose coefficients separately must
be set equal to zero. We are now interested in the terms which are inde-
pendent of e. These results were worked out in 129, where the parts of
x 2 and y 2 independent of e were derived. The explicit results were given
in equations (103), the relations in the present notations being
_( _,< > ,<< 3 > />
ttl,0,0 Tl >
a3,0.0 73 J
a3,0.0 /3
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266 PERIODIC ORBITS.
The condition for the consistency of the two expressions for y," is equation
(106), which determines 7"' = Co except as to sign. Therefore, by 132,C and 0gj are also determined except as to sign, and Co is defined.
And by 133 it is seen that the
C, #" (p^O) are all determined except
as to sign, while the C, remain as yet undetermined. The equations corre-
sponding to (103) determine 73" uniquely, but 7"' remains so far arbitrary.
136. Coefficients of e. We shall write explicitly only the terms whichinvolve those coefficients C, SSi tT ft f . and flfij which, at the successive
steps, are unknown. The quantity z 3 is involved in the right memberof (127) under the integral sign, but since this term is multiplied by e, it
introduces at this step only C -0 as an unknown coefficient, and this unde-
termined constant enters linearly. It is determined from the terms whichare independent of e when v = 2.
Consider first the parts of the coefficients of e tuV=It and e iWy/=lt whichare independent of sint and cost. We find from (127) that
4^4m^--^&2 +/&?> -24eCC =2Aa{ o,wM+/{,, (130)'
where /jfr" and f^ are linear functions of C which is the only un-
known that they involve. Since bjJ = Co ,the condition that equations
(130) shall be consistent is a condition on their right members which a
detailed discussion shows uniquely determines the coefficient C - Then
equations (130) uniquely define Co-Now we set
equalto zero the coefficients of ^ u" /=rit
s\nt,e*"*^' s \ n t
f
e*" v=T'cos<, and e*
Uy/:=:i 'cost. The explicit expressions are found from equa-tions (127) to be, respectively,
V=l VI [+ a*, + 4 VJ ,- C. a, J
- * ,
V=T VA [- aSa
- 2 VJ C {& ]- * ,
VA[+VAalja?l,+<^ffi,-*y>ottl,] =<P 3 ,
VJ[-2VA C &, - a", ft" l ]=
<p t ,
(131)
where <p t ,. . .
, <p< are functions of known quantities and the arbitrary
Co,which enters linearly. This constant remains undetermined until
the equations are derived for v = 2. The unknowns in the left members of
(131) are a"},, $'},, d, and 0j& A , which enter linearly. On making use of
the fact that C" = C. ,the determinant of their coefficients becomes
A=-A(C) 4
[4A-1], (132)
which is distinct from zero. Therefore these quantities are uniquely deter-
mined as linear functions of the arbitrary C, .
This illustrates sufficiently the method of determining the coefficients
from the integral. The complexity of the details makes it unprofitable to
carrythe
explicitresults further.
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 267
DIRECT CONSTRUCTION OF THE TWO-DIMENSIONALSYMMETRICAL PERIODIC SOLUTIONS.
137. Terms in Xv \ It was shown in 116 that the periodic solutions
exist and are expansible as power series in X7 '. It is found from equations
(13) that the terms of the first degree in X7 * are defined by
x"-2y[-[l+2A+6Aecost+ ]x 1 -[6Aesmt+ !
lfc-0, 1
yi+2x[-[6Aent+ ] 1 -[l-A-3ilecost+ -Jyi-O. J
The general solutions of equations (133) are known from the general
theory to have the form
x,= a?V v~%+a? )
e-* v= 1, u i +a?)
e p, u 3 +a'?)
e- pt u i ,
a =a +r, e'+ ,
tt.-tf-Hipe + (i=i, ... ,4),
vi =vl
m +v (
i
l)e +
where a{, . . .,a 4
l)
are arbitrary constants, and where the u, and the v t are
periodicfunctions of t with the
period2ir.
In order that the solutions shall be periodic we must first impose the
conditions
a ,
1) = a 4
1) = 0. (135)
The constant a is a continuous function of n, n , and e. It will be supposedthat these parameters have such values that a- is a rational number. Then,since u[
j) and v(
,
J)are periodic with the period 2t, the solution at this step
is periodic with the period T, where T is a multiple of 2x and 2x/<r.
In the symmetrical solutions, x' (0) = y (0) = 0. Since these relations
are identities in X*, we have x[ (0) = y 1 (0) = 0. Since in the symmetricalorbits y 1 changes sign with a change of sign of t while its numerical value
remains unaltered, we have
- a?ea^%( -t)- oJV'p.C -t)- a^'v^ -
1) .
It follows from this identity that
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268 PERIODIC ORBITS.
Without restricting the generality of the results, we may suppose that
Vy (0)= v t (0)
=v, (0) = v t (0) = 1 . Therefore we have
a}" = Ojl>
( = in case of periodic orbits),
*i = +< [eaV ~'
w,- e-"^ ' u t
~]+ a
[>'u 3- e' pl
wj ,
. (136)
We shall suppose that the initial conditions are real as well as such
as to give the symmetrical orbits. Then, since changing the sign of V 1
in (133) does not alter these equations, with the same initial conditions
the solutions will be identical with (136). Therefore we have
m1 (-\^^T)=-w 2 (v^T), u 3 (-V=i)=u 3 (V=T), w4 (-v^T)=w 4K^T),v l (-V~=l) = -v i (^i), v 3 (-V=T)=v 3 (V=T), t (-V=D -(V^D.
If we change the sign of both t and y x , equations (133) are unaltered.
With the same initial conditions as before, which this transformation does
not affect, since y x (0) 0, we have an identical solution except that y x is
changed in sign. Therefore
t*i(-0 - -u t (t), w,(-0 = ~u.it),
,(-<) -+*C0,v
s {-t)=+v i {t).
Now if V l, t, and ft are changed in sign the differential equations
are unchanged, and hence it follows that
Mi(-v^ r l,-0 = +Mi(v^ r T,0i i(-v^=T 1 -0--i(v^ :T,i)j
^(-v^ =T,-<) = +w 2 (vA
=T,0, >i(-V=T,-0 = -t> 1 (v'=T,0,
w3 (-v / ^T,-<) = w4 (a/^T,0, ^(-*Vr
=T,-4)+# 4 (V^T,0,
w4 (-v^T,-0 = -w 3 (v/ ^T,0; 4 (->/ :r T,-<) = +y3(v
r
=^T,0-
It follows from the last three sets of relations that w4 , . . . , u k ,
v, ,. . .
,v t ,
when expressed as Fourier series, have the form
u l= '2[+ajCOSJt-\-\^^ibjsinjt], u t
= 'S[+CjC08Jt+d J sin jt],
w, = S [ djcos jt+ V^T 6 ; sin,/*], w4= 2 [
- Cjcos^+d; sin jt],
Vi= S [ + V^T ^ cos;'* + & sin #] , B, 2 [+ %cos i< + 5, sin jt] ,
u,= 2 [ + v^T a, cos jt A sin ji] ,
v t= 2 [ + y, cos jt 5; sin jt] ,
(137)
where the a,, b,, c Jy d,, a,, ft, y,, and 5, are real constants and powerseries in e.
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 269
In the case of the periodic orbits we have simply
,= <' [
r * =al -e- #y=B
ttJ , y^a \fv=li v l -e~' v=it
v^\. (138)
In the case of the periodic orbits it follows from equations (137) that the
numerical coefficients of the cosine terms in xx are real, and that those of
the sine terms are purely imaginary; and the opposite is true in y x .
138. Coefficients of X. It is found from equations (13) that these
coefficients are defined by
x" -2y' -[\+2A+Aecost + -fo -$Aemat+ -]y 2= X2 ,
y 2 +2x 2 -[ GAesint+ -]x 2 -[l-A- SAecost+ -\y 2= Y2 ,
(139)
Xt=[-3B-12Becost+ ]x]+[-24:Besmt+
+ [f + 6ecos+
Yt=[-12Besint+ ]x
2
+[3fl+125ecos+
+ [9flesin*+
r _ + 1 Mo + MoP + (ox zm'
-]zi*/i
-]yl,
-]yl,
s. r
(140)
The character of the solutions of the equations of the type to which
(139) belongs was determined in 30, where it was shown that they consist
of the complementary function plus terms of the same character as X2 and
F, . Hence the periodic solution of (139) is
x 2=
a{2) *=
1 +a sa>r"*= u 2 +f 2 ,
y t= a?
>
e,aV=li
P, +af -**=* v 2 +g 2 ,
(141)
where oJ2) and aj" are constants which are as yet undetermined, and where
/ 2 and g t are the particular solutions.
We shall need certain properties of / 2 and g 2 . It is evident from (139)
and (140) that they are homogeneous of the second degree in a{ and in
e *v=n an( j e -#v=* it follows from (137) and (138) that in x\ and y\ the
coefficients of those cosine terms which are multiplied by e 2aV=il and e~ 2 " v=Jt
are real and identical, while the coefficients of the sine terms are purely
imaginary and differ only in sign. The terms which are independent of
e <rv=rt an( j jr*v=t con si st only of cosines whose coefficients are real. In the
product Xj y x the coefficients of those cosine terms which are multiplied bye 2<rv=ii are purely imaginary and differ from the coefficients of those cosine
terms which are multiplied by e~ aay/=lt only in sign, while the coefficients of
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270 PERIODIC ORBITS.
the sine terms are real and identical. The terms which are independent of
e <rv=7, anc j e -<rv/=r< are on iy g j ne terms and are real. Therefore it follows
from (140) that X2 and Yt have the form
x t= x^'^' +x;-
2> e-^' + xr,
Yt= fjv**^ + f<-
!)e- 2*^' + r;
o>.
In X and X l
t
~ v the coefficients of the cosine terms are real and identical,
and the coefficients of the sine terms are purely imaginary and differ onlyin sign. In XJ
0)there are only cosine terms and their coefficients are real.
In Fj2) and F,~
2)the coefficients of the cosine terms are purely imaginary
and differ only in sign, and the coefficients of the sine terms are real andidentical. In Fj
0)there are only sine terms and the coefficients are real.
It follows from the properties of X2 and F 2 which have just been derived,and from the form of equations (139), that/ 2 and g t have the form
g,=
)2
<tf' #"=* +)V-; 2) r"*= + )2C (142)
where ff, /,~2)
, /, g, g(
t
~ 2)
, and g are periodic with the period 2r.
It follows from the properties of X2 and Yt which have been found that
equations (139) are not changed if in them the sign of V l is changed.Therefore this property is true of the particular solutions, and we have
/i2, (-^T) =/<-
2
'(v^T), /<0,
(- V^T) -/T(V=D,
It also follows from the properties of X, and F 2 that if we change the
sign of t and y 2 , equations (139) are not altered. Therefore
/,(- = +ft* (o, /.(-<) - +f?\t),
9?(-t) = -g (
t- i)
(t), g?(-t) = -g?(t).
It can be proved similarly, from a consideration of equations (139), that
ST (-V=T,-*) = +/f (V=T,0, g? (-V=i,-t) = -g? fy=T,0,
/i"2,
(- v^T, -0- +/i-2)
(v^T, 0, ^-2)
(-v^T, -0 - - *(V=I, 0,
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 271
It follows from these three sets of relations that when f, . . ., g
are written as Fourier series they have the form
ff =2[ajCOS.#+A/^r&,sin7<], gf =2[+v^a,cos.#+ftsin#],
/<-" = S [aj cos jt- \T=T bjsinjt], gf
2) = 2 [- v^T ajcosjt+^sinjt],
f = 2 c,cosj7, gf =2y>smjt,
(143)
where the a, ,b Jt c Jt a t , ft , and 7, are real constants and power ser ies in e.
It is seen from equations (142) and (143) that g 2 (0)= 0. Therefore,
since y t {0)=0, we have a= af, and equations (141) become
(144)
where both a"' and a}2)
are so far undetermined constants.
139. Coefficients of XVl. It is found from equations (12) and (13) that
these terms are defined by
x"-2y' - [l+2A+6Aecost+ ] x 3 - [64esin*+ ] y 3 = X3 ,
y i
y+2x' -
[64esin* + ]x 3- [l-A-SAe cost+ ] y 3
= Y3 ;
X3 =+[-2K-6Kecost+ -]x,+ [-6Ke sint+ -]y,
+ [-65-245ecos<+ ]z 1x 2 +[-24flesinH ^{x^+x^y,)
+ [3fl+12fle cos* + ]2/ 1 2/ 2 +[4C+ ].r?+[-6C+ -Jrtfjjl (
Y3 =+[-6Kesmt+ ]x 1 + [K+3Ke cost+ -]y,
+ [-24Besmt+ ~]x l x 2 +[SB+12Be cost+ ^(x^+x^,)
+ [l8flesin*+-- ]fcfc+ [- 6C+ ]xly 1 +['3C+ -]y\;
jyr fl1 Mo I Mo
tV (0>8 (0)3' ^ (0)6 T (0)5
'
'l '1 '1 's
It follows from the results of 29 that in general the terms of the first
degree in e"*^ 1 ' and e~" vz:T ' will introduce non-periodic terms into the solution.
We must determine a"1
,if possible, so as to make their coefficient vanish.
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272 PERIODIC ORBITS.
The general solution of the first two equations of (145), when X, and F 3
are zero, is
" j
v x +a? e\+a
ept
Ui+ae~
p'u t ,
v 2 +a?ep'v 3 +a?e-
f"v
t ,
(146)
where a {
'\ . . . , af are arbitrary constants. Now, on supposing they are
variables and subjecting them to the conditions that equations (142) shall
be satisfied when their right members are included, we get
e'^' u 1 (a?y+e-a
^\(a?y+ep'u 3 (a?y+e-
p'u i (a?y= 0,
f[W=r ,+;]
(a?y+e"xv = TI
[-<rv=Tui +u' ] (dry.
+e"0 3+<] (a?y+e-p
'[- pu 4 +<j )'= X3 ,
**=* 1 (a 1
w)'+-rV=I '
v,{ay+<rv,{ay+e-',
v l (fi?y =0,
,*= [rV=itk-H<] (afO'+e"'^' [-* </=Tvf <] (,*)'
+e" [pk + (af )'+e-p '
[- p 4 + tQ ()' - F 3 ,
where (aj3)
)'> . . ., (af)' are the derivatives of af, . . .
,a with respect tot.
The solutions of these equations for (a{3)
)', . ., (a 4
3>
)' are
(147)
A(a?)' =[A. X.+A, F 3 ]
<f"^
, AW)' =[/>
X3 +Z> 3 , F 3 ]"*
,
A(ar)' = [A 1 X3 +DM Fje*^ , ACO'^Z^X.+A, F 3 ]e+P
',
where
(148)
A =
if.
*>--Mj w, i
A,= +Uj , u, ,
w4
^2 f *>3 f 4
oV^T W, + Wj, pMj + Ug, -pW 4 + tt 4
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 273
Dn and Dn are obtained from Da and D12 , respectively, by changing the sub-
script 2 to 1 and by changing the sign of V^T and of the whole expression ;
ZX. and D3i are obtained from Dn and Da , respectively, by changing the
subscript 3 to 1 , p to a V 1, and by changing the sign of the whole expres-sion; and D41 and D42 are obtained from Du and Du , respectively, by changingthe subscript 4 to 1, p to <rV l, and by changing the sign of the whole
expression. It follows from the discussion of 18 that A is a constant, andin this case it is a power series in e.
In order that the solution shall be periodic it is necessary that the
right members of (148) shall contain no constant terms. We shall show these
conditions are sufficient. When they are satisfied the general term of the
right member of either of the first two equations has the form
[a lk cos jt+b lk sin jt] e uaVZ~u ,
where j and k are integers distinct from zero and where a Jik and b jk are con-
stants. Consequently aj3) and a are sums of terms of the type
a,j, k (2ka V- 1 cosjt+jsinjt) + bj, k ( -jcosjt+2 ka \ r ^\ sinjTJ ,_ 4p - (149)
The right member of the third equation of (148) never has any terms
which are independent of t, but contains terms of the type
[a Jtt cos jt+b Jjk sh\jf\
e[rat+ixrV- 1 -p]<
where j and k arc integers. There can be no exception to this form. There-
fore of is a sum of terms of the type
fp -i "I Wk+l)<rV=Tt -pi
,, t
|L(2*+lW-- 1 -pJ eoBJt+jmajt ^ +[i l h+ 1)y j ^^
I"|-
-, "1 (2*+l)<rv/ =T( -pt
^cos^-[(2/c
+l),V^-p]sin,7| ?+[(2
^ +i)gv/ ^ T _pr
(150)
The type terms for a" differ from those for a," only in the sign of p. There
is an additive constant of integration with each of the af\ It follows, from
the form of (146), (149), and (150), that if we put the constants of integra-
tion associated with a"' and af}
equal to zero, the resulting expressions for
Xj and y 3 are periodic with the period T. They may be written in the form
Xt -a?f=*u l +a*, e-= It v t +f; y*
= a?)
e''^'v1 +a?
)
e-<r^%+g i , (151)
whereaf
and o are constants which so far are undetermined.
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274 PERIODIC ORBITS.
It remains to show that a[" can be so determined that the right membersof the first two equations of (148) shall have no constant terms. Let us
consider the first of these equations. We are to set equal to zero the con-
stant part of the coefficient of e"^ 1 ' in DnX3 +D l2 Ys . It follows, from the
form of X3 and Y3 , equations (145), that the term which must be made to
vanish does not depend on af. It also follows that the conditional equa-tion which must be imposed has the form
W-Q.W-O, (152)
where P, and Q^ are power series in e, the former coming from those terms
of X, and Y3 which are linear in x x and y x and independent of x 2 and y 2 ,
and the latter coming from those terms which are of the third degree in
Xj and y l ,or which involve x 2 and y 2 .
Thesolutions of
x = y = 0, and(152)
are0^
=0,
which leads us tothe
trivialresult
(153).
The significance of the double sign was discussed in 116-118 in connection
with the existence of the solutions. The expressions for P l and Q1 are
power series in e and both of them contain terms independent of e, as wasshown in Chapter VI in the discussion of the corresponding problem for
e = 0. Therefore a[l)
is a power series in e having an absolute term.
It remains to be shown that this value of a{" also satisfies the equation
which is obtained when the constant term of the right member of the secondequation of (148) is set equal to zero. We shall show that the constant
part of the coefficient of e aV^ Tl in Dn Xi -\-D n Y3 is identical with the con-
stant part of the coefficient of e~ a,yzr:i in D2l X3 +D 22 F 3 . Let us first consider
the term [2K6Kecosl-{- ]x 1 oi X3 which contributes to P, of equa-tion (152). So far as this term is concerned, we have
A )'= Dn [
- 2K - 6Kecost+
A(a(
2
3)
)'= D21 [-2K-6Kecost+
}x l e-' rV =' u,
-]x 1 e + ' v ^ ri.
(154)
On referring to (138) and the values of Dn and D2l ,we see that the constant
parts of the right members of these two equations are respectively the
constant parts of
-aw, u t
oV=\v 2 +v' , pv.+v' , -pv t +v[
[-2K-QKecost-\ ],,
-a (i)
u, W
+(^f=lv i
+v[, pt>,+t>3, -pvt +v'
[-2K-6Kecost-] ]u 2 .
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p n in
OSCILLATING SATELLITES, ELLIPTICAL CASE.
These expressions are equivalent to
f^U, ,U3 Ut , 11, + Ui
u t v 2 , v 3 -i\ , v 3 +v i
W^T l vMM>i, pfa+pJ--H*-i p{v 3 -v<)+v' +v[
u^ , u 3 u t ,w3 +w 4
u 2 v t ,v 3 -v t ,
v 3 +v t
+<rV=Tu 2 v l +u 2 v' 1 , p(v 3 +v A)+v' -v' 4 , p(v 3 -v t )+v' +v'
275
_aj_
F 1 (t),
Fi(t),
where
F(t) = [-2K-6Kecost+ ]
Theparts
of theseexpressions containing
uyu
2as a factor are identical
and need no further consideration. The parts multiplied by x^v 2 and u 2 v t , so
far as they appear in the second lines of the determinants, are respectively
,d)
+ --u 2 v l
u 3 -u t , u 3 +u t
p(v 3 +v t )+v' -v' t , P {v 3 -v i )-]rv 3 +v'
u 3 -u t ,u 3 +u t
p(v t +v 4 )+v' -v' t , p(w, -)+!+!
[-2K-6Kecost-\ ],
[-2K-6KccoaW ].
It follows from (137) that u 3 u t is a sum of cosine terms, that u 3 -\-u t is a
sum of sine terms, that v 3 +i\ is a sum of cosine terms, that v' 3 v\ is a sumof cosine terms, that v 3
vi is a sum of sine terms, and that v' 3 -\-v[ is a sum
of sine terms. Therefore the determinant parts of these two expressions
are sums of sine terms, which, multiplied by a cosine series on the right,
are sums of sine terms. Hence, to get the constant parts of these expres-
sions, we need only the sine terms of the products utv 2 and u t v l . It is seen
at once from (137) that the sine terms of these products are identical, but
that the cosine terms differ in sign. Therefore the constant terms comingfrom the parts of the two expressions which are multiplied by u v
v 2 and u 2 v lf
so far as they come from the second lines of the determinants, are identical.
The parts of the expressions which contain u^v 2 and u^, so far as theycome from the third lines of the determinants, are respectively
a+ -fc*y/~=lu lv t
--\
<s s/~=\u 2 v x
U3 u i ,
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276 PERIODIC ORBITS.
they are identical Therefore the constant parts of the two expressions, so
far as they arise in this manner, are identical.
It remains to consider only the constant parts of the two functions
[-2K-6Kecost+ I
in W, W4 , W3+ W4
V,-
*>4 , ^3 + ^4[-2K-Keco&t+
It follows, as before, that we need only the cosine terms of u y v 2 and v^v[ .
We see from (137) that the coefficients of the cosine terms in these productsare identical. Therefore the constant parts of the right members of equa-tions (154) are identical.
The discussion for the other terms of X, and Y3 which are linear in xl
and y l is made in a similar manner, and it is thus proved that the P, which is
obtained from the second equation of (148) is identical with the one which
depends on the first.
It is now necessary to consider those terms of X, and Y3 which are not
linear in x t and y x. Let us treat in detail the term in X3 which contains
XjX 2 as a factor. So far as this term is concerned, the first two equationsof (148) become
A(af)' = Ai[-6P-24Becos<+
A(af)'=Ai[-6P-24PecosJ+ J x xx 2f!l+oV=it
(155)
On referring to equations (138), (144), and the expressions for Dn and Dsl ,it
is seen that the constant parts of the right members of these equations are
respectively identical with the constant parts of
-Crt a) V(a?)*
u, u
aV^\v 2 +v 2 , pv 3 +v' 3) -p+4[-6B-24Becost+ ][i/,
a9-w,/,
e)],
-(a?) v,
u A
a
+ <rV^Tv l +v[, pV 3 +v' , -pVt+v'i
[-6P-24fiecos*+ ][,/,"> -uJJ-*]
Since/ is a cosine series, and the product of it and [ 6P 24Pecos+ ]
is also a cosine series, the discussion for the terms multiplied by w, /j0) and
Uj /J0) does not differ from that
givenabove for the terms
multiplied byx x
.
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*
OSCILLATING SATELLITES, ELLIPTICAL CASE.
We have now to find the constant parts of
277
2
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27S PERIODIC ORBITS.
Let us consider, for example, the term of X3 which is multiplied by xx y\.
Then, so far as this term alone is concerned, the first two equations of
(148) become
A(a)'-D u [-6C+ -U^e-^AA(aiT = A,[-6C+ J JfcfB ?"=.-}{ V
The constant parts of the right members of these equations are respectivelythe constant parts of
+W)
+W)
u, , u 3 , u t
-ff-s/^Tvj+tfj, pv 3 +v' , -pv t -\-v'
[-6C+--'][2u 1 v l v,+u t v[],
[-6C+---][2iw,+u 1 ifl.
(158)
+ ffV^Tl) 1 + 1', Pt>j + t>s, -P^ + *>4
Since t^f, is a cosine series having real coefficients, the discussion for the
the terms multiplied by 2m 1 ?;,i> 2 and 2u 2 v lv 2 does not differ from that given
above for that part of X, which is multiplied simply by x x.
If we refer to equations (137) and (138), we see that v\ and v\ have the
properties of /" and ff%
\ as regards the relations existing between their
respective coefficients. Therefore the discussion of these terms of (158) is
identical with that of (156), for which the proposition was established.
In a manner similar to this the identity of the constant parts of the
right members of the first two equations of (148) can be established for all
of the elements of which X3 and Y3 are composed.
140. General Proof that the Constant Parts of the Right Members of
the First two Equations of (148) are Identical. We shall treat first the
parts which depend on X3 . We shall need the following properties of Xs :
(1)
(2)
(3)
(4)
It is a polynomial in x 1} y x ,x t , y 2 .
Those terms which are of even degree in y x and y 2 taken togetherare multiplied by cosine series having real coefficients.
Those terms which are of odd degree in y r and y 2 taken together aremultiplied by sine series having real coefficients.
If the general term is x[l
x'i y\> y 2
-
,then ^1+2^ + ^+2^ is an odd
integer (in the present case one or three).
The parts of the first two equations of (148) which depend on A' 3 are
A(a?)' = Dn X3 e- A(ary = Dn X3 e'^'. (159)
It is obvious from (137) and properties (2) and (3) that those parts of
X3 e~ av ^ JI and X3 e"^^' which are independent of the exponentials e~ aVZrJt
and e" s ~' are sums of cosines having real coefficients and of sines having
purely imaginary coefficients, and that the real coefficients in the two ex-
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 279
pressions differ respectively only in sign, while the imaginary coefficients are
respectively identical. Hence, referring to the expressions for Dn and Dn ,
we may write these parts of equations (159) in the form
ilj ,u 3 -u x ,
u 3 +u t
,v 3 -v t ,
v 3 +v,
-oV^\~v 2 +v 2 , pfo+0j-H-J, p(t>,-0+-H>4
Wj ,u 3 -u 4 , u 3 +u t
h , v 3 -v t ,v 3 +v t
2 [A j cos jt
+ V^i B j sin jt],
2[A j cos jt
l Bj sin jt].
It easily follows from these expressions, as in the discussion in 139, that their
constant parts are real and identical.
Now consider the terms depending on Y3 ,which has the properties
(1) and (4) belonging to X3 ,and
(2) Those terms which are of even degree in y x and y 2 taken together are
multiplied by a sine series having real coefficients.
(3) Those terms which are of odd degree in y l and y 2 taken together are
multiplied by a cosine series having real coefficients.
The parts of the first two equations of (148) which depend on Y3 are
A(a?y = D12 Y3 e-<rv^T< A(a?y =DY2 e (160)
It follows from (137) and properties (2) and (3) that those parts of Y3 er a ^ rli
and F, e <rVTrT' which are independent of the exponentials e-"^ =li and e ffvCrT '
are sums of cosine terms having purely imaginary coefficients, and of sine
terms having real coefficients, and that the purely imaginary coefficients are
respectively identical while the real coefficients differ respectively only in
sign. Hence, using the explicit values of Dl2 and DM ,these parts of the
right members of (160) are found to have the form
I/, ,u 3 -u, ,
u 3 +u t
v 2 , v 3 -v, , v,+ v t
- <tV^Tu 2 +u 2 , p(u i +u i )+u 3 u+, p{u 3 u^+u' +u[
W, ,W3 -W 4 ,
U3 + U,
1 , 0.3-^4 > 03+04
+ (rv / ^Tw 1 +^, p(w,+tt 4 )+Ws-wI> P (m 3 -W 4 )+W3+M 4
2 [\/~-iA, cos jt
+B, sin jt],
2 [V^TAj cos jt
-Bj sin jt].
It follows from (137) that the constant parts of these two expressions are
real and identical. Therefore the constant parts of the right members of the
first two equations of (148) are identical, and when one of them is made to vanish
by a special determination of a[l)
the other one also vanishes.
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I'M) PERIODIC ORBITS.
141. Form of the Periodic Solution of the Coefficients of X' A . It follows
from the form of X3 and Y3 , given in equations (145), that/, and g 3 of equa-tion (151) have the form
/,-aJ 8 /"^ ~ l +a?f - i) e-^^'+a/+/f <?
lav^n/ (1) .,<7v/=7+/r I +/;(-1)
~"+tt<-3)
/> -30\'=7<
fc= < ft
08 e^'+af </<- e-^+a ^+g? (T^'+g? e^<+g (
r e-'^'+g^ e~
(161)
where / 3
2)
,. . .
, <73
_s>are known functions of Z . We need certain properties
of these functions. It follows from the properties of X3 and Y3 and of the
left members of the differential equations, and from certain considerations
of changes of sign of v^i , t, and y 3 ,in both the differential equations and
the solutions, that
g?(-V=J)
= /^'(V^T),/!<-**<y=T)i
/r, (-o=+/r ,
w,
1,-0-+/,' (V=ri,0,
(</=T,<) (j= 0, 1, 2, 3, -1, -2, -3).
It follows from these relations that the /"' and the g^ have the form
/<3)
=-2[a? cosjt+V^I b? sin jt], g? -2{+y=Taf cosjt+pf sinjt],}
gl- = 2 [-V^Taf cosjt+(3?
sinjt],
gf =2[+V=iaf coBJt+$f sinjt],
g<- .2 [- V^Ta* cosj<+,8? sin#],,-
T<cos.;7+,3<I,
sin#],
Ta"' cos jt+Pf sinjt],
gT =2/3;n)
sin?7,
(162)
/<-" = 2[<
cos./*- v^T 6 sinjt] ,
/ = 2 [a* cosjt+V^ib sinjt],
/<-2) =2 [a? cosjt-V^lbf sinjt],
/," =2[<cos^+v^T6;,,
sin#],
/ J- = 2 [a;1 '
cos/*- V~^\bf sinjt] ,
/f =2 a? cos jt,
where the af,..., $f are real constants.
It follows from equations (161) and (162) that g 3 (0)=0. Therefore,
since t/ 3
(0) =0,we have
af=
af,and
equations(151) become
9 3
"=2[+
x 3= a [e*^ 1
m,- e~" v
~~" uA + afW e 2 ^-'+/<-
2) e-*^'- r -/f ]
+/V^'-'+/<-s)
e-3) c -3ff>
"-+/,"<1) o-, "+/^ 1)
e-^-",
, i= a'
1
3l
[e^-"v l-e- -'iu] + a?[g?e '+g<- e-*
+g?e*'^'+g'- 3)
e- +gle^-+g
711+g?\
(-D -ff\-i<
(163)
142. Coefficients of X2. It is seen from (13) that the coefficients of
X2 are defined by the differential equations
x"-2y' -[] + 2A+()Aecost + &-[6Aesin*+ -k = A'4 ,
1
y"+2x[-[6Aesint+ ] xt -[l-A -3Aecost+ -]y<=Y t ,
(164)
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 281
where
Xt= +[-2K-6Kecost+ ~\x 2 +[-6Ke sint+
+ [-3B-12Becost+ ][xl+2x 1 x i ]
+ [-24J5esin+ ] [x 2 y 2 +x l ij 3 +x 3 y 1~\
]Vt
+ [#+6ecos*+Yi =+[-GKesint+
+ [-12Besmt+
+[3B+2Becost+
][yl+2y l y 3 ]+X i!
~]x 2 +[K+3Kecost+
][aZ+2x 1 x t~\
][x 2 y 2 +x l y 3 +x 3 y l']
]Vi
(165)
+ [9Besint+ ][yi
2 +2y l y 3 ] + Yif
where X4 and F 4 are independent of x 3 and y 3 and linear in x 2 and y 2 . In
X, the terms which are of even degree in y x and y 2 are multiplied by cosine
series having real coefficients, while those which are of odd degree in y x and
y 2 are multiplied by sine series having real coefficients. In the case of Yt
the cosine series and sine series are interchanged. If x\ x 2 x 3 y\l
y"%
yk
3
'
is the general term in Xt or F 4 ,then i 1 +2 i / 2 + 3j 3 + ft
1 +2A- 2 +3fc 3= 4 or 2.
When the right members of (164) are set equal to zero, the general solution
of the equations is
x i= a^
)
e aV ^ l 'u l +a^)
e -"w 2 +c44Vu 3 +< )
e- p 'w 4 ,
y 4= < e'^ 1 '
Vt+a? e- v =~>' v 2 +a? ef v 3 +a? e'"1
v x ,
(166)
where a[*}
,. . .
, af are arbitrary constants. Now, on varying them and
subjecting them to the conditions that (164), including the right members,shall be satisfied, we find
A )'= [A> Xt +D a Yt ] e~"=, A (aJ)' = [Ai Xt +D a YA e-,
*(a!*y = [D n Xl +Dn Yt ]g* A(a?y = [D 4l Xi +D i2 Yi ]e+>,(167)
where Dn ,. . .
,Z> 42 are the same as in 139.
Necessary conditions that the solution shall be periodic at this step are
that the constant terms in the right members of the first two equations of
(167) shall be zero. It follows from the form of Xi and Y4 ,as given in (165),
and from their properties, that these constant terms are independent of
a'i' and involve a linearly. Therefore the condition that the constant
term of the right member of the first equation shall be zero has the form
P 2 <+& = 0, (168)
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282 PERIODIC ORBITS.
where P, and Q2 are power ser ies in e. It was shown in Chapter VI, in the
treatment of the case where e = 0, that P 2 has a term independent of e whichis distinct from zero. Therefore for |e| sufficiently small af is uniquely
determined by (168) as a power series in e.
The equation obtained by setting the constant part of the right memberof the second equation of (167) equal to zero is of the same form as (168);it is, in fact, identical with (168), as will now be shown. It follows fromthe properties of XA and Yt that the parts of the right members of the
first two equations of (167) which are independent of the exponentialsc -,v=Ti an j e ov=ii have the form
+Ai 2 [Aj cosjt+V^TBj sin j^+D^ [+V^T C} cos jt+Dj sin jt],
-D tl 2 [A } cos jt-V l B, sinj^-D^ [- v^T C, cosjt+Dj sinjt].(169)
These equations are of exactly the same form as those encountered in 140
in the preceding step of the integration, and the conclusion follows in the
same manner. Consequently if af is determined so as to satisfy (168), andif the additive constants arising with the integrals of the last two equationsof (167) are taken equal to zero, then the solutions of (164) are periodic. It
follows from the properties of Xt and F 4 that they have the form
x i= a i * )
[efxV - 1 'u 1 -e-
y^d^le'^'Vi =it _ ,
(170)
where the f\" , g\J>
, f\" , and g\S) have properties exactly analogous to those
of equations (162).
143. Induction to the General Term of the Solution. We shall supposethe Xj ,
. . ., x-i) y t ,
. . ., y n -\ have been computed and that their
coefficients have all been determined except a{"-2) and of -0
,which enter in
z2 , Vn-2 , Zt , and ?/_, in the form
x n _ 2 =+a?-t)
[ev =~ 'u 1 -e-<'
v =~uUi l ]+ ,
y n - 2= +<- 2V v= I( v l -e-'^ =il v 2 ]+ ,
+a?-i)
[f?- i ^ v=n +7^? e- t ^'+f (
:L 1 ) +
+r,
[^i 1 *r
^+fc? e--^'+^l,] +
(171)
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OSCILLATING SATELLITES, ELLIPTICAL CASE. 283
We shall suppose that x v and y (p =1, . . .
,n 1) have the properties
-2 / y,= 2j/:
( -*> J<rVil
/;"= 2[a^ )
cosW+V^T6?-')
sin<*] (jVO),
/J-^SK'" cos^-V^Tft?" sin?*] (jVO),
pj"= 2[+^3Ta?-'
)
cos^+^'sin^] (j^O),
g(
fJ) = 2[-%A=7 a-" cosrt+jS*" sin?*] (jVO),
/i0) =2 a^
0)
cos?*, d"-2 #* on**.
(172)
The differential equations which define x n and ?/ are seen from (13) and
(14) to be
(173)
(174)
xl-2y' -[l+2A+6Aecoat+ -]x n -[6Aesmt+ -]fe-Xj
y:+2x:-[6Aesin<+ J* +[l-A-34ecos+ -]y n =Y n ,
where
X.= +[-2.K-6Ji:ecosH- ] x n_z+[-Kemit+ -}y n _ 2
+ [-3 JB-12ecosJ+ ][2x l x a ^ 1 +2x l x.. a ]
+ [-242tesinH- ][x l y- 1 +x 2 y n - 2 +x n - 2 y t +x n - 1 y 1 ]
+ [3 + 12ecos*+ '][*V-t+1h1l-d+Xn
Y=+[-QKesint+ ] x n _ 2 +[K+3Kecost+ ] y n _ 2
+ [-12Besin*+ ] [2x t x. l +2x t x- 2 ]
-f-[35+2ecosJ+ ][x l y- l +x 2 y a -2+x tt - 2 y 2 +x n - 1 y 1 ]
+[9flesin*+ ][2y l y m. l +2y,y n - a ]+T t .
The functions Xj, and Yn do not involve x n -i or y n ^ , and are linear in _
and y n - 2 . In Xn the terms which are of even degree in ft , ... , t/_ 2
are multiplied by cosine series having real coefficients, while those which are
of odd degree in y x ,. . .
, y- 2 are multiplied by sine series having real
coefficients. In the case of Y the cosine series and sine series are inter-
changed. If x'i a;i"__i t/J' yt-'i is the general term of Xn or Yn ,
then
h+2j t + +(*-l)i._ 1+M-2+ +(n-l)^_ 1= n or n-2. (175)
Necessary conditions that the solutions of (173) shall be periodic are
that the right members of
^aT)' = [D n Xa +D ii Yn \e- *{aly~[D n Xu +Dn Yu \ *<=* (176)
shall contain no constant terms. It follows from (174) that these constant
terms are independent of a~ and involve ajn_2>
linearly. The coefficient
of a'"~ v is distinct from zero for \e\ sufficiently small, for in Chapter VI it
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284 PERIODIC ORBITS.
was seen to be distinct from zero for e equal to zero. Therefore a{"~2)
is
uniquely determined as a power series in e by setting the constant term of
the right member of the first equation of (176) equal to zero.
It can be shown, precisely as in the discussion when w=
4, that theconstant parts of the right members of equations (176) are identical. There-fore a ( "~ a
is uniquely determined by the conditions that the solutions
of (173) shall be periodic. It follows from the properties of x iy . . ., _,;
y, ,. . .
, y- u and from (175), that when these conditions are satisfied the
solution of (173) has the form
x n = af e*^' wx+< e-"^ 1 ' ^+a<B)
e" l +a 4w e"" u,
yn =
<)
(f
+aJM, [ff> %e
+-
l(+/<-
e -*rv^ +/<0>
]+ S f e >*V=i
v2 +ae>"
v3 +a?
e~
++<-" [ gf f^'+g*-* e-*=-+g?] + 2 g e"^' ,
(177)
where a[n)
,.
(n-i) are undetermined, and where the /, /^_2)
, /(0)
n ;
gf , ft-*, gT,fn\ and rf have the properties of (172).
In order that (177) shall be periodic it is necessary and sufficient to
impose the conditions a = a=0. Then it follows, from the propertiesof v lt v 2 , g?, glr", g? that y n (0)-a?-a? = 0.
that a =aj
n)
,and equations (177) become
Since y (0) =0, it follows
x n = a [e" v=Xl
1*1 - e~' v=l ' u 2 ]
+ a (-.)[f
V ^v^I, +/ (- 2)g-Sffv^Ti
+a?-l)
[g? <? aV=T '+? n-* e~*^
+- u +g?\ + 2 g e"^-
J=-n
(178)
and equations (172) are satisfied for p = n.
In picking out the constant part of the right member of the first equationof (176), in general only those terms in Xn and Yn which contain e rV=0
as a factor to the first degree will be used. But because a is a rational
number there will eventually occur, in the higher powers of the exponentials,
multiples of <r which are integers, and constant terms in the right memberof the first of (176) may occur from these terms, but their presence does
not prevent the determination of the constants so that the solutions shall be
periodic. After such terms once appear, they occur in general at each
succeeding step of the integration.
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Chapter VIII.
THE STRAIGHT-LINE SOLUTIONS OF THE PROBLEMOF fl BODIES.
144. Statement of Problem. In his prize memoir* on the problem of
three bodies, Lagrange showed that, for any three finite masses mutuallyattracting one another according to the Newtonian law, there are four
distinct configurations such that, under proper initial projections, the ratios
of the mutual distances remain constant. The bodies describe similar conicsections with respect to the center of mass of the system, the simplest case
being that in which the orbits are circular. In three of the four solutions
the masses lie always in a straight line, and in the fourth they remain at the
vertices of an equilateral triangle. This memoir is one of the most elegantwritten by Lagrange, and its mathematical form does not seem capable of
improvement. But the method which he employed can not be extended
conveniently to the case of more than three bodies, and it has not led to
practical results in applied celestial mechanics.
This chapter is devoted to the solution of two closely related problems :
I. The number of straight-line solutions is found for n arbitrary positivemasses; that is, the ratios of the distances are determined so that under
proper initial projections the bodies will always remain collinear. This is
the generalization of Lagrange's straight-line solutions to the problem of nbodies. For each straight-line solution of n finite masses there arc n-fl
points of libration near which there are oscillating satellite orbits of the
types treated in Chapters V -VII. Therefore the results of this chapterlead to generalizations of those of the preceding three chapters.
II. The problem is solved of determining, when possible, n masses such
that, if they are placed at n arbitrary collinear points, they will, under proper
initial projection, always remain in a straight line.
The first problem, in a somewhat different form, has been considered
by Lehmann-Filhes.f The method of treatment adopted here J, though
originally developed independently, has considerable in common with that
of Lehmann-Filhes, and the discussion completes in certain essential respects
the demonstration of the German astronomer. It is shown that whatever
real positive values the masses may have, and whatever the rate of their
revolution, there are %n\ real collinear solutions.
Lagrange's Collected Works, vol. VI, pp. 229-324. Tisserand's Mecanique Celeste, vol. I, chap. 8.
\Aslronomische Nachrichten, vol. CXXVII (1891), No. 3033.
{Read before the Chicago Section of the American Mathematical Society, December 28, 1900; abstract
in Bull. Am. Math. Hoc, vol. VII (1900-1901), p. 249. 285
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286 PERIODIC ORBITS.
In the second problem it is proved that when the number of arbitrarily
chosen real collinear points is even, the n masses are, in general, uniquelydetermined by the condition that it shall be possible to place them at these
pointsand to
givethem initial projections so that
theywill
alwaysremain
collinear and revolve in orbits of specified linear dimensions. Or, if it is
preferred, the period of revolution can be taken as the arbitrary in place
of the linear dimensions of the orbit. In general, the masses will not all be
positive, and therefore the problem will not always have a physical inter-
pretation. When the number of points is odd, it is not possible to determine
the masses so as to satisfy the solution conditions unless the coordinates of
the points themselves satisfy one algebraic equation. When they satisfy
this condition, any one of the masses may be chosen arbitrarily, after which
all of the others are, in general, uniquely determined.
I. DETERMINATION OF THE POSITIONS WHEN THEMASSES ARE GIVEN.
145. The Equations Defining the Solutions. Let the origin of coordi-
nates be taken at the center of gravity of the system, which will be supposedto be at rest. This point and the line of initial projection of any massdetermine a plane. All the other masses must be projected in this plane,
for otherwise they would not be collinear at the end of the first element of
time. All the bodies being initially in a line and projected in the same
plane, they will always remain in this plane. Consequently, if solutions
exist in which the n masses are always in a straight line, the orbits are
plane curves.
Let the plane of the motion be the i^-plane. Let the masses be de-
noted by m1( mj, . . . ,m K , and their respective coordinates by (,, 77,),
(&> V2), > (, Vn)- Then, choosing the units so that the Gaussian
constant is unity, the differential equations of motion are
(Ph 1 dUdt
2 mt d&'
n n
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STRAIGHT-LINE SOLUTIONS FOR W BODIES. 287
In case the n bodies remain collinear, the line of the resultant
acceleration to which each one is subject always passes through the origin.
Therefore, in collinear solutions it follows from the law of areas that, for
each body separately,d6
miT*~dl
=Ci ( *" 1 ' '.*)
But when the bodies remain collinear we have also
dd 1 = d61= = djh.dt dt
'
dt'
from which it follows that
i Kc,(3)
j \m t Cj'
where the a tJ are constants. That is, if any collinear solutions exist, the
ratios of the distances of the bodies from the origin are constants, and it
easily follows from this that the ratios of their mutual distances are also
constants. They are therefore of the Lagrangian type.If the n masses remain collinear, the ratios of their distances from the
origin being therefore constants, the ratios of their coordinates are constants.
Therefore in all collinear solutions
<= Z,, Vi
= X,v (i=l,...,n), (4)
where the x, are constants. Upon substituting in equations (1), we have,as necessary conditions for the existence of the collinear solutions,
dt2 t
lx t [(x i -x J )T' r
3 (J *' l *>
de % xiLix,- xj)T' r"
r " v * + '
(5)
In order that $ and rj as defined by their initial values and equations
(5) shall be the same for all values of i, the coefficients of /r* and rj/r3 must
be set equal to a constant independent of i. Letting -co 2 represent this
constant and rij=
-\/(x i Xj)i
, these conditions, which are sufficient as well
as necessary for the existence of the collinear solutions, become
-a run p-
, mi (x xi)|nh(x n Xi) . m3 (x n x 3 ) n _. x_
i 3~-\
^3
--1
zr~"r -r u idj,
'nl ' m ' Hi
(6)
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288 PERIODIC ORBITS.
Suppose the notation is so chosen that in any solution z, < < x;
then the terms of the left member of the last equation are all positive.
Since the origin is at the center of gravity, x is positive, and therefore or is
positive in all real solutions. For every set of values of a;, , . . . , x n
satisfying equations (6) the solutions of (5) are the same for all values of t,
and these solutions substituted in (4) give the coordinates in the collinear
configurations.
Since equations (5) have the same form as the differential equations in
the two-body problem, it follows that in the collinear solutions the orbits are
always similar conic sections. In case the orbits are ellipses, the coefficient
of %/r3 and 77/r
3is the product of the cube of the major semi-axis of the
orbit and the square of the mean angular speed of revolution. If the
undetermined scale factor be so chosen that x t is the major semi-axis of the
orbit of m, , the mean angular velocity of revolution of the system is co.
The hypothesis is made that w2 and mx ,. . .
, mn are real positive
numbers, and the problem is to find the number of real solutions of (6) for
any value of n. For each of these solutions there is a six-fold infinity of
collinear configurations, the six arbitraries being the two which define the
plane of motion, the one which defines the orientation of the orbits in their
plane, the one which determines the epochs at which the bodies pass their
apses, the one which determines the scale of the system, and, finally, the
eccentricity of the orbits.
146. Outline of the Method of Solution. The method of solutioninvolves a mathematical induction and consists of the following steps:
Assumption (A). It is assumed that for n = v the number of real
solutions of (6) for x t , , . .,x v is Nv ,
whatever real positive values o>5
and m, ,. . .
,m may have. It is known from the work of Lagrange that
when v = 3 the number is AT, = 3 = \ 3 ! .
Theorem (B). If to the system mx ,
. . .,w of positive masses an
infinitesimal mass mv+l be added, then the whole number of real solutions
is (H-l)tf,.Theorem (C). As the infinitesimal mass ni r ^ increases continuously
to any finite positive value whatever, the total number of real solutionsremains precisely (v+l)N .
Conclusion (D). From successive applications of theorems (B) and
(C) it follows that the number of real solutions of (6) for n = v+n is
Nv+lx =(v+n){ v +n-\) ( v +2)(v+l)N v .
Since Ar
3= 3!
,it follows that A" 3+M
= | (/x+3)! . Let n+3 = n and we have
Nn =~nl. (7)
To complete the demonstration of this conclusion it remains only to
prove theorems (B) and (C).
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STKAIGHT-LINE SOLUTIONS FOR % BODIES. 289
147. Proof of Theorem {B).~ When there are v finite bodies ml} . .-. ,m,and the infinitesimal body m +l , equations (6) become
Vi =-tf Xi+ Q+ *byx*) + . . . + m,(x x -x,) + rn,+ L(x 1 -x ]!+J ) ^^ ^ X2 |
fniJXi-Xi}|_q_|_
. . ,i
my (Xit -a; y ) mH-ife-rc+i) _IS '2c '
2, f+1~3 T ~3'2w ' 9
(8)
,_
.,2 r |mi(XyXi) m^rc-rcg)
i
, n ,
/n y+1 '(a; y a; y +i) n#v w x-r -f- 3 t tui is- =u,
'lv 'SK 'iM'+l
, - -j* , wiiC^+i-xO.rwjC^+i-Xs), , w(x +1 -x,) , n n<Pv+i= w ^+i+ ,.3 .1 3~ -h -f -3 hU = U.
The last column of these equations is zero because w +1=
(is infini-
tesimal). Consequently the first v equations, which involve x 2 ,. . .
, xalone as unknowns, are the equations defining the solutions when n = v.
By (A), it is assumed that there are precisely Ny real solutions of these
equations. Let any one of these solutions be x 1= xf\ . . .
,x = x
Then the last equation of (8) becomes
,.2-v 1 7n\\X v +\ Xx )1
TThK Xy+iXi ) 1mv [x v +i x v ) n / n -,
Vi/+i=~~ u x v+i~r (o)3 I w)3
_-r 1 zws V. (V)
'1. r+l '
2. c+1 'v, J/+1
The number of real solutions of this equation is required.
Consider <p v+l as a function of x +1 . It is easily verified that
lim*v + i(+<)=-5, ""o^ +1 (*r+)=+*> y-i. .
lim*r+l(->) = +>, , = 0^-*)= -00 ti-1, ...,")
(10)
Since ^ +1 is finite and continuous except at x y+1= xf\ . . .
, xf,-\- 00, 00,
it follows that there is an odd number of real solutions in each of the
intervals 00 to xf, where x is the smallest xf, x to x (
<0)
, where
xf and x,<0>
are any two x'f which are adjacent, and xf to +00, where
re is the largest xj". But we find from (9) that
d<p+i _ 2 2m t 2/na 2?n y
a~ = CO (0)2 ZMi' ' '
r 'o>2 '
"Cy+l 'l, v+1 '2, c+1 'v, v+1
which is negative except at x +1=
Xj ,. . .
,x +1
= x,where it is infinite.
Therefore p +1 is a decreasing monotonic function in each of the intervals,
andconsequently
vanishes once, and but once, in each of them. Since
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290 PERIODIC ORBITS.
there are v+l of these intervals, there are, for each real solution of the
first v equations of (8), precisely v + 1 real solutions of the last equationof (8). Since the first v equations have, by hypothesis (A), Np real solu-
tions, equations (8) altogetherhave
precisely (v + l)N,real solutions.
This completes the demonstration of Theorem (B).
148. Proof of Theorem (C). Let x } =xf fj-1, . . .,
v + \) be anyone of the {v-\-l)N v real solutions of equations (8) which are known to exist
when w +l= 0. It will be shown that as m +1 increases continuously to any
finite positive quantity whatever, the xf can be made to change continu-
ously so as always to satisfy equations (8), and that they remain distinct,
finite, and real. From this it will follow that there are at least (v + l)Nreal solutions of (8) for every set of finite positive values of m, ,
. . .,mv+l .
It will also be shown that no new solutions can appear as m, +1 increases
from zero to any finite value. Consequently, it will follow that the numberof real solutions of (8) is exactly (V + \)N V for all finite positive values of
the masses ml ,. . .
,mv+l .
The roots of algebraic equations are continuous functions of the coef-
ficients of the equations so long as the roots are finite and the equationsdo not have indeterminate forms. Consequently, the xf are continuous
functions of mv+l if no xf becomes infinite and if no xf = xf . The real
roots of algebraic equations having real coefficients can disappear only bypassing to infinity, or by an even number of real solutions becoming
conjugate complex quantities in pairs. Therefore we have to determine
(1) whether any finite xf can become equal to any xf, (2) whether anyxf can become infinite, and (3) whether any two real solutions can become
complex for any finite positive values of m, ,. . .
,ra +1 .
(1). The masses ml ,. . .
,mp+1 , by hypothesis, are all positive. Let
the notation be so chosen that for any values of m, ,. . .
,mv+l for which
the xf are all distinct the inequalities xf <xf < < xf < x^ are
satisfied. Suppose that as some mass is changed the difference xf xfapproaches zero in such a way that xf and xf remain finite; that is, r
tj ,
which occursonly
in the expressions <p, and <Pj , approaches zero. Supposei < j. Then the term involving r u becomes negatively infinite in <p t and
positively infinite in <p s. Consider <p t
= 0. Another r (t must approach zero
in order to restore the finite value of the function <p ( , and the term
involving r lk must become positively infinite as r u approaches zero. There-
fore k<i. But r rt enters besides only in <p t ,and similar reasoning shows
that rm ,where I < k, must also approach zero. In this manner we are
driven to the conclusion finally that an r M ,where one of the subscripts is
unity, approaches zero. Then consider ^ = 0. All its terms except w2
.Tj
are negative, and since one of its r(J ,
viz. rvt , approaches zero the first
equationof
(8)can not be satisfied.
Consequentlythe
original assumption
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STRAIGHT-LINE SOLUTIONS FOR H BODIES. 291
that some rt] can approach zero for finite values of xf, . . .
, xf +l andfinite positive values of m, . . .
,my+l leads to an impossibility, and it is
therefore false.
(2). On multiplying equations (8) by mx , m2 ,. . .
, m +1 , respectively,and adding, it is found that
cf(m l x 1+mt z t + +m,,x y +m, +1 x v+1 )=0.
It follows from this equation that no xf alone can become infinite, and that
if one of them becomes negatively infinite then some other one must become
positively infinite.
Suppose the notation is again so chosen that
Z<X?< <*<*&.
Then, if any xf becomes negatively infinite xf must also become negatively
infinite, and from the equation above it follows that x +l must become
positively infinite. Now suppose this occurs and consider <p x= 0. In order
that this equation may remain satisfied, xf must also become negativelyinfinite in such a way that xf xf shall approach zero. But now it follows
from <?,= 0, since orxf and m^ixf xf)/r\ are both positive, that xf
must also become negatively infinite in such a way that xf xf shall
approach zero. Then it follows similarly from <p 3 that xf must become
negatively infinite in such a way that xf xf shall approach zero. This
reasoning continues until it is found that x|0>
,. . .
, xf +l must all becomenegatively infinite. But xf +1 at least must become positively infinite.
Therefore xf can not become negatively infinite, and similarly xf +l can
not become positively infinite. Hence no xf can become infinite.
In order to prove now that, as ra +I approaches zero, equations (8)
and their solutions remain always determinate, and that there are accord-
ingly no solutions besides those obtained in theorem (B), consider a solution
Zi,..., x v+l ,in which the x } are all distinct for a set of positive values
of ?!,... ,ra +1 ,
and then let m, approach zero as a limit.
In the first place, if x, approaches neither x i+l nor x i ^ l as a limit as mt
approaches zero as a limit, then by the reasoning of (1) and (2) above noXj can approach any x t as a limit.
In the second place, x t can not approach x i+l as a limit as mt approacheszero as a limit unless x t ^i approaches x t+l as a limit, for otherwise cp {
= can
not be satisfied. But if x,_i approaches x i+ i as a limit as m{ approaches zero
as a limit, then <p,_i= and <p, + i
= can not be satisfied unless ( _ 2 and x (+3
respectively approach x t as a limit. This shifts the difficulty to ^,_ 2=
and <p, +2= 0, and so on until <pi
= and <p +1= are reached, which can not
be satisfied under the hypotheses it has been necessary to make.
In the third place, x, can not become positively infinite as w, approaches
zero, for then <p t = can not be satisfied unless ,_i becomes infinite in such
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292 PERIODIC ORBITS.
a way that x, x,_i approaches zero. Continuing through <p l _ 1=
0, . . .
we are led to the conclusion finally that xx ,
. . ., x v+1 all become posi-
tively infinite, but then the center of gravity equation can not be satisfied.
Consequently the solutions all remain regular as mt approaches zero asa limit.
(3). Since the solutions of (8) are continuous functions of m +1 ,it
follows that no two solutions which are real for m +l= can ever become
conjugate complex solutions for any real value of m, +l without having first
become equal; and, conversely, no two solutions which are complex for
my+l= can ever become real for any real value of m +1 without having first
become equal. Consequently, if a multiple solution of (8) is impossible for
every set of finite positive values of mt ,. . .
, m +1 ,it is impossible that
any real solutions should disappear by becoming complex, or that any
complex solutions should become real.
The conditions that x = x m shall be a multiple solution of /(#) =0 are
that /(a;co>
) =0 and df(xw)/dx = 0. The corresponding conditions that a
set of simultaneous algebraic equations shall have a multiple solution are
that a set of values of the variables shall satisfy the equations and that the
Jacobian of the functions with respect to the dependent variables shall
vanish for the same set of values. That is, the conditions that
Xj = xf (j =l, . . . ,i/+l),
shall be a multiple solution of (8) are that these values shall satisfy (8) and
also the equation
A =
dXi
dXi dXt
dx 2'
dx+i
d<f>2
dx, e+l
dXi dXt'
d<p tV+i
dx v +i
= 0. (11)
Consider two solutions of a set of algebraic equations having real
coefficients. As they change from real to conjugate complex quantities, orfrom conjugate complex to real quantities, for some value of a continuously
varying parameter, then for this particular value of the parameter they arc
not only equal but they are real. Consequently, it is necessary to examineA only when all of its elements are real. It will now be shown that it can
not vanish for any set of real values of the x } when mt ,. . .
,m +1 are
positive, and consequently that it can not vanish for any particular set
which satisfies equations (8). When this is established, it will have been
proved that all the solutions of (8) which are real for w>+1= remain real
when mr+l increases to any positive value, and that those which are complex
remain complex.
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STRAIGHT-LINE SOLUTIONS FOR U BODIES.
From equations (8) and (11), it follows that
293
v*
A=
M, ,
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294 PERIODIC ORBITS.
same form as the original determinant, and the sum of the elements of its
tft row i s (J mjr\ x. Consequently every term in the expansion of the
minor which does not vanish will contain at least one of the w2
mji* u
as a factor. But theseelements
appear onlyin the main
diagonalof the
minor. Hence all terms in the expansion of the minor which do not vanish
depend upon at least one element of the main diagonal. In considering our
particular term it may be supposed, without loss of generality, to depend
upon the first main diagonal element of the minor. In the expansion of the
original determinant the product of these two diagonal elements will be
multiplied by the co-factor of the minor of the second order of which theyare the main diagonal. This co-factor has the same properties as the first
minor just considered, and in the same way it is proved that at least one of
its diagonal elements must be involved in the term in question; that is,
the term under consideration depends upon at least three elements of themain diagonal. On continuing in this manner it is proved that any term
in the final expansion depends upon all the elements of the main diagonal,
which are all of the same sign in every one of their terms. Consequently,all the terms which do not cancel out in the expansion of the determinant
have the sign ( l)v+l
;and it has been seen that there is at least one such
term, viz. ( 1)"+1 w2( " +1)
. Therefore the determinant not only can never
vanish, but it can never be less than a>2( " +1> in numerical value.
Since A can never vanish for real distinct xf when all the m, are real
and either zero or positive, it follows that no real solutions can ever be lost
or gained as the nij vary so as not to become negative, and therefore thatthe number of real solutions of (8) is (v-f- l)N y
= % (V+ 1)! for all positive
finite values of ml ,. . .
,w +1 ,
and w2.
149. Computation of the Solutions of Equations (6). There are well-
known methods of finding the roots of a single numerical algebraic equationof high degree, but they are not readily applicable to simultaneous equationsof high degree. However, when the order of the masses has been chosen,
equations (6) will become polynomials in x t ,. . .
,x after they have been
cleared of fractions. Then by rational processes n 1 of the x t can be
eliminated from these equations, giving a single equation in the remainingunknown. The solutions of this equation can be found by the usual
methods and the results can be used to eliminate one unknown. By repeated
application of this process to the successively reduced equations, the solu-
tions can all be found. The one satisfying the conditions of reality of
x t ,. . .
,x n and their order relation is the one desired.
The solutions of (6) can also be found by a method closely related to
that by means of which their existence was proved above. Suppose for
mi= raf (t
=l, . . .
, ft) a solution x,= xf of equations (6) is known.
The mf are supposed to be zero or positive. Suppose it is desired to find
the corresponding solution, that is, the one in which the masses are arranged
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STRAIGHT-LINE SOLUTIONS FOR n BODIES. 295
on the line in the same order, for mi=
ra,0> + ^ . Let the corresponding
set of the x( satisfying (6) be x, = xf + & ,
where the 4 are functions of
Mi,*-. . ., m to be determined. On substituting #,
= xf + & a,ndm (= mf+ij, (
in (6), making use of the notation of (8), expanding as power series in the
tand n t (which is always possible, since it has been shown that no xf can
become infinite and no xf can equal any xf), and remembering that x t= xf
is a solution of (6) for mt= mf ,
the resulting equations are found to be
where the
(13)
are the symbolic powers used in connection with the power-series expan-sions of functions of several variables.
The determinant of the terms of the first degree in the %t in equations
(13) is the A of equation (11), which has been proved to be distinct from
zero in this problem. Therefore equations (13) can be solved by the method
explained in 1, and by 2 the solutions converge for |ju|>0, but sufficientlysmall. Suppose they converge if
|/z<| ^ r. Keeping the fi t within this limit,
a solution x(= x l
t
x)
is computed. Then this can be used as a starting-point
for a second application of the process, which can be repeated as many times
as may be desired.
Hence, to find the solution in which the bodies mt ,. . .
,mn have any
finite positive values and lie in a determined order on the line, we may start
with m1 , m2 , and m3 and solve the Lagrangian quintic* which defines their
distribution on the line. Then an infinitesimal body m4 is added and its
position is found by solving the single equation (9), in which v = 3. This
infinitesimal mass mx is made to increase, step by step, to the required finite
value and the corresponding values of x, ,. . .
,x 4 are computed. It
follows from the fact that the dy^/da;, are less than fixed finite quantities
depending upon w2 and m1 ,. . .
,m
,while A is not less than w8 in numerical
value, that any finite value of m4 can be reached in this way by a finite
number of steps. After the required value of ra 4 has been reached, the
process can be repeated for m5 , etc., to any finite number of bodies. Not-
withstanding the fact that this would be very laborious if the number of
bodies were large, we must regard the problem as completely solved both
theoretically and practically.
Tisserand's M&canique Celeste, vol. I, p. 155, or Moulton's Introduction to Celestial Mechanics, p. 216.
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290 PERIODIC ORBITS.
II. DETERMINATION OF THE MASSES WHEN THEPOSITIONS ARE GIVEN.
150. Determination of the Masses when n is Even. Suppose co and
the n distinct points on a line, x l ,. . .
, x n ,are given, and consider the
problem of determining w, ,. . .
,mn so that the circular solutions shall
exist. There will be no loss of generality in selecting the notation so that
,<,< <,. With this choice of notation equations (6), which are
necessary and sufficient conditions for the solutions, become
8*-+ o +^ +'12 '23
'l.n-l
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STRAIGHT-LINE SOLUTIONS FOR ft BODIES. 297
m1 x l +m 2 x i + + m x = 0, which is a consequence of (14), to the set
of equations. There are then n -f- 1 linear homogeneous equations in
if,m1
,...,m.In order that
theyshall be consistent their eliminant
E=
o,
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298 PERIODIC ORBITS.
Suppose x 2 and x 3 are chosen and consider F as a function of x t. Then
it follows at once that
From the inequalities x 2 < x 3 and r ]2 < r 13 ,it follows that dF/dx l
is positive
for oo <x t <x 2 . Therefore there is but one solution of (17) for x,<x 2
when x t and a positive x 3 are chosen. By symmetry, there is but one
solution of (17) for x 3 >x 2 when x 2 and a negative , are chosen.
If Xi is negative and x 3 is positive, but both otherwise arbitrary, Fconsidered as a function of x 2 gives
S5Jte+o-+, Lm F(x 3 -e)=-^, ||=2 *i ?*<<). d9)
U^/j / 2J / 13 /jj
Therefore there is but one solution of equation (17) for x 2 which satisfies
the inequalities x L <x 2 <x 3 .
Suppose a negative x : , a positive x 3 ,and m2 are given arbitrarily and
that x 2 is defined by (17). Then equations (14) give
m-* B [+>*-j?} m^rl^-^x-7
^]- (20)
If w2 is negative, ml and m3 are necessarily positive. If m2 is positive and
sufficiently small, both m^ and m3 are positive. As m2 increases, m^ and m,
decrease. Suppose a; 3 > x, . Then, for a certain positive value of ra 2 themass m3 vanishes while m2 is still positive. For a certain greater value of
mj ,the mass ml
is zero and m3 is negative. For still greater values of m, ,
both mj and m3 are negative. From the fact that x, must be negativeand x 3 positive, and from equations (20), it follows that not all three of
the masses m,, m2 ,and m3 can be negative simultaneously.
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Chapter IX.
OSCILLATING SATELLITES NEAR THE LAGRANGIANEQUILATERAL-TRIANGLE POINTS.
By Thomas Buck.
153. Introduction. This chapter is devoted to an investigation of
certain periodic orbits which an infinitesimal body may describe whenattracted according to the Newtonian law by two finite bodies revolving in
circles about their center of mass. It has been shown by Lagrange thatthree bodies placed at the vertices of an equilateral triangle can be givensuch initial projections that they will retain always the same configuration.The orbits here considered are in the vicinity of the equilateral-triangle
points defined by the two finite bodies. The infinitesimal body is displacedfrom the vertex of the equilateral triangle, and its initial projection is deter-
mined so that its motion is periodic with respect to that of the finite bodies.
The existence of the solution is established by the method of analytical
continuation. The construction is made by the method of undetermined
coefficients, using the properties obtained in the discussion of the existence.
The solutions are given in the form of power series which converge for
sufficiently small values of the parameter employed.
154. The Differential Equations. The motion of the infinitesimal bodywill be referred to a rotating system of axes, the origin being at the center of
mass, the ^77-plane being the plane of the motion of the finite bodies, and the
rate of rotation such that they remain on the |-axis. The masses of the finite
bodies will be represented by /j. and 1 n so taken that m^s> their distance
apart will be taken as the unit of distance, and the unit of time will be
so chosen that the proportionality factor k 2is unity. Then the equations
of motion for the infinitesimal body are
<tl_ drL _dU d?v_ l9 c% = dU tn = dU m
df dt"
dt'
dt2 "'
dt"
dr,'
dt2"
df' K J
where
r^Viz+nr+f+F,
r, and r, being the distances from the infinitesimal body to the bodies 1 n
and m respectively.299
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300 PERIODIC ORBITS.
The Lagrangian equilateral triangle-solutions are
I.
&-J-M, *- + jVS, f,-0.
II. Z =2~ fl , W *"yVf, f = 0.
The two points in the rotating plane defined by these solutions will be
referred to as point I and point II respectively. The question of the exist-
ence of periodic solutions of equations (1) in the vicinity of these points is
to be investigated. For this purpose the origin is transferred to the pointin question by means of the transformation
g-i-H-sr, n=\Vz+y, f-#. (2)
After the transformation is made the right members of the equations are
expanded as power series in x, y, and z. The region of convergence of
these series is determined by the singularities of the functions \/r x and l/r 2 .
When point I is considered, the region of convergence is given by the values
of x, y, and z satisfying the inequalities
-Kx2
+f+z--{-x-\-\^y<+l, -Kx2
+yi
+z--x+Vsy<+l.This region consists of the common portion of two spheres, excluding their
centers which are at the finite bodies, each of radius V2. For point II the
region of convergence is defined by the inequalities
-Kx'+y'-hz'+x-VsyK+l, -Kx-+y 2-\-z'
i -x-Vsy<+l.
Since the origin in this case is at the second point it follows that this region
is the same as that found for the first point.
As the two cases differ only in the sign before the Vs, it is necessaryto consider in detail only one of them. The discussion will be given for
point I with the understanding that by changing the sign of Vs the corre-
sponding expressions for the point II are obtained.
Two parameters, e and 5, are introduced as in Chapter V. Then,
denoting derivation as to r by accents, the differential equations become
x-"-2(l + 5>/ = (l + 5)2
[X 1 + X2 e+X 3 e2 + ... J,
y"+2(i+d)x'=(i+8y [y 1 +y, +y,y+ . . .
],
z" = (l-H) 2 [Z 1 +Z,+Z l *+ ],
(3)
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OSCILLATING SATELLITES, EQUILATERAL-TRIANGLE CASE. 301
where
X1= + iaH-|V3(l-3M)z/, F 1= + fV3(l-2 M)o;+fz/,
X2= + ^ [7(1 -2 M)x
2
+2V3x2/- 11(1 -2 M)2/2 +4(l-2 M)2
2
],
Y2= + ^[V3x?+22(l-2iJL)xy+3V3tf-4:Vzz%
Z, = -z, Z 2= + f[(l-2 M)xz+V%2],
(4)
X3= + ^ [
- 37z 3 + 75V3(1 - 2n)x2y+ 123x^
2 +45V3(l -2/x)*/
3
-12x2 2 +6V3(l-2 M)t/z2
],
F 3= + ^[-25V3(l-2M)a;
3
-r-123x2
2/ + 135V3(l-2/i)a;?/2
+37/3
-60\/3(l-2/x):cz2
+132i/z2
],
^ 3= -
1 [x2
z+lly2z-4:Z >
+10Vsa-2v)xyz].
For sufficiently small values of x, y, z, and e these series are all convergent.
Equations (1) admit the integral
\ [(!)'+*+ (I)'] :**-*The corresponding integral of equations (3) can be expressed as a powerseries in e. The terms independent of e are
4[x,2
+2/'2
+z'2
]-(l-r-5)2
[3a;2+%2 -4z 2
-r-6V3(l-2 i u)a;t/]= const. (5')
These terms will be found useful in the existence proofs which follow.
155. The Characteristic Exponents. For e = 5 = equations (3) become
x"-2y'- f x- f Va(l-2^ = 0,
y"+2x'-\ V3(l-2/x)x-| 2 /=
0,
z"+z = 0.
(6)
The last equation, being independent of the first two, can be integrated
immediately, giving
z = c t sin t+Cj cost.
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302 PERIODIC ORBITS.
To integrate the first two equations, let
x = Ke XT, y = Le x
\
On substituting in the first two equations of (6) and dividing out eXr
, we have
[x=-f]#-[2X + f V3(1-2 M)]L=
0,
[2X-| V3(1-2 M)]^+[X2
-|]L = 0.
(7)
In order that these equations may be satisfied by values of K and L different
from zero, the determinant of thesystem
must vanish. Thisgives
for the
determination of X the characteristic equation
V+X 2
+f M(l-M)=0. (8)
Each of the four values of X satisfying this equation gives a particular solu-
tion of equations (6). The corresponding K and L must satisfy equations
(7). Since these equations have a vanishing determinant the ratio only of
the K and L is determined. In what follows K will be considered as arbi-
trary, and L will be determined in the form L = bK.
In order that a solution shall be periodic, the corresponding X must be
a purely imaginary quantity. Upon solving (8), we have
v= -lVl-27 M(l- M).2
For small values of n the roots of (8) are pure imaginaries; the limiting
value of ju for which this is true is given by the equation
1-27m(1-m)=0.
The root of this equation which is less than \ is /i= .0385 .... For
(i4* .0385 . . . the values of X are purely imaginary and the corresponding
particular solutions are periodic. Let o-xand <r 2 be two numbers defined by
2 _ 1+Vl-27 M (1 -n) ,_ 1-Vl-27 M(1- M)1 ^ O t r.
It follows that <t x and <r 2 do not exceed unity for n ^ .0385 . . .,and that
oy^aj. Then the roots of (8), which are the characteristic exponents of
the problem, become (r,V 1 and <r a \/ 1-
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OSCILLATING SATELLITES, EQUILATERAL-TRIANGLE CASE. 303
156. The Generating Solutions. The general solution of (6) is
x = a 1 e"v - lT + a 2 e- aiV - lT + a 3 e aW- iT + a t e"" /=lr
,
y=6, a, e* v = Ir +^ 2 e-r '^ 1T +&3 a 3 e*^ T+6 4 a 4 e
- ff ^ T,
z=c 1 sinT+c 2 cosr.
The quantities a x ,a 2 ,
a 3 ,a t ,
c t ,and c 2 are arbitrary, while b x ,
b 2 ,6 3 ,
and6 4 are determined by equations (7) when the proper values of X are sub-stituted. Thus it is found that
, _. +8(r 1 A/^T- 3 V3(l- 2M) h +8<y/^I-3 V3(1-2m )
h = ^ lV^l3V3(l 2 fi) 7 -8o- 2 \/^ri-3V3(l-2M)4<rf+9
'-
4a|+9
Various periodic solutions are obtained from this general solution by assign-
ing suitable values to the arbitrary constants and to the quantity n- For
H < .0385 ... we have two distinct periodic solutions :
I. x = a l e'* v - lT +a i e-' riV - lT, y = b 1 a l e lV - lT +b 2 a 2 e- asV - 1T
,2 = 0.
II. x = a 3e aW~ lT +a i e-'" v - lT
, y = b 3 a 3 ea ' v - lT +b i a i e- (nV - lT
,z = 0.
These equations represent ellipses in the xy-pl&ne with centers at the origin.
The major axes of the ellipses coincide and make an angle 6 with the positivex-axis defined by
tan20=-V3(l-2M),
with cos 20 positive. The major and minor axes of the second are greaterand less respectively than those of the first. The periods are 2ir/<r 1 and
2r/<r 2 respectively. If /x= .0385 . . . it follows that <r x
= <r t ,and solutions
I and II coincide.For all values of m we have the periodic solution
III. x = 0, y = 0, z = Cj sin t+Cj cost.
This solution defines an oscillation on the z-axis with the period 2t.
It is possible to give n values such that <rx and <r 2 are relatively commen-
surable. Let m2 (r1= mI
(r 2 ,where ml and m, are integers. Then, by using
the definitions of <r x and cr 2 ,we find
_ ml ml
Vl-27 M(l- M)= m2 + w2
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304 PERIODIC ORBITS.
For m ^ .0385 . . . the expression on the left takes all values on the interval
from to 1. By choosing m, and m2 so that 0< {m\ m2 )/(w2 +m 2 ) < 1, and
solving the equation for n, we have a value of n making <r, and <r 2 com-mensurable. For such values of /x we have the additional
periodicsolution
IV.
x = a t e' 1 v-* * + Oj e- l v~ l T + a 3 e"1^ T + a A e~" ^~ l T
,
y = b 1 a 1 e' w- lT +b t a 2 e-" lV =lT +b 3 a 3 e'*v - lT +b i a A e""
v ^ lT,
the period of which is 2ra 1 7r/<r 1= 2ra 2 7r/<r 2 .
By an argument precisely similar to the preceding it can be shownthat for special values of /x, the characteristic exponents (r
t and a 2 separately
may be commensurable with unity. We have then the periodic solutions
V.
c'iv-it + a 2 e-" lV - 1T
y = b 1 a l e"*=* r + b 2 a 2 e""^-
1 ',
z=c x sin t-\-c 2 cos r;
VI.
<r% Vl t -ffiv'-^Irx= a 3 e u,v- 1T + a 4 e~
y = b 3 a 3 e" ^ T + b t a 4 e- *= T,
z=c 1 sinT+c 2 cosT.
The periods are 2?mr = 2m 1 ir/<r 1 and 2mr = 2n 2 ir/a 2 respectively.
Finally, a x and a 2 may be commensurable with unity for the samevalue of n.
Let aa t= b and c<r 2
= d; then
26 2 -cr <?-W\/i-"27m(1-m) =
or
From this relation it follows that a 2d? = c 2
(a2 b 2
). If now we give a, b, c,
and d such integral values that this relation is satisfied and such that
(262
a?)/a2
lies between and 1, the corresponding value of fi will make a x
and <r 2 commensurable with unity. For example, such a choice is
a = c=13, 6=12, d = 5.
For such values of p. we have the periodic solution
<r,V-ir_|_ ^g-nv-lr^. ^^.V-lr^. a4e-<"V=lT
VII.
* 3.
3 a 3'y = 6, <*! e <r,v/=i T+ &, a 2 e"''^
1 T + 6, a 3 e*^ T +b t a t e-"v~ l T
,
z = c t sin T+c 2 cosr.
The period of this solutionis
2rmr=2m l Tr/(r l
=2m t Tr/<r 2
,
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OSCILLATING SATELLITES, EQUILATERAL-TRIANGLE CASE. 305
These periodic solutions are the generating solutions for the general
problem. We shall now suppose that e is not zero and consider the questionof the existence of the continuations of these solutions with respect to the
parameter e. The period in the variable r will in all cases be taken the sameas that of the generating solution. The period in t of the solution is foundfrom the relation i=(l+5)r.
157. General Periodicity Equations. For e = the general solution
of equations (3) is
(9)
(10)
x= a e Via+i)V~ lT + a g-i+J)v'-ir a e <ria+&w-iTi a e -<na+S)ViT
y = b^e* a+5) v ~' T+ b 2 a 2 e~ai il+s > v~ l T+ b 3 a 3 e
ai +*v=l ' + b^e'"1 a+ *> vZZi T
,
z =c 1 sin(l + 5)r -fc 2 cos(l + 5)r.
Normal variables are introduced by the transformation
X Xj ~t~ X2 ~1~ X3 ~j~ X4 ,
y = b 1 x 1 +b 2 x 2 +b 3 x 3 -\-b i x i ,
'-y 1 (l+)v^T( I -a^)+r f (l+.*)'v^I(aB l -* 4 ) l
y'=
<r, (1 + 5) V~=I (6i x,- b
2x
2 ) +a
2 (1 + 5) a/^T (b 3x
3
- 64x
t ).
The differential equations then become
x[- Vl (l + S)y^Tx l= A l (X 2 e+X 3 e'+ )+5 1 (F 2 e+F 3 e
2 + ),
x' +v 1 (l + 8)V=ix 2= A2 (X 2 t+X 3 f?+ )+B 2 (Y 2 e+Y 3 S+ ),
xg-r t (l+l)V=T%- A 3 (X 2 e+X 3 e*+ -)+B 3 (Y 2 t+Y 3 e+ ),
x 4 + cr 2 (l + 5)v^ix 4= A i (X 2 t+X 3 t
i + -)+5 4 (F 2 e+F,eJ + ").
z"+(l + 8Yz =(l + 5y[Z 2 e+Z 3 e+ },
where
A t =(l + 5)*%,B t =(l + 6)*%,
A being the determinant of the transformation (10), and AJt the minor of an
element in this determinant. The first subscript indicates the row and the
second one the column.
(11)
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306 PERIODIC ORBITS.
For e = 5 = the initial values of the variables x t ,x t ,
x 3 ,and x t are
Od <h> a t> and o respectively. In the general problem we take as the initial
conditions
^,=
0,4-0,, x 3=
a s +a 3 , 2=0, j
x, = a,+o 2 ,3J 4= a 4 + a 4 ,
z' = c+y. j
Since there is a component of force always directed toward the xy-pl&ne, it
is clear that at some time 2 must be zero. Hence we have supposed that
2 = at t = 0.
According to 14 and 15 equations (11) can be integrated as powerseries in the parameters a 1} 04, a 3 ,
a 4 , 7, 5, and e, which converge for
|aj| ,. . .
, |e| sufficiently small, and for O^t^T, the value of T, which in
this case is the period, being given in advance. These solutions have the
form
x^ia.+a,) e+"< 1+s >^ i
'+ep 1 (a,, a,, a 3 ,a 4 , y, 5, e; r),
x 2= (0,+Oj) e-" a+S)V=iT +epz (a,, a,, a 3 ,
o 4 , y, 5, e; t),
z 3= (a 3 +a 3 ) e+'^+b^+ep, (a,, Oj, a 3 ,
a 4 , T , 5, e; r),
x A= (a 4 +a 4 ) e-^'^'^^+tp* (o ,, a 3 ,
a 4 , 7, 5, e; r),
z =(c+t) 8^(1+5)7+6^^0!, Oj, a 3 ,a 4 , 7, 5, e; t),
2' = (l + 5)(c+7)cos(l + 5)T+ep, (a 4 , Oj ,a 3 ,
a 4 , 7, 5, e; t).
(13)
The general periodicity equations for the period T are
x 4 (77
)-x 1 (0)=0 (t=l, ... ,4),
z(T)- 2(0) = 0, 2'(T)-2'(0)=0.(14)
These equations are sufficient conditions for the periodicity of the solution.
On solving them for the arbitraries a t , a, ,a 3 ,
a 4 , 7, 5 as power series in e,
a determination of these quantities is obtained such that the correspondingsolution is periodic. Hence, on substituting these series in (13), the resulting
expressions for the x t , 2, and 2' are periodic. These expressions can be
rearranged as power series in e which will converge for e sufficiently small,
and for all <! r ^ T, provided the values of a t ,. . .
,o 4 , 7, 5 obtained
from (14) lie in the domain of convergence of (13). The convergence can
be secured by imposing the condition that the expressions for the arbitraries
obtained from (14) shall be power series in e, which vanish with e. Thesolutions are then analytical continuations of the generating solutions.
The periodicity equations will now be set up for each of the generating
solutions, and the possibility of solving them for the arbitraries in the
required form will be considered. The equations will be written for the
point I only. The conclusions are the same in all cases for the point II.
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OSCILLATING SATELLITES, EQUILATERAL-TRIANGLE CASE. 307
158. The First Generating Solution. The explicit form of equations
(14) is now to be determined for T = 2ir/a 1 and a 3= a 4
= c = 0. On accountof the existence of the integral (5), one of the equations is redundant. Forif we let
x l= a l e
+n ' =lT +y l ,
*b% t*2*
z 3= 0+2/ 3 ,
*4= 0+2/4,
Z = Q+ W,
z' = 0+w',
where y 1 (Q)=y a (0)=y 3 (0)=y i (0)=w(Q)=w'(0) =0, we find that the partial
derivative of the integral (5') with respect to y 2 is
Mcr\(2<j\-l)a x
(4^+93
for r = 2ir/<r 1 ,and y l
= y i= y l
= y i= w = w' = 0. The integral can therefore be
solved uniquely for y 2 in terms of y l; y 3 , y i} w, and w'. If the latter
quantities are periodic it follows that y a also is periodic. Therefore the
second equation is redundant and can be suppressed. On computing the
necessary terms of (13), it is found that the remaining equations have the
form
8 (2*V-li)+
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308 PERIODIC ORBITS.
from zero. If the coefficients of a, and a, in the second and third equations
respectively are also distinct from zero, it follows that there is a unique
solution for a 3 ,a t ,
and 5 as power series in e, vanishing with e. It is neces-
sary, therefore, that<r
x be suchthat the
expressions
jWEI ,-*r<r,v=i
. 27T 2ir .
e" l
1, e *'1, sin
,cos 1
shall not vanish. The first two vanish if <T^= ma xl where m is an integer.
Since <r, and <r 2 are positive and <r x ^ <r 2 ,this occurs only when <r,
= <r 2 .
This case will be treated later in the discussion of the commensurable
cases. It will be shown that orbits exist in this case also. The last two
expressions vanish only if a x is the reciprocal of an integer. Suppose, then,
that <rl= 1/w. On solving the first three equations of (15) for a
3 ,a 4 ,
5
and substituting in the last two, there remains, after dividing out e2
and y,
a term in each independent of the arbitraries. There can be, then, no solution
of these equations in the required form. They can be satisfied only by
putting 7 = as before. Hence equations (15) have a unique solution of
the same form in this case also.
The question of the existence of an orbit re-entering after m revolu-
tions will now be considered. The period in this case is 2mx/o- 1. The
periodicity equations have a unique solution, as before, except when n is
such that m^ a^. In this case the second and third equations do not
admit solutions for a 3 and a 4. This case will be treated later in the discus-
sion of the commensurable cases. It will be shown that orbits exist for
these values of m also. Hence there is a single orbit re-entering after mrevolutions. But one such orbit is obtained \>y m repetitions of the orbits
re-entering after one revolution. It follows, therefore, that this orbit is the
only one.
The periodicity equations have now been satisfied with a x and a? still
remaining arbitrary. Since we now have z=0, one relation between these
arbitraries is obtained by fixing the origin of time. It will be supposedthat x' = at r = 0. This gives the relation
<tx (a x Oj + a x a,) + <r, (a 3 a 4 )
=0.
The same choice of the origin of time in the generating solution gives ,= a 2
.
We have then
o-i (i - o) + <r 2 (a,- a 4 )
= 0.
This equation may be regarded as determining c^ in terms of the arbitrary a x.
There will then be in the final solution the two arbitraries a, and a x besides
the parameter e. Since a, and a, occur always in the combination a, + a,
they can be replaced by a single arbitrary. When the solutions of the
periodicity equations are substituted in (13), the desired continuation of
of the firstgenerating
solution is obtained.
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)+(**+
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310 PERIODIC ORBITS.
second equation of (14) can be suppressed because of the integral. The
required terms of the series for the x t are found from equations (11), andthe explicit forms of the periodicity conditions are
(a 1 +a l )5[+2m 1 7rv^T+ ]
+ (a l +a 1)?^[8 li (a 1 +a l )(a i +a t )+d u (a i +a i )(a i +a t )]<?+ =0,
(a,+a,)5[+2w 2TrV^T+ ]
+ (a s +a 3)^[^(a 1 +a 1 )(a J +a,)+033(a 3 +a,)(a 4 +a 4 )] es + =0,
(a 4 + a 4 ) 6 [-2ra 2 7rV^T+ ]
+ (a 4 +a 4)^^[0 42 (a 1 +a 1 )(a 2 +a 1 )+^(a,+a3)(a 4+a 4 )] 2+ =0,
where the d tJ are functions of m which will not be given explicitly.
The first equation of (17) is solved for 8 and the result substituted in
the other two equations. After dividing them by e2
, they have the form
(17)
(a,+a 3 )[^ 11 (a 1 + a 1 )(a s +a s )+^ ls (a 3 +a 3 )(a 4 +a 4 )] +[ ] =0,
(a 4 +a 4 )[^, 1 (a 1 +a,)(a s +a 2 )+^ 22 (a3+a 3 )(a 4 +a 4 )]+e[- ]=0,(18)
where the A u are functions of /x. In order that solutions of the required
form shall exist, it is necessary that
a a [A n a 1 a t +A ls a t a i ]= 0, a l [A 2l a l a i -\-A 32 a 3 a l ]
= 0. (19)
These equations are satisfied by a 3= a 4
= 0. When these conditions are
imposed, equations (18) can be solved uniquely for a and a, as power series
in e, vanishing with e. The generating solution is reduced to that con-
sidered in 158, and it is now possible to supply the proof for the exceptionalcases which were not covered by the previous discussion.
For an orbit re-entering after one revolution the proof did not include
the case when cr^o-, . When m1= m2 =l the discussion just given supplies
this deficiency. For an orbit re-entering after m revolutions, the case whenm<Tt=<r l was not included. By putting m, = m and mi
=1, we have the
desired proof.
The corresponding cases arising from the second generating solution
can be treated by so combining the periodicity equations that the equations
(19) carry the factors at and a 2 respectively. The discussion is then the
same as that just given.In order that equations (19) may be satisfied by values of the a t different
from zero, it is necessary that the determinant of the A u should vanish.
This determinant can be developed as a power ser ies in Vju. If it is identi-
cally zero in ju, each coefficient of this series must vanish. The coefficient
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OSCILLATING SATELLITES, EQUILATERAL-TKI ANGLE CASE. 311
of V/x was computed and found to be different from zero. For the special
values of fi under consideration here it may be possible to make this determi-
nant vanish, but because of its complicated character this possibility has not
been considered. The question of the existence of these orbits is thus left
open, but it seems improbable that the necessary conditions can be satisfied.
161. The Fifth Generating Solution. The values of ft in this case are
such that the period of the generating solution is 2rmr = 2m 1 7r/o' 1 . The last
equation of (14) is suppressed. The remaining equations have the followingform :
(a 1 +a 1 )5[+2m 17rV^=T+ ] + e[a u (a 1 + o I )a 3
+a n (a 1 +a 1 )a i +a n (a i +a i )a 3 +a u (a i +a i )a i +a ls al+a li alj
+t*[a il (a 1 + a1 )
i +a K (a 1 +a 1 )\a 2 +a i )+a n (a 1 +a 1 )(a 2 +a i )
2
+a 24 (a 2+a ;! )3+a !!6 (a 1+a 1 )(c+7)
2 +a M(a 2 +a 2 )(c+7)2H ]h =0,
(a t +a t )d[-2m 1Tr\f=T + ']+e[b n (a 1 +a 1 )a 3
+ b a (a l +a l )a i +b u (a t +c^)a 3 +b u (a i +a i )at+b li al+b 16 ai]
-r-e2
[6 il (a 1 + a,)3 +6 1B (a 1 +a 1 )
2
(a !! +a 2 )+6 23 (a 1 +a 1 )(a 2 -|-a 2 )2
+6 24 (a 2+a 2 )3
+&26 (a 1+a 1 )(c+T)2
+&M(a s +a 2 )(c+7)2
+ ]+ =0,
i t [{<fi~*"=*-l)+ ]+e[c u (a 1 +a0 2 +c 12 (a 1 + a
1 )( 2 +a 2 )
+ c 22 (a 2 +a 2 )2 +c 23 (c+7)
2 +a 3 (- }+,< >]+ -0,
a 4 [(e-*<w^-l)+. .
.]+ e [d u (a 1 + a1 )*+d M(a I + a 1 )(a,+o 1 )
+d 2i (a i +a 2Y+d i3 (c+yy+a 3 ( )+*< )]+ =0,
[2wirH ]+
e[e,a3 + e 2 a 4 +5(e 3 H
)]+[e,i(a1 + a
1 )*
+f 22 (a 2 + a 2 )s +e !! 3(a 1 +a 1 )(a J +a 2 )+e 24 (c+7)
2
+ a 3 (---) + a 4 (---) + 5(---)] + -'-=0,
where the a tJ ,. . .
,e u are functions of n which are readily determined.
The last three equations are solved for a 3 ,a 4 , and 5, and the results thus
obtained are substituted in the first and second equations. After dividing
by e2
, these equations have the form
4(a 1 + a 1 )2
(a 2 +o 2 )+(a 1 +a 1 )(c+7)2
+*( )=
0, 1
CCa. + a.) (a 2 +a 2 )2 -fZ)(a 2 +a 2 )(c+7) 2 +<( )=0. J
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312 PERIODIC ORBITS.
In order that solutions of (21) of the required form shall exist, it is
necessary thata 1 [Aa l a t +Bc*} = 0, a 2 [Ca 1a s +Dc i
]= 0. (22)
These equationsare satisfied
by e^= a
2
= 0. With a x and a 2
havingthis
value,equations (21) then have a unique solution for a^ and o 2 . But the generating
solution has reduced to that considered in 159. The orbit obtained is,
therefore, the continuation of the third generating solution re-entering
after m revolutions, and moreover the value of n is such that ma^ = m1 ,
where in, is an integer. Thus we have a proof of the existence of an orbit
in one of the exceptional cases omitted in discussing the third solution.
In order that equations (22) may have a solution for which (^,0^, and c
are different from zero, it is necessary that the determinant of the A, B,
C, and D shall vanish. This determinant can be developed as a power series
in Vyt. Ifit is
identically zero,each coefficient in this
development sepa-rately must vanish. The coefficient of Vju was computed and found to be
different from zero. For special values of /x it may be possible to satisfy
(22) by values of a lt Oj, and c which are distinct from zero. Because of the
complicated character of the coefficients, this possibility has not been
established. As in the preceding case, the existence of orbits of this type
seems improbable, but complete proof is lacking.
The discussion for the sixth generating solution differs only in notation
from that just given. No new orbits are found, but a proof is obtained of
the existence of the continuation of the third generating solution re-entering
after m revolutions, when n is such that ma 2 m2.
This is another excep-tional case not treated in the discussion of the third generating solution.
162. The Seventh Generating Solution. The values of /x for this case
are such that the period is 2mir = 2m 1 Tr/<r 1= 2m 2 ir/<r 2 . As in the previous
case, the last equation of (14) is suppressed. The remaining periodicity
equations have the following form:
(a l + a1 )5[+2/W 1 7rV^T+ -]+ 2wMr [pn(<h + *i)
,
(fl*+t)
+ Vu (a 1 + a 1 )(a 3 +a 3 )(a 4 +a 4 ) + ^ 1 3(a 1 + a 1 )(c+7)2y+ =0,
(a 2
+a a
)a[-2ra1 irv^T+
]+2mir[<p2l (a i + a 1 )(a 2 + a 2 )-
+V? 22 (o 2 +a 2 )(a 3 + a 3 )(a 4 -|-a 4 )-|-v 2s(a 2 +a 2 )(c-|-7)2
]e2 + =0,
(a,+a 3 )5[+2m 2 7r\^rr+ 0+ 2 * [?n(i+i)(i+ i) (* + )
+ <P 3i (az+a 3Y(at+a l )+<P 3i (az+* 3 )(c+yyy+ ' =0,
(a 4 +o 4 )5[-2w 2 7rv/ "=T+ ]+2w-[^ 41 (a 1 -f-o 1 )(a 2 +o s )(fl 4 -|-a 4 )
+V(a3 + a 3 )(a 4 + a 4 )2 +?(a4+a 4 )(c-|-Y)
2y+ . . . =0,
2mir5 + mir [<p a (a, + Oj) (a 2 +<%) + <p bi (a, -f a,) (a 4 + a 4 )
+<p*(c+yyy+ =o,.where the <p tJ are functions of m-
(23)
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OSCILLATING SATELLITES, EQUILATERAL-TRIANGLE CASE. 313
The last equation of (23) is solved for 5, and the result obtained substi-
tuted in the first four. After dividing by e2
, the equations have the form
(a l +a 1 )[^ n (a l +a 1 )(a i +( h )+^ 12 (a 3 +a 3 )(a i +a i )+^ n (c+y)f\+ [ ] =0,
(a 2 +a i )[^P il (a 1 +a 1 )(a 2 +a i )+^(a 3 +a 3 )(a i +a i )+^P 23 (c+yy]+e[^ ]=
0,
(a3+a 3 )[^i(a,+a 1 )(a s +a 2 )+^ 32 (a3+a3)(a. 4 +a 4 )+^3 3 (c+7)2
]+ e[- ]=
0,
(a 4 +a 4 )[^ 41 (a 1 +a 1 )(a 2 +a 2 )+^ 42 (a 3 +a 3 )(a 4 +a 4 )+^ 43(c+7)2
]+ e[- ] =0,
(24)
where the ty iS are functions of ju- In order that solutions of these equationsof the required form shall exist, it is necessary that
aibPnaia 3 +^na i a l +^uC 2}=
0, a 3 [yp 3l a l a 2 -\-^ 3i a 3 a i -\-^ 33 ci
]=
0, 1
a i [t 2l a 1 a 2 +\(, 22 a 3 a t +f 23 c*]=
0, a4 [^ 4I a 1 a 2 +^ 42 a 3 a 4 +^ 43 c
2
]= 0.
J
These equations are satisfied by a t= a 2
= a 3= a 4
=. With these values
equations (24) can be solved in the required form for a lt <%, a 3) and a 4 . Theorbit obtained is the continuation of the third generating solution, andre-enters after m revolutions. Moreover, n is such that ma 1
= m1 andm<r 2
= m2 ,where m, mx ,
andm,
are integers. This is the onlyremaining
exceptional case not considered in discussing the third generating solution.
It has now been shown that the continuation of the third generating solution
re-entering after m revolutions exists for all values of n.
In order to obtain the continuation of the seventh generating solution,
it must be possible to satisfy (25) by values of the a t and c which are different
from zero. On eliminating these quantities, two functions of the \pu are
obtained which must vanish if the non-vanishing solutions exist. Thesefunctions can be developed as power series in VjL If they are identically
zero each coefficient must separately vanish. The coefficient of V/Z was
computed for one of the developments and found to be different from zero.
It follows, then, that equations (25) can not in general be satisfied in the
required way. For special values of p this may be possible, but on account
of the complicated character of the \p t) the possibility has not been proved.
163. Construction of the Solutions in the Plane. In constructing the
orbits in the plane it has been found convenient to use the normal variables
which were introduced in the discussion of the existence. The differential
equations are the first four of (11), and the solutions are given by (13),
when the quantity y has been put equal to zero. It has been shown that
the quantities a 2 , a 3 , a 4 , and 5 can be determined as power series in e,
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314 PERIODIC ORBITS.
vanishing with e, so that the corresponding solution shall be periodic,
while the quantity e^ still remains arbitrary. When these series are sub-
stituted in (13), the expressions obtained for x 1} x t ,x 3 ,
x i can be rearranged
as power seriesin e
which convergefor
sufficiently small valuesof e.
Wehave then
x 1 Xjo -(- Xn 6 ~r x l } fi' * **j x 3 x
3l) -T x 3l e + X& e -p
x t= xM~r Xji c .~r #22 T * * *
j *?" ^40 ~r ^41 ' t %a ~\
5 = 5,6+ 5, e2 +
(20)
where the a^ are functions of t.
It has been shown that the series (26) have the following properties:
(a) They satisfy the differential equations identically in e.
(b) Each x i} is periodic with the period 2^/0-! for one set of orbits, andwith the period 2ir/a 2 for the other set.
|
(c) We have supposed that x' = at t = 0, and therefore it follows that
<Tl (x 1 x i )-\-<T i (x s x i )=0 at t = 0.
(d) The arbitrary a a occurs always with the arbitrary a, in the combi-
nation a t + a, ; there will be no loss of generality if a t is specialized.
It will besupposed,
then, that a t is taken so that at r =
^1+^2+^3+^4 = Gi+a, = a.
The differential equations in the normal variables are
x[-<r l (l + 6)yf=Tx l= A l (X i e + )+B l (Y t e+ ),
^+(7,(1 + 5) xT=^x 2= A i (X i + )+B i (Y i e+ ),
^-<r 2 (l + 5)V^Z 3= ^ 3 (X 2 e+---)+ 3 (F 2 e+---),
x' i +a i (l + 8)yf=Tx i =A i (X 2 e+---)+B i (Y 2 e+---).
(27)
The series (26) are now substituted in these equations, and the coefficients
of the corresponding powers of e are equated. The resulting equations are
solved for the x tJ , and the periodicity conditions are imposed.The construction will be made first for the orbits with the period 2-iv/a^ .
The terms independent of e are given by the equations
^io 'i^ / Tx 10= 0, x[ o- 2 a/3Tx 30
= 0,
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OSCILLATING SATELLITES, EQUILATERAL-TRIANGLE CASE. 315
The general solution of these equations is
^lO w10 e} ^70 ""20 e
J ^O u 30 cJ ''40 u 40 c
On applying condition (6), it is found that a 30= a 40
=0, and from (c) and (d)
that a 10= a 20
= a/2. The solution satisfying the conditions is, then,
t'lo~
2 ' 20 2 ' 30 ' 40 '
which, expressed in terms of the original variables by (10), becomes
x = acos ffl r, 2/ = -3
^~Q Mlq coser.T- ^fg sin^r. (28)
The coefficients of the first power of e are given by the equations
x'n-^V^Tx^+a.V^T 8 1 x 10 +A 1 X?+B l Yf ,
a4+*i V^T .t 21= -a.V^T S iXm +A % X*>+B t Ff,
4 - ^ / "="T z 31= + r, V^T 5l x 30 +^3 Xf + 3 Ff,
a4+.r,V=Ta- -WV^I5x^0+^,
Xf +Bt
Yff
where X and F 2
0)
represent the expressions obtained by substituting x
and y for x and / in X2 and F 2 . In order that the solution of the first
equation shall be periodic, the coefficient of e' lV=iT in the right membermust vanish. Otherwise non-periodic terms of the type Te aiV ~ lT will be
introduced. For the same reason the coefficients of e~ aiV ~ lT,
e"' v ~ 1T,and
e -nv=ir m ^g r ight members of the second, third, and fourth equations
respectively must vanish. All the terms of this type come from the first
terms of the right members, since the other terms are of the second degree
in x w , x 20 , x 30 , and x m . Since x 30 = x i0 = 0, these conditions are satisfied inthe third and fourth equations. Since we have at our disposal the undeter-
mined quantity d a ,the desired result is obtained in the first and second
equations by putting Oj= 0.
The equations are now integrated and conditions (6) , (c) ,and (d) are
imposed. The details of this work will not be given. The results expressedin the original variables are
#i= 0io+ a ii cosayr+a^ sinoyT+ai 2 cos 2<t{r -\-a' u sin 2^7,
Vi=
Oio + b n cos oyr -f b' n sin o^t + b 12 cos 2oyr + b' i2 sin 2a \r,
*,-0,
(29)
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316 PERIODIC ORBITS.
where _ 9 ^+8^8(1-2,0*,,
,}
. = _ ,
"10 27u(l u)u n \ u io i "is/ >
wn ^ u u>
_ (l6<il+9)ii 1I +3VS(l-2M)g,i+16<r 1 gMa,!
~ ~
12rf(5o?-l)/ _ _ (!6ql+9)AM-16r 1 g M+3V5(l-2M)^
ai2~ "
12<r2(5<rf-l)
, _ , 3V3(l-2M)^ 10 -3.g 10 . 3V3(l-2/x)o 11 +8(r 1 a(,10_i ""
27/*(l-M)"~ "
4*5+9. / 8(T 1 a 11 -3>/3(l-2M)an
,
0,1" "
4cr2 +9
h _ , SV3(l-2.)A ii -16a l A' 2 -(l6o>+Z)B l .
12~~
t~
12<r2(5cr
2-l)
fc# _,_ 16<r 1 A 11 +3V8(l-2/i)ii- (l6g
2
+3)^( 312 "*"
12a2
(5<r*-l)
^,o= + -|f[7(l-2M)-2V3(6 1 +6 2 )-44(l-2M)& 1 &s ]'
^ 12= + ^[7(l-2 M)-2^(& 1 +6 2 )-22(l-2 M)(^+6D]'
^; 2= +
i |ff 9 [V3 + H(l-2M)(& 1 +b 2 )]'
5 10= -
|f^[ V3+66(l-2 M)(6 1 +6 2 ) + 12V3& 1&2 ]
-
B 12=
-^[V3+66(l-2M)(6 1 +& 2 )+6V3(6
2 +6 2
)]
^' 2= +
ii[+ 9 [H(l -2m)+3V3(6 1 +& 2 )]
It will now be shown that this method of obtaining the coefficients of
(26) is general. Suppose xv and 8j (i=l, . . ., 4; j = 0, . . .
,n 1) have
been determined and that the x tJ are periodic. For the determination of
x in and b we have equations of the following form:
/=o
+i
_/=o
1+1
x^-^V^T x 3= +<r 2 V^T 5 n x 30 + 2 [0 ****+*, e-*'v=ir] f
J=0
"+1 __ _xL+^x^J x in = -a.V^i 8 n x i0 + 2 [6 t ,e
1" v - lr+r, ti e-
1 '^- lT] ,
where the 6 (J and tj<> are known constants. Since x 30= x to
= 0, no non-
periodic terms can enter the solutions of the last two equations. In order
that the solutions of the first two equations shall be periodic, it is necessary
that the coefficients of e"'v ~ 1T and e~" lV
~1T in the first and second respectively
(30)
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OSCILLATING SATELLITES, EQUILATERAL-TRIANGLE CASE. 317
shall vanish. This gives for the determination of 5, the only undetermined
constant, two equations c^V 1 a5+20 u = O, a^V 1 a5 2rj 21= 0. Since
the existence proof has shown that 5 is uniquely determined, it follows that
these equations must give the same determination of 5 .
An additional proof is obtained by means of the integral (5'). Theterms of this integral which are independent of e are first expressed in the
normal variables. Then the variables are replaced by their expressions as
power series in e and the terms are rearranged so that the integral remains
a power series in . Since the integral is an identity in r and e, it follows
that the coefficient of each power of e must reduce to a constant identically
in r. We will consider the coefficient of e". When the expressions for the
x l} as functions of r are substituted, this coefficient consists of a sum of
linearly independent functions of r. The coefficients of each of these
functions must then vanish.
Let <p y and ft denote the coefficients of e" lV=iT and e~ ffW ^ T in the first
and second equations of (30) respectively. Then, on integrating these
equations, the terms in x ln and x 2n carrying ft and <p 2 are found to be
The terms in the coefficient of e" in the integral which carry x ln and x 2n are
-8o*\_(X 10 -X 20 ) (x ln -X 2n ) + (M M) (Mi. -
Ms..)]-
6(a; 10 +a; 2 o) (x ln +x 2n )- 18(Mio+M2<>) (b x x ln +b 2 x 2n )
- 6V3(1 - 2m) [Or 10 +z 20 ) (Mi+M2) + (x ln +x 2n ) (b 1 x 10 +b 2 xj~]
When the expressions for x 10 ,x t0 ,
x ln ,and x 2n are substituted, terms carrying
r, Te ia ' v~ lT and T e~ iaW=iT are obtained. All other terms entering this
coefficient contain only x tJ (i=
1, . . ., 4; j =
1, . . .,n 1), and are
consequently periodic. Hence the total coefficients of the above non-
periodic terms are obtained from (31). Since the coefficients must vanish,
relations are obtained which ft and v 2 must satisfy. The coefficients of
Te 2<r.v^iT anc i Te -taw-iT van i s h identically. The coefficient of r gives the
relation 32o-j(4o-' 1) (ft + ft)/(4o-J + 9) = 0. Since <r\>\, the coefficient
of(P!+<p 2
does notvanish,
and we haveft+ft
= 0. Both^
andft carry
5
linearly. Hence if 5 is determined so that either of them vanishes, it follows
that the other must vanish also. The determination of d is therefore unique.
Equations (30) are now integrated. By means of conditions (6), (c),
and (d), the new arbitrary constants are uniquely determined in terms of the
original arbitrary a . The results when expressed in the original variables
have the formn+i n+1
x n= 2 t a > cosjr+a nJ sin jr] , y n =^[b n) cos jr+ b' n} sinjV ] ,
x _ 2fl u _ 21721
(31)
(32)
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318 PERIODIC ORBITS.
From the character of the differential equations it is readily shownthat x n and y n carry the factor a n+1
,that & carries the factor a", and that
a enters in no other way. Recalling the transformation by which t was
introduced,we have the final series
x = x e+x 1 '+ , y = y e+y l f?+ ,6 = a i e + 5 2 e
2+ , (33)
the Xj , y s ,and dj being given by (32). From the way in which a enters
the series, it is seen that the arbitraries a and e occur always in the combi-
nation ae . Therefore we can put a = 1 without loss of generality, and the
final series contain the single arbitrary e.
Y
^ \^ ^--_--
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OSCILLATING SATELLITES, EQUILATERAL- TRIANGLE CASE. 319
164. Construction of the Solution with Period 2-k. The discussion of
the existence has shown that the quantities a 1} a 2 ,a 3 ,
a 4 ,and 5 can be
determined as power series in e, vanishing with e, so that x t ,x 2 ,
x3 ,
x A ,and z
will be periodic with the period 2 w. When the power series obtained in this
way are substituted in (13), we have, after re-arrangement, x x ,x 2 , x 3 ,
x i ,and z
expressed as power series in e,which converge for e sufficiently small. By the
use of equations (10) , x and y can be expressed in the same way. Therefore
the solution has the form
x = x +x 1e+x i ei + ,
z = z +z l e+z 2 (?+ ,
V = Vo+Vi*+y^+ ,5 = 0+5,6+5^+
The series (34) have the following properties:
(a) They satisfy the differential equations identically in e.
(6) Each X},yj, and z, is periodic with the period 2-k.
(c) z(0)=0; therefore z,(0)=0 (j=0, 1, 2, . . . oo).
(d)z'(0)=c; therefore z' (0)=c, J(0)=0 (j=l, 2, . . . oo).
(34)
(35)
The last property follows from the fact that the arbitrary y occurs alwayswith the arbitrary c in the form c+y, and can be put equal to zero without
loss of generality.
The differential equations are
x''-2(l + 5)y'-(l+6yQ x+\ V3(l-2 M)</]=
(l + 5)2
[X 2 e+X 3 e2 + ],
j/"+2(l + 5)x'-(2+5)2
[|V3(l-2M)x+fi/]=
(l + 5)2
[F s e+F 3 e2
+--.],
z" + (l + 5)2z=(l + 5)
2
[Z 2 e+Z 3 eM----}
(36)
The series (34) are to be substituted in these equations and the coefficients
of the powers of e equated. The x )t y t ,and z t are determined by solving the
equations thus obtained and imposing on the results the conditions (35).
The terms independent of e are given by the equations
4-2y' -f x -
f V3 (1-2 M) ifc-0,
y':+2x'-
J V3 (1 -2 M) x,-
f y,-0,
The general solution of these equations is
x - a 01 e<" *=* T+ a 0l e~ =* T+ a M e> ^ T+ a' 02 e~" *=* T,
y=b 0l e"^ + b' 0l e->^+ bm e"^= iT + b' 02 <T
z = c 01
'
[cos T-fCo, sin t.
(37)
-<72 \/ 1 T(38)
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320 PERIODIC ORBITS.
By condition (6), we must have
Ooi=
Oqi=
a,*.= <4 = K = Ki = K = b' - 0,
and conditions (c) and (d) give respectively
c 01=
u, c 01 c.
The solution satisfying the given conditions is, then,
Zo="0, 2/o=
0, z = c sin t. (39)
The coefficients of the first power of e are given by the equations
3 /5M_9..W_ = 3V3V n __ c9 ^ f (40)
8 V
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OSCILLATING SATELLITES, EQUILATERAL-TRIANGLE CASE. 321
For the general terms we proceed by induction. Suppose that x s , y, ,z s ,
and Sj(j =0, 1, . . .
,n 1) have been determined, and that for j even
it has been found that
to*#-0, Vj = 0, z ] ='[c t coB(2k+l)T+c*8\n(2k+l)T];
while for j odd, it has been found that
O+l) /J
X]= 2) [a * cos 2 kT+a t sin 2 krl,
t=i
a+ti/s
2/>= 2) [&*cos2A;r + &t sin2A;r],
*=i
z, = 0, 1,-0.
It can be readily shown that when n is even the coefficients of en
are
given by the equations
x:-2y' n -\x n -\Vz{l-2>x)y n= 0,
y:+2x' n -\Vz{l-2,x)x n -\y n= 0,
2:+ Z =-2^ n sinr + 2[^ ,
+1 cos(2i+l)T+C^ 1 sin(2.;-+l)r].1=0
(44)
In order that the last equation shall have a periodic solution we must imposethe conditions
-2c8 n +C'(n) =
0, CT-0.
The first relation serves for the determination of 5 . Since by the existence
proof the periodic solution is known to exist it follows that the expression
C? is zero.
An additional proof that Cf is zero is obtained by considering the
integral (5')- The series for x, y, z are substituted and the terms are
re-arranged as a power series in e. Each coefficient of this series must reduce
to a constant identically in t. Consider the coefficient of e". The terms
of this coefficient which carry z are
2z'J+2z,z n . (45)
Suppose x a ,. . .
,x n ; y ,
. . ., y n ;
z,
. . ., i ; S
l ,. . .
, 5_i have
been determined and that the x, , y s ,and z, are periodic. The equation
for the determination of z has the form
z'+z^ysinr+C co&T+
On integrating, the following non-periodic terms are obtained
z =\t\t
cost| Cf
t sinr.
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322 PERIODIC ORBITS.
When the expressions for x J} y,, and z, as functions of r are substituted in
the coefficients of e" in the integral, the only non-periodic terms obtained
come from z n . They are of the form t, t sin2r, and r cos2r. Since the
coefficient of t is a constant identically in t, it follows that the coefficients
of these non-periodic terms are zero. Those for r sin2r and r cos2t vanish
identically. The coefficient of r gives the relation c C = 0. Since Ct*Q, it
follows that CT = 0.
Upon integrating (44) and imposing conditions (35), we find
z n = Cm cos t + c' nl sin r,^gs
The quantities Cj"Vi and CJJJ, are known from the differential equations.The constants of integration c nl and c' nl are found by (c) and (d) of (35) to
have the values
c _ y CSV, ,'<>_ y (2,/+i)0 .
When n is odd the equations obtained from the coefficients of e" are
(n+l)/2
x:-2i/:-Jrc -fV3(l-2M)2/ fl
=
2 [4
(B>
cos2ir + 4'< >
sin2ir])
.7=0
Cn+l)/2
l/:+2*:-fV3(l-2/i)x.-fy.= 2 [Brcos2ir+5;w
sin2ir] ,
+* " -25 n csinr.
From the last equation it follows at once that 5 =0, for otherwise z will
not be periodic. Integrating these equations and imposing the periodicity
equations, we find
1 16/(4/-l)+27 M(l- M)-Sin^r
-(!6J
2+9) Af + 3 V3 (1
-2ju) Bf + 16,/g'"" 9
. "I
"[" l6/(4/-l)+27 M(l- M) COS^rj,
* Zf L I6j(4/-1)+27 M(1- M)-sm2jr
3 V3 (1- 2 M) A? ~
16j AT -(16 j
2 + 3) Bf 9. "IT
i 6 /(4/-l)+27 M(l- M) COS^rJ,
z n = 0, 5 = 0.
(47)
(48)
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OSCILLATING SATELLITES, EQUILATERAL-TRIANGLE CASE. 323
If we make use of the transformation by which the parameter e was
introduced, we have for the final series
x = x e + Xie2
+ x 2 e 3 +y = y e+y l e
2 +y 2 e3 +
Z = Z e+Zie 2 +Z 2 e 3 + (49)
the Xj, y u Zj, and Sj being given by (46) and (48). It is not difficult to
show that x, y n ,and z carry the factor c" +1
, and 5 the factor c", and that c
enters these expressions in no other way. Consequently in (49) the arbi-
trages c and e occur always in the combination ce . Therefore we may putc = \ without loss of generality. The final series then contain only the
arbitrary e.
An approximate idea of the shape of
the orbit can be obtained by consideringthe first two terms of (49) . These terms
were computed for /*= 0.01 and e = 0.5.
The projection on the x?/-plane is an ellipse
of small eccentricity whose center is on the
negative y-axis and whose major axis cuts
the positive z-axis. This projection is
shown in Fig. 5. The projections on the
xz and 2/z-planes are shown in Fig. 6 and
Fig. 7 respectively. The orbit thus con-
sists of two elongated loops, one above
Fk;. 5. Fig. 6. Fig. 7.
and the other below the xi/-plane, the double point being in the fourth
quadrant of the xy-plane.If r is replaced by t+7t in the expressions for Xj, y,, and z u then x s
and y, remain unchanged while it is seen that z } changes sign. It follows
that the loops are symmetrical with respect to the zy-plane, and that each
loop is described in half the period. For positive values of e the upper loopis described first, and the motion is such that the projection of the infini-
tesimal body on the xi/-plane moves in the clockwise direction. The period
of the motion is given by the relation T=
27r(l+5).
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324 PERIODIC ORBITS.
Orbits about the Point II. To each of the orbits about point I there
corresponds an orbit about point II. The proofs of the existence of these
orbits were omitted, since they are similar to those for the first point. Theseries for these orbits can be obtained easily from the corresponding series
for the first set of orbits. The differential equations for the orbits about
point II are obtained from those for the orbits about point I by changingthe sign of V3- The periodicity conditions to be imposed are the same
in both cases. It follows, therefore, that the solutions for the one case
can be obtained from those for the other by changing the sign of y/%. There-
fore, in order to get the series for the orbits about the second point we makethis change in the series already obtained for the first point. On referring
to equations (1) and (2), it is seen that the differential equations for point
I reduce to those for point II if the signs of y and r are changed. Hence
this transformation can be madegeometrically by
a reflection in thexz-plane
and a reversal of the direction of motion. Thus, it is easy to get an idea
of the shape of these orbits from those already discussed.
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Chapter X.
ISOSCELES-TRIANGLE SOLUTIONS OF THEPROBLEM OF THREE BODIES.
By Daniel Buchanan.
165. Introduction. This chapter treats of periodic solutions of the
problem of three bodies, in which two of the masses are finite and equal.
Thethird
bodyis
started at theinitial
timet
from the centerof
gravity ofthe equal masses, and the initial conditions are so chosen that it moves in
a straight line and remains equidistant from the other bodies. In I the
third body is assumed to be infinitesimal and the initial conditions are so
chosen that the equal bodies move in a circle about the center of mass.*
In II the third body is considered infinitesimal and the initial conditions
are so chosen that the equal bodies move in ellipses with the center of massas the common focus. In III the third body is considered finite and the
solutions derived have the same period as those obtained in I, and reduce
to those solutions when the third body becomes infinitesimal.
I. PERIODIC ORBITS WHEN THE FINITE BODIES MOVE IN ACIRCLE AND THE THIRD BODY IS INFINITESIMAL.
166. The Differential Equation of Motion. Let ml and ffi, be twofinite bodies of equal mass, and n an infinitesimal body. Let the unit of massbe so chosen that m1
= m2= 1/2; the linear unit so that the distance between
m, and m2 shall be unity; and the unit of time so that the Gaussian con-
stant shall be unity. Let the origin of coordinates be taken at the center
of mass, and the ^-plane as the plane of motion of the finite bodies. Let
the coordinates ofw,,
r 2 ,and
mbe
& ,ij
u 0; 2,
%, 0;and
0, 0, f respectively.If m1 and mt are started at the points 1/2, 0, and 1/2, 0, 0, respectively,
so that they move in a circle, then
Si= -It" |cos(*-0, Vi= -Vi= ^sin(t-t ).
*When the finite bodies move in a circle, the motion of the infinitesimal body can be completely deter-
mined by means of elliptic integrals. The problem was first solved in this way by Pavanini in a memoir,"Sopra una Nuova Categoria di Soluzioni Periodiche nel Problema dei Tre Corpi," Annali di Malematica,Series III, vol. XIII (1907), pp. 179-202. The elliptic integrals obtained by Pavanini were later developedindependently by MacMillan in an article, "An Integrablc Case in the Restricted Problem of Three Bodies,"Astronomical Journal, Nos. 625-626 (1911). MacMillan further developed the solution as a power series
in a parameter, the coefficients of which are periodic functions of t. The solution obtained in I has aclose relation to MacMillan's solution. 325
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326 PERIODIC ORBITS.
The differential equation for the motion of the infinitesimal body is
i i+4r2)s/2 w
where the accents denote derivatives with respect to t. The integral of (1) is
where C is the constant of integration. If C is positive, the particle n recedes
to infinity. If C is negative, the particle /x does not pass beyond a finite
distance from the origin. From a consideration of (2) it can be shown that,
if C is negative, the particle crosses the ij-plane. Hence the initial time
t can be chosen, without loss of generality, as the time when the particle
is in the ^rj-plane. It can also be shown from (2) that if C is negative,
the motion of the particle is periodic, and that the period can be expressed
as a power series in the initial velocity of n, whose limit is 27rA/8 as the
initial velocity approaches zero. We shall, however, prove the existence of
a periodic solution of (1) by a different method.
167. Proof of Existence of a Periodic Solution of Equation (I). In
this proof it is convenient to make the transformation
*-< d= VV8(1+8)t, (3)
where 8 is to be determined so that the solution of (1) shall be periodic in t
with the period 2t. At t = let
f = 0, t = a, (4)
where f = d{/dr. Let us make the further transformation
r~a*j (5)
then when (3) and (5) are substituted in equation (1), we obtain
_ (1+S)g /fi
x
(l+4aW* Wwhere z is the second derivative of z with respect to t. The initial conditions
for z become, as a consequence of (4) and (5),
Z (0) =0, i(0) = l. (7)
For \a\ sufficiently small the right member of (6) can be expanded into
the series
2=-(l +5)z{l+(~ 1
3/2)4aV+ +(
_t
3/2)(4aV)*+ }>
(8)
where
/-3/ 2 \ m \2)\~2j' ' '
\ 2~)
\ i / 1 2 i
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ISOSCELES-TRIANGLE SOLUTIONS. 327
Equation (8) can be integrated as a power series in a 2 and 6 which convergesfor 0^t^27t, provided |o| and \b\ are sufficiently small. Let us write this
solution in the form
z = 2 S Zu Sl
a". (9)
The initial values of the zit , as determined from (7) are
m(0)-0, (i,j =0, ... oo), 1
4,(0) -1, KM=o (i+j>o). j
Upon substituting (9) in (8) and equating the coefficients of the various
powers of 5 and a 2
,we obtain differential equations from which the z tJ can
be determined so that the initial conditions (10) shall be satisfied.
The differentialequation
for the termindependent
of 5 and a 2is
and the solution of it which satisfies (10) is
The differential equation for the term in 5 alone is
*u+H~ -sinr,
and the solution of it which satisfies (10) is
Zi,o=--|sinT + JcosT.
The solution of equation (8) is therefore
z = sirjT + 5[ttCOSt -sinr] + terms of higher degree in a 2 and 5. (11)
With the initial conditions (7), the variable z is an odd function of t,
and therefore a sufficient condition that it shall be periodic with the period2x in t is z(t) =0. With the value of z given in (11), this condition becomes
= \ 5+ terms of higher degree in a 2 and 5. (12)
Since the coefficient of 8 is different from zero, this equation can be solved
uniquely for 5 as a power series in a 2
, vanishing with a 2. Let us denote
this solution for 5 by
5= 2 Lo". (13)
When (13) is substituted in (9), we obtain
z=2z 2J a il, (14)
/=0
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328 PERIODIC ORBITS.
which converges for \a\ sufficiently small. Since the periodicity condition
has been satisfied, z is periodic in r with the period 2ir. Hence
Zll {T+2Tr)=Z tl {r)=
0, . . . oo). (15)
The initial values of z u as determined from (7) are
%/0)-0 (i=0, ... oo),
J
2o(0)=
l, 2,,(0)=0 0=1, ...oo). J
168. Direct Construction of the Periodic Solution of Equation (1). Let
us substitute (13) and (14) in (8) and equate the coefficients of the various
powers of a". Since the result is an identity in a 2
, there is obtained a series
of differential equations from which the coefficients of the solution (14) can
be determined.
The differential equation for the term independent of a 2is
z +z =0,
and the solution of it which satisfies (15) and (16) is
z = sinr.
The differential equation for the term in a 2is
z'j+z 2 =-5 2 z +6z2=(-5 2 +|)sinT-|sin3T.
The term sinr gives rise to a non-periodic term in the solution, and, in
order that (15) shall be satisfied, its coefficient must be zero. Hence
x _ 9Oo = '
2 2
and the solution for z t satisfying (16) is
z 2=
(sin 3r 3 sin t) .
The differential equation for the term in a* is
3+24=-(S4+^)sinT+6sin3T-gsin5T.
In order that (15) shall be satisfied, 54 must have the value
1418 *= ~
32'
and the solution for z 4 is found to be
z t=
^[431sinr-192sin3T+29sin5r],
where the constants ofintegration have been determined so as to satisfy (16).
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ISOSCELES-TRIANGLE SOLUTIONS. 329
So far as computed, it has been found that the 52y are uniquely deter-
mined by the periodicity and the initial conditions, and that each z 2j is a sumof sines of odd multiples of r, the highest multiple being 2./+1. We shall
now show by an induction to the general term that all the 8 2J are uniquelydetermined by the same conditions, and that all the z v have the propertieswhich have been stated. Let us assume that 5 2 ,
. . .,
52i_ 2 ;
z,
. . .,z v - 2
have been uniquely determined, and that each z n (k = 0, . . ., j\) is a
sum of sines of odd multiples of r, the highest multiple being 2A;+1. Fromthese assumptions and the differential equations it will be shown that
5 2] and z 2J are uniquely determined, and that 2^ is a sum of sines of odd
multiples of r, the highest multiple being 2j + l.
Let us consider the term in a". The differential equation is
Z2> + Z2J
= hlZo + Zi)) (17)
where Z 2j is a known function of z 2< (i=0, . . . ,j 1) and 8 it (k=
1, . . ., j 1).
The general term in Z 2J has the form
T =z"' z"* h"
k
where Xu . . .,Xt ; p,, . . .
, n t ; p, and q are positive integers (or zero)
having the following properties:
(a) Mi+ +Mt is an odd integer,
(6) Hi\+ +/x*X t +Mi+ +Ptl+qp = 2j,
(c) q is or 1.
Since each z,
. . .,
z 2 _,_ 2 is a sum of sines of odd multiples of t, it
follows from (a) that Zti is a sum of sines of odd multiples of t. Thehighest multiple is
tf = Mi(Vfl) + + l x t (\ k +l)=2j+l-qp.The highest value of N1} is obtained when q = and is, therefore, 2,7 + 1.
Hence (17) has the form
'4,+z s ,= [-5 2 ,+aHsinT+arsin3r+ +a 2 1 sin(2j+l)r, (18)
where a?", . . . , ati are known constants. From (15) it follows that
o 2y -ai .
The solution of (18) satisfying (16) is therefore
z il=Asm T+Asm3T+ +AZ J
l l sm(2j+l)r,
where
A? - tScm>+ a&.,a/>
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330 PERIODIC ORBITS.
Hence the periodic solution of (1) in terms of the variable r is
29siniv7-l
(19)f = ^ = asinT+^a
1
[sin3T-3sinT] + ^[431sinT-192sin3r+29sin5r]1
16 i"'""' J '
256
+ +a tJ+l [A? n 8\nT+A? i>smZT+ +A^sm(2j+l)r] +It is a power series in odd powers of a with sums of sines of odd multiples
of t in the coefficients. The highest multiple of t in the coefficient of a 2t+1
is 2k +1. In the sequel we shall call such a series a triply odd power series.
The period in t is 2tt, and in t it is
2tt r-. , 9 . 1414,35 6_i_ I1
II.
SYMMETRICALPERIODIC ORBITS WHEN THE FINITE
BODIESMOVE IN ELLIPSES AND THE THIRD BODY IS INFINITESIMAL.
169. The Differential Equation. Let m, and m^ represent the twofinite bodies and fx the infinitesimal body. Let the system of coordinates be
chosen as in 166. Let the unit of mass be so chosen that m1= mi
= l/2,
and then let the linear and time units be so determined that the mean dis-
tance from m1 to m2 and the gravitational constant are each unity. Withthese units the mean angular motion of the bodies also is unity.
Let n be started from the center of gravity of ra x and m2 perpendicu-
larly to the plane of their motion when they areat
apsidesof
their orbits,which can be assumed to lie on the -axis. From the symmetry of the
motion with these initial conditions, it follows that
i= h %=
^i
Let the motion of the finite bodies be referred to a system of axes
rotating about the f-axis with the uniform velocity unity. The coordinates
referred to the rotating axes are defined by
X|=^cos<-|-r7 < sin^ y t= ^sinZ+^cosJ, z = f ({=1, 2).
The Xi and y { are determined by the conditions that mx and mt shall movein ellipses and be at apsides at t = t
,which in this case is put equal to zero.
Then it follows, from the properties of elliptic motion, that
where
x x= x 2
= rcos(t> t), y t= y 2
= rsm(v t), (20)
r =m[l ecosf+y (1 cos2<) ]>
v =< + 2esinH-Je2sin2<+ ,
m= m1= mi
= > e = eccentricity of ellipses.
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ISOSCELES-TRIANGLE SOLUTIONS. 331
The differential equation for the motion of /x is
mz mz 2mz /01 s2=
~T"T="^T' (21)
wherer
1 =r i= Vx\+ yl+z
2 ^ Vxl+ yl+z2
.
When we substitute the values of x t and y t from (20), equation (21) becomes
2mz
[m2
{l-2ecos + |-(3-cos20+ } +z 2
]
r , 2T , tU"(22)
Where m occurs in the denominator we shall substitute its value 1/2,but in the numerator we shall make the substitution ra = ra +X, and con-
sider X as a variable parameter while m and m both remain fixed. In orderto obtain the solution of the physical problem we must put \ = m ma in
the final results. With these substitutions, (22) becomes
2" = -(m +X)2#2, +1 2w+1, ( 23 )
J=0
where
# 1 =+16[l+3ecos*+-^(l+3cos20 + ;
*
]
E s =-96[l+5ecost+ ]>
and where each E 2J+l is a power series in e with cosines of integral multiples
of t in the coefficients, the highest multiple being the same as the exponentof the eccentricity e.
170. Determination of the Period by a Necessary Condition for a
Periodic Solution of (23). If the motion is periodic, let the period be denoted
by T. Since the period of motion of the finite bodies is 2t, we must have
T = 2vtt, (24)
where v is an integer which denotes the number of revolutions made by the
finite bodies in the period T.
Let us take the initial conditions
2(0) =0, z'(0) = a. (25)
With these initial conditions it can be shown from (23) that z is an oddfunction of t. Hence if fi is started from the ^-plane when ml and m^ are
at apsides of their orbits, a necessary and sufficient condition that z shall be
periodic with the period T is
2(772) =0. (26)
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332 PERIODIC ORBITS.
In order to determine the period T from the condition (26), we inte-
grate equation (23) as a power series in a and X, but only in so far as the
term of the first degree in o is concerned. The differential equation for
this termis
< +m lZl ,
= 0. (27)
This equation belongs to the class of differential equations with periodiccoefficients which was treated in Chapter III, where it was found that the
character of the solutions depends upon whether or not 4 Vm is an integer.
Since m depends upon the way in which m is separated into ra +X, and since
|X| must be taken small in order that certain solutions appearing in the
sequel shall be convergent, the value of ra is in the vicinity of 1/2. Wemay therefore regard 4vw^ as not an integer, and when it is not an integerthe
generalsolution of
(27)is
zli0= AT e'^^'u.+Af e-^ 1 u 2 , (28)
where u t and u 2 are conjugate complex functions of the form
m, = 1+ 2J 25 f + x/ ~ T a *B)
sin ^+C ( cos ^ -1)1 e ">
n=l *=i
oo n
M,i+2! 25[~^- Ta iB>sin ^+ 6 iB)
( cos ^- i ^ e "'
a / .3Vmi(l I6m ).
,
The A and Af are constants of integration; the af and b are real con-
stants which depend upon the coefficients of the various powers of e in E 1 ;
and a is a power series in e with real constant coefficients, determined bythe condition that u y and u^ shall be periodic with the period 2r.
From (25) we have
Af+Af^O, WV^T+u[(0)]Af) + [-<rV^T+u' (0)]Ar = a. (29)
Since wj(0)=
wi(0), the determinant of the coefficients of ^4{
0)
and A in(29) is
A = - 2[<rV=T+u[ (0)] = V=T A, , (30)
where A, is real, and it is different from zero because A is the determinant
of a fundamental set of solutions at t = 0. The solutions of (29) for A and
A? are
Am = A =A,
Since A vanishes with o, and conversely, it is convenient to integrate
(23) as a power series in Af and X.
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ISOSCELES-TRIANGLE SOLUTIONS. 333
The form of the solution of (23) arranged as a power series in A 1
is
z = A?)
[ev= T 'u 1 -e-'
rV =i t u 2 ]+A?)
P(A?)
, X,e; t),
where P is a power series in Af, X, and e. Upon imposing the condition(26) that z shall be periodic with the period T, we obtain
O^r^Dly^ - e-" /=t
^]+A?P(Ar i X, e; f-)- (31)
This equation is satisfied by Af =0, but this value of Af leads to the trivial
solution z=0. In order, then, that (31) shall have a solution for ^4.{0) which
is different from zero, the coefficient of A must be zero. Now
uA)=
1, or 1+a power series in e,
according as v in (24) is even or odd respectively. Hence u x {T/2)t for |e|
sufficiently small, and we have
e "^T- e-"^= i i=0. (32)
In order that this condition for the existence of a periodic solution may be
satisfied, T must have the value
T= 2Nir>
where N is an integer which denotes the number of oscillations made bythe infinitesimal body in the period T. Then it follows from (24) that
N= v*. (33)
If a is a rational fraction, v can be so chosen that iV will be an integer. Inas-
much as a is a continuous function of e, it can be made a rational fraction
by a proper choice of e less than any value of \e\ which will insure the
convergence of the power series. The numerical values of N and v can be
obtained when a has been determined as a rational fraction.
171. Existence of Symmetrical Periodic Orbits. Let us consider the
terms in (23) of higher degree in Af and X. The differential equationwhich defines the term in Af\ is
3&+ro, E l Zu = Z l =-\A? ) E l if=
m,- e~ *=* u 2 ) . (34)
The complementary function of (34) is the same as that of (27), viz.,
ha = of ^ aV='1 ' i+i u
e- av=lt u 2 .
On using the method of variation of parameters, we obtain
>)' Ota/^T^-K) e'^'+ia?)' (-aV^Tu.+ui)e~ a ^ i = Z
l.
'
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334 PERIODIC ORBITS.
The determinant of the coefficients of (a{l)
)' and (c4l)
)' is a constant, by18, and is the same as (30), viz., A = V 1 A^O. Therefore
(a|T. + V=lr"m . _ V=T*A?B,
(_ e -,.^,
^(a l)y = _ V^Ie+^Z, = _ V=l\ATB l
(MiW2_ e+s ^, MD.
Ai Ai
(36)
The integration of (36) gives non-periodic terms as well as periodic terms
having the period T. We shall be concerned only with the non-periodicterms. Let the constant part of V 1 E xu x ujb>. x be denoted by P,; it is a
power series in e with constant coefficients which are purely imaginary, the
absolute term of which is found to be 2V l/Vm^. Hence
a{ =A (
1
l) +\A? )
[P x Aperiodic terms],
a2i = A?+\A [P x Aperiodic terms],
where A and A are constants of integration which are to be so determined
that z u (0)=
z[, x (0)= 0. Then
z u = XA P x t [ea **> u x + e v ^'
u,] + periodic terms. (37)
It is necessary to obtain in addition to this only the term in (Af)'.
This term is obtained from the differential equation
z'^+m E x zM= Z s= - m E 3
z3
. (38)
On forming the equations analogous to (36), we have
(39)
The terms not written in (39) carry the exponentials e* iJaV=It =1, 2) as
a factor multiplied by the fourth power of u t and u t considered together.
The integration of these terms gives periodic terms with the period T.
Let the constant part of 3V lm- E t u\ u\/k x be denoted by P 2 , a power seriesin e with constant coefficients which are purely imaginary, the absolute term
of which is found by computation to be 36V lVm . Then upon integrating
(39), we obtain
a? = A?) + (A?
)
Y [P t Aperiodic terms], 1
a? = A?+(A'0)
y[P i t+periodic terms],(40)
where A and A are constants of integration which are to be determined
so that z 3 , (0) =zj, (0) = 0. Hence
z
M= (A'
0)
yP1 t[e''
v~ t u l +e-''v ~'u % ]
+periodic terms. (41)
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ISOSCELES-TRIANGLE SOLUTIONS. 335
Now imposing the condition (26) that z shall be periodic with the period
T, we obtain from (37) and (41)
0=f \\ATP l +{ATYP^\ea
^^u 1 (^)+e-a
^^u 2 (^)]+ (42)
The expression [eaV=Tr/2 u 1 (T/2)+e-"
v =~ T/2 u 2 (T/2)] is different from zero
for e = 0, and therefore remains different from zero for |e| sufficiently small.
Equation (42) is satisfied by Af =0, and hence the right side carries Af
as a factor. In order to find a solution of (23) other than z=0, it is necessaryto consider Af ^ ;
therefore the factor Af can be divided out of (42) . There
remains a power series in X and Af and, since P x and P 2 are different from
zero for \e\ sufficiently small, the terms of lowest degree are X and C4i0)
)2
.
There are no terms in A^e and e alone, and the coefficient of (A^Y has aterm independent of e, viz., 36V 1 Vm . Hence, after Af is divided out,
equation (42) can be solved for A* as a power series in X*, the coefficients
being power series in integral powers of e.
Two periodic solutions of (23) therefore exist having the period T.
They have the form
2 = \JQ(X*;*),
where Q is a power series in X } whose coefficients are power series in e.
In the practical construction of the solutions it can be shown that z is a power
series in odd powers of X4
. This fact follows also from the dynamical natureof the problem, since the motion of m is obviously symmetrical with respect
to the xy -plane. The two solutions are therefore of the form
z=z 2i+1 \, (43)1=0
where each Z2 J+ i is periodic with the period T.
In 117-118 it is shown by a discussion, which is applicable in this
problem, that if v is even in (24), the orbits obtained by taking the two signs
before X}
are geometrically the same, but in the one the infinitesimal body is
half a period ahead of its position in the other. If v is odd, the orbits for
+X } and X 1 are geometrically distinct.
By an argument similar to that in 115, it can be shown that it is pos-sible to choose X > so that the solutions (43) will converge for all
=t^ T.
172. Direct Construction of Symmetrical Periodic Solutions of (23). Let
us substitute (43) in (23) and equate the coefficients of the various powersof X*. The constants of integration occurring at each step are determined
by the conditions that the orbits shall be symmetrical and periodic with
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336 PERIODIC ORBITS.
the period T. The condition to be imposed in order that the orbits shall
be symmetrical is z(0) = 0, from which it follows that
'21+\ (0)=0 (j=0, . . . ao). (44)
It is necessary to consider the terms up to X 6/2 before the induction to the
general term can be made.
The differential equation for the term in X jis
z" + 171^ = 0,
and the solution of this equation is (28). When (44) is imposed
z,= AJ" [e
av=liu,- e~' v=Tl
m,],
where A is an undetermined constant.
The differential equation for the term in X3/2 is
z' i '+m E 1 z 3= Z 3
= -E 1 z 1 -m E 3 z31 . (45)
When expressed in terms of t, the right side of (45) has the form
Z 3 =A?d?HA?) 3
e?, (46)
where the 6 3t+l)
(i=0, 1) are homogeneous and of degree 2t+l in e +irvCrT ' ande -<rv=r< The undetermined constant A"' is written explicitly so far as it
occurs. In ^"(t-O, i) the coefficient of [***=]* [e^*^* differs from
the coefficient of [eaV=ll
}
h [e~ aV:=li\
honly in the sign of V 1, /, and j t
being positive integers (or zero) such that./ 1 +,7 2 = 2t+l. These coefficients
are power series in e with V 1 s\n jt and cos,;7 in the coefficients, k beingthe highest multiple of t in the coefficient of e*. If the exponentials in d 3
i+l)
are expressed in trigonometric form, it is observed that the dfi+l) are power
series in e in which the coefficient of e" has the form
V=T2 cf+,,
sin[(2t+l)r+j]<,
where the cft+l) are real constants. Hence the 0f
+ "are purely imaginary.
In order that z 3 shall be periodic, the constant parts of the coefficients
of e +aV=T ' and e~ aV=~1 ' in u 2 Z 3 and u^Z 3 respectively must be zero. From
the form of u lt u^, and Z 3 it follows that, when we equate to zero the
constant parts of these coefficients, we obtain only the one equation
-AA;i)
[P 1 +(A{1,
)2
^] =0, (47)
where P x and P 2 are the power series which appear in (42). Equation (47)
is satisfied by A"' = 0, but this value of A"' leads to the solution z =and is excluded. The solutions of (47) for A{" which are different fromzero are
A?=
V=i Pl , (48)
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ISOSCELES-TRIANGLE SOLUTIONS. 337
where p tis a power series in e with constant coefficients which are real since
P t and P 2 are both purely imaginary and their absolute terms have the same
sign. The absolute term of p t is found by computation to be 1/Vl8ra .
Since A is
purely imaginarythe expression for z 1 is real. When the
signof V 1 is chosen in (48), the periodic solution of (23) which satisfies the
initial condition 2(0) =0 is unique.The general solution of (45) is
z 3= A? f^'^+A" e-^'^+V^t^+^f], (49)
where Af and Af are the constants of integration. The particular integrals
<pf and <pf are respectively of the same form as Of and Of in (46). Fromthe form of <pf and <pf it follows that
and imposing the condition (44) on (49), we obtain
Af+A? = 0.
The solution (49) therefore becomes
z 3= A? [e
a ^<u-e-"^'u 2 \ + V^i[<pt)
+<pr\, (50)
where ^4{3) remains undetermined at this step. If Af is found to be purely
imaginary this solution for z 3 is real.
From a considerationof
the termsin X5/2
, and then by an inductionto
the general term, we shall show that Af and all the remaining constants of
integration are uniquely determined, after the choice of the sign in (48) has
been made, by the conditions that the orbits shall be symmetrical and peri-
odic with the period T.
The differential equation for the term in X6/2 is
z':+m <>E l z i
= Z i =-[E l z i +Zm E i z\z 3 ]-{E iz i
l +m a E i z\]. (51)
In order that 2 5 shall be periodic the constant parts of the coefficients of
e +v=Tt an( j e -<rv=ri m Ul Z i an( j Ui z & respectively must be zero. From the
form of u lt u 2 , and Z 6 it follows that, when we equate to zero the constant
parts of these coefficients, we obtain only the one equation
AfPf + V^TP 3=
0, (52)
where Pf and P 3 are power series in e with real constant coefficients which
are unique after the sign of V 1 in (48) has been chosen. The absolute
term of P{3>
is found to be 32, and therefore the solution of (52) is
4T*V=Tft, (53)
where p, is a power series in e with real constant coefficients.
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338 PERIODIC ORBITS.
With A? determined as in (53), the general solution of (51) is periodic
and has the form
Zi= A? e'^'u.+Af e-'^'Ut+V^lW+tpT+cp?], (54)
where A and A are the constants of integration and where the ^ 2i+1)
(i= 0, 1, 2) are of the same form as the 0f'
+l)(i=
0, 1, 2), respectively. It
follows from the form of <p, ipf, and <pf that, when the condition (44) is
imposed, (54) becomes
z i= A?[e + ^>u-e-<'^u 2 \ + V=iW?+ lp?+<p?\ (55)
where Af remains as yet undetermined. This solution for z 6 is real if Afis purely imaginary.
We shall now make the induction to the general term. Let us supposethat A, . . .
, Af"~3) have all been uniquely determined as power series
in e with constant coefficients which are purely imaginary. Let us also
suppose that z x ,. . .
,z 2a _j have been uniquely determined and that they
are of the formi
e^-A*** [e'^%-e^^%] + V^T^<p^ tjfe-l, . . ., n-1), (56)
1=0
where the <pf Xi are f the same form as the ^2 ' +1) in (46) t = 0, . . .
, k. It
will be shown that AfB_1)
is purely imaginary and is uniquely determined
by the condition that z in+1 shall be periodic; also when the condition
z tn+l (0) = has been imposed, that z in+1 has the form
=0
where the <p 2^ are of the same form as the <p+" in (56).
Let us consider the term in \ (2 "+ 1) / 2. The differential equation for this
term is
z" n+i +m E l z 2n+1 = Z 2n+l =-[E l z 2n - 1 +3m E s zlz 2n _ l ]+ (57)
The part of Z 2n+1 not written explicitly involves z x ,. . .
,z 2 _ 3 to odd
degrees when considered together. The only undetermined constant which
enters Z 2n+i is A J
2 " -0,and it has the same coefficient in Z 2 +1 that A has in Z 6 .
In order that z 2n+i shall be periodic, the constant parts of the coefficients
of e +aV=r ' and e~ aV=lt in u 2 Z 2n+l and u l Z 2n+x respectively must be zero.
Now since Z 2n+l is similar in form to Z 6 , we obtain only one equation whenthe constant parts of these coefficients are equated to zero. The form of
the equation is
A e-i>p<3> + ^CT ?|H = 0> (58)
where P in -\ is a power series in e with real constant coefficients. Thesolution of this equation for Af~ l)
is
AT^-V^Ta*-,, (59)
where p 2n -i is a power series in e with real constant coefficients.
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ISOSCELES-TRIANGLE SOLUTIONS. 339
In general, there are no other terms in UjZ 2 +1 and u x Z 2n + x which yield
non-periodic terms in z 2 +i. But since <r = N/v, N and v being integers,there are values of n for which other non-periodic terms than those alreadydiscussed can occur. It follows from the
propertiesof Z
2n+lthat u
2 Z 2n+lcontains the term
(60)
K (A?)
)in+l
e<*"+ *= II
[a +a 1 cos+ +a t coskt+
+V=Tb 1 sinf-t- + v^T&* sinkt+ ],
where K is a constant. Now
e (2,.+l)<r V=It = e<r V=Tl
[ cog 2 n<T t + V^T Sill 2U(Tt}.
Consequently these non-periodic terms arise if k = 2n<r, k an integer, or
if kv=2nN. This relation is satisfied if 2w becomes a multiple of v.
Suppose v is odd. Since v and N are taken relatively prime, the smallestvalues of n and k for which the non-periodic terms in question can arise
are n = v and k = 2N. If v is even, N is odd; and the smallest values of nand k are n = v/2 and k = N. The terms in which these non-periodic terms
first arise are multiplied by x (2v + 1) /2 e2JV
or x'+'e*, according as v is odd or even.
After these terms first appear they in general occur similarly at all subse-
quent steps. When they are present, the equation analogous to (58), in
so far as the terms in u 2 Z 2 +i are concerned, is
^r- ,, ^ 3) +v^TP 2n _ 1 +^ 2n _ 1 (A?r +,=o, (6D
where K2n - t is a constant multiplied by e
2JV
or e
Naccording as v is odd or
even. The terms in u x Z 2n + x corresponding to (60) differ from (60) only in
the sign of K and V l. Non-periodic terms arise from these terms in the
same way as from (60). The equation analogous to (58), in so far as the
terms in u t Z 2n+l are concerned, is the same as (61). This equation can be
solved uniquely for Afn ~ l) and the solution is of the same form as (59).
Hence in all cases A{2n_1> can be determined by the symmetrical and the
periodicity conditions.
With Afn ~ l) determined as in (59), the solution of (57) is periodic. The
general solution of (57) is
z 2n+l = Ar +l) e'^'u l +Ar +1) e-' r
^t u 2 +V^T I rfjf.
<-0
From the form of pSJj? it follows that
<Cff(o)=o,
and when the condition (44) is imposed, z 2n+l becomes
where Af +1) remains undetermined at this step. This solution is real if
Af n+l)is purely imaginary. This completes the induction.
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340 PERIODIC ORBITS.
III. PERIODIC ORBITS WHEN THE THREE BODIES ARE FINITE.
173. The Differential Equations. We shall now consider the question
of the existence of orbits whichare
periodicwhen
mis
finite,and which
have the same period as those obtained in I. The question is one of deter-
mining initial conditions for mt ,ra 2 , and /x so that the motion of the system
shall be periodic when it is finite, and shall have the same period as when
M is infinitesimal.
The origin of coordinates will be taken at the center of mass of the
system. The plane passing through the center of mass and perpendicular
to the initial motion of n will be taken as the i^-plane. Let the coordinates
of m1 , m, ,and /* be & , to > f 1 J & > to ft ;
and |, r?, f respectively. Let the
values of , t\, f, ', v', to, to, &, and f 2 be zero at t = t . Further, let
i(0 = -&(0 =\, (4)--(4), *U)--i(0.
Under these symmetrical initial conditions
&" &, to" -to j fi"fi- ( 62 )
On making use of (62) in the center of gravity equations, which are
m1 1 +m 2 ! +/z =0, m1 i? 1 +m i! t? 2 +A"? = 0, m^+m^+nt =
0,
we have
e*o, r,+Mr=o. (63)
Hence /x always remains on the f-axis.
With the units chosen as in 166, the differential equations are
* Sr 8[^-{-(I+m)^
2]372 '
"- _!2i_ Mtoto- 8/ .s
[,
J + (1+/i)*r .]i/i'
(1+M)f-d+M) 2
f
where r* = t* -\-rj>1 . Let us transform (64) by the substitutions
f "[r
2 +(l+ M)2 rf 2
(64)
i= rcost>, Tj^rsin^ f = Vi/8(1+5)t, (65)
where 5 has the value determined in I. Then equations (64) become
r rift |
(1 + 8) _ M(l + g)r
rti+2ft> =0,
_ (i+M)d+8).rf
8[^+(i+M)rf
(66)
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ISOSCELES-TRIANGLE SOLUTIONS. 341
For /x= these equations admit the solutions
= vV8(1 + 8)t, r=\, r f,
where ^ is the function defined in (19) and is periodic in r with the period 2-ir.
Now let
r-f(l-fp), =VV8(i+)t+m, f*#+w, (67)
where p, u, and w vanish with /x= 0. When equations (67) are substituted
in (66), the differential equations for p, u, and w are found to be
(i+p)ii+2p(Vi/8(l+)+M)=0,
,v, _ (i+a)(i+/i)Ofr+ttO , (i-h[(l + p)
2
+4(l4- M)2
(^+;)2 F 2
"t
"[l+4^r-
The second equation of (68) admits the integral
=(T^p
- VV8(i + 8), (69)
where d is an arbitrary constant. Since u and p vanish with /x, we substitute
where do = WsU + S), and X is an undetermined constant. On substi-
tuting (69) in (68), we obtain
4; , (1 + S)P _ 2doX/*+XV _ m(1 + 8)(1+p)Pi ~8(l+p)
3
(l+p)3
[(l+p)2 +4(l+ M)
2
(^+w;)2
]3/2
'
_ (1 + 8)(1+m)(*+u>) (1 + 5)^U> "
[(l+p)2 + 4(l+M)
2(/'+^)
2r +[H-4^] 3/2
'
(70)
174. Proof of Existence of Periodic Solutions of Equations (70). For
M= equations (70) admit the periodic solutions p = p = w = w = 0. It will
now be proved that if|/x| is not zero, but sufficiently small, equations (70)
admit solutions expansible as converging power series in /x, which vanish
with fj. and which are periodic in r with the period 2t.
Let us take the initial conditions
p(0) = a M p(0)=0, tc(0) =0, w(0) = a 2 . (71)
With these initial conditions it can be shown from the properties of (70), bythe usual method, that p is even in t and that w is odd in t. Therefore
p(7r)=p(-7r), ()(-),and if the conditions
P(tt)=w(t)=0, (72)
are satisfied, p and to will be periodic in t with the period 2w.
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342 PERIODIC ORBITS.
Equations (70) will now be integrated as power series in at , a, ,
and n
in so far as the a, and a, enter linearly in the solutions. If the terms of the
solutions in which the aj and o, enter linearly are denoted by p, and wl ,
then the differential equations defining p t and wt are
p 1 +i(l+3)p 1= fi = 2VV8(l + 8)X/x
-( f j
1^wMi+s)[i+.^(~]
/2
)(2j+i)(4^y]w 1 =w 1
| (73)
- - (f^ -(i+^( 2 ^+ W) S ("J/, )iW^
The first equation of (73) is independent of the second equation. The
complementaryfunction of the first
equationis
p 1= ^ 1 cosVV8(l + 5)T+5 1 sinVi/8(l + 5)T,
where A v and B x are constants of integration. The function P x involves ^to even degrees, and is therefore a power series in a 2 with cosines of even
multiples of t in the coefficients. The highest multiple of t in the coeffi-
cient of a 2iis 2k. In the sequel, such a power series is called a triply even
power series. The particular integral arising from P t is a triply even powerseries unless WsU + S) is an even integer. If Wsd + S) is an even integer
the left side of the first equation of (73) has the same period as certain
termsof the
right side,and the solution will therefore contain
non-periodicterms. When V 1 /8(l + 5) is an even integer, the period of the motion of
m^ and mt is an even integral multiple of the period of the oscillations of m-
The mutual attractions of the three bodies will then have a cumulative
effect and produce non-periodic motion. We therefore exclude from our
consideration those values of a for which Wsd + S) is an even integer.
With this restriction upon a, the solution of the p r equation satisfj'ing the
initial conditions (71) is
p 1= [a 1
-t xC l (0)] cosWsU-r-^T+M^r), (74)
where C^t)is
a triply even power series, and it contains X as an undeter-mined constant.
When (74) is substituted in the i^-equation, all the terms of Wx are
known. With the left side simplified and Wl= 0, the equation becomes
iv1 +[l+I18 t ,a
v~\w 1
= 0, (75)
where
0*= ~
f + 9 cos2t, e t- g - 48 cos2t + ^ cos 4t,
and where each 2 * is a sum of cosines of even multiples of t, the highest
multiple being 2k.
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ISOSCELES-TRIANGLE SOLUTIONS. 343
Equation (75) is one of the equations of variation, and the expression ^(t)or \{/\(t t )/V 1 /&(l + 5)\, obtained in (19), is the generating solution. Twoarbitrary constants, viz., t Q and a, appear in its generating solution, and
accordingto 32
and 33 the two fundamental solutions of (75) are obtainedby taking the first partial derivatives of \p with respect to these constants.
One solution is therefore
wii=|-*tt*-o)/Vi/8(i+8)} --;*(*)>
and it is periodic in r with the period 2tt. This solution contains the
factor a, and since it is multiplied later by an undetermined constant
the factor a may be absorbed by the undetermined constant. This solu-
tion can then be expressed as (see page 330 for \f/)
wn = <p= 2 <p i]a 21 =cosT + a 2
(cos3r cosr)+ , (76)
where a<p = d\p/dr. Therefore the (p 2Jare sums of cosines of odd multiples of
t, the highest multiple being 2,7 + 1. The initial values of this solution are
wn (0) =l, t%(0)=0. (77)
The other solution of (75) is obtained by differentiating the generatingsolution with respect to the constant a; hence this solution is
w- = Ya-\da) + ^Ya' (78)
where (-^r) denotes that the differentiation is performed only in so far as a
occurs explicitly. Now
(H)=sinr+^a
2
(sin3T-3sinr)+ ,
dr drdb f 9 . 465 3 , "I
Ta=
dlTa= T l-2 a+ aJr - ' T
and therefore the solution (78) is
w12 =[sinT+Ja2
(sin3r-3sinr)+ .]+,-*[- a*+^a* ]
The initial values of this solution are
wu (0)=0, ti; 12 (0)=fi =l-|a
2+f a<+
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344 PERIODIC ORBITS.
Since it is more convenient for computation to have a solution wu in which
the initial values are
wu (0) = 0, wu (0) = l, (79)
we take as the second solution of (75)
w^=x+At<p, (80)
where x and A are found by computation to be
X= S XijaM=
sinr+^a2
(5sinT+sin3r)+ ._*,
i>Hk+f*+From the way in which x has been derived, viz.,
B\daJ
it follows that each x 2 jis a sum f sines of odd multiples of t and that the
highest multiple is 2j+l. Further, since
adrp= --^>
it follows from the character of \p that the coefficients of the cosines and sines
of the highest multiples of r in <p v and w respectively are equal numerically
andhave the same
sign.The solutions (76) and (80) constitute a fundamental set of solutions,
since their determinant is unity, and hence the general solution of (75) is
Wl = nrcp+r>ll)
[x+AT<p], (81)
where wf and n^ are constants of integration.
When (74) is substituted in (73), Wx becomes an odd power series in a
with two types of terms in the coefficients.
(1) There are terms not multiplied by cosV 1 /8(1 + 5)t which enter
through p x and they form a triply odd power series. They have n as a
factor and will be denotedby
n Mt .
(2) The remaining part of W, consists of terms which are multiplied
by cos\/ 1 /8(l + 5) t. As we have already excluded those values of a for
which W8(l + 5) is an even integer, and as we subsequently exclude those
values of a for which V 1 /s(l + 5) is an odd integer, these terms in W1 do
not have the period 2ir. In the direct construction of the solutions such
terms do not appear in the right members of the wyequations. They appearas the complementary functions of the p r equations and, since they do not
have the period 2w, they are excluded by assigning zero values to the con-
stants of integration. We can, therefore, disregard the terms in Wt whichdo not have the period 2x and consider WY to have the form fiM x
.
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ISOSCELES-TRIANGLE SOLUTIONS. 345
By varying the parameters wf and w"' in (81), we obtain
<V+n[x+^r^] =0, ^Htflx+^rH?)]^!,. (82)
The determinant of the coefficients of wf and w"' in (82) is a constant
[18], and from (77) and (79) it is seen that the value is unity. Equations(82) can therefore be solved for hf and h 2 \ and the solutions are
tij*-pM t \x+ATfl, n? = ixM ltp . (83)
Upon integrating these equations, we obtain
nt) = y] t
) -n[aMTr+Mt ) +ArR l ], n^^^+ftR,. (84)
The r;"' and iff are the constants of integration. The M is a powerseries in a
2
with constant coefficients. The M is a power series in oddpowers of a with sines of even multiples of r in the coefficients, the highest
multiple of r in the coefficient of a 2i+1being 2k+2. The i? t has the same
form as the M except that it has cosines instead of sines. Since the
coefficients of a wcos(2j + l)r and a"sin(2j + 1)t occurring in ip and x
respectively are equal, so also the coefficients of a M+1 sin (2j + 2) r anda u+1 cos (2j + 2) r in Mf and R t respectively are equal. When (84) is
substituted in (81) the solution of the second equation in (73) is found to be
w^^v+tfix+ATd+^S-aMTTv}, (85)
where S 1 is a power series in odd powers of a with sines of odd multiples of
r in the coefficients. From the form of the <p, x, M, and Rt it follows that
S 1 is a triply odd power series. The expressions <S\ and Mf carry X as anundetermined constant. When the constants of integration are chosen so
that (71) shall be satisfied, the solution (85) becomes
w l= [a 2
-t j.\'S 1 (Q)-aM?\][x+AT<p]+ l*[S l -aMrT<p\. (86)
Now let us impose the periodicity conditions (72) on the solutions (74)
and (86) . In so far as the linear terms in a! and a 2 are concerned, we get
p (tt)= - - VV8(1 + S) [o x
- ft C, (0)] sin vV8(l + 5) *
+ terms in fia 2 and higher degree terms in a u fta^, and ft,
w(ir)=0= a 2 Aw
+ terms in a, , ft and higher degree terms in ft, a t ,and a 2 .
(87)
These equations are satisfied by a,=
a.,=
ft=
0, and X arbitrary. The deter-
minant of the coefficients of the linear terms in a^ and o 2 is
D = A tt Ws(l + 5) sin VVs(l + 8) tt.
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346 PERIODIC ORBITS.
We now exclude those values of a for which WsU+S) is an odd integer,
and as we have already excluded those values of a for which WsU + S)
is an even integer, sinVVsU + S) w is not zero and D can vanish only
when a is zero. In order to obtainsolutions
which are not identically zero,a must be distinct from zero. Hence the determinant D is distinct fromzero and, by the theory of implicit functions, equations (87) can be solved
for a t and a 3 as power series in n, vanishing with /*. The coefficients of the
various powers of n are power series in a and contain additional terms in
l/a\ The X enters the coefficients of the solutions as an arbitrary, but the
solutions are unique if X is assigned. Hence periodic solutions of (70) exist
and are of the form
p = 2p t n', w= Sw,/*', (88)
where each p, and wt is periodic in t with the period 2ir.
175. Proof that all the Periodic Orbits are Symmetrical. Let us
suppose that the condition is no longer imposed that the equal bodies shall
be at apsides of their orbits when the third body crosses the ^-plane, andlet us consider the question of the existence of periodic solutions of (70),
with the period 2ir in t, when the initial conditions are
p(0) = ai , p(0) = a 2 , w(0)-0 J w(0) = a,. (89)
Sufficient conditions that p and w shall be periodic with the period 2ir are
p(2x)-p(0)=0, p(2tt)-p(0)=0, w(2t)-w(0)=0, tb(2ir)-tb(0) = 0. (90)
These four conditions can not be satisfied by the three constants a, unless
one condition is a consequence of the other three. We now show that the
last condition can be suppressed when the first three have been imposed.The original differential equations (64) admit the integral
() !
+0h)2 +M(l+M) (f')
2 = ^ +[r, + (l
2
^ M)y])+const.
When the substitutions (65) and (67) are made and u is eliminated by meansof (69), this integral takes the form
P F
+7j|p5i+4|*(l+/)(*+w), =
(1 + 8)
LTlVpl+
[(1 +p)-+4(l +,,)(*+ )]+ const
]
(91)
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ISOSCELES-TRIANGLE SOLUTIONS. 347
Let us make in (91) the usual substitutions
p = p(0)+p, p=p(0)+p w = 0+w, w = w(Q)+w, (92)
where p, p, w, and w vanish at r = 0, and let us denote the resulting equation
by (91a). By putting r =0, we obtain from (91a) an equation (91b) con-
necting the terms in (91a) independent of p, p, w, and w. When this equa-tion (916) is substituted in (91a) there results an equation of the form
F(p,p,W,w)=0, (93)
in which there are no terms independent of the arguments indicated. Thelinear term in w enters
(93)with the coefficient
8*(l+ M)&H-tK0)],
which, we shall show, is different from zero at t = 2ir. Since the third bodyis assumed to be finite, 8/z(l +m) is distinct from zero. The coefficient of w is
therefore different from zero at t= 2x unless w(0) =\j/{2ir)
= a. Nowthe third body, when assumed to be infinitesimal, has the initial speed
a/y/i/%(\ + 8), and w(0)/V 1 /8(l + 5) is the additional initial speed to be so
determined that the orbits shall be periodic in t with the period 2ir when the
third body becomes finite. If this additional initial speed is a/ VVsIT+S),then the whole initial speed is zero, and the third body remains at the center
of gravity since there is then no force component normal to the ^77-plane. In
order therefore to obtain solutions in which f is not identically zero, we mustconsider w(0) 7* a. Hence the coefficient of the linear term w in (93) is dis-
tinct from zero at t = 2ir, and therefore (93) can be solved for w(2ir) as a
power series in p(2ir), p(2ir), and w(2t) which vanishes with p(2ir), p(2w),and w(2t) . This power series is unique if X, which appears in the coefficients,
is assigned. Hence if the first three conditions of (90) are imposed, then
p(27r)=p(27r)=M;(27r)=0; and since w(2w) vanishes with p(2x), p(2ir), and
w(2ir), it follows that w(2tt) =0 or w(2t) w(0) = 0. Therefore the fourthequation of (90) is a consequence of the first three and may be suppressed.
Equations (70) will be integrated as power series in n and a x {i=\, 2, 3),
but only in so far as the a, enter linearly in the solutions. The differential
equations from which these linear terms in a, are obtained are the same as
(73). Their solutions satisfying (89), in so far as the linear terms are con-
cerned, are
a 2
p^^cosVVsd + ^T+ ^/^^^ sinVVsd + ^r,(94)
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348 PERIODIC ORBITS.
When the first three conditions of (90) are imposed on p and w, we have as a
consequence of (94)
0= a, [COS V/8(l + ) 2* ~1] + Vl/ 8 (l + a)
Sln V'
i /8( I +^ 2?r
+ terms in n, /xa 3 ,and higher degree terms,
(95)0= -VV8(l + 5) a,sin\/V8(l + 5) 27r+o 2 [cosA/V8(l + 5)27r-l]
+ terms in /x, /za, ,and higher degree terms,
= 2cl 3At+ terms in a 1; a,, /z, and higher degree terms.
The determinant of the coefficients of the linear terms in a, in (95) is
4 Air [1- cos vVs(l + 5) 2tt].
This determinant does not vanish when a is notzero,
andconsequently
(95) can be solved for o, as power series in /x, vanishing with n. These
solutions are therefore unique if X, which enters the coefficients, is assigned.
Hence the periodic solutions of (70) for p and w, with the initial conditions
(89), are of the same form as those obtained in (88) for the symmetricalorbits. The unrestricted and the symmetrical orbits are unique for a not
zero and for any value of X, and therefore, since the unrestricted orbits
include the symmetrical orbits, all the periodic orbits are symmetrical.
176. Direct Construction of the Periodic Solutions of (70). In order
to construct theperiodic
solutions of (70), we substitute (88) in(70)
and
equate the coefficients of the various powers of m- The arbitrary constants
of integration are to be so determined that w(0) =0 and that each p t and w,
shall be periodic in r with the period 2ir.
The differential equations for the terms in /x are
ft+i(l+*)Pi- ^i =2<U- (T+^) T/ij
The general solution of the p r equation is
p 1= ^
1 eosVV8(l+5)T+5 1 sinVi78TT : H)T+^iW,
where C l (j) is the same triply even power series as in (74). Since the
complementary function does not have the period 2t when \/i/8(l + 8) is
not an integer, the arbitrary constants A x and B Y must be zero in order
that p, shall be periodic with the period 2ir . Hence the desired solution of
the ^-equation is
Pi-GM.(97)
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ISOSCELES-TRIANGLE SOLUTIONS. 349
When (97) is substituted in Wt all the terms of W1 are known. Thegeneral solution of the ^-equation is the same as (85) except that the
particular integral does not carry the factor (x. Hence
>i = *?V+if [x+Ar<p] +S l -aM? )
T<P . (98)
Since all the periodic orbits are symmetrical we impose the condition
ttf(O) =0, from which it follows that
w,(0)=0 (t=
l, ... . oo). (99)
As a consequence of (99), the constant ijj is zero. In order that to, shall
be periodic, the right member of (98) must contain no terms in r exceptthose in which it occurs under the trigonometric symbols. The non-periodicterms in (98) disappear if the constant y^ is so determined that Aitf =aMf\from which it follows that
^> = ipV),
where P a\a?) is a power series in a 2 with constant coefficients. When these
values of r?"} and r?"' are substituted in (98), the solution for w t becomes
viJBflr* ** 1, (100)
where each S^ +l)is a sum of sines of odd multiples of r, the highest mul-
tiple being 2/ + 1.
The differential equations for the terms in /x
2
are
(a) P s +|(l + 5)p 2 =F 2 , (6) w2 +[l+ le 2j a^w 2= W2 . (101)
The first equation is independent of the second, and the terms in P 2 are
completely known. The complementary function of (101a) does not havethe proper period and is excluded from the solution by taking zero values
for the constants of integration. Since P 2 has the period 2-w the particular
integral has the same period and is the solution desired. The function P t
is a triply even power series multiplied by 1/a2
,and since the particular
integral has the same form as P 2 it is denotedby
Vi ^2<rv, ( 102 )
where each C is a sum of cosines of even multiples of r, the highest
multiple being 2j.
When p 2 has been obtained, all the terms in W2 are known. That partof W2 which is independent of p 2 and wx is a triply odd power series. Theterm p 2 is multiplied by a triply odd power series and yields a triply odd
power series multiplied by 1/a2
. The terms wx and w\ are multiplied by triply
even and triply odd power series respectively and together yield a triply odd
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350 PERIODIC ORBITS.
power series multiplied by 1/a*. The lowest power to which a enters a* W2
is found from the term w\\}/ to be 3. Hence the form of Wt is
where each W +1)is a sum of sines of odd multiples of r, the highest multiple
being 2j+l.The complementary function of (1016) is the same as that of (75), viz.,
Wt = n?<p+n?[ x +AT<p]. (103)
The general solution of (1016) has the same form as (85) and is denoted by
i?<P+i?[x+AT<p]+S t -\-M?T<p, (104)~a
where S 2 has the same form as W2 and where Mj
0)is a
powerseries in a 2 with
constant coefficients. The jj{
2> and jjj
2)are the constants of integration.
The constant t]f must be zero to satisfy (99), and in order that w2 shall
be periodic rjf must have the value
C=^ = ^ (2V), (105)
where P (2)
(a2
) is a power series in a 2 with constant coefficients. With these
values of t\f and t^2
', the solution (104) becomes
^=^2J^w+1, w+l
, (106)a 7=0
where the S?1+1) have the same form as the 8?* in (100).
The form of the w} is apparent from (100) and (106). The form of the
P>0>2) is not apparent from (97) and (102), and, before the induction can
be made, it is necessary to consider the term in // in so far as p 3 is concerned.
The differential equation for p 3 is
R+!(1+*)A-1V, (107)
where all the terms of P 3 are known. That part of P s independent of the
Wj is a triply even power series multiplied by 1/a2
. The Wj enter P 3 multi-
plied by power series which are triply odd or triply even according as the
Wj, considered together, enter to odd or even degrees respectively. These
terms form a triply even power series multiplied by 1/a4
. Since P z contains
the term m^, the lowest power of a 2 in a 4P 3 is unity. The complementaryfunction of (107) does not have the period 2x, and the solution desired is
the particular integral. This solution has the same form as P 3 and is
denoted by
V* = jrStcroP, (108)
where theC have
thesame form as the C in (102).
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ISOSCELES-TRIANGLE SOLUTIONS. 351
Let us suppose that the p if w{ , iff (*>-l, . . ., n-1; j = l, 2), have been
computed and that
y=i
*?! Wj ^2 ^nri-r (.a J,
(109)
where the C, ?'+1)
, and P (0(a
2
) have the same form as C?\ S?j+l)
, andP m
(a2
) respectively. From these assumptions and the differential equations
arising from the coefficients of n" it will be shown that equations (109) hold
when i = n. The differential equations for the terms in fj." are
(a) p.+|(l+*)p.-P (b) \b n +\\+lj v a?>\wn= Wn . (110)
As in the previous steps, the first equation is independent of the second.
Since the right member of the first equation in (70) carries n as a factor,
the Pj and w}which enter P n have j < n. Hence all the terms of P are com-
pletely known. The general term of P n has the form
pK . . .
pKw [ . . . tfi (111 )
multiplied by a triply odd or by a triply even power series according as
Mi+ +M^ is odd or even respectively. The X,, \[, . . ., y.j, v'j are
positive integers (or zero)such that
\K+ +\K+ihlH+ +MjM'^tt-l. (112)
From the form of the general term it follows that P n is a triply even powerseries multiplied by 1/a', where
i=(2\-2)K+ +(2X t -2)\:+2( Ml Mi+ +/*,/#. ( 113 )
This expression is even and has its highest value when X(= = \' t
= 0, i. e.,
in the terms of P in which only the Wj appear. Hence, from (112), the
highest value of i is 2n 2. Since P contains the term wn _ 1 ^, the lowest
power of a2
in atn ~ 2
P n is found to be unity. Therefore the form of P is
OO
P _ j 'y pew n vr n n tn ~ i j '
where the P (21) have the same form as the Cf. The only solution of
(110a) which has the period 2x is the particular integral, and it has the form
CO
l J n~
ritn-i Zj ^n u 1
where the C (2i)are of the same form as the CJ
(2/)
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352 PERIODIC ORBITS.
When p n has been determined, all the terms in Wn are known since
they arise from Pj(j^n) and wk (k<n). The general term of W has the
same form as (111), but it is multiplied by a triply odd or triply even
power series according as m>+ +/*J is even or odd respectively. The\,\[, . . .
, Hj, n'j are positive integers (or zero) such that
\\[+ +n J n' ^n. (114)
From the form of the general term it follows that W is a triply odd powerseries multiplied by (1/a)', where
l=(2\-2)\[+ +(2\ t -2)\; + 2 MlM ;+ +2 Mj m;.
This expression is even and has its highest value, 2n, in the terms of Wn in
which only the w} appear. The lowest power to which a enters a in Wn is
obtained from the term in which ^ enters to the lowest power. This termis WiW^ip, and therefore the lowest power of a in a in Wn is found to be 3.
Hence the form of W is
*.- 2 *!*?*.
where the W J+1) have the same form as the Sf J+l).
The complementary function of the second equation of (110) is
and by varying the parameters nJB) and n we have as the complete solution
wn= ^<p+r j ?[ x +AT<p]+S n --k= i MTr<p, (115)
where S a has the same form as Wn ,and where Mf is a power series in a 2 with
constant coefficients. The i^"* and r^ are the constants of integration.The constant Tjj
n) must be zero to satisfy (99), and in order that wn shall be
periodic, itf* must have the value
_() _ Mn _ 1 pd)/2\
where PM(a
2
) is a power series in a 2 with constant coefficients. With these
values of the constants of integration, the solution (115) becomes
B=^S^ +1,aW+1',
where the S +1) have the same form as the S? J+l). This completes the
induction.
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ISOSCELES-TRIANGLE SOLUTIONS. 353
177. The Periodic Solution of Equation (69). When the solution for
p is substituted in (69), u can be obtained by a single integration. Since pis a power series in /x with coefficients which are triply even power series
multiplied by l/a2J
, the right sideof
(69) will contain terms independent of t.After the integration, u will therefore contain a term in t with X appearingin the coefficient. We shall now show that X can be so determined that
this coefficient shall be zero. When X has been so determined u will be
periodic in r with the period 2-ir.
The solution for p x obtained in (97), in so far as the term in X is con-
cerned, is
p^vm+ry (116)
When(116)
is substituted in(69)
and d is
replaced byd Q
-\-\fi,the constant
terms appearing on the right side of (69) are
3 X/z+ higher degree terms in X/x and fx. (117)
Since X carries the factor n, we may replace X/x by a, and then the expres-sion (117) becomes
3o--f- higher degree terms in a and n. (118)
If (118) is equated to zero, the resulting equation can be solved uniquely for
a as a power series in /x, vanishing with n (and converging for |ju| and \a\
sufficiently small) . Let this series be denoted by
00
from which the value of X is found to be
x= xy, (ii9)1-0
the <tj and X^ being constants. With this value of X the u will be periodic,and since X and p are power series in it, the u is also a power series in n andis denoted by
u= 2 Ujijl', (120)
where each Uj is separately periodic for \n\ sufficiently small. The solution
(120) converges for \a\ and |m| sufficiently small.
In order to construct the Uj directly, we substitute in (69) the series
(119) and (120), and the solution already obtained for p, in which X is to be
replaced by(119). The various X, are determined so that the right side of
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354 PERIODIC ORBITS.
(69) shall contain no constant terms in the coefficients of the ju'- The con-
stant term appearing in the coefficient of m is
-3[X - 2 XTa"L
the X"" being known constants. This term is zero if
a = 2j a <X .
1=0
It is necessary to consider the terms up to ju3 before the induction to the
general term can be made. The constant term appearing in the coefficient
of m2 has the form
-d[\- S Xfa"],)=0
wherethe
Xfare known constants. This term vanishes if
\has the value
\= 2 Xf'V'.i-0
The constant term in the coefficient of jt has a similar form; that is, it
can be written
B y=o
the X^ being known constants, and this term vanishes if \ is so determined
that00
\ = ~2j\ a1=0
Suppose X, \ ,
. . ., X_j have been uniquely determined in the same
way and that
\= 2 XfV', X^-^jgxfV' e =l, , -D,
the XfA
being known constants. From the form of p tin (115) it follows
that the constant term in the coefficient of nn+l has the form
CO
where the \ are constants derived from the j> } and Xf (t
=l, . . .
, w-1), andare therefore known. This constant term is zero if
00
\ * ^ \ aJ) n 2 >" a 2 " -2 & " *
The same process of determining the X, obviously can be indefinitely con-
tinued.
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ISOSCELES-TRIANGLE SOLUTIONS. 355
With X thus determined as a power series in n, the integration of (69)
yields periodic terms only, and from the form of the j> s it follows that the u,
have the form
^pa^U?* (*=
1, ... >),
where the U are sums of sines of even multiples of r, the highest multiple
being 2j. The periodic solution of (69) is therefore
-S^S^v+tf,where U is the constant of integration. Since the mass ra x is started from
the point 0, 0, 1/2, at t=
t or at r=
0, it follows that v=
at r=
and there-fore, from (67), the value of u at t = is zero. Hence the constant U is zero.
178. The Character of the Periodic Solutions. When the periodic solu-
tions for p, u, and w have been determined, the solutions for &, % , f 4 (t=
1, 2)
and f are obtained by means of the equations (67), (65), (63), and (62).
These solutions are all periodic in t with the period P = 2 r vVsU + S) .
Three arbitrary constants appear in the solutions, viz., a, n, and i .
The expression a/VtysU + S) represents the initial speed of the third
body in I, n the mass of the third body, and t a the epoch. The mass /x is
restricted in magnitude but can be increased step by step by making the
analytic continuation of the solutions already obtained, provided the series
do not pass through any singularities in the intervals. This can be done
by the process already developed. The parameter a is restricted in mag-nitude and so that VVsU + S) is not an integer. As already stated in
166, it can be shown from equation (2) that the motion of the infinitesimal
body will be periodic if the initial conditions are chosen so that the constant
C is negative. With the initial conditions chosen as in (4), the constant Chas the value 4{2a
2
/(l + 6)-l[. Now if 2a 2
/(l + 6) = l or >1, the infini-
tesimal body recedes to infinity with a velocity which is zero or greaterthan zero respectively, and therefore the motion will not be periodic.
Hence a must be restricted so that 2a 2
/(l + 5)<l. The epoch t is arbi-
trary and may be chosen to be zero without loss of geometric generality.
Hence for given values of \n\ and of \a\ sufficiently small and such that
VV8(1 + 8) is not an integer, there exists one and only one set of periodic
orbits which are geometrically distinct with the period P in t, and which
reduce to those obtained in I for ju= 0.
In proving the existence of the periodic solutions of (70), if the period
were chosen to be 2vw in r, v an integer, it could be shown by the same
process that periodic solutions would exist under the same restrictions on
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356 PERIODIC ORBITS.
a and m- The constant X could be determined so that the solution for uwould have the period 2vir, and hence the periodic solutions of (64) would be
unique as soon as v were chosen. Therefore, for given values of a and of /x
sufficiently small,and such that
Vrfsil+d)is not an
integer,there exists
one and but one set of orbits which re-enter after v synodic revolutions.
Hence the result is unique for every v, and since the orbits reentering after
v revolutions include those reentering after one revolution, there are noorbits which for m= Q reduce to those obtained in I, having the period 2vt
in t which do not have the period 2ir also.
Since ti , fs > and f are odd in a and in t, the orbits are symmetrical with
respect to the i^-plane, both geometrically and in t. The two masses m1
and m2 move in the same orbit and always remain 180 apart.
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t
Chapter XLPERIODIC ORBITS OF INFINITESIMAL SATELLITES
AND INFERIOR PLANETS.
179. Introduction. This chapter is devoted to the discussion of certain
periodic orbits of the problem of three bodies in which two of the masses are
finite, while that of the third is infinitesimal. The finite bodies are assumedto revolve in circles, and the infinitesimal body to move in the plane of their
motion, relatively near one or the other of the finite bodies. The periodic
orbits which are obtained are those in which the periods of the solutions are
equal to the synodic periods of the bodies. The nearer the infinitesimal
body is to one of the finite bodies the less its motion is disturbed by the moredistant one. The orbits under discussion reduce to circles as the disturbance
from the more distant body becomes zero, and they are therefore of the class
called by Poincare "Solutions de la premiere sorte."*
The results of this chapter are of direct practical application, particu-
larly in the Lunar Theory. They are coextensive in this domain with the
Researches^ of Hill and the first memoir by Brown. $ When the masses of
the two finite bodies have the ratio ten to one the problem reduces to that
which Sir George Darwin treated by numerical processes, and the results
obtained include the orbits called by him "Satellites A" and "Planets A."Darwin's "Satellites B" and "Satellites C" are imaginary for small values
of the disturbing forces and belong to values of the parameter for which the
series obtained in this chapter do not converge. It appears from the present
investigations that there are three families of satellites and of inferior planetswhose motion is direct. It follows that Darwin's search for them was exhaus-
tive so far as the satellites are concerned; but he found only one family of
planets whose motion is direct. The work of this chapter shows that there
is an equal number of retrograde orbits; that is, three families of real or
imaginary periodic orbits around each of the finite bodies.
As compared with previous work on this subject, the methods of this
chapter are characterized by the fact that the validity of all the processes
employed is proved. They have the merit of generality, not only in showingthe number of orbits that exist, but also in being applicable to any ratio of
masses of the finite bodies. They have the disadvantage that the series do
not converge for all values of the parameter. The numerical processes
*Les Melhodes Nouvelles de la Mecanique Cileste, vol. I, p. 97.
turner. Jour. Mathematics, vol. I (1878) pp. 5-26, 129-147, 245-260, and Hill's Collected Works, vol. I.
%Amer. Jour, of Mathematics, vol. XIV (1892), pp. 141-150.
\Acta Mathemalica, vol. XXI (1897), pp. 99-242; Mathemalische Annalen, vol. LI (1898-9), pp. 523-583.
357
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358 PERIODIC ORBITS.
employed by Darwin are easily applicable to the cases where the series fail.
Instead of arranging the solutions as Fourier series, as was done by Hill and
as has been customary among astronomers, power series are employed, and
the work has the simplicity which is characteristic of processes involvingpower series. For example, every coefficient is determined by a single step
and is modified by no subsequent operations.
180. The Differential Equations. Let m, and m2 represent the masses andtake the origin at m, for the determination of the motion of the infinitesimal
body. Then the differential equations of motion in polar coordinates are
r"-r(*/)2+~^ = &2 m2 ^, rv"+2r'v' =k 2
m^,tj = 1 rcos(vV)
(1)
[R2
-2rRcos(v-V)+ri
] i R2
where r and v are the polar coordinates of infinitesimal body, and R and Vare the polar coordinates of m2 .
In the present discussion those orbits are considered in which r is small
relatively to R. In this case U is expansible as a converging power series in
r/R, and equations (1) become
U=^[l+ \^[Wcos2(v-V)]+l^[3cos{v-V)+5cos3(v-V)]+ },
r "_ r(t ,T+
^= +
^^jl+3c o S 2(.-y) + g[3cos(
? ;-F)
+5cos3(y-F)]Hj,
rv''+2r'v' = -^^{d Sm(v-V) + ^[ S in(v-V)+5smZ(v-V)]+}
It was assumed in the beginning that the relative motion of mx and mt
is circular. Hence we have, from the two-body problem,
(2)
R = A = constant, V= ky/n^ nh(t -Q=N(t- 1 ) , (3)
where Af is the angular velocity of the relative motion of the finite bodies.
When the right members of the second and third equations of (2) are
put equal to zero, that is, when the infinitesimal body is supposed to revolve
about m, without being disturbed by m2 , they have the particular solution
r = a = constant, v = 3^ (t t )=n(t t ), (4)
where n is the angular velocity of the infinitesimal body with respect to m, .
In the periodic solutions of (2) when the right members are included, the
mean angular velocity will be kept equal to n and a will be defined by the
equation
n2
a' = /i;
2
mj. (5)
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INFINITESIMAL SATELLITES AND INFERIOR PLANETS. 359
Since mx is given, a has three determinations, two of which are complex. In
the physical two-body problem there is interest only in the real value of a,
and it is immaterial whether a or n is regarded as given. It will be observed,
however, from the purely astronomical point of view, that n can be determined
by observation much more accurately than a, because the former is an angularvariable and any error in its determination causes the theory to deviate
secularly from the observations; while the latter is a linear variable and the
discrepancies which arise from errors in its determination do not accumu-late. But in the three-body problem all three values of a must be included
in determining orbits whose period is defined by n, because all the orbits
may become real for certain values of the parameters which occur in the
right members of the differential equations. In fact, Darwin found three real
orbits for certain values of the Jacobian constant.*
In the Lunar Theory, as developed by de Pontecoulant, for example, the
solutions were forced into the trigonometric form by various artifices. Butit will be noticed that the method of procedure was the opposite of that
adopted here, so far as comparison can be made, in that the mean motion
was continually modified by de Pontecoulant in order to preserve the
trigonometric form; while here the mean motion is regarded as being given
arbitrarily in advance by the observations or otherwise, and it is kept fixed.
New quantities p, w, t, and m will be introduced by the equations
r = a(l+p), v = n{t-Q+w,N N{l+m) ; An ,. , s N f , . n
=^7n> n=^> r={n-N){t-Q = -{t-Q,
A Kmi+rrhs Vl +m/ \mi+m 2 / \ > /
(6)
(7)
For brevity a/ A will be written in place of its expression as a power series.
As a result of these transformations, the last two equations of (2) become
p-_ (1+p)(1+m+ ^ + ^^|!_^_( 1+p
)j[1+ 3cos2(x+;)]
+!j(1+p)[3cos(t+w)+5cos3(t+w;)]+
. . . 1
(l+p)^+2p(l+m+tb) =-^^^-(l +
p)J3sin2(r+^)
+ f-f (l+ P )[sin(r+w;) + 5sin3(T + w)]+ I,
where the dots over p and w indicate derivatives with respect to t. These
equations are valid for the determination of the motion of the infinitesimal
body as long as a(l+p) is less than A.
*Acla Malhematica, vol. XXI (1897), pp. 99-242.
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360 PERIODIC ORBITS.
It follows from equations (6) that ap and w are the deviations from uni-
form circular motion due to the right members of the differential equations.
They are functions of m, and the initial conditions are to be determined so
that they shall be periodic in t with the period 2t. Since the right membersof (7) contain m* as a factor, the periodic expressions for p and w will contain
ms as a factor.
181. Proof of the Existence of the Periodic Solutions. For m=
equations (7) admit the periodic solution p = w = 0. It will now be provedthat for m distinct from zero, but sufficiently small, equations (7) admit a
periodic solution which has the period 2x in t, and which is expansible as a
power series in ml/3
, vanishing with m. It is the analytic continuation of the
solution p = w = m= with respect to m as the parameter. Since t enters
explicitly in the right members of (7) in terms having the period 2x, it follows
that the period of the solution must be 2w, or a multiple of 2ir.
Equations (2) are not altered if we change the sign of v, V, and t. It
easily follows from this that, if v (t )= V (t )
= p' (t )=
0, the dependent vari-
ables p and v are even and odd functions respectively of t t a . An orbit in
which these conditions are satisfied is symmetrical with respect to a line
always passing through ml and m2 . Such an orbit will be called symmetrical.
Expressed in the variables of (7), p and w are respectively even and oddfunctions of r in the case of symmetrical orbits; and w(0) =p(0) =0.
The existence of symmetrical periodic orbits will be established, and then
it will be shown, in connection with the construction of the solutions, that the
condition that the solutions are periodic implies that they are also sym-metrical. It will follow from this that all of the periodic orbits of the typeunder consideration are symmetrical.
Suppose that the initial conditions are
r = a(l+p) = a(l+o)(l-e), to = 0,
r = ai> = l+m+w= ( l + m)^+e' ,
i P > -r/rc-t-u>(i-j-o)'
/s(l e)
s/ *
(8)
If the right members of equations (7) are neglected and the transformation
r =a(l+p), 0=(l+?n)r+M>
is made, then equations (7) reduce to the ordinary polar equations of the
two-body problem for the motion of the infinitesimal body with respectto ml . In this two-body problem with the initial conditions (8), it is foundthat aa is the increment to the semi-axis a, and e is the eccentricity of
the orbit of the infinitesimal body. Since the properties of the solution of the
two-body problem in terms of the major semi-axis and eccentricity are
known, we can at once write down the properties of the solution of (7) in
terms of a and e, so far as they are independent of the right members of the
equations. It was precisely for this reason that p and w were given the
peculiar initial values defined in (8).
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INFINITESIMAL SATELLITES AND INFERIOR PLANETS. 361
Equations (7) are now integrated as power series in a, e, and m}' 3.
The results have the form
P = Pi ( a > e, ml/3; t), w = p 3 (a, e, ml/3
; r),
p = p 2 (a, fl,ra 1/3
; t), w = p i (a, e, m1/3; t) .
J
The moduli of a, e, and m1/3 can be taken so small that the series converge while
t runs through any finite range of values starting from zero. The periodmust be a multiple of 2w, say 2kir, and consequently it will be supposed that
these parameters have such small moduli that (9) converge f or ^ t ^ kir.
It follows from the symmetry of the orbits under the initial conditions
(8) ,that if, at r = kir, the three bodies are in a line, and if the infinitesimal body
is crossing perpendicularly the rotating line which joins mA and m2 , then the
orbit necessarily re-enters at 2kir, and the motion is periodic with the period2kir.
Conversely, if,at t =
0,the orbit crosses
perpendicularlythe
rotatingline which joins m1 and m2 ,
and if the motion is periodic with the period 2ir,
then the orbit at r = kir necessarily crosses perpendicularly the rotating line
which joins mx and m2 . Therefore, necessary and sufficient conditions for
the existence of symmetrical periodic solutions having the period 2 kir are
p 2 (a, e, mV3; kw)=0, p 3 (a, e, ml/3
; kir)=0. (10)
Since p 2= p 3
= at t = 0, it follows that equations (10) are identically satis-
fied by a = e = ra 1/3 = 0. The problem of solving them for a and e as power series
in m1/3, vanishing for m= 0, will now be considered. All the terms of the solu-
tions which do not carry m2 as a factor are obtained from the solutions of the
left members of (7) set equal to zero. Since these terms belong to the two-
body problem equations (9) become, when these terms are explicitly exhibited,
p = p 1 e^osi'T +|(cos2j/t 1)+ '~\+mt
q 1 (a, e, mI/s
;T),
P = p 2 =+ e[
I 'sin vT+ev sin2^T + ] +ra 2
q t (a, e, w}' 3
; r),
= p 3=
\_-(l+m)T+vT + 2esmvT+ ]+m2
g 3 (a,e, m1/3; r),
= Pi= [ (l+m)+i>+2i>ecosvT + ]+m2
g 4 (a, e,m1/3
;T),
V!
t'r
(11)
where 1+ra / 1oN
"-$+&' (12)
The series q l ,. . .
, q i depend upon the right members of the differential equa-tions (7). All terms in the [ ] which are not written contain e
2 as a factor.
The conditions, (10), for the existence of symmetrical periodic solutions
become as a consequence of (11)
Ptikir)=
ev\smvkir-{- esm2vkir+ ] +m}q 2 (a,e,ml/3
; kir) =0,
p 3 (kir)=
[ (l-t-m)kw-t-vkir+2esm2vkT+~^+m i
q 3 (a,e,ml/3
;kir)=
0,
(13)
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362 PERIODIC ORBITS.
where all the unwritten terms in the [ ] carry e2 as a factor and are linear
functions of sines of multiples of vkir. On substituting the value of v from
(12), it is found that
sin>A;ir = sin[l4-w |o+ ],/*=( 1)* sin [m |a+ -~\jkir }
where j is an integer. Every term in the [ ] contains either m or a as a factor;
therefore every term of the second equation of (13) contains either moraas a factor. The coefficient of a to the first power is 3/2 kit, which is
distinct from zero; therefore the second equation of (13) can be solved for
a as a power series in m and e, vanishing with m= 0. The term of lowest
degree in m alone is the second, and its coefficient depends upon the right
member of the differential equations (7). Therefore the solution of the
second equation for a carries ra 2 as a factor, and has the form
a = ra 2
p(ra1/3
, e). (14)
Suppose a is eliminated from the first of (13) by means of (14). After
the elimination, a factor m can be divided out. The result then contains
a term in e alone to the first degree, and its coefficient is ( 1)*. Therefore
the resulting equation can be solved for e as a power series in ra 1/3, vanishing
with m= 0, of the forme = mq(m 1/3
). (15)
As a matter of fact, the expression for e contains m2 as a factor, as can
easily be shown. Suppose equations (7) are integrated as power series in ra1/3
,
and let the initial conditions be p = P = w = w =0, in order to get the terms
which are independent of a and e. The series will have the form
p = p 1 m-r-p 2 ra2 +p 3 wi
8/3 + ''?* w = w1 m-)rW i mi-\-w i mm -\-
-
The explicit result of the integration is
p = m-t
[ 1+2cost cos2t],
p 3= ^r ^ + f cost -cos2t 2rsin7-l,3 mi+Wj L2 3 6 J'
w3= ^r \ r
~ TsinT+^sin2T 4tcost],
From these equations it follows that p 2 (kr) has m2 as a factor, but that it
does not have m3 as a factor. Therefore the expression for e as a powerseries in mv * contains m2 as a factor. Then (15) and (14) together give
a =w
2
P 1 (w1/s
),e =
m2
P i (ml/
'), (16)
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INFINITESIMAL SATELLITES AND INFERIOR PLANETS. 363
where P 1 and P 2 are power series in m1/3 which converge for the modulusof m sufficiently small. Therefore the symmetrical periodic orbits exist, and
equations (16) and (8) give the initial values of the dependent variables in
terms of the parameter mm which is
defined, exceptfor the cube root of
unity, by the data of the problem and the third equation of (6) .
182. Properties of the Periodic Solutions. The periodic orbits whoseexistence has been proved re-enter after the period 2 for, where k is anyinteger. Those orbits for which k is greater than unity include those for
which k equals unity. Since, according to the discussion which has justbeen made, the number of periodic orbits is the same for all values of k, it
follows that the period of the solutions is 2-k. When m= the infinitesimal
body makes a revolution in 2t, and m can be taken so small that the orbit
is as near this undisturbed orbit as may be desired. Therefore a synodicrevolution is made in 2-k for all \m\ sufficiently small.
If the expressions for a and e given in (16) are substituted in (9), the
result becomesP = 2 pMm" 3
, w=Sw,(T)m ,/J, (17)
i-e i=*
where the summation starts with j = 6, because the expressions for a and e
have m2 as a factor, and p and w have no terms in m alone of degree less thanthe second.
The p and w are periodic with the period 2w for \m\ sufficiently small
because the conditions for periodicity have been satisfied. Therefore
2 pj(T+27r)m'/3 = 2 Pj(T)m
i/3,
2 Wj{T+2ir)m m= 2 w^^m" 3
;
1-6 1-6 1=6 1-6
whenceP] (t+2w)= Pj (t), Wj(t+2t) = w,(t) =
6, ...oo). (18)
Since p(0) =w(0) =0, it follows that
2 hi (0) m" s =0, iwj (0)m'
/3 = 0;1-6 1-6
whencep/0) = 0, ^(0)=0 (j=6, ... oo). (19)
If the orbit of the infinitesimal body is retrograde, n is negative and mhas the definition ._ t ,
,._
ra= -N/(n+N)for a given sidereal period. Therefore, for a given numerical value of n, the
parameter m is smaller in retrograde motion than it is in direct motion. Fora given sidereal period the deviations from circular motion are less in the
retrograde orbits than they are in the direct. The physical reason is that
the disturbance of the motion of the infinitesimal body by m^ is greatest
when the three bodies are in a line, as can be seen from (7) or by graphically
resolving the disturbing acceleration; and in retrograde motion this approxi-mate condition lasts a shorter time than in direct motion.
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364 PERIODIC ORBITS.
DIRECT CONSTRUCTION OF THE PERIODIC SOLUTIONS.
183. General Considerations. It has been proved that equations (7)
have solutions of the form (17) whichsatisfy (18)
and(19).
The solutions
are in m1 "only because a/ A is a series in mI/3
, given explicitly in (6). Theexpression for a/A can be modified by writing
j= Mm, (20)
where M is to be regarded in the analysis as a constant independent of m.This amounts to a generalization of the m as it appears in certain places in
the last equation of (6). The particular transformation (20) is made in
order that the right members of (7) shall be in integral powers of m. Theproof of the existence of the periodic solutions can be made precisely as
before, because the transformation (20) affects only the higher terms whichwere not explicitly used. While there is nothing essential* in the trans-
formation, it will be made for the sake of convenience, after which equations
(7) become
p-(l+p)(l+m+^)2+^^ =
^7?(l+p){[l+3cos2(r+ti;)]
+ fMm(l + p)[3cos(T+w) + 5cos3(7+w)]+ "} (
(l+ P )w+2i> (l+m+w) =~Yv(l+p)hsm2(T+w)
+ fMm(l+p)[sin(T+w) + 5sin3(T+w)]+ j,
whereirk
(21)
v =mi+nk
In the right member of the first equation of (21) the coefficient of m' is
a sum of cosines of integral multiples of r+w, the highest multiple being j;
the coefficient of m' in the right member of the second equation is a sum of
sines of integral multiples of (t+w), the highest multiple being j.
In a closed orbit around mt there are two points at which w is zero.
The arbitrary t will be so determined that w(0)=0. The first condition
of (8) will not be imposed in advance, and it will be shown that it is a conse-
quence of the others. It will follow from this that all of the periodic solutions
of the type under consideration are symmetrical. Equations (21) will there-
fore be integrated in the formCO oo
p = 2 pjtn', w=2 Wjtnf (22)
subject to the conditions (18) and the second of (19).
*A different transformation was made in Transactions of the American Mathematical Society, vol. VII,
(1906), p.542.
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INFINITESIMAL SATELLITES AND INFERIOR PLANETS. 365
184. Coefficients of m2. These terms are denned by the equations
p i -3p 2 -2w 2=
^-q (1+3 cos2t) , w2 +2'p 2=
-|r?sin2r,
the general solution of which is
P 2=
\n + 2 cf + cf cos r + cf sin r - v cos 2r,
w2= cf -
(ij + 3 cf ) t- 2 cfsin r + 2 cf cos r + sin 2r,
(23)
where c{2)
,. . .
, cf are the constants of integration. By conditions (18) andthe second of (19), it follows that
c?=-~v, cf 2c (24)
Therefore the solution (23) becomes
p 2= -7j + cf cos r -f c sin r 77 cos 2r,
w2= - 2 cf - 2d? sin t + 2c* ' cos r + ^ sin 2r,
where c and cf are constants which remain so far undetermined.
(25)
(26)
185. Coefficients of m3. The differential equations which define the
terms of the third degree in m are
P 3 3 Pi 2w 3 6p 2 +2w 2 + -zM-q [3 cos r +5 cos 3t]
= -7,+ (2cf + |Mr ? )cosr + 2cf sinr -|7,cos2r+^r7Mcos 3r,
ib 3 +2p 3= 2p 2 |M?7[sinT+5sin3r]
= (2cf - Mr?) sin r - 2 cf cos r - 4j sin 2r - f Mr; sin 3r .
From the second of these equations it is found that
w3= -2p 3 +cf -
(2cf -|M7,)cosr-2cf sinT+2rjcos2T+|M7?cos3r, (27)
which substituted in the first gives
P,+P 3= -7?+2cf
)
-(2cf-^Mr;)cosT-2cfsinT+|7 7 cos2T+|iW"7 7 cos3r. (28)
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366 PERIODIC ORBITS.
In order that the solution of this equation shall be periodic the coefficients
of cost and sinr must be zero; whence
<f-gtf 9 , tf-0, (29)
after which the general solution of (28) becomes
p l =-i;+2c 1
a)+cf cosr+cf sinr-
Jijcos 2r-|M7; cos 3r, (30)
where c, c, and cf are undetermined constants. The result of substi-
tuting this expression in (27) is
w3=
2r7-3c;
3,
-(2cf +|M7j)cosr-2c<
3,
sinT+j77COs2r+ |iWr,cos3r. (31)
In order that the solution of this equation shall be periodic the right membermust contain no constant term; whence
<f-ff. (32)
With this value of cf it is found, upon integrating (31) and imposing the
second condition of (19), that, so far as the computation has been made,
p 2= ~\n + l
^Mr] cost- r, cos 2t, w2= - |i) sinr + ~ri sin 2t,
P 3= + \ v + cf cos t + c sin r -
J 17 cos 2r -|jMr, cos 3t,
w3= - 2c? + 2c? cos r -
(2 cf + 1 Mr,
)sin r + ^ q sin 2r + Mij sin 3r,
where c"' and c? are constants which are as yet undetermined.
186. Coefficients of m1
. Theintegration
will be carried onestepfurther and then the induction to the general term of the solution will be
made. The differential equations which define the coefficients of w4 are
p\-3p4-2itf 4= 6p3-3p2+2pju> 2 +2tb,+3p,+u>2+!i;p !! +f PjCos2t
- 3 w2 sin 2t + ^ M'r, [9 + 20 cos 2t + 35 cos 4t] ,16
aiv t + 2 p, = pj wt 2 p 3 2 p 2 wt
- p 2 sin 2r 3m, cos 2r
_16^M
27 7 [2sin2r+7sin4r].
(34)
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(35)
(36)
INFINITESIMAL SATELLITES AND INFERIOR PLANETS. 367
Upon developing the explicit values of the right members of (34) bymeans of (33), it is found that
P 4 -3p 4 -2w 4
=[-^ ^+|^-||MV+^M^]+2
CfsinT
+ [2^ + 2
f^-S^]cosr + [-f^-|,-||MV+ f M*ij
]cos2r +f Mv COS 3r + [|+f| MV,]
COS At,
w4 +2p 4 =-2(f cosr+[2c<3)
-|Mr,2
]sinr
+ [-f^-^-iM V -fM^]sin2r
+[f^2
-l^] sin3 -+[-S"2
-lHsin4r -
The first integral of the second of these equations is
w, = - 2 Pl + c<4) - 2 cf sin r -
[&-g Mtj
2
]cos r
+ [^+^+l^Y+^]cos2r-[f M*- ! M ]
cos3 ' + [f *+S W| cos4 ^
where cj4)
is an undetermined constant. Then, on substituting this value
of iv t ,the first equation of (35) becomes
p 4 +p 4 =[2cf-f v2+^-f MW+f- M2
v]-2crsmT
+ [-2<? +f M'-M]cosT + [-2i,'+f
+ ||MV+fM^]cos2r+[-f M^ + |M,]cos3r
+ [f"2+f W|cos4t.
In order that the solution of (37) shall be periodic, the conditions
Cf = fM^-|Mr,, ef-0 (38)
must be satisfied. Then its general solution becomes
Pl= 2cr
, -f Tf+f^-gMy + ^M^+^'cosr+^sinr
+[+! r>
2 -,? -S MV-f M2,
']cos2r
+ [+I M"2
-l^] cos3 - + [-|"2 -iH COs4T
'
whereCj
4>
,
c,and c
4)are as
yetundetermined constants.
(37)
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368 PERIODIC ORBITS.
If (39) is substituted in (36), it is found, by using (38), that
^=-3cr+3
f^-3^+||MV-|M^+[-2 Cr-fM^ + ^^]cosr
-2crsinr+[-7,f +^ + ^|MV+j|M
2
7,]cos2r (40)
+ [-i^+S*']*"* + [f*+g*V|cos4r.
The periodicity condition determines c{fi
by the equation
cT = +^-r,+^My-\M\ (41)
Then the integral of (40) satisfying the second condition of (19) is
w,= -2eP+2ebwCQST- ^cf + ^iWy-^M^Jsinr
+ [-^2 + i" + ii MV+lM^]sin2r [
(42)
The results so far obtained are
p 2= -7j + Mtjcost T7COs2r, wt
=Mrjsmr + rism2T,D Id o o
P*= +?*+ \S Mr?- |^]cosr- |7,COs2r -fjMijCOsSr,
W3 =-[^M 7 7
2
+fiM7 7 ]sinr + ^r 7 sin2r+|Mr ? sin3r,
^ = +i^-i^+li Ar ^ 2
-^ ikf2r7 + c ^ C0SX + c ' 4>sinr
+ [+ff-f,-iMV-|M^]cos2r I (43)
w4= - 2<f +2ejJ
cos r -[2c<
4) + ^ ^V -^ Mi,] sin r
+ [-^2+^ + li MV+lM^]sin2r
where iy ? '
, and where c,4) and c^ are so far undetermined.
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INFINITESIMAL SATELLITES AND INFERIOR PLANETS. 369
It is observed that, so far as the variables are completely determined, the
Pj and Wj are sums of cosines and sines respectively of integral multiples of r,
the highest multiple being j. At the jth
step of the integration one of the
four arbitrary constants which arise at that step is determined by the perio-dicity condition on the wJ} and another by the initial condition on the w,.
The other two constants remain undetermined until the next step, but twowhich arose at the preceding step are determined by the periodicity condition
on pj. It will be shown that these properties are general.
187. Induction to the General Step of the Integration. Supposep 2 ,
. . ., p_i; w2 ,
. . ., w_i have been computed and have the properties
expressed in the following equations:
Pj ,
= af + a^COST + a? COS 2r+ ' +af COSJt (j=
2, . . ., n-2),
Wj= j8 1
0>sinT+i8 1
0>sin2T+ +p? sinjr =
2, . . ., n-2),
Pn-i= +4', - 1)
sinT+aJB - ,)
+c*-1)
cosT+a<n - 1)
cos2T+ +a (B
:l)
cos(n-l)r,| (44)
wn^= -2cr-1)
+2c<n - ,>
cosT+[-2c;''-1)
+6<B " 1>
]sinT+/3f-1)
sin2T
+ ---+/3t
::;)
sin(n-l)r,J
where the a", &?, and 6j"~" are known constants, and Cj"-1 ' and cf
- " are
undetermined constants.
In writing the differential equations which define p and w all unknown
quantities will be given explicitly. The terms involving these undetermined
coefficients are the same at every step. It is found from equations (7) thatthe coefficients of m" are defined by
p n -3p n -2w n =+2c?-1)
cosT+2c(r l) sinT+P n (pj,Wj,w j ; r), ]
(45)wn + 2 p -+2cf sin r-2df cos t+Q(pj, p J} Wj, iv; r).J
where F and Q are polynomials in p Jt p } ,Wj, and w, (j =2, . . .
,n 2) and
the known parts of p_i , p_i ,wn _ x ,
and w_i ,and where r enters in the
coefficients only in sines and cosines.
It follows from (7) that, aside from numerical coefficients, P has terms
of the types
P" } = p h Wj, O'i +j 2= n, or n - 1 ) ,
P=p h Wh Wh (h+J2+h = n),
PT =p)\
'
p)' GU,+ ; +hj v=
?i, n-l,orn-2),
P^ = M'p)\ &V& <^Tn 7r= 0, 1, , n-2; k,+ +k^j+l;
ig:j+2; j+hjt+ +K] v +\Vi+ +\p li= n-2).
If \+ +Xiseven, the term is multiplied by costV; and if \+ +XM
is odd, the term is multiplied by sinir. The terms P"', Pf, and Pf come
from the left member of(7),
andP?
comes from theright
member.
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370 PERIODIC ORBITS.
It follows at once from (44) and the conditions on the j t and k t that
P\ P?> and P are sums of cosines of integral multiples t, the highest
multiple being n at most. If \+ +X M is even, the product w^,1
w% is a
sum of cosine terms, and it follows therefore that in this case P? is a sum ofcosines of integral multiples of t. The highest multiple of t is
N = k 1 j l + +kj,+\p l + +\ ll p ll +i,
which becomes, as a consequence of the relations to which the exponentsand subscripts are subject,
N= n-2+i-jSn-2+j-{-2-j = n.
If \+ +X is odd, the product vftl m* is a sum of sines of
integral multiples of t. Therefore, in this case also, Pf is a sum of cosines
of integral multiples of t; and it is shown, precisely as before, that the highest
multiple is n. Hence the general conclusion is that P is a sum of cosines
of integral multiples of t, the highest multiple being n.
By a similar discussion it can be proved that Qn is a sum of sines of
integral multiples of t, the highest being n. Hence equations (45) can be
written in the form
(46)
(47)
p B -3p fl -2w n =+2cr- 1)
sinr+^r) + [2cr
i)
+^in)
]cosr
+^rcos2r+ +A? cos nr,
ii>n+2p n= -2c<"- l)
cosT+[2c<"-1)
+J
Bn sinr+ <B) sin2r+ +^ n)
sinnr,
where the Af and the Bf = 0, . . ., n) are known constants.
The first integral of the second equation of (46) is
Wn= - 2 Pn +c[n) - 2 g- sin t - [2 ef +{">] cos r - 1 B cos 2r
_ . . . gf> COS TIT,
where c{n)
is an undetermined constant.
On substituting equation (47) in the first of (46), it is found that
Pn + Pa = - 2 cf-1 '
sin r+ [2 Cf" +A?] +[ - 2c?~ +A - 2 B?] cos r
4-[<>-|5r)
]cos2r+ + [iT-jj B?] cos nr.
In order that the solution of this equation shall be periodic, the conditions
c?-" - 0, 2c<"-" - A? - 2B (49)
must be imposed. They uniquely determine the constants c^-1 ' and cj
n_1),
whichremained undetermined at the preceding step of the integration.
(48)
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INFINITESIMAL SATELLITES AND INFERIOR PLANETS. 371
After equations (49) are fulfilled, the general solution of (48) is of the form
P = c^ sin r + C + cf cos r + of cos 2t+ + a cos nr, (50)
where of and cf are arbitrary constants, and where
<>= l[A <">
2R""~l r-_ uA. 2 B)"} / o \ (51)
If equations (49), (50), and (51) are substituted in (47), the result is
wn = -[3cP+24T]-2P fflHT-[2<f+iir-Br]co8T
-[2o{"
)
+|^)
]cos2t [2af+iBf >
]cosir-
-^ar' + i^rJcosnT.
In order that the solution of this equation shall be periodic, its right membermust contain no constant term; whence
c?=-\A?. (53)
Then the integral of (52) satisfying the condition w(0) =0 is
wn = -24*+2cf oosr- [2cf +Ar)
-r,
]sinr
-|[2C + |Br]sin2r 4[2ar+jBr]sinir
I[ 2a r + ^r]cosnr.
The results obtained at this step are
p^cr'sinr + C+cf cosr + of cos2r+ + af cosjr
+ + <C COS flT,
wn =-2 c? + 2 rf cos r - [ 2 <f + A - Bf ] sin r +& sin 2r
+ +/3ln>sin nr,
w-_i/l("> -w_ [j A/ 21?/] ,.2
s
(54)
..'" , ) _ a?-2b? w . r2jAr -(/+3)Br](
.
2 n)* ~2 ' M'
"~ r ~
fCf-D
cr=o,
wherecT
andcf
are asyet
undetermined constants.
(55)
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372 PERIODIC ORBITS.
Since the results expressed in the first two equations of this set are
identical in properties with the equations (44), with which the discussion of
the general step was started, and since the properties of (44) were fulfilled
for the subscripts 2, 3, and 4, it follows that the induction is complete. Theprocess of integration can be carried as far as may be desired.
The hypotheses under which the discussion has been made are that the
solutions are periodic and that w(0) = 0. Solutions satisfying these properties
and p(0) = were known to exist from the existence discussion, and therefore
they could certainly be found because the assumed properties are included
in those of the symmetrical orbits. It appears in the construction that
the hypotheses adopted imply also that p(0) = 0. Therefore all periodic
orbits of the type under discussion which are expansible as power series in mare symmetrical orbits. It can be shown by direct consideration of the series
that there are no others expansible in any fractional powers of m.
188. Application of Jacobi's Integral. The differential equations admitJacobi's integral, of which no use has yet been made, and it is the only
integral not involving the independent variable. Upon transforming the
integral to the variables used in this chapter, it is found without difficulty
that its explicit form is
\2
p*+(l+p)2
(l+w)2 -m 2
(l+p)2 - 2(1 +m)
l+P
-|(l+p)cos(r +^)}=-C,
(56)
where C is the constant of integration. It will be shown that this integral
can be used as a searching test on the accuracy of the computations of the
solutions, or to replace the second differential equation of (7).
Since the periodic solutions are developable as power series in m, the
integral can be expanded as a power series in m and written in the form
Fo+Fxipj, p Jf wJf Wj] r)m+ +F u (pp i ,Wj,w i ;T)mn + =C. (57)
In the F n thehighest
value ofj
is n. Since theintegral converges
for all
\m\ sufficiently small, each F n separately is constant, and therefore
F(pj ,Pj,Wj,Wj; t) = C . (58)
It follows from the form of (55) and (56) that F n is a sum of cosines of
integral multiples of t. In F the sum of the products of the exponents and
subscripts of the factors of any term not involving cosjr or sinjr can notexceed n; and in any term involving cos jr or sinjr the sum can not exceed
ni. Therefore the highest multiple of r in F is n, and (58) can be written
in the form
Fn=y?+y?
)
cosT+ + y Cosjr+ +y?coanT = Cn . (59)
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INFINITESIMAL SATELLITES AND INFERIOR PLANETS. 373
Since this relation is an identity in t, it follows that
* J?~C mj 7r=0 (j = l,...,n). (60)
These relations are functions of a, . . ., J*; ^\ . . .
, j8, and their
fulfillment iox j= 1, . . . ,n serves as a thorough check on the expansion of
U and on all of the computations.It will now be shown how the relations (60) can be used in place of
the second equation of (7). If (56) is expanded as a power series in m,it is found that
^n = 4p B+2w+4p_ 1 +(r B (pj, f>j, wj} Wj] t), (61)
where Gn is a polynomial in p h ph wJ} and ivj and involves r only in sines
and cosines. Moreover, the greatest value of j in Gn is n 2. Suppose
that p 2 ,. . .
, p_ 2 ; w2 ,. . .
, w_ 2 are entirely known, and that p B_ t and wB_!
are known except for the undetermined coefficients c 2
" -1);
it will be shownthat equations (60) and the third and fourth equations of (55) define the aj
B)
and /3]
n)
uniquely.It follows from the properties of F n and equation (61) that this func-
tion can be written in the form
F n = [4 C + CT] + [4c 2
" _2) - 2A+2B + Cf] cos r
+ [4or+4#' + Cr cos2r+ +[Aa?+2j0+C?. ] cos jr
+ +[4a Bn>
+2w/3 B" ) + CB
n>
]cosnr.
Consequently equations (60) become
7r, =4c 2
n - i) -2^r ) +25r , +^r=o, (62)
7 r=4ar+2^r+c )= o 0-2, . . ., >.
,
It follows from equations (55) that
4c*, - ,)
-2Aj")
+4fl I
w = 0,
4ar + 2i/3<'',
+ |JBr = Ci-2, . . . ,).
I (63)
Upon comparing equations (62) and (63), it is found that
2B?~jC* (n =2, ... oo
; j=l, ..'.., n). (64)
Therefore the third to the seventh equations of (55) can be written
a? = -\A? t df = Al ~ Cl, of'" =0 (i=2, ...,),
w ur-cfi) flW _ 4A,"-(iM-3)c r
f.
,^-(/-I)
^ '
2i(/-l) -*.;/l
(65)
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374 PERIODIC ORBITS.
These equations express the coefficients, which are determined at this step of
the integration, uniquely in terms of constants which depend only upon the
first equation of (7) and upon the integral. In practical computation it is
more convenient to make the determination of the coefficients depend uponthe A and Cj
n) than upon the A and Bf\ for the former have manyterms in common, except for numerical multipliers, and both are coefficients
of cosine series, which are easier to check than are the sine series on which
the Bjn)
depend. But the chances of error in lengthy computations are so
great that if the developments are to be made to high powers of m, the onlysafe method is to use both the second equation of (7) and the integral, or,
what is the same thing, to secure the fulfillment of equations (64).
In order to illustrate the process the expression for F t will be developed.It is found from equation (5G) that
Fi = 4p 4 +2w 4 +4p 3 + i>\ pj4-^+4p 8w2 -p J 7j -3p 27/cos2r
+3v? sin 2t+M 2
v [9+20 cos2t+ 35 cos4t] .
(66)
Upon developing the right member explicitly by means of the first four
equations of (43), it is found that
F t-C? + [4 <f
-1 Mr,*]
COS r
+ [4o*
+4A-
f r?
-%r,
- S Mv -1 MYl cos 2t + [
4a "' + 6# 4> + f M
-|M^]cos3r + [4ar + 8^-f t 7
2
-|M2
7 7 ]cos4t.
It is found in the notation of (62), and by comparing with the right memberof the second equation of (35), that
C? = +c?-'gMr? = 2B?,
2Cj -f2
-f u-i! Af V-i M^ = 2Bf,
3C=+fW- f^M-n= 2Bf,
4C = - ^ rf- ? M\ = 2Bf,o 8
(67)
exactly fulfilling equations (64).
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INFINITESIMAL SATELLITES AND INFERIOR PLANETS. 375
189. The Solutions as Functions of the Jacobian Constant. It follows
from equation (57) that when the periodic solution is given, the constant C is
uniquely defined. The relation between C and the constant of the Jacobian
integral, when it is expressed in terms of the variables in more ordinaryuse, will be found. If the origin is taken at the center of gravity of the
system, the differential equations of motion in rectangular coordinates are
X =-r > y -= ' u= >
dx dy n n
V(x-x 1y+(y-y 1y, r t= V(x-x,y+(y-y,y
(68)
These equations admit the integral
x'2
+y'2
-2N(xy'-yx')=2U-C , (69)
where C is the constant of the Jacobian integral and N is defined in (3) . Therelation between C and C of equation (56) is required.
The variables x and y are expressed in terms of the polar coordinates,
r and v of (1), by the equations
x = rcosv ^ AcosNt, y = rs\nv ^AsinNt;
from which it follows that
x'2 + y'
2 - 2 N(xy' -yx') = r
' 2 +rV 2 -2Nr 2v'
2
AN2
rcos(v-Nt)-m\A2N2
.
mi+rrii
(70)
Upon making the transformations (6) and referring to (3) and (5), it is easily
found that
k 2
m, = n 2 a 3 =A\+mya?, Fm,= N2A 3 -n 2a s =N 2A3 ^ ,
/2 = a2 m2p2 'r
2v'
2 -2Nr 2
v'=^{l+p)2
{l+w)2 -a 2N2
{l+p)2
,
(71)mI AN2
rcos(v-Nt)=-^-aAN2
(l+ P )cos(T+w),
k 2mi . k
2m _ N2
{l+m)2a 2
n r t m2
(l +p)
Upon substituting (71) in (70) and (69) and comparing with (56), the relation
between C and C is found to be
C-m 2 A 2N2 -^-C- f-2M 2/3 A*N*
C ( 72)
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376 PERIODIC ORBITS.
It is seen from (56) that when C is expanded as a power series in m, or
in mv *if a/ A is eliminated by the last equation of (6), it starts off with a
term which is independent of m. Therefore C, the Jacobian constant for
the integral in the ordinary form, is expansible as a power series in ml/s
and is infinite for m = 0. The three periodic orbits, of which two are
complex for \m\ sufficiently small, corresponding to the three determinations
of a/ A in (6), coincide and branch at m= or C = oo . Since the coordinates
in the periodic orbits are analytic functions of mVi, and m1/3
is an analyticfunction of C through the inversion of (72), it follows that the coordinates
in the periodic orbits are analytic functions of C . One branch-point is at
C = oo . In the special problem treated by Darwin,* in which the ratio of
the finite masses is 10 to 1, he found by computation in the case of the
orbits around the smaller finite mass that there is another branch-point
for a certain value of C , at which the complex orbits first become real andcoincident, and then real and distinct.
190. Applications to the Lunar Theory. In the development of the
Lunar Theory the differential equations have been so treated that the
resulting expression for the distance, or its reciprocal, is a sum of terms whichinvolve the time only under the cosine and sine functions. The longitudeinvolves terms of the same type and the time multiplied by a constant factor.
Considering the problem in the plane of the ecliptic, there are terms whose
period is equal to one-half the synodic period of the moon.They
are knownas the variational terms. Now the period of the periodic orbits which havebeen found above is the synodic period of the revolution of the infinitesimal
body, or twice that of the variational terms. The terms of the solutions
which are of even degree in n have the period of half the synodical period of
revolution. The variational terms in fact belong to the class of periodicorbits treated here. The detailed comparison, up to m9
, with the work of
Delaunay was made in the Transactions of the American Mathematical
Society, vol. VII (1906), p. 562, and perfect agreement was found except in
the coefficients of the higher powers of m, where errors are almost unavoid-
able inDelaunay's complicated
method.Hill wrote a remarkable series of papers on the Lunar Theory in the
American Journal of Mathematics, vol. I (1878), in which he proposed to
start from the variational orbit, instead of from an ellipse, as an intermediate
orbit for the determination of the motion of the moon. The elliptic orbit
as an intermediate orbit came down from Newton and his successors, andthe inertia of the human mind is such that it was retained for over a centuryin spite of the fact that it has little to recommend it. Hill has the greathonor of initiating a new movement which, it seems certain, will be of the
highest importance.
*Acla Mathematica, vol. XXI (1897), pp. 99-242.
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INFINITESIMAL SATELLITES AND INFERIOR PLANETS. 377
The results obtained by Hill are coextensive with those given here if
we put M= and 77= 1 in the latter series. The method employed by Hill
was entirely different from that of this chapter. It was convenient in prac-
tice, but its validity can not easily be established. The same method wasextended by Brown to include terms which contain M as a factor* to the
first, second, and third degrees. A comparison of the results obtained bythe methods of this paper with those of several writers on the Lunar Theory,especially in the coefficient of a/A which converges most slowly, will befound in the Transactions of the American Mathematical Society, vol. VII
(1906), p. 569.
191. Applications to Darwin's Periodic Orbits. In Darwin's compu-tations,f the ratio of the masses of the finite bodies was ten to one. It is
found from the definition of 77 and the lastequation
of(6)
that for the
motion around the smaller of the finite bodies
vm^+m* ii' A Kmt+mJ \n) W \l+m) * ;
Darwin defined his orbits by the value of the Jacobian constant, andtheir periods were found from the detailed computations. In comparingwith his work it is simpler to take the periods which he obtained and to find
the orbits from equation (43). The comparison will be made first with his
"Satellite A" for the Jacobian constant in his notation equal to 40.5, loc. cit.,
p. 199. The synodic period was found to be 61 23'= 61.383, where the
period of the finite bodies is 360. Therefore
m= 6L|3 =0 17051>a ,
(l)
1/3
(_^_)
2/3
= .12449. (74)
The m for this orbit is more than twice that occurring in the Lunar Theory.With these values of the constants substituted in the series of 186, it is
found that
(75)
r = 0.12427+0.00652 cost -0.00420 cos2r+ 0.00004 cos3r
-0.00006 cos4t+
w= -0.12062 sinr+0.05079 sin2T-0.00184 sin3r
+ 0.00095 sin4r+
The infinitesimal body is in a line with the finite bodies and betweenthem when t = 0. The value of r at this time is found from (75) to be
r(0) =0.12657. The corresponding value given by Darwin is 0.1265. Theinfinitesimal body is in opposition at t = t, and it is found from (75) that
r(ir) =0.11345. Darwin's value is 0.1135. These agreements show that
the "Satellites A" are of the class treated in this chapter.
*American Journal of Mathematics, vol. XIV (1891), pp. 140-160.
]Acta Mathemalica, vol. XXI (1887),pp.
99-242.
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378 PERIODIC ORBITS.
In a retrograde orbit having the same sidereal period, the expressionfor m is
mn+N l+N/n {7b)
In this case
N 613.83
n 61.383+360'
thereforem= -0.12715.
The value of a/ A is the same as before, and the series for r gives
r-0. 12412 - 0.00057 cos r - 0.00158 cos 2r - 0.00000 cos 3t
-0.00001 cos 4t+W- 0.00820 sin t+0.01654 sin 2t+0.00047 sin 3t
+0.00010 sin 4t+
(77)
which are seen to converge somewhat more rapidly than the series of (95).
No retrograde orbits were computed by Darwin in his memoir in the Acta
Mathematica.
Comparison will also be made with one of Darwin's "Planets A." In
this case
*,-10, .-!, 1-1, t-^-frThe orbit will be taken for which the Jacobian constant is 40.0. The period
given by Darwin (loc. tit. p. 225) is 154 13'. Therefore
m=^||p
= 0.42838,-|
= 0.43404. (78)
With these values of the parameters, the series for r gives
r = 0.43373+0.00776 cost- 0.01286 cos2r- 0.00104 cos 3r+ (79)
From this series it is found that
r(0) = 0.42759, r(x) = 0.41415.
Darwin's results in the respective cases were r(0) = 0.423 and r(ir) =0.4140.
The agreement of these results shows the identity of his "Planets A"and the orbits covered by the analysis of this chapter.
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Chapter XII.
PERIODIC ORBITS OF SUPERIOR PLANETS.192. Introduction. The preceding chapter was devoted to the consid-
ation of the motion of an infinitesimal body subject to the attraction of twofinite bodies which revolve in circles. The periodic orbits whose existence
was there proved inclose only one of the finite bodies, and they are more
nearly circular the smaller their dimensions and the shorter their periods.The present chapter also will be devoted to the consideration of the
motion of an infinitesimal body subject to the attraction of two finite bodies
which revolve in circles; but the periodic orbits now under discussion inclose
both of the finite bodies and are more nearly circular the larger their dimensionsand the longer their periods. There are three families of orbits of this class
in which the motion is direct, and three in which it is retrograde. For small
values of the parameter in terms of which the solutions are developed, onlyone family each of the direct and of the retrograde orbits is real.
The mode of treatment of the problem of this chapter is similar to
that of the preceding. A certain parameter fj. naturally enters the problem.When n is zero, the problem reduces to that of two bodies, which admits a
circular orbit as a periodic solution. The existence of the analytic continu-
ation of this orbit with respect to the parameter n is proved, and direct
methods of constructing the solutions are developed. It is shown also howthe integral can be used as a check on the computations, or as a substitute
for one of the differential equations in the construction of the solutions.
The results of the preceding chapter were directly applicable to the
Lunar Theory; those of this chapter have no direct bearing on the practical
problems of the solar system, at least as they are at present treated. Their
chief value at present is that they cover a part of the field of the problemof three bodies in which one is infinitesimal and in which the finite bodies
revolve in circles.
193. The DifferentialEquations.
Let theorigin
of coordinates be at
the center of gravity of the finite bodies m,y and ra 2 ,and take the x?/-plane
as the plane of their motion. Suppose the infinitesimal body moves in the
z?/-plane. Let the coordinates of m,, m2 ,and the infinitesimal body be
( x i> yd> ( x i> 2/2) i and (x, y) respectively. Then the differential equations of
motion for the infinitesimal body are
d?x__dU d?y_ = dU TT _ Jim , Vm*d? dx' df dy' n "*"
r,'
rl= V{x-x 1 y-\-{y-y l Y, r^Vix-xtf+iy-ytf
379
(1)
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380 PERIODIC ORBITS.
Then, in polar coordinates, equations (1) become
<Tr /dvV dU <Fv in drdv l dU
(2)
aer\dtJ
'
dr dt2 ^
dt dt~ r dv wThe potential function U will now be developed. From (1) and (2) it
is found that
(4)
(5)
=
^^ + ^lll [1+3cos2(v_
2;1)]
+ ^~~- 2
[3 cos (-,)+ 5 cos 3(t>-tO] +]
Then equations (3) become
d*2 r UJ +
r2
"m.+m, rU L 4 1+dC0S ^ yJ J
+ fil~ fia
[3cos(>-t0+5cos3(>-t; 1 )]+ }>
r <fc2 +Z dtdt~ m1+m i r* \* 8mZ {V Vl)
+ ~ i
-^~*1
[sm(v-v l )+5smZ(v-v 1 )]+ ]
If the orbits of ml and m2 are circles, which is assumed to be the case,
equations (1) admit the Jacobian integral
{%'+(%' -*<*dl-y%)-- c ' >-** w
It follows that Wj is the mean motion of the finite bodies and that v x ^n^tQ.In polar coordinates the integral becomes
r -"+rV 2 -2n 1 rV = 2[/-C, (7)
where the primes indicate derivatives with respect to t.
When the right members of (5) are put equal to zero, the equationsadmit the particular solution
r = a, v= kVmJ+
m*(t-t B)=n(t-t ), (8)a 3 '*
where n is the angular velocity of the infinitesimal body in its orbit and t is an
arbitrary constant. It will be supposed that n is given by the observations,or that its value is assumed, and that a is determined by the second equation
of (8). The constant a has three values, only one of which is real.
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PERIODIC ORBITS OF SUPERIOR PLANETS. 381
New variables, p, 6, and t, and new constants, p. and M, will be introduced
by the equations
r = a(l+p), v = n(t-Q+d, (n-n){t-Q =r, - M,
7~^h=M-
&It follows from (6), (8), and (9) that
f- m
2/3
; (10)
and equations (5) become
i--<+')[T^+*]'+ft^?(W-
-(T^ ?( lS-.{![l+3coS 2(r +6 )]
2/3 1
+SSfe)[ 3cos(T + s)+5cos3(r+8) ]+ ) I <n >
(1+p)s+2 ;[_JL_+
a]= _^ _^ y< ll
sin2(T+9)
+^ i ^[-Dfr++8A,3(r+)]+ },
where the dots over the letters indicate derivatives with respect to r. These
equations are valid for the determination of the motion of the infinitesimal
body provided |m|<1. The right members of equations (ll) involve onlycosines and sines respectively of integral multiples of t+0. The parts in the
brackets proceed according to powers of p 2/3,the coefficients of even powers
of p 2/3 in the first and second equations being cosines and sines respectivelyof even multiples of t, and the coefficients of odd powers of p
2/3
being cosines
and sines respectively of odd multiples of r.
194. Proof of the Existence of Periodic Solutions. Suppose p = /3, p = 0,=
0, 6 = 7 at r = 0, and let the solution of (11) be written in the form
P=/(&t;t), 6 = <p(fi,y,T). (12)
Now make the transformation
P = Pi, 6=~6 lt T=-T^ (13)
The resulting equations have precisely the form (11). Consequently their
solutions with the initial conditions p l=
(3, ^ =0, 6 = 0, 6 x
= y are
Pi=KP,y,T 1 )=K(3,y-T)=p, e i= <p(P,y,T 1 )=<p((3,y;-T) = -d. (14)
Therefore, with these initial conditions, p is an even function of r, and 6 is
an odd function of t. The orbit is symmetrical with respect to the p-axis
both geometrically and in t. Such an orbit will be called symmetrical,whether it is
periodicor not.
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382 PERIODIC ORBITS.
Now consider the conditions for a closed symmetrical orbit. Since the
right members of (11) involve only sines and cosines of integral multiplesof t, sufficient conditions that in symmetrical orbits p and shall be periodic
with the period 2jir are
P-/(fc y; h) =o, o= v (p, r, j*) -0; (15)
and these conditions are necessary, provided they are distinct.
In order to examine the solutions of (15), it is convenient to use par-ameters other than /3 and y. Suppose that, at r = 0,
r = a(l+p)=a(l+a)(l e), r = a'p=
0,
(16)M . a_ M Vl+el-/i
+*~l-/*(l+a)*(l-e)*
It follows thata(l+a)
and e are themajor
semi-axis andeccentricity
of the
elliptic orbit which would be obtained if the right members of equations
(11) were zero. Because of the well-known properties of the solutions of
the two-body problem in terms of these elements, the properties of the
general solutions, so far as they do not depend upon the right members of
(11), are known. These properties will be important in solving the conditions
for periodic solutions.
Equations (11) are regular in the vicinity of p = 0, p = 0, p = 0,= 0,
= for all values of t. It follows that the moduli of a, e, and pI/3 can be
taken so small that the solutions will be regular while t runs through anyfinite
preassigned rangeof values.
Weshall
chooseas the interval for
r therange
=:t = 2jt and integrate (11) as power series in a, e, and p
1/3
, vanishingwith a = e = p I/3 = 0. That is, the results will be the analytic continuation
with respect to these parameters of the particular solution r = a, v = nt, which
exists when p = 0. The results may be written in the form
P = p 1 (a,e,p1/3
;r), = p 3 (a,e,p1/3
;T), 1
P = 2> 2 (a,e,p1/3
;r), 6 = p t (a, e,pt/3
; t),J
where p u . . ., p t are power series in a, e, and p 1/3
, with r in the coefficients.
The conditions for a periodic solution, (15), become
Pi (o, e, pl/3
; jr) = 0, p, (a, e, S/3
;jt) - 0. (18)
It will be shown that these equations can be solved for a and e as powerseries in p
1/3, vanishing with p
,/3 = 0, which converge if the modulus of p1/3
is
sufficiently small.
Since the right members of (11) carry p10/3 as a factor, the part of the
solution depending on the right members will be divisible by p10/3
. If
the right members of (11) were zero and if the solution were formed with
the initial conditions (16), the mean angular motion of the infinitesimal
body in its orbit would be
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PERIODIC ORBITS OF SUPERIOR PLANETS. 383
Consequently, from the solution of the two-body problem, it follows that
equations (18) have the form
PtW^ \sin fJTr+esin2vjir+ |+M
10/3fc(a, e, ii>
3
; jv) = 0,
p 3 (jir)=-^j^jir-pJT-2esmvJT- i-\-n
mq 3 (a,e, f x
l/3
;JTr)=0,
(20)
where the unwritten parts in the brackets are sines of multiples of vjir, and
carry e2 as a factor.
Upon referring to (19), it is observed that the first equation of (20) is
divisible by m2
> and the second by n. After dividing by these factors the
equations are still satisfied by a = e = n = 0; moreover, the determinant of
their linear terms in a and e is
A =, >
= + fiW0. (21)
Therefore, besides the solution m= 0, equations (20) have a unique solution
for a and e as power series in ///3
, vanishing with ju,/3 = 0, which converge for
the modulus of ix'3
sufficiently small. These power series carry ni/3 as a
factor, and can be written in the form
a = M4/3 P l (M
t/3
), e = MV3
W/3
)- (22)
Upon substituting these series in the right members of (17), which vanish
with a = e = m1/3 = 0, the result is
p=m 4/3&(/x
1/3; r), 0=m 4/3 Q2 (m
1/3
; t). (23)
The series Qx and Q2 are periodic in t with the period 2j't because the con-
ditions that the solutions shall have this period have been satisfied. Since
(17) converge for all O^t^jV if the moduli of a, e, and mV3 are sufficiently
small, and since the expressions for a and e given in (22) vanish for n = 0, it
follows that the modulus of n1/3 can be taken so small that the series (23) con-
verge for all t in the interval; and since they are periodic with the period 2Jx,
the convergence holds for all finite values of r.
The integer j has so far been undetermined. When j is unity, the
periodic solutions exist uniquely and their period is 2w. When j is greater
than unity the periodic solutions also exist uniquely. Since the periodic
orbits for j greater than unity include those for j equal to unity, and since in
both cases there is precisely one periodic orbit for a given value of nVi
,
it follows that all the symmetrical periodic orbits of the class under consideration
have the period 2ir in the independent variable r.
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384 PERIODIC ORBITS.
It follows from (6) and (9) that t-\-B = vv x . Since in the periodic
solution is periodic with the period 2v, the period of the solution is the
synodic period of the three bodies. Hence, if the motion of the infinitesimal
bodyis referred to a set of axes
havingtheir
originat
the centerof
gravityof the system and rotating in the direction of motion of the finite bodies at
the angular rate at which they move, and if the z-axis passes through the
finite bodies, then the periodic orbit of the infinitesimal body, which has
been proved to exist, will be symmetrical with respect to the z-axis. Since,
by hypothesis, a > R, it follows from (6) and (8) that n^ > n. Therefore, evenif the motion of the infinitesimal body is forward with respect to fixed axes,
it is retrograde with respect to the rotating axes.
It is supposed that the period of the finite bodies, and therefore n is
given in advance and remains fixed. The variation of the parameter /i1/a
corresponds to a variation of the period of the infinitesimal body definedby n. If the motion with respect to fixed axes is forward, n has the same
sign as r^, and /x1/3 has three values, one of which is real and positive while
the other two are complex. If the motion is retrograde, /tV3 has three
different values, one of which is real and negative while the other two are
complex. Therefore, for a given period, there are six symmetrical orbits,
three direct and three retrograde; and for small fx one direct orbit is real andone retrograde orbit is real, while in the others the coordinates are complex.This means, of course, that the corresponding solutions do not exist in the
physical problem. The coordinates of the complex orbits are conjugate in
pairs. For a certain value of /x
1/3
they may become equal, and thereforereal, and, for larger values of /*
1/3, real and distinct.
Upon transforming the integral (7) by (9), it is found that
+ a-X+p)^ 1^^ (24)
+5cos3(t+0)]+ }-(?
where C = n\(l m)2 o 2
Ci- It follows from this equation and (23) that Cj
can be expanded as a power series in ///3
, vanishing with ju1/3
- The term of
lowest degree in n1/3
, after substituting (23), is nm. Therefore, n Vi can be
expanded as a power series in C1
/3
. For Ci = 0, the three branches of the
function are the same. Since a = R/n Vs, the relation between C and (7, is
n n?/? (!-/*)*& n]R2
n .i nunC = v
Vt*' - =
yf- [1 + power series in n] . (25)
From this it follows that C=oo for juI/3 = 0. Therefore the periodic orbits
branch at C = oo , and there are two cycles of three each.
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PERIODIC ORBITS OF SUPERIOR PLANETS. 385
195. Practical Construction of the Periodic Solutions. It has been
proved that the symmetrical periodic solutions under discussion are expres-sible in the form
p=S P(M ,/3, d=2e,n t/3
, (26)1=4 (=4
where the p, and 0, are functions of r. Since these series are periodic and
converge for all |p1/3
| sufficiently small, it follows that each p, and d, separatelyis periodic; that is,
P( (r+27r) = Pl (r), 0,(t+2tt) m8&) . (27)
In every closed orbit there are points at which dp/dr = 0. Suppose t
of (9) is so determined that this condition is satisfied at r = 0; it will follow
from this and the convergence of (26) for all |m1/3
| sufficiently small that
Pf = (i=i, ... co). (28)
In the symmetrical periodic orbits the value of 6 is zero at t = 0. Butthis condition will not be imposed, because the general periodic orbits, whoseinitial conditions are not specialized, include those which are symmetrical;and in the construction it will appear that the conditions for symmetry are
a consequence of those for periodicity. Hence all the periodic orbits of the
class under consideration are symmetrical.
Equations (26) are to be substituted in (11), arranged as power series
in p.m
,and the coefficients of the several powers of /x
1/3 set equal to zero.
The coefficients of nm set equal to zero give the equations
P 4 = 0, 4 = O. (29)
The solutions of these equations satisfying (27) and (28) are
p t= a t , 4
= &4 , (30)
where a A and 6 4 are so far undetermined constants.
The coefficients of p5/3
,. . .
, nm are the same as (29) except for their
subscripts, and their solutions satisfying (27) and (28) are similarly
Pl = aj, 6j = b } (j=5, ...,9), (31)
where all the a, and b s are so far undetermined constants.
The coefficients of m10/3give the equations
p I0= 3a 4 -jM[l+3cos2r], IO
= -|Msin2r. (32)
In order that the solution of the first of these equations shall be periodicthe condition
a,= \M (33)
must be imposed, which uniquely determines the constant a 4 . Then the
solution of (32) satisfying (27) and (28) is
Pio= a 10 +^Mcos2r, lo
= 6 lo +|Msin2r, (34)
where a 10 and 6 10 are as yet undetermined.
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386 PERIODIC ORBITS.
The coefficients of /x11/3 are defined by the equations
Pn = 3a 6 , (?,i=
0,
from which it follows that
a 6 = 0, fti-Ou, flu = 6,1, (35)
where a n and b n are as yet undetermined.
The coefficients of m12/3 in the solutions satisfy the equations
Upon imposing conditions (27) and integrating, it is found that a, = 0, and
Mjjni nh)
2(mj(36)
ftt-Oa- *">,Wl
, ^ r3cosT+|cos3rl,2 12
2(,Wi+/as) L '9 J'
i 3Af(mi ms)r,
5 1g " = 6 - mi+m, L sinr+ ? sm3rj,
where a I2 and 6 12 remain so far undetermined.
In a similar way it is found from the coefficients of nli/ * that
a 7= 0, p I3
= a 13 +|Mcos2T, 0i 3= 6 13 +|jMsin2T, (37)
where a 13 and b n remain undetermined at this step.
So far all the b } have remained arbitrary, and it is necessary to carry the
integration one step further in order to see how they are determined. Thecoefficients of n li/3 are defined by
Pi4= 3a 8 -3a
2
+jM[36 4 sin2T+2a 4 +6a 4 cos2T],
(38)
fli4= -^flio 3-M" [6 4 cos2t 2a 4 sin2-r].
Upon substituting the values of a 4 and 1O from (33) and (34), imposing the
conditions (27) and integrating, it is found that o 8= t^M*, and
(39)
Pi4=
fli4 |64-Wsin2T ^Af2
cos2r,
flu=
6,4 + f&4Mcos2r - ^ M2sin 2r.
The condition (28) for j = 14 gives the equations
64 = 0, Pl4= a 14 -^M= cos 2r, 14
= 6 14 -|M 2 sin2r. (40)
It is found in a similar way from the coefficients of ju15/8 that b 6
= 0, and
'-"'-^SftH'+l 83*]'
9 ''= 6 '>+i J,f SfS[ 98inr+ S sin3 4where a 15 and 6, 6 remain so far arbitrary.
(41)
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PERIODIC ORBITS OF SUPERIOR PLANETS. 387
It will be observed that, so far as the computation has been carried, the
coefficients of the p t are cosines of integral multiples of t, and that the coeffi-
cients of the 6 j , except for the undetermined additive constants, are sines of
integral multiplesof t. In the
computationof
pjthe
periodicityconditions
have uniquely determined a,_ 6 ,and the condition p s
= at t = has
required that &,_ 10= 0. It will now be shown that these properties are
general. Suppose p 4 ,. . . , p; $4t . . . , have been computed and that
the coefficients are all known except a_ 6 ,. . .
,a n ,
which enter additivelyin p_ 4 ,
. . ., p respectively, and &_ 9 ,
. . .,b
,which enter additively in
0_ 9 ,. . .
, B respectively. The differential equations for the determination
of p n+1 and n+1 are
PB+l= 3a n _ B+|M6 B_ 9 sin2r+F n+1 (r), B+l
= Gn+1 (r) , (42)
where F B+1 (t) and (j b+1 (t) are entirely known functions of r. It follows from
the assumptions respecting p 4 ;. . .
, p; 4 ,. . .
,and the properties of
equations (11) that F b+1 (t) is a sum of cosines of integral multiples of t, andthat G +1 (t) is a sum of sines of integral multiples of t. Hence they may be
written in the form
F n+i (r) -2 Af +l) cos jr, (? n+1 (r)= 2 5< n+1)
sinyr.
In order that the solution of the first equation of (42) shall be periodic
the condition3a n _ 6
+^r+I> = (43)
must be imposed, and this condition uniquely determines a_ 5 .
After equation (43) has been satisfied, the solution of the first equationof (42) is
P +1= a n+1 -|M6_ 9 sin2r+S a^" cos jr, a** - -kA*+. (44)
J
The condition p = at t = makes it necessary to take
&- 9= 0. (45)
Thenp B+1
iscompletely
determinedexcept
for the additive constant an+i ,
and it is a sum of cosines of integral multiples of r.
The solution of the second equation of (42) is
+ i= 6 B+ i+S/3r
+,)sinjr, ff+ = -B? +i)
. (46)
Hence 0+i is a sum of sines of integral multiples of r, except for the unde-
termined constant 6 B+ i ,which must be put equal to zero in order to satisfy
the condition on p +u . These results lead, by induction, to the conclusion
that the p ; and 0/= 4, . . . oo) are sums of cosines and sines respectively
of integral multiples of t whose coefficients are uniquely determined.
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388 PERIODIC ORBITS.
From the properties of the solutions which have just been established,
it follows that not only is p = at t = 0, but also 0(0) = 0. Therefore these
periodic orbits are the symmetrical orbits whose existence was established
in 194. In the construction it was not assumed that the orbits were
symmetrical, and since this property is a necessary consequence of the
periodicity conditions, it follows that all periodic solutions which are expan-sible as power series in p
,/3 are symmetrical. It is easily shown, by direct
consideration of the construction of periodic solutions, that they can not be
expanded as power series in p1/3
except when,;' is a multiple of 3, and that then
they reduce to those found above.
196. Application of the Integral. The differential equations admit the
integral (24), which, for brevity, can be written in the form
F(p,p,M,T,/z,/3
)=0.
It follows from the form of (24) and the expansions (26) that the left memberof this equation can be developed as a power series in p}
/3, giving
F = F,+F^+F^+ +V+ =0. (47)
Since the p, and 9, are sums of cosines and sines respectively of integral
multiples of t, and since p enters in (24) only in the second degree and onlyin even degrees, it follows that the Fj are sums of cosines of integral multiples
of t. Equation (47) is an identity in p.1 ' 3
,whence
FB= SCfcos;V=0 (n-0, ... oo).
Since these equations hold for all values of t, it follows that
C<"' = (n-0, ... oo; j=o, ... ,). (48)
The Cf are functions of the f, . . ., af and #\ . . .
, # n). Hence
equations (48) can be used as check formulas on the computation of the
coefficients of the solutions.
Equations (48) can be used in place of the second equation of (11) for
the determination of the j8j
B)
, the coefficients of the trigonometric terms in
the expression for 0. Suppose p 4 ,. . .
, p- tand 4 ,
. . .,6 n - r have been
determined except for additive constants in p_ 6 , . . . , p n - t . It follows from(24) that F n is
Fn = -26 n + Pn ( P j, Pj ,e h Ot) (i=4, . . .
, n-l),
where P is a polynomial in the arguments indicated. Consequently
equations (48) are of the form
Cr--2j^+D(ar,/5T)-0 Cr-4, .... n-l),
which uniquely determine the #"\
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Chapter XIII.
A CLASS OF PERIODIC ORBITS OF A PARTICLE SUB*
JECT TO THE ATTRACTION OF n SPHERESHAVING PRESCRIBED MOTION.
By William Raymond Longley.
197. Introduction. The restricted problem of three bodies furnishes
naturally the starting-point* for the consideration of the periodic orbits
of an infinitesimal body, or particle, which is subject to the Newtonianattraction of certain finite spheres whose motion is supposed to be known.The two finite bodies are supposed to revolve in circles about their commoncenter of mass, and the motion of the particle is restricted to the plane in
which the finite bodies move. One class of orbits occurring in this problemis that in which the particle revolves about one of the finite bodies, and for
the consideration of these orbits it is convenient to refer the motion of the
particle to a plane rotating with the angular velocity of the finite bodies.
All of the known periodic orbits of this type possess one and only one line
of symmetry, namely, the line joining the finite bodies, and this propertyof symmetry plays an important part in the proof of their existence and the
construction of series to represent them.
The purpose of this chapter is to generalize the restricted problem byintroducing into the plane of motion more than two finite bodies. Thecoordinates of the finite bodies (spheres) are supposed to be known functions
of the time, that is, the motion of the spheres is prescribed. For the analysiswhich follows the nature of the forces producing this motion is unimportant.The spheres are supposed to attract the particle according to the Newtonian
law. Besides involving additional terms in the disturbing function, this
generalization modifies the original problem by introducing cases where
the periodic orbits have no line of symmetry, and cases where there are
more lines of symmetry than one. This modification necessitates some
changes in the details of the analysis which must be worked out. In order
to avoid cumbersome notation, the analysis will be developed for simple
particular cases of the motion of spheres under their Newtonian attraction;
with slight changes it is applicable to more general types of prescribed
motion of the finite bodies, which are indicated in 207.
*See papers by Hill, American Journal of Mathematics, vol. 1 (1878), p. 245; Darwin, Ada Maihemalica,
vol. 21 (1897), p. 99; and Moulton, Transactions of the American Mathematical Society, vol. 7 (1906), p. 537.389
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390 PERIODIC ORBITS.
198. Existence of Periodic Orbits Having no Line of Symmetry. It
was shown by Lagrange* that an equilateral triangle is a possible configu-
ration for three spheres revolving in circles about their common center of
mass. This motion of three finite bodies will serve to illustrate the case
when the periodic orbits of the particle about one of the bodies possess no
line of symmetry. Let the masses of the three finite bodies moving accord-
ing to the equilateral-triangle solution be denoted by M, Mlt Mt , and sup-
pose the particle P revolves about the mass M. Suppose also that the
masses Af, and Mt are unequal. f
With reference to M as origin and an axis having a fixed direction in
space, let the polar coordinates of Af,, Mt ,and P be respectively (R x , FJ,
(R t , Vt), and (r, v). The coordinates of the bodies are expressed in terms
of the time, t, as follows:
R^R^A, F.-F.-f-M, (1)
where
Here N denotes the angular velocity, A the length of a side of the triangle,
and fc is a constant depending upon the units employed.The differential equations of motion of P are
dt2 r
\dt)+
r2 ~dr'
rd?
+ Zdt dt~ rdv' (Z)
where
&-*[ + f-^rcos(,-7 1 )-frcos(,-F 2)]
!
(3)
h-V? + A 2
-2rAeos{v-V 1 ), r t= Vr i +A i
-2rAeos{v-V i ).
Let us define m and a by the relations
mv = N, vW = k 2
M, (4)
where v is a quantity to be assigned later.
By the substitution v = w-\- Vx= w + Nt the motion is referred to an
axis rotating with the angular velocity of the finite bodies and passingalways through Mt ; and factors depending upon the units employed are
eliminated by the relations r = ap, vt = r. On making these substitutions in
equations (2) and dividing by v*a, the differential equations of relative
motion become
Prize memoir, Essai sur le Problbne des Trow Corps, 1772; Coll. Works, vol. 6, p. 229.
tlf M,=Mi the periodic orbit of P about M lias a line of symmetry, namely, the median of the trianglefrom the vertex M, which is the line joining M to the center of mass of the system. For the treatment of
this special case it is convenient to make use of the property of symmetry and to employ analysis similar
to that developed in 202 and 203.
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PARTICLE ATTRACTED BY W SPHERES. 391
We can expand Q as a power series in ap/A which is convergent for all
values of w provided the distance MP= ap is less than A ; and in all thatfollows this condition is supposed to be satisfied. The expansion has the
form
Qsb VM; 1[l+ Kl p )
2
{1 + 3cos2t 4+i(i p )
S
{3cosw+5cos3wU ]
+^t+Kf#+3<--f)} ,
+Klp)'{s~*(-f)+s<3(-f)j+ ]
Let \ and X2 be defined by the relations
M.^CAft+M,), Af.-X^+A/,). (6)
From equations (1) we have
kXMt+M,) _ M1+Mi ytA3
~ M+Mt +M,Then, on setting Ml +Mt jrM+M1+Mt
A '
it follows that the second members of equations (5) have the form
y2
1
-^T= JK:m 2
p[|x1[l+3cos2^j+|x 1
(-jp)|3cosw;+5cos3w;JH
+Jx s
|l+3cos2(w-|)}-r|xfQp)|3cos(tt;-|)+5co83(ii;-|)}+
iiCm2
p| X!sin2w+-X t (-jp )jsinw;+5sin3w) [+2
ap dw
+|x t
sin2(;-|)+|x,Qp){sin(M;-|)+58in3;(w-|)}+
(7)
It is convenient to introduce a parameter p. into the differential equations
(5) by the relations
m=p, X, = Xp, T"** 1 ( 8 )
wherever the degree of a/A is higher than the first. The quantities X and r?
are numerical constants. By relating X2 and p the existence proof is madeto depend only upon general properties and certain terms of the differential
equations which involve \ ;that is, upon terms in the disturbing function
which are due to the body M\ . We shall consider the solution of equations
(5) as power series in the parameter p. The differential equations, and
consequently also the solution, do not represent the physical problem under
consideration for any value of the parameter except the one satisfying the
relations (8). But if the solution is valid when this particular numerical
value of p is
substituted,then it is a solution of the differential
equations
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where
392 PERIODIC ORBITS.
representing the physical problem and therefore has a physical interpre-
tation. The generalization of the parameter a/ A is merely for convenience
in having finite expressions in the equations which determine the coefficients
at the various steps in the solution.
On introducing the parameter p as indicated, equations (5) become
f= Kp\\\^(l+3cos2w)+^^ p(3cosw+5cos3w)i+pF(p,p.,cosjw,smjw)\>
g= Kp\\A- sin2w+ x -r-p (sin w+ 5 sin 3 w)[+nG(p,n ) smjw, cosjw) I
where F and G are functions of the indicated arguments.
Equations (9) are periodic in w with the period 2ir and do not involve texplicitly. Suppose that
P = &(t), w = ^{t)
is a solution. Sufficient conditions that the solution shall be periodic with
the period 2pw (where p is an integer) are
fc(2px)-2p,r=fc(0), tf(2p>)-tfty), J
where ^ and \f/' denote derivatives of ^ t and \f/ 2 with respect to r.
When p=
a periodic solution,which will
becalled the undisturbed
orbit, is known, namely,p = l, W=
T, (11)
and the initial conditions are
p=l, p' =
0, w =0, w' = l. (12)
It will be shown by the process of analytic continuation that, for values of
n different from zero, but sufficiently small, there exists a periodic solution
which, for m = 0, reduces to equations (11). For this purpose we consider
the solution of equations (9) subject to the initial conditions
P-1+&, p'=&, w-A, tf-l+Aj (13)
where ft , /3, , /3 3 , /3 4 , are to be determined as functions of p, vanishing with
p, so that the conditions of periodicity (10) shall hold. It follows from the
differential equations that the solution is expressible as power series in ft ,
ft > ft > ft >an d p and that, for sufficiently small values of the parameters,
the series are convergent for all values of t from to 2pir. We supposethat this condition on the moduli of the parameters is satisfied.
For the determination of those terms in the series which involve the
initial conditions but not p2
, it is possible to use the known solution of the
two-body problem, since for p = equations (9) reduce to the equations of
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PARTICLE ATTRACTED BY ft. SPHERES. 393
motion of a particle P when subject to the attraction of M alone. Hence,instead of the additive increments ft , ft , ft , ft ,
it is convenient to introduce
new parameters a, e, 0, <p defined by the relations
,_o _ esinflp ~ Pi ~
Vl+a(l-ecos0)'=l+p l =(l + a)(l-ecosO),
a I" COS0 C "|w =ft= arc cos s3 Ll-ecos0j
arc costi
cos(0 y) e"
ecos(0 <p).
, _ , . a Vl e2
W -1+fc-( 1+a )3/2 (l
_ ecos0 )2
(14)
By introducing /x in the fourth of equations (14) it is possible to use the two-
body problem for determining all terms of the solution which are independentof ii
2. In terms of the parameters a, e, 0, <p the properties of the solution
P is the position of the
particle at t = 0.
W, is the longitude of the
particle at t=0.
Wt
Wt is the longitude ofperihelion.
W- aacis
Fio. 8.
are well known, and the conditions of periodicity can be easily discussed.
The geometric meaning of the angles 0, <p, W ,and Wt ,
which occur below, is
shown in Fig. 8.
On making the substitution w = u /jlt and then setting n = in equa-tions (9), we obtain the differential equations of the two-body problem, of
which the solution is
r = (l + a)(l-ecostf), cos(tt+ Wx- W)
=t
C0Sjg 1 >x
1 ecosE
where E is defined by the relation
sin (u+W -W)- Vl ~ e2 sin ^.in xM+w1 w )
l _ ecosE>
(15)
(l+a)V2 +0 esin0 = '
esini?.
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394 PERIODIC ORBITS.
All other terms in the solution involve ps
. To find the terms in p1 and pV
we will write
p=l+p,p* + ppV + , W= T-TH+<p + Wt nt+WH*<p+
On substituting these expressions in the differential equations, there results
for the determination of p, and wt the following set of equations:
^-2^-3p s=^(1+3cos2t)+^-|(3cost+5cos3t),
-57+2^= - sm2r
^(smr+SsinSr).
These equations are integrable, and the result of integration is
p 2=J R:x 1 [-l-2| + (2+f f)cosr + J|^rsinr-cos2r-||cos3r].
^ r/5 ,,o\ ( .,
201 a\,
15 a . n,
135 a ~|wi =K\ 1 \_(j+3j)T-(4:+ jjsinr + jrcosT + -sin2r + jsm3rj.
By a similar computation it is shown that
p=XX 1 [-(4+f^)sinr+J||rcosr+2sin2r + ||sin3r]>
V\ T21 i 288 a ( Q i333 Ol\ 15 a ,11 , 45 a 1
The terms independent of p.2 are obtained from equations (15) by Taylor's
expansion and the relation w = u pt; and the solution becomes
p=l + a ecosr aefcosT -Tsinr) +e0sinr
+ ae0(sinr | r cos r) +p 2 p! +ppV + ' ' '
t
w = t -Ta+2esinT+^ rp 3aerCOST ed(l 2cosr)
+e^>+oe0 (5cosT+3rsin t)+w2
p
s
+wpV+' ' '
Applying the conditions (10) that the solution shall be periodic, we have
(a) O=-3pxa60+^ P T/*V+ ,
(b) = 3 P 7rae+|-|pV+ "',
(c) 0= 3pira 2pjrp 6p7rae+ ,
(d) = 6piraefl-~p7rpV +
(16)
(17)
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PARTICLE ATTRACTED BY ft. SPHERES. 395
The conditions (17) involve the four quantities a, e, 0, <p, and, if independent,would determine them in terms of li. But the differential equations (9)
do not involve r explicitly and hence admit the integral of Jacobi. This
furnishes a relation of the type
F(a, e, 0, <p, /x)= constant,
and equations (17) are not independent.* It follows that if (a), (6), and (c)
are solved for the three quantities a, e, and in terms of ju and <p and the
results substituted in (d), the equation is satisfied identically in <p. In this
problem the dynamical interpretation is simple. Since the finite bodies
move in circles the origin of time is arbitrary.! The most convenient
choice is t = when w = 0, which is equivalent to choosing <p= 0.
Consider the solution of equations (a), (6), and (c) for a, e, and 0. The
equations have the following properties:(I) There are no terms independent of a and n. This follows from the
fact that, in the two-body problem, the period does not depend upon e and 0.
(II) There are no terms involving n to the first degree except the oneterm 2 pirn, which occurs in (c).
(III) There are no terms in independent of e, since does not enter
the initial conditions independently of e. It follows from these propertiesand the particular form of the first terms of the equations that a, e, andare determined uniquely as power series in n by the following steps :
(1) From (c) we obtain
ci = m[ 3+ + function (fx, e, i)\
(2) This value of a when substituted in (6) permits a factor /z to bedivided out. We can then solve the result for e as a power series in /x andwhich contains /tasa factor, and obtain
e==/x [i6Z~r-+ function
(m,0)].
(3) When the values of a and e are substituted in (a) a factor n* can be
divided out and obtained as a power ser ies in n alone, vanishing with fi.
(4) By the substitution of the value of thus found in the expressionsfor e and a, we obtain finally
o = mPi(m), e = np 2 (n), = np 3 (n).
The preceding operations are known to be convergent for all values of
a, e, 0, and n which are sufficiently small. Hence, for a given value of n
sufficiently small, it is possible to determine the initial conditions (14) as
power series in /x such that the solution of the differential equations (9)
shall be periodic in r with the period 2pir.
*See Poincarfi, loc. cil., p. 87.
fWhen the finite bodies do not form a fixed configuration in the rotating plane the integral of Jacobi
does not exist and the origin of time is not arbitrary. In this ease it is necessary to determine the four
parameters from the conditions of periodicity. The case of the triangular solution when the finite bodies
move in ellipses has been treated by Longley in a paper in the Transactions of the American Mathematical
Society, vol. 8 (1907), pp. 159-188.
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396 PERIODIC ORBITS.
When the values of a, e, in terms of /* are substituted in equations
(16) the periodic solution is obtained. The period of the solution is 2pir
in t, where p is an integer, and from the conditions of periodicity (10) it
is
apparentthat the
particlemakes
prevolutions in the rotating plane
during a period. The process by which the periodic solution was obtained
yields a unique result; therefore, for an assigned value of n, there exists one,
and only one, orbit having the period 2pir. Since the orbits having the
period 2pir, p> 1, include those having the period 2ir, it follows that all the
orbits of this analytic type are closed after one synodic revolution*
Since t = vt the period of the solution in t is 2v/v, and the quantity v,
which is so far arbitrary, can be determined by assigning the period of the
solution. The parameter n is then determined by the relation n = m=N/v,that is, the numerical value of /z is the ratio of the mean motions of the
finite bodies and of theparticle.
If the direction of revolution of thepar-
ticle is the same as that of the finite bodies that is, if the orbit is direct
v and N have the same sign and m is positive; if the orbit is retrograde, v
and N have opposite signs and n is negative. Since for an assigned value
of n there exists one, and only one, periodic orbit, and since values of ju
which are numerically equal, but opposite in sign, give orbits having the
same period in t, it follows that for a given period there exist two, and only
two, real orbits of the type under consideration. In one the motion is
direct, and in the other it is retrograde.
We may now state the result as follows : The period 2-k/v of the solution
maybe
assigned arbitrarilyin
advance, subject onlyto the condition that the
ratio N/v is sufficiently small, where 2ir/N is the period of the motion of the
finite bodies. Then there exis t two, and only two, real periodic orbits of the
particle having the required period. In one the motion is direct, and in the
other it is retrograde. All the orbits of this type are closed after one synodic
revolution.
In deriving this conclusion no use was made of the explicit values of
those terms in the disturbing function which are due to the body M2 . The
proof depends entirely upon the form of certain terms of the solution which
involve \. Hence the analysis and conclusions are applicable without
change to the case where n finite bodies revolvein circles in
such a way asto form in the rotating plane a fixed configuration.
199. Construction of Periodic Orbits Having no Line of Symmetry. It
is possible to construct the periodic solutions of the differential equations
(9) by the method indicated in the existence proof, but the process is labo-
rious. A method will now be given by which the solution to any desired
number of terms can be conveniently constructed. It is not necessary to
determine the initial conditions explicitly in advance, and the computationinvolves only algebraic processes.
Since no new orbits are obtainedby
takingp>
1 we will assume hereafter thatp= 1.
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PARTICLE ATTRACTED BY n SPHERES. 397
It has been proved that the periodic solutions are expressible in the
form
\ (18)
The series (18) satisfy the differential equations (9) uniformly over a finite
interval in p, and hence, when the series are substituted in the differential
equations, the coefficient of each power of p must vanish. Furthermore,the series are periodic with period 2w in r; and, because the periodicity
holds for a continuous range of values of p, each coefficient p, and wt sepa-
rately is periodic with the period 2x in r. It has been shown also that wecan choose w = when r = 0, and because this holds identically in p, it follows
that wx (0) = for every i.
Let the solution (18) be substituted in the differential equations (9)
and arrange the results as power series in p. The terms of the first membershave the following forms, where the accents indicate derivatives with
respect to r:
Q> P iti
" 2 i" 3 i I n *
I
^2 =PiP + P 2 M +P 3 M + +p t H+ ,
P (^+m)2
= 1+ [ Pl +2(;;+l)] p
+ [p 2 +2w 2 +2 Pl K+l) + K+l) 2
]p2
+[p 3 +2^
+2(w' + l)w' +2p 1 w' 2 +2p 2 (w' + l)+p 1 (w' +iy] t i.
3
+ + [ Pt +2w' +2(w' + l)w' - l + 2 Pi _ 1 (w' + l)
+ 2p 1 wJ_,+ ]p + ,
i = l-2p lM -(2p 2 -3pDM 2 -(2p 3 -6p 1 p 2 -4p3
)M3
P
+ +(2^-6*.^+ )m'+ ,
p|j =<p+ +pX) m'+K'+^K+^Om 3
+ +(w:+ Pl w'u+ +p 1- 1<v+
^(^+m)=p;m+[p;+p;k+d]m2
+[p;+p 2 (^+i)+pX] m3 +
+t^+pU(5+r)+ +p>;_JM
< +
(19)
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398 PERIODIC ORBITS.
The second members have no terms independent of p2
. Therefore, on
equating to zero the coefficients of the first power of p, we have for the
determination of p x and wl
<^_ 2 <^!-3 Pl= 2, ^1+2^=0. (20)dr 2 dr dr 2 dr v
It follows from these equations that
p^a+O+Ccosr+cfsinr,J
Wl = c> -
(4 + 3 ef j r - 2 <f sin t + 2 ej cos r,
where cj", c,", c"', c"' are constants of integration. Since p, and w, are
periodic, the coefficient of t in w^ must vanish. This condition determines
the constant cf, namely, c{= 4/3. Since ^ = when t = 0, c"'= 2c"\
The constants c and cjl)
are so far undetermined.On equating to zero the coefficients of the second power of p, the fol-
lowing set of equations is obtained :
^_2d^-3p 2
= (w[+iy+2 Pl (w 1 +l)-3p2
l +f ,
^ + 2^=- Pl <-2p>;+l)+0 o ,
(22)
where / and g are obtained from / and g respectively by writing p = 0,
w=t, p
= l. The second members are known functions of t and theequa-tions are explicitly
(23)
^-2^-3p J= yl
, + (^"+^)
)cosr+fc>
sinr
+A?co82t+A? cos3r,
^? + 2^ =(| c ;
i
>+i)f)
)sinT-fc)
cosr+i)fsin2r+Z))
sin3r.
On integrating the second equation, we have
^+2p,= cr
)
-(| C< 1) + I>i
2)
)cosr-fcr)
sinr-'^cos2r-^cos3r. (24)
On eliminating ?p3from the first of equations (23) by means of equation
(24), there results
^+ Pi= A?+2c? )
+(-2c!i)
+A?-2D?)cosT-2cl smT
[ (25)
+ (^B-|Z);
2
>)cos2r + (^)
-|i)r)cos3T.
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PARTICLE ATTRACTED BY tt SPHERES. 399
In order that the solution of equations (25) shall contain no non-periodic
term, the coefficients of cost and sinr must vanish; hence c and c"' are
determined by the conditions
2c^=AT-2Df, ef-O.
With these values of c{ " and c"
} the solution becomes
p 2= ^;
2)
+2cf)
+ci2)
cosr+cf sinT+a^2>
cos2T+^)
cos3r,
where
On substituting this value of p 2 in equation (24) and integrating, we obtain
for w2 a solution of the form
wt = cf - (2 A? +3 ef ) r - 2 c<2)
sin t + 2 c<2)
cos r + 5,2)
sin 2 r + If sin 3 r,
where
hT=-UDT+2jaT) = 2,3).
Since wt is periodic, c is determined by the condition
2A?)
+3c?) = 0.
Since w, = when t = 0, c is expressible in terms of c, namely,
cf+2c = 0.
Of the eight constants of integration which have been introduced in thefirst two steps, five (c, c, c, c, c) have been determined uniquely;c has been expressed uniquely in terms of c; while the remaining two
(c, c) are still arbitrary.
By equating to zero the coefficients of the third power of p the followingset of equations is obtained:
^-2^-3 Pl= 2( Pl +w[+l)w' +2 Pi (wl+V
+ Pl K+l) s -6p 1 p t +4 P ;+/ 1 ,
^?
+2^=-p1<-pX-2p;k+d-2p>;+^ 1 ,
(26)
where f x and g x denote the coefficients of p. in / and g respectively. Thesecond members are known functions of t, and the equations have the form
(Fp 3 dw 3
|^-2^-3p,= ^ )
+(^ C;
2
'+^Iw)cosr + (fcf
,
+fir)sinr
+A? cosjr+BfsmJT 0=2, 3, 4).
^ ?
+2^=(fc+i>r) sinr +(-f c" ) + c " ,
) coST
+D?mnJT+C?9t*jr.
(27)
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400 PERIODIC ORBITS.
The treatment of equations (27) proceeds by steps similar to those
employed in the solution of equations (23). Four new constants of inte-
gration are introduced, namely, cj", c\ cf, cf, while cf , cf, and cf are
uniquely determined bythe conditions
2c? = A?-2D?, 2c? = B?+2C?, 3ef- 2ilf.
The solution has the form
ft= af+cf cost +cf sinr+ 2 (af cosiT+/3f sin/Y,)>
+ 2 (^smJT + yfcosjr)-
From the condition that w3= when r = 0, it follows that cf is expres-
sible uniquely in terms of cf by the relation
cf+2cf+2 T f = 0.
The two constants cf and cf are determined in the next step.
It can be established by complete induction that the preceding process
can be carried as far as is desired. Suppose p l} wt ; p l} wt ; . . . ; p^ x , wt _ x
have been determined by this process. The expressions have the following
form:
p,- of + 2 (a? cos jr + 0" sinjr) .
wt= y?+ 2 (5f sin jV+T^cosir) (1=1, 2, . . . , i-2),
p 1_ 1=
oi'-1)
+c<'-1) cosT- r -c<
, - 1)
sinr+ 2 (a-l)
cos jr + ft'-" sinj T )>J-2 V '
^_ 1 =cr ,,
+(-2 Cr-^r+|Dr-i,
)sinr+(2cr)
+fj5r-i)
+|cr-i)
cosr)
+ 2 (^y-^sinir + Tr^osir)-
The constants of integration have been uniquely determined except cf",cj'~
l)
, and cj<-1)
. The first two are so far arbitrary, while c""" is expressiblein terms of c""
l>
by the relation
ci, - ,) -
r -2cri, + 2 7;'-
1) = 0.
/si
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PARTICLE ATTRACTED BY n SPHERES. 401
The equations for the determination of p, and w, have the form
Hi|* -2^-3Pt=A?+ (^
c<<-
+ ^>)cosr + (^cr'+J
Br
,
)sin
+ s(^ ,
cosir + ^ ,
sinir)
^ +2^'=
(^c
- +I><<')
s in T+(- ^- + cr>)cosr
+ Sl
(Dj"sinyr + C<,,
cosir)-
\ (28)
The coefficients A, B, C, D are known constants and equations (28) are
solved by the steps employed in the solution of equations (23) . During the
process four constants of integration are introduced, namely c"\ c, cf, c,and four are uniquely determined by the conditions
2 c- l) = A? - 2 Z){ ,2 c*'-" =
JBJ" + 2 C,
ci1 "" = - 2 c'*"" - 2 Ty
_1>, 3 <f = - 24 >.
1=1
(29)
(30)
The solution of equations (28) is
p.=a'+f cost +ef sin t+ S (&9MJr+tfmnJT\J=2 V '
u ,i= c ;-f-6
)
sinT + 7icosr+ 2 (Jsin./T+7ji)
cos< /T),
where the coefficients are given by the formulas
^^Or+yC), Ti,, =2 Cr
,
+|5r,
+|c,
The formulas (31) together with the conditions (29) are sufficient to
construct the periodic solution of equations (9) to any desired degree of
accuracy; the computation is entirely algebraic. In order to determine the
constants of integration entering in the last (ith
) step, it is necessary to
compute the coefficients Af** t 5{ 1+n , C[ ,+l) , D[ ,+1) of the next following step.
(31)
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402 PERIODIC ORBITS.
200. Numerical Example 1 . For the purpose of illustration, we assign
numbers to the constants involved in the preceding analysis and construct
an orbit. In this and the other numerical examples which occur later, it has
not been shown that theprocesses
are valid for the numerical values which
are employed and which have been selected for convenience in graphical
representation. It is probable that the series are convergent, although it
ha3 not been found possible to determine the true radii of convergence.The differential equations of motion are equations (5). On putting
m= n and writing the second members explicitly as far as terms of the
second degree in a/ A, we have
^" p (^ +M ) 2+ ^=
y^[K1+3cos2M;
}+ fl />
{3COSM,+5COS3M;
}
+^(^-)VJ9+20cos2w+35cos4w|+ ]
+ ^[i{l+3cos2(.-|)} + |^3cos(.-|) +5cos3(.-|)}
+^(2)V{9+20cos2(w-|)+35cos4(;-|)}+ }
cPw, n dp/dw ,
\ JfMipfs
+ f jp[sin;+5sin3wj+^( j)V{2sin2^+7sin4w|+ ]
+H(i)V{*!!
(-|)+^*(y-|)}+";J
We select Mfor the unit of mass and suppose Mx= 10, Mt
= 5. For the unitof distance we take the distance between the finite bodies, that is, A = i;and the unit of time is selected so that N = 1. The period of the solution is
assigned so that v = 5, whence
Nv
(32)
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PARTICLE ATTRACTED BY U SPHERES. 403
The constant k 2is determined from the relation
whence
&2 = 0.06250, k 2 M,= 0.62500, k 2M2= 1.56250m.
The constant a was defined by vi a i = k 2 M, whence
== a = 0.67860m-
With these numerical values equations (32) become
? -P$r +mY+ K - (0.31250+0.93750cos2m;)pm2
ut \ar / p
+ (0.78125-1. 17188 cos 2w+ 2.02977 sin 2w)pm3
+ (0.47714 cos w+ 0.79523 cos Sw) pV
+ (0.59643 cos w+ 1 .03308 sin w - 1 .98808 cos 3w)pV
+ (0.16188+0.35974cos2w;+0.62954cos4w)pV+
p^f +2^(^ + M)= -(0.93750sin2M;)pM
2
+ (1.17188sin2w+2.02977cos2w)pp3
- (0.15905 sinw+0.79523 sin 3w)pV
+ (-0.19881sinw+0.34436costf>+1.98808sin3w)py
- (0.17987 sin 2m; +0.62954 sin 4 w)pVH
The periodic solution of equations (33) is
p =l-| M+(0.45139+0.39762cosr- 0.62500 cos2t)m2
+ (- 0.47647 + 1 .04542 cos r +0.86090 sin r + 0.46875 cos 2r
-1.35318sin2r-0.16567cos3r)/x3 +
'
,
w = r+(-0.79524sinT+0.85938sin2T)M2
+ (0.13882-3.25720sinr+1.72180cosr+0.27995sin2r
-1.86062cos2T+0.19881sin3T)p3
H
(33)
(34)
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404 PERIODIC ORBITS.
Substituting the numerical value m= 0.2, equations (34), if convergent,
are the equations of motion of the particle P. The orbit is shown in Fig. 9.
In this and the figures of the following numerical examples the comparison
circles are not the circular orbits which have been called the undisturbed
orbits. The undisturbed orbits are referred to fixed axes while the draw-
ings are made with reference to rotating axes. The comparison circles
represent orbits in which the particle would make a complete revolution
-M t
with respect to the rotating axes during the period. The points which are
numbered 1, 2, . . ., 8 represent positions of the particle in the periodic
orbit at intervals of t = x/4. The corresponding positions in the comparisoncircle are indicated by the numbers 1', 2', . . .
, 8'.
201. Some Particular Solutions of the Problem of n Bodies. Theexistence of symmetrical periodic orbits of the particle depends upon the
masses and motion of the finite bodies. So far as the analysis is concerned,
this motion may be arbitrarily periodic, without reference to the nature
of the forces producing it. It is required only that the motion of the finite
bodies shall be known and that they shall attract the particle according to
the Newtonian law. It will be interesting, however, in developing the
analysis, to prescribe motion for the finite bodies, which is possible under
the law of the inverse square of the distance. For three finite bodies the
two solutions of Lagrange are well known. In the case of the equilateral-
trianglesolution the
periodicorbits of the
particleabout one of the bodies
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PARTICLE ATTRACTED BY 71 SPHERES. 405
have a line of symmetry if the other two masses are equal. In the'case of
th^* straight-line solution the periodic orbits of the particle about any one of
the bodies are symmetrical with respect to the line joining the finite bodies.
The particular solutions which Lagrange has given for three bodies havebeen extended to some cases of more than three bodies,* and we shall
consider two examples: (1) in which there are five bodies, and (2) in whichthere are nine bodies.
Let the masses of n finite bodies be represented by Mt , Ms ,. . .
, Mn .
Suppose that the bodies lie always in the same plane, and that their coordi-
nates with respect to their common center of mass as origin and a system of
rectangular axes which rotate with the uniform angular velocity N are,
respectively, (x x , yj, {x 2 ,y^, . . ., (x n ,y n ). Supposing that the bodies
attract each other according to the Newtonian law, the differential equationsof motion are
>=i
df dtyi
jJ rfj
i,
u = V(x t -x J y+(y t -y 1 )t
(t-1, 2, . . ., n; j9*i).
(35)
If we assume that eachbody
is
revolvingin a circle about the common
center of mass of the system with the uniform angular velocity N, its coordi-
nates with respect to the rotating axes are constants and the derivatives of
the coordinates with respect to the time are zero. Equations (35) therefore
reduce to the following system of algebraic equations
-ivit t +y2;M' (
%Xi) = -*. * j ***>^^ '
t.i
-N%+I- M r yi) =0.
i=i
(36)
It follows from these equations that
Ml x l +M2 x i + +Mn x n=
0, MlVl +M2 y 2 + +Mn y n=
0, (37)
which express the fact that the origin of coordinates is at the center of mass.
These equations may be used instead of two of (36) .
*See Hoppe, "Erweiterung der bekannten Special ltisungen des Dreikorperproblems;" Archiv. der Math.und Phys., vol. 64, pp. 218-223. Andoyer, "Sur l'equilibre relatif de n corps;" Bull, astron., vol. 23 (1906),
pp. 50-59. Longley, "Some particular solutions in the problem of n bodies;" Bull. Amer. Math. Soc, vol.
13 (1907), pp. 324-335.
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400 PERIODIC ORBITS.
This system of 2n simultaneous algebraic equations involves the squareof the angular velocity, N, the n 1 ratios of the masses, and the 2w 1
ratios of the distances x, and y t . Accordingly n 1 of these quantities maybe chosen arbitrarily and, if the resulting equations are independent, theremaining 2n quantities are determined by the relations (36). In order to
admit physical interpretation, the quantity N2 and the masses must be real
and positive, while the coordinates must be real. With these restrictions
it is not easy to discuss the general solutions of equations (36), but some
interesting results can be obtained by a study of special cases. If, in the
problem of three bodies, the assumption is made that the triangle formed
by the three bodies is isosceles, it can be shown that equations (36) can besatisfied only if the triangle is also equilateral.
On supposing the number of bodies to be five, the system of equations
to be satisfied is
(a)
(b)
(c)
(d)
(e)
(/)
(g)
(h)
(0
C?)
M, Z, +Mt x t +M3 x 3 +Mt x 4 +Mb x b= 0,
_ N\ -i. Mt (x t -x t ) , Mfa-x,) ., Af 4 fa -s 4 ) , Mt fa -z,) _ n
1L2 X1~ ^ "1 ~S"~
~3 ~T Tj W,"> ' 1.1 '
1.3'
1.4'
1,5
_#! r -\.M1 (x t -x 1 ) , Mzjxs-Xs) . Mfa-xJ , _ Mb (x t -x b ) _ n
"> '2.1 '2,3 '2,4 '2,5
_ Nl r _i_ Mx {x 3 -xj ,
M2 Qr 3 -s 2 ) .
M4 (s 3 -a; 4 ) _,_ Mb (x 3 -x b ) _ n2*3 I
~3 V,
1,1 '3,2 r 3'3,5
_ Nl r _i_M 1 fa-g 1 ) a M2 (x t -x t ) , M3 (x t -x 3 ) ,Af t (s 4 -s,) _ n
2 ''(T^j
T ^j ^3 U,K, 1.3 1, B
M,y 1+M1 y 1+MI j/ 1+M4 y 4+M5 y 6=
0,
-ft*,, -1- M*(yi-y*) . Mt (y t -yJ , M^y.-y,) ... M.fa-y,) _ A"' '1.* '1.3 '1,4 M.5
-Wy+ M^-yJ M,(y t -y t ) , M<(y t -y<) . Mfa-yJ nwftT r i ^ I s r ^ =U,"' '2.1 '2,3 '2,4 *2,5
- jg 4. ^1(1/1 -yi) 4 M2 {y 3 -y,) Mt (y,-y t ) , Mt (y t -y t ) _ n* '.l '3,2 '3.4 3,5
_ jgtf 4 Mx (y<
-y,) , M2 (y 4
-y,) , M3 (i/ 4
- y 3 ) Mb (y t- yj
Jc2 y *
r 3 r* 1.3 1.5
(38)
Let us suppose that Mb lies at the origin of coordinates and that the
other four masses, which are equal in pairs, lie on the coordinate axes at
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PARTICLE ATTRACTED BY 71 SPHERES. 407
the vertices of a rhombus with the equal masses opposite each other (see Fig.
10\ This is equivalent to the relations
Xi J\j X2 U, 3 Ay X^ Uj x 6=
0, M1= M, =M/
,'
(39)fc =
0, th-KA, y t=
0, y,= -KA, y,-0, M^M^M".On substituting the assumed values (39) in equations (38), we find that
Fia. 10.
()> (c), (e), (/), (</), and (t) are satisfied. Equations (6) and (d) become
identical, yielding2M '
+ <*J + %> (40)^*(Vlfir)
3+ A!
k* (2A)
and equations (h) and (j) become identical, yielding
W 2Mk 2
2M" M6
(41)A 3
(Vl+i^) 3
(2KA)3 ^
(KAY
When M' , M", Mb , X, and A have been chosen or determined, these equa-tions insure a positive value for N2
.
On eliminating N2
/k2 between equations (40) and (41), we obtain for the
relation between the masses,
8K 3 -(Vl + ICy J
M, = i W-K'iVi+K'y i^ (4(i-jp)(vr+g)'i ,
8K 3 -( VT+K 2
)3 (42)
The choice of the constant K, which is the ratio of the diagonals of the
rhombus, is limited by the condition that the resulting ratio of the masses
must be positive. To investigate this condition we set Mb= l. Then,
regarding K as a parameter, equation (42) represents a straightline in the
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408 PERIODIC ORBITS.
M'M" plane. Only those pairs of values (M',M") which represent a
point in the first quadrant are admissible. This condition will certainly be
satisfied if the slope is positive, that is, if the coefficient of M' is positive.
This condition is easily found to be
-^<K<V3. (43)
If the slope of the line is negative, it may still lie partly in the first quadrantif the intercept on the 3f"-axis is positive that is, if the coefficient of Mh
in equation (42) is positive. It is easily verified, however, that values of
Fig. 11.
K which" make the slope negative, also make the intercept on the M"-axis
negative. Hence the choice of the ratio of the diagonals is limited by the
condition (43) ; otherwise it is arbitrary.The conditions for the rhombus configuration with a fifth body at the
center may be summarized as follows :
Suppose a value of K satisfying condition (43) is assigned; then two of the
masses M', M", and M6 can be chosen arbitrarily* and the third is determined by
equation (42). The length, A, of one semi-diagonal can be selected at pleasureand the angular velocity is then determined by equation (40) or equation (41).
In the following discussions this configuration will be referred to as
configuration (A).
The second configuration, (B), will consist of nine bodies, arranged as
shown in Fig. 11. Four bodies M1} Mt , Mst Mt lie on the coordinate
axes and on the circumference of a circle of radius A; four others, M6 , Mt ,
Except when K = 1 and the rhombus becomes a square. Then M' and M" must be equal.
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PARTICLE ATTRACTED BY U SPHERES. 409
M7 , Ms lie on the bisectors of the angles between the coordinate axes andon a circle of radius KA, while the ninth body is at the center. Supposingthat all the masses on the same circle are equal, we have the following con-
ditions:
x,= A, 2/i
=0,
x a= 0, y, = A
f
x 3 =-A, 2/3=
0,
x 4= 0, y k =-A,
x,= \V2KA, y b
= \V2KA, .
x 6 =-V2KA, y 6= \V2KA,
x,= -i V2KA, y 7= - i V2KA,
x, = \V2KA, y,= -lV2KA,x 9=
0, y* = o,
M1=Ma
=M3=M4
= M', ilf 6= M6
= ilf 7=M8
= M".
Corresponding to equations (38) of the five-body problem, there is a set
of equations, which, upon the assumptions (44), reduce to
1 J- 1 4- KN\A* m,( l
, l\ , n A V2,
Ai "V2k
(44)
iM'
(v/2+ 4) +2M ( (V1-V2 K+K*)
3 +(V1+V2K+K*) 3
)+M"
FA 3 A V2K ^V2K I M"(l . IN,k? [(Vl-V2K+ICy
^ (V1+V2K+K 2)3
)^ K3 W^^iJ i "
45)
Eliminating JVMVA; 2 from equations (45), there results the followingcondition on the masses M'
, M", M9 ,and the ratio, K, of the radii of the
circles,
M \ (J_ + - V- + KV^2 KV2+2 )
'
\\V2 T a) K3 t (Vl-VSK+K*) 3 ^(Vl + V2K+K 2
)3
]
~
( 1 1 2K-V2 2K+V2 j j / IN
(46)
The choice of the ratio K and two of the masses is limited by the condi-
tions that the third mass, as determined by equation (46) and the squareof the angular velocity, iV
2
,from (45), shall be positive. The limits within
which the choice can be made have not been determined, but for the purposeof application in numerical example 3, the following set of values satisfying
the conditions has been computed :
K =2, M' = l, A =
\, M9 =l, M" = 8.2526, A/2 = 1.6399 A;
2.
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410 PERIODIC ORBITS.
202. Existence of Symmetrical Periodic Orbits. For the developmentof the type of analysis applicable to symmetrical orbits we shall use configu-ration (A) of the preceding section, the notation being unchanged exceptthat the mass at the center will now be denoted
by Minstead of
Mt.
Withreference to M as origin and an axis having a fixed direction in space, let
the polar coordinates of Ml} M% , M3 , Mti and P be, respectively, (#i,F t ),
(R t ,V t ), (R 3 ,V 3 ), (# 4 ,F 4 ), and (r,v). The coordinates of the bodies are
expressed in terms of the time as follows :
R^R 3= A, R2
= R t =KA, F.-F.-f= V3 -* = V- ^ = Nt,
where N is given by equation (40) or equation (41).
The differential equations of motion of P are
^T_ /^Y_i_ JIM- 9 <fo, 9 drdv_l<m ,, 7 x
dt* \dt) r i*'
dr' r dt i
'
tZ dtdt~ rdr' K*' }
where
n= km+ Mi + Mi + MA _M1{v_ V)L r, r, r s n A* K
~j^rcosiv-VJ
-f|rcos(t;-
V3 )- ^r Cos(v-
V<)]>
r 1= Vt'+A 2 -2rA co^p-VO, r 3 =Vr*+A 2 -2rAcos(v-V t ),
r 1= Vr*+A 2 -2rKAcos(v-V i ), r i =Vr*+K iA i -2rKAcos(v-Vd-
We now define m and a by the relations
mv = N, v2 a 3 = k 2
M,
where, as in the preceding case, v denotes the mean angular velocity of P.
The motion is referred to an axis rotating with the angular velocity 2V and
passing always through Mx by the substitution v = w-\-V 1 w+Nt, andfactors depending on the units employed are eliminated by the substitution
r = ap, vt = t. We obtain then the differential equations of relative motion
d*p fdw . V . 1 1 dft d?w , dpfdw, \ 1 3Q rAQ ^
di- p KTr +m )+?=7~aWiy
P d? +2 drKd7 +m)=
7a^d^-(48)
These equations have the same form as the set (9) and the analysis
and results of that problem are applicable in this case. We know, then,
that there exists one, and only one, orbit in which the particle P moveswith direct motion and with a preassigned period. We shall see that there
exists one, and only one, such orbit which is symmetrical to the line joining
Mand Mx (also to the line joining Mand M2 ) ;hence it will follow that there
are no unsymmetrical orbits of this type. Furthermore, all the periodic
orbits which are given by this analysis are closed after one revolution in
the rotating plane; hence in case symmetrical orbits exist, they are also closed
afterone revolution.
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PARTICLE ATTRACTED BY U SPHERES. 411
We can expand ft as a power series in ap/A which is convergent for all
values of w so long as the distance, ap, of the particle from M remains less
than the distance from M to the nearest finite body; and in all that follows
this condition is supposed to be satisfied. The expansion has the form
a _kiMl" A
+
[1 +Kl' ,
)
2
{1 + 3cos2w
}
Vjp) J3cosw+5cos3w4+
+ 8^(2 p )
3
{3cos ( w~f) +5cos3 (
t/;
~f)} +]
+ i(-jp)
3
{3cos(w-7r)+5cos3(w-a-)j+.1
-
[1 +4t(l^{ 1+3COS2 (-f)}
. k 2M.+ KA
+Mjp)3
{3co < w - ?)+ 5cos3 <>-)}+ I
From the conditions of the configuration (A) we have
Let X, and X2 be defined by the relations
M'-xfW, 2M" \ M^ _ (W , 2M" \M ~Al U +
<V1+jK*)V' iT~ A2 V4 + (Vi+X )v"
From equations (40) we have
(M'
2M" \
l~+(Vi+g)V _ [MVd+DM-SAfjFA 3 (M'+4M)v
/
(l+X 2)3 +8M' /
'
On setting the coefficient of 2V 2 in this equation equal to k, it follows that
the second members of equations (48) have the form
= Km2
p[X 1 (l+3cos2u;)+X 2 (l-3cos2w)v^a d(ap)
+ terms involving only cosines of even multiples of w],
1 r)Q = <cm 2
p[3X!sin2t(; 3X 2 sin2w;v
ia
2
p dw
+ terms involving only sines of even multiples of w] .
(49)
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412 PERIODIC ORBITS.
On introducing the parameter of integration p by the relations m = pand a/ A =
r;p, where rj is a numerical constant, equations (48) become
g -,(+)+ ,W/, ,+2$( +)-*, (50)
where
/=Kp[X,(H-3cos2w)+X 2 (l-3cos2w) + ],
0= kp[3(X, X,)sin2w+ ]
Suppose
p = ^( T ), W= fc( T ) (51)
is a solution of equations (50) such that p'= w = at r = 0. Then it follows
from the form of the differential equations that , is an even function, and
that ^, is an odd function of t. Hence, if the particle crosses the w-axis
orthogonally, the orbit is symmetrical with respect to the line w = 0, and
with respect to the time of crossing this line.* Suppose that when r = v the
particle crosses this line (or, what is the same thing, the line w = t) again
orthogonally; the orbit will be symmetrical with respect to this line and the
time of crossing, and the particle will have again its initial position andrelative components of velocity at the end of the period t = 2tt. Hencesufficient conditions that the solution (51) shall be periodic are
p'(tt)=0, w(t)-tt = 0. (52)
For p the equations have the form which occurs in the problem of
two bodies, and a symmetrical solution having the required period is known,
namely,
p=l, W= T.
The initial conditions for this solution are
p=l, p'=
0, w = 0, w' = \.
Consider the solution for values of p different from zero but sufficiently
small, and let the initial conditions for t = be
p=l+ft = (l + a)(l-e), . p' = 0, w =0,
/=!+& = vr= I
(l + a)"'(l-e)2
(53)
*It may be remarked also, in this particular example, that if the solution (51) is subject to the initial
conditions, p' = 0, w = r/2, (t = t 1 ), then ^ is an even function, and i, t/2 is an odd function of t t,
and hence, if the particle crosses the line tc = x/2 orthogonally, the orbit is symmetrical to this line andthe epoch t = ti. The sufficient conditions of periodicity (52) might be replaced by the conditions
>'(I)=- <!)-i-*since the orbit would then have two lines of symmetry. For the purpose of covering more general cases it
isbetter to base the existence proof on only one
lineof symmetry.
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PARTICLE ATTRACTED BY U SPHERES. 413
The solution is symmetrical with respect to the epoch t = 0, and can be
expressed as power series in a, e, and n, which are convergent for an interval
in r including the interval to ir, if the parameters are sufficiently small.
If a and e can be determined in terms of n, vanishing with /*, so that the
conditions (52) are satisfied, then the solution will be periodic with the
period 2-k. All terms of the solution which are independent of m2 can be
obtained from the two-body problem by making the substitution w = ufj.T.
These terms are given in finite form by the expressions .
p=(l-\-a)(l ecosE), w = arccos(- s ) =arcsin( ^-)>\iecosE/ \ l ecosE J
where E is defined by the relation
jp^yr t =E-esmE.
On returning to the variable w, writing the terms in a and e as powerseries by Taylor's expansion, and applying the conditions (52), we obtain
the equations
0=-|7rae+ , 0=-|7ra-7r M+ (54)
It follows from the known properties of the series that there are no terms in
e alone, and there are no terms involving n to the first degree except the
term irn. The equations are satisfied by a = e = n = 0, and in the second
the coefficient of the first power of a is not zero;hence the second equation
can be solved uniquely for a as a power series in e and fi, which contains n as
a factor. The result has the form
;(-f+ ) (55)
When this value of a is substituted in the first of equations (54), a factor /x
can be divided out, leaving = \ ire + ' This equation is satisfied
by e=n =0, and since the coefficient of the first power of e is not zero, it
furnishes a unique determination of e in terms of n, vanishing with fi. Whenthis value of e is substituted in equation (55), we have a expressed uniquelyin terms of n, vanishing with ix.
Hence for a given value of n sufficiently small it is possible to determine
the initial conditions (53) as power series in n such that the solution in fx is
symmetrical with respect to the line w = t and the epoch t = t. Since it is sym-metrical also with respect to the line w = and the epoch t = 0, it is periodic in t
with period 2ir.
The orbit is symmetrical with respect to the line joining the bodies
M and M,. If we take for the initial line the line joining M and M it
follows from the same analysis that the orbit is symmetrical with respect to
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414 PERIODIC ORBITS.
this line and the time of crossing it. Since for a given value of p there is
only one periodic orbit of this type, it follows that it is symmetrical with
respect to both the lines joining M with Mx and with Mt . For the con-
figuration (A)of the finite bodies the
periodicorbits of the particle have two
lines of symmetry; there are four apses and the apsidal angle is ir/2. For
the configuration (B) (see numerical example 3) the periodic orbits havefour lines of symmetry, there are eight apses, and the apsidal angle is 7r/4
In establishing the uniqueness of the periodic solution of equations (54)
it is to be noted that no use was made of the explicit values of the terms
from the second members of equations (50). Hence, if the second membershave forms which permit symmetrical solutions, the preceding analysis is
applicable without change to show the existence of symmetrical periodic
orbits. // the masses and motions of the finite bodies are such that there can
exist orbitsof
theparticle
about oneof
the bodieshaving
a lineof symmetry,
wehave established the existence of periodic orbits having this line of symmetry.
As in the problem treated in 198, the period of the solution in t may be
assigned arbitrarily (that is, when the finite bodies form a fixed configurationin the rotating plane). There exist then two, and only two, symmetricalclosed orbits having the required period; in one the motion is direct, andin the other it is retrograde.
203. Construction of Symmetrical Periodic Orbits. The method of
constructing symmetrical periodic solutions is similar to that explained in
199. There is a slight difference in the conditions which determine theconstants of integration, and the calculation is simpler because p contains
only cosines of multiples of t, while w contains only sines. It has been
proved that symmetrical periodic solutions of equations (50) exist, and that
they are expressible in the form (18).
Since the solution is periodic for a continuous range of values of p, each
coefficient p, and w, is periodic with period 2w in r. Also, since the initial
conditions p'(0) =0, w(0) =0 hold identically in p., it follows that p' (0) =0,and wt (0) =0 for every i.
The left members of equations (50) are the same as the left members
of equations (9), and therefore, when the solution (18) is substituted in
equations (50), the terms of the left members have the form (19). The
right members have no terms independent of p2
, and the equations for the
determination of the coefficients of the first power of p are
<fpi dwi _ d?Wi , dpi _ ndV~ 2
~dr~~ 6pl ~ Z'
~drT+Z 'dh~
{)'
of which the solution is
p 1= 2(l+cl
,>
)+cJ1)
cosT+c;i)
sinr,
Wi= c
' -(4 +
3 CJ)
T - 2 <' sin r
+2<f
cos t.
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(56)
PARTICLE ATTRACTED BY n SPHERES. 415
Since wx is periodic the coefficient of r must be zero; whence c{"= 4/3.
The constants c and c"' are determined by the conditions
ft '(0)=0, Wl (0)=0.
Therefore c,1 ' = c"' = 0. The constant c,
1 '
is determined in the following
step of the integration [see equations (23)].
This process is applicable to all the succeeding steps. The differential
equations (50) have a particular form which admits a symmetrical solution,
and it can be established by complete induction that the equations for the
determination of p, and wt have the form
f|* -2^ _3 A =iC+(ifc$-
u + 4)cosr+4J"cos2t+ +jifcdBtr;
^ <
+2^ =(fc-
1,
+Z)r,
)sinr+D,
sin2r+ +D>ntY,
where c"-1 '
is determined by the condition [compare equations (29)]
2cf*-ir-2Df:
The solution of equations (56) is
p < = a)
+c")
COSr+c4"cos2r+. . .
+ a" >
costr,
wt= Csinr+^sin2T+ +if
>
mn%T,
where
r- nb (a t- 2J
f)> r = -A (i>r+2i.f) o= 2,
. .., *
(57)
(58)
204. Numerical Example 2. As a first example of a symmetricalperiodic orbit we consider three finite bodies revolving in circles accordingto the straight-line solution of Lagrange. We suppose that the mass M,about which the particle revolves, is between M2 and Mt . Choosing M for
the unit of mass, we select Mx= 10, M2
= 5. The unit of distance is MMX ;
and it follows from the solution of the quintic equation of Lagrange* that
the distance M2M is R2= 0.77172 .... The unit of time is selected so
that JV =1, and the period of the solution is assigned so that v = 5; whence
M= m= JVy = 0.2.
*See Moulton, Celestial Mechanics (second edition), p. 312.
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(59)
416 PERIODIC ORBITS.
The differential equations of relative motion of the particle are
+|apj3cosuH-5cos3w]+-aV J9+20cos2uH-35cos4w]h ]
+^ ? M2
p[^l+3cos2(^-1
r)}+^p|3cos(;-7r)+5cos3(w-7r)J
+^(|-)V{9+20cos2(m>-7t)+35cos4(w -*)!+],
p ^F +2 3?(^ +M)= -fc 2iW
^p[fsin2w+faJsinw+5sin3t4
+^aVJ2sin2w; + 7sin4M;]+ ]
^3J^[fsin2(w;-7r) +
|^-pjsin(M;-7r)+5sin3(w-7r)[-
+^(^)V{2sin2(w-x)+7sin4(u;-T)}+ ]>
where a is given by the relation v2a" = k 2
. The constant k? is determined by
M+Mt +M,R v ^(1+Rl))
'
whence
fc* = 0.23763, fclM> 2.37630, ^=2.58518,t
a = 1.05914/i, -=1.37222 p.
The differential equations of motion become
gjS-
P (^ +m/+ ^ = (2.48074+7.44222 cos2u;)p M2
- (1.15938 cosw+ 1.93230 cos3w)pV
+ (4.23765+9.41700 cos2w>+16.47975 cos4 w)pV +
p ^ +2 5? +M) =-(7 - 44222sin2
^PM2
+ (0.38646 sin w+ 1.93230 sin3w)/>V- (4.70850 sin2w+ 16.47975 sin4w)pV+
The periodic solution of equations (60) is
(60)
p = 1 -| M- (0.27136+0.96615 cosr+4.96148 cos2r)p
2
+ (0.62584-19.13740cost-2.47963cos2t+0.40256cos3t)m3+
w = t+(1.93230 sin t+6.82204 sin 2 t)/x2
+ (41.10885 sinT+10.74793sin2T-0.48307sin3r)p3H
(61)
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PARTICLE ATTRACTED BY 11 SPHERES. 417
On substituting the value p.= 0.2, the orbit represented by equations
(61) is shown in Fig. 12. The points which are numbered 1, 2, . . .,24
represent positions of theparticle
in theperiodic
orbit at intervals of
r = 7r/12. The corresponding positions in the comparison circle are indi-
cated by the numbers 1', 2', . . .,
24'.
8 7
21 19
18 1920
Fia. 12.
205. Numerical Example 3. For a second example of a symmetrical
periodic orbit we use the configuration (B), 201, of nine finite bodies, the
numerical values being those given. The unit of time is selected so that
N =1, and the period of the solution is assigned so that V =
5, whence p = 0.2.
The differential equations [corresponding to equations (50)] of relative
motion of the particle are
->( +<)*+ ? - %5? '"'[if1 + 3< * 2 <
-*>}
1 | pJ3
cos (io-W t )+ 5 cos3(w-TV,)|
^(^-)V[9 + 20cos2(u;-lf i )+35cos4(^-lF ( )]-f]>
j|p{sin(u;-W,)+ 5 sin
3(w-F,)j
(|-)
2
p2
[2sin2(tt-lf( ) + 7sin4(u;-lV (
)j+---]-
+
+
+
+
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418 PERIODIC ORBITS.
On taking account of the relations (44) ,and choosing the unit of distance
so that A = l, the preceding set of equations takes the form
^- p (^+m),
+ ^=A; 1
MVp[2+ iay{9+35cos4w}+. .
]
+ ^ M.p[2+i(5)V[9-35cos4u>]+
. .
.],
^+ 2 tXtr + )= -^Vp[?aVsin4. + ' '
]
+ k*Mf-Vp[!V^+ }
From 201 we have the following values:
K = 2, M' = \, M" = 8.2526, JV= 1.6399 A*.
Since iV =l, the last equation gives A;
8 = 0.60994. It follows that
^|P =0.62920, a 2 = 2.10300 M2
, (J?)
*
= 0.52575 M*.
On substituting these numerical values, the differential equations become
fi -p(t?+mV+A = 2.47828 p/x2+[3.63039+8.32914cos4m;]pV+ar \ar J p
p ^ +2 ^(^ +M) =[-8.32914sin4 W]pV+
The periodic solution of these differential equations is
P= 1-!/z-0.27054m*+1.70909ms-3.43127p
4-0.83291m<cos4H ,
w = T+0.93702p4 sin4H ,
On substituting the value p = 0.2, the final result is found to be
P = 0.86403-0.00133cos4t+ ,
w = T+0.00150sin4r+
The orbit has four axes of symmetry, namely, the lines connecting the
central body with the others. It differs from the orbits of the other numer-ical examples in one respect that is, it lies entirely inside the comparisoncircle (see 200). In terms of p the radius of the comparison circle is about0.8855. Fig. 13 is not drawn to scale, but the characteristic properties of
the orbit, which are readily seen from the numerical values of p and w, are
exaggerated to make them apparent in a small drawing. The inner circle
is drawn merely to indicate the direction of the deviation of the orbit from
acircle.
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PARTICLE ATTRACTED BY 71 SPHERES. 419
206. The Undisturbed Orbit Must be Circular. In the proofs of the
existence of periodic orbits ( 198-201) it was assumed that the undisturbed
orbit is circular. It remains to be shown that this assumption is necessary.The proof will be made for the case of symmetrical orbits [equations (50)]
and is applicable also to the orbits of 198. The undisturbed orbit is given
by the solution of equations (50) when p.= 0. For p = 0, the equations are
the equations of motion of a particle subject to the attraction of a central
force varying inversely as the square of the distance. The undisturbed orbit
is therefore a conic; and, since we
are concerned only with closed
orbits, must be an ellipse. Since
the period in t is 2x, the majorsemi-axis of the ellipse must be
unity (in p). The eccentricity,
which will be denoted by e, is,
however, arbitrary; that is, for
p = the differential equationsadmit an infinite number* of
symmetrical periodic solutions.
Starting now with an ellipse for
the undisturbed orbit, it will be
shown that the eccentricity mustbe zero in order to fulfill the
conditions of periodicity.
Forp=0thesolutionof equa-FlGURE 13 -
tions (50) representing an elliptic orbit of eccentricity e is
cosE-7,7 /cosis e\ /vi e"sinii\ fpn \p= 1 ecosE, w = arccos( ;
= s ,)=arcsin( = ^r)' (o2)M ecosE' V 1 ecosE '
where E is defined by the relation T = E esinE. The initial conditions for
r = are
P = l e, p' = 0, w =0, w' =
(1-i)2
Consider the solution for values of n different from zero, but sufficiently
small, and let the initial conditions be
ic
P =l-e+^ = (l+a)[l-(c+e)],
VT-Xe+e?
p' = 0, w =0,
Vl-e(l-e)=
+ft =(l+a) 3/s
[l-(i+e)]2
If a and e can be determined in terms of n, vanishing with p, so that the
conditions (52) are satisfied, then the solution will be periodic with the
*The generalcase when the differential
equations (for >i=0)admit a
periodicsolution
containingan
arbitrary parameter has been mentioned by Poincar6, loc. cit., vol. 1, p. 84.
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420 PERIODIC ORBITS.
period 2x. All terms of the solution which are independent of n* may be
obtained from the two-body problem by making the substitution w = u ht.
These terms are given in finite form by the expressions
p = (l + a)[l-(e+e)cos],
W=arccos( 1
c- os |f? +4) = arcsin (^?^)'where E is defined by the relation
= E-(e+e)sinE./2(l+o)
On returning to the variable w, writing the terms in a and e as powerseries by Taylor's expansion, and applying the conditions (52), we obtain
the equationsn 3 e 3 l+e ,0= -
2^(131)^-7^(1-=^+(63)
It follows from the known properties of the series that there are no terms in
e alone, and there are no terms involving n to the first degree except the
term pw/x. Hence the second of equations (63) can be solved for a as a
power series in e and ju in which /i is contained as a factor; the result is
/ 2 (l-e)' , I
When this value of a is substituted in the first equation, a factor n can be
divided out, leaving
This equation can be solved for e as a power series in n, which vanishes with n,
if and only ife = 0. Since only those solutions are under consideration which
are the analytic continuations with respect to // of those for m= 0, the con-
dition e = must beimposed.
The condition e = means that the undisturbed
orbit must be circular.
207. More General Types of Motion for the Finite Bodies. This sec-
tion contains some remarks upon possible extensions of the analysis whichwill permit applications to practical problems of celestial mechanics, and is
followed by an illustrative example.The particular problems treated in the preceding articles have no appli-
cation in nature because the configurations assumed for the finite bodies donot exist. But a glance at the details shows that these configurations are
not essential to the proofs. The possible generalizations of the motion of the
finite bodies can be made in three ways:
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PARTICLE ATTRACTED BY n SPHERES. 421
(1) In 198 the existence proof depends only upon certain terms of
the disturbing function which are due to the body Mt . If Mx retains the
motion there prescribed, we may add other bodies to the fixed configuration
in the rotating plane provided the operations with the power series are valid.This merely increases the number of terms in the second members of the equa-tions of motion
; the existence proof and method of construction are unchanged .
(2) In the examples treated the finite bodies form a fixed configurationin a plane rotating with constant angular velocity. This is not necessaryfor the type of analysis used. If Mx moves in a circle with uniform angularvelocity, the other bodies can have any periodic motion, provided alwaysthat the convergence conditions hold. In this case the differential equationsof motion of the particle involve t explicitly and are periodic in r. Twopoints of difference occur in the analysis: (a) Suppose the period in r of
the differential equations is T; then the assigned period of the motion ofthe particle must be a multiple of T. (6) The differential equations donot admit the integral of Jacobi, and hence no use can be made of this in
the existence proof. This is equivalent to saying that at t = we can not
assume w = 0, but must determine the initial longitude of the particle bythe conditions of periodicity. The method of determining the constants of
integration in the construction of the solutions is explained in a paper in the
Transactions of the American Mathematical Society, vol. 8 (1907), pp. 177-181.
(3) A further generalization of the motion of the finite bodies is pos-sible by permitting Mx to move in a path which is not circular. It is possible
to show that the analysis can be used if the motion of Mx is subject only to
the mild restrictions that the expression for the radius vector shall contain
only cosines of multiples of t while that for the longitude shall contain
only sines. The case when the orbit of Mx is an ellipse is treated in the
article referred to above. For this generalized motion of the finite bodies
there may exist symmetrical orbits of the particle. In equations (50) the
first contains only cosines of multiples of w, and the second only sines of
multiples of w. The periodic orbit of the particle may be symmetrical if
the first equation contains also sines of multiples of w multiplied by oddfunctions of r, and cosines of multiples of w multiplied by even functions
of t, and the second contains cosines of multiples of w multiplied by oddfunctions of t, and sines of multiples of w multiplied by even functions of t.
From these remarks it is apparent that the treatment can be madesufficiently general to permit applications in the problems presented by the
motions of the solar system. For example, suppose P is a satellite of one of
the planets M,and that Mx is the sun. This implies that the disturbing effects
of the satellite upon the other bodies are neglected, since we assume that its
mass is infinitesimal. The conditions upon the motion of Mx are fulfilled
if we neglect the perturbations of the other planets upon M; that is, if we
suppose the orbit of Mx relative to M is an ellipse. If we neglect the incli-
nations of the orbits of the other planets, and suppose that their motion is
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422 PERIODIC ORBITS.
periodic (that is, we assign a periodic motion which is approximately correct),
it is possible by the methods given to treat the periodic motion of the satel-
lite in the plane of the planetary orbit, when subject to the attraction of the
sun and all the planets. The following numerical example is a simpleillustration of the general idea.
208. Numerical Example 4. The mass of M is taken as the unit of
mass and Mx ,of mass 10, is supposed to revolve about M in a circle of unit
radius with uniform angular velocity N. A third mass, Mt= M=
1, is
supposed to revolve about Mx in a circle of radius A t with uniform angular
velocity Nt . The unit of time is chosen so that iV =l, and the period of
the motion of the particle is assigned so that v = 5, whence
M= m=^=0.2.v
With reference to M as origin and an axis passing always through Mx ,the
coordinates of Ml} Mt , and P are, respectively (1, 0), (R t , Wt ), and (r, w).
The differential equations of relative motion of the particle [correspondingto equations (50)] are
^- P (^+M)2
+^ = A-JM1M
2
p[i|l-f3cos2^|+|apJ3cosw+5cos3w;|
+
(64)
^aVJ9+20cos2w+35cos4w|+. . .1
23-V P
[||l+3cos2(w-^2
)}+|| s pJ3cos(w-^s )+5cos3(w;-Pr 2
)j
~(J-)VJ9+20cos2(w-TF,)+35cos4(w-TF s )JH ]i
dp +2 ^(^+p) =-A; 2MlM2
p[|sin2u'+|ap|sint<;+5sin3^j
+ ^ay-|2sin2w+7sin4w;j+ ]
-Vp[| sin2(io- Wt ) + f |-
P{sin(ti>-
Wt) +5sin3 (w-W t )j
+ ^6 (f)V{2sin2(t
<;-Hg+7sin4( W
;-^)j+]
The constant &2is given by the relation Ni = k 2 (M+M^, whence
A;2 = 0.09091, k 2M, = 0.90909, h?M t
= 0.09091.
From the relation fa* = k 1
M, it follows that
a = 0.76910p.
The angular velocity of Mt about Mx will be selected so that its period with
respect to the rotating axis MMX is one-half the period assigned for P.
HenceNt -N = 2v, or #,= 11.
The radius, A t , of the circular orbit of Mt with respect to Mx is determinedby the relation ~N\A\ = k %
{M x -\-M^, whence A t= 1.01200p. On assuming that
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PARTICLE ATTRACTED BY U SPHERES. 423
at t = the finite bodies are in conjunction in the order M, Ml} Mt ,
the coordi-
nates (R t , Wt ) of Mt with respect to M are given by the expressions
R t= V\+A\+2A i cos2vt = ^l+A\+2A 1 <iosi2T,
.T 4,sin2T ttt 7214-1 A\
sin Wt= ^ ' cosW t
= '
2i ^
On substituting the values of the constants and the coordinates Rt , W2
in equations (64), we obtain for the numerical differential equations of
relative motion
S ~p(f^ +m)*+^ = (0.50000+ 1.50000cos2w)pm2
+ (0.86523 cosu>+1.44205cos3w)pV+ (0.33273+0.73940 cos 2w+ 1.29395 cos 4w)pV
+ (0.276008^278^2^-0.13800008210.4140000821-0082^/*'+ (0.10474+0.17456 cos4t+0.31422cos2w+0.10471cos4tcos2m>
-0.55864sin4rsin2w)pM4
+ (0.07960sin2Tsin^-0.31840cos2rcosw-0.53068cos2TCOs3w;
+0.39801 sin2rsin3w;)pV+ ,[ (65)
p~ +2^(^+ M)= -(1.50000sin2w)p M
J
- (0.28841 sinw+1.44205sin3u;)pV-
(0.36970 sin 2w + 1 .29395 sin 4w)pV
+ (0.27600 sin2rcos2w;+0.41400cos2rsin2w))pM3
-(0.3 1422 sin 2w+0. 10741 cos4r sin 2;+0.55864 sin 4r cos 2w) Pt x
l
+ (0.02653 sin2TCOsw+0.10613cos2rsinw+0.39801sin2rcos3w;
+0.53068cos2rsin3w)pV4 + "'
The right member of the first equation contains only, (1) cosines of
multiples of w, (2) cosines of multiples of w multiplied by cosines of multiplesof t, and (3) sines of multiples of w multiplied by sines of multiples of r.
The first equation is then unchanged if we replace w by w, and r by t.
The right member of the second equation contains only, (1) sines of mul-
tiples of w, (2) sines of multiples of w multiplied by cosines ofmultiples
of r,
and (3) cosines of multiples of w multiplied by sines of multiples of t. Hencethe second equation is also unchanged if we replace w by w and r by t.
Now let us suppose that
is a solution of equations (65) satisfying the conditions p'(0) =w(fl) =0. It
follows from the form of the differential equations that \f/ l is an even function,
and ypi is an odd function of r. When r = the finite bodies are in conjunctionin the order M, Mlt Mt . Therefore, if the -particle P crosses the line MMX
orthogonally when the finite bodies are in conjunction in the order M, Af,, Af,, the
orbit in the rotating plane is symmetrical with respectto this
line and this epoch.
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424 PERIODIC ORBITS.
On constructing the solution of equations (65) by the formulas (58), we get
p = 1-.66667m+(0.38889+0.72102cost-co82tV
+ (-0.02616+2.09168 cos t- 0.45400 cos 2t- 0.30043 cos 3t
+0.03450cos4tV+ ,
w= t+(- 1.44204 sin r+ 1.37500 sin 2t) m*
+ (- 6.29838 sinr+2. 12066 sin 2t+0.36051 sin 3r
-0.03881 sin 4t) m'+
(66)
The
M,
The orbit represented by equations (66) is shown in Fig. 14.
points which are num-bered 1, 2, . . .
,8 rep-
resent the positions of
the particle in the peri-
odic orbit at intervals
of t = ir/4. The corre-
sponding positions in
the comparison circle
are indicated by the,
numbers 1', 2', . . .,8'.
"
With reference to
the differential equa-tions (65) we can makethe same statement
concerning the unique-ness of the solution
that was made in 198.
These equations were
written on the assump-tion that, at t =
0, the finite bodies are in conjunction in the order M, Mlt Mt .
Without this assumption the expressions for R t and Wt contain a parameter
indicating the position of M% in its orbit at the origin of time. Withreference to the physical problem, therefore, we can not affirm in this case
that, for a preassigned period, there exists one and only one direct periodicorbit. It is necessary here to add a condition on the form of the configu-ration of the finite bodies and particle at the origin of time. For example,we might obtain another symmetrical periodic orbit having the preassigned
period if the particle crosses the line M, Mx when the finite bodies are in
conjunction in the order M, Mt , M^, and we might have still other orbits
with the preassigned period if P crosses this line when the finite bodies are
not in conjunction.
Fro. 14.
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Chapter XIV.
CERTAIN PERIODIC ORBITS OF k FINITE BODIESREVOLVING ABOUT A RELATIVELY LARGE
CENTRAL MASS.
By Frank Loxley Griffin.
209. The Problem. For a given system of k finite bodies, moving in a
given plane relative to another given body, there is a 4A;-fold infinitude of
possible orbits the variations which the configuration of the system under-
goes and its orientation in the plane being determined jointly by the mutualattractions of the bodies according to the Newtonian law, and by the values
at any instant of the 2k relative coordinates and their first derivatives with
respect to the time. The differential equations admit no algebraic or uni-
form transcendental integrals,* aside from the two fundamental integrals of
energy and areas, even when the masses of all the bodies except one are
very small; nevertheless, by restricting the initial values of the coordinates
and their derivatives, in a manner to be shown below, it is possible to find
an extensive class of periodic solutions.
In fact, for arbitrary values of the masses (save that one of them, M,shall be large in comparison with the others, Mx , Mt ,
. . ., Mt )rf there
exists a k-fold infinitude of distinct periodic orbits of the system, having an
arbitrarily ^reassigned period T. In these orbits (which, for small finite
values of Mlf Mt , ... , Mt} depart but little from a set of concentric
circles about the planet) the k satellites come periodically into a "symmet-rical conjunction," that is, they are all momentarily in one straight line with
the planet and moving at right angles to that line. These conjunctions may,or may not, always occur at the same absolute longitude; in the latter case
the motion is periodic with reference to a uniformly rotating line.
Besides the demonstration of the existence of such periodic orbits, this
chapter contains: Amethod of constructing the solutions without integration,
a single application of that process having provided formulas which reduce
the problem to one of algebraic computation; a numerical application to the
case of Jupiter's satellites I, II, and III; a proof of the non-existence of cer-
tain other types of orbits; and a brief consideration of some related questions.
See memoirs by Brims and Poincard, in Acta Mathematica, vols. 1 1 and 13.
tin other words, the distribution of masses is such as is presented by the sun and any number of planets,or by a planet and any number of satellites. For convenience, in what follows, a single expression, planetand satellites, will be used with the understanding that it covers also sun and planets.
425
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426 PERIODIC ORBITS.
The problem may be formulated thus: Let quantities n, /3,, . . ., & be
defined byMfrv =Mt (i
=l, . . . ,*), (1)
where one of the /S'sis to
beselected
arbitrarily. Leta
systemof
positiveor negative integers without common divisor, p f (t
=l, . .
,k 1), and a
number q t ^0 be selected arbitrarily, save for the restriction mentioned
below, and let v , n, , and a ( be defined by
9 7 * p,(*-i,...,*-i), (2)
nJaJ^M (i= l, ...,*), (3)
where k* denotes the gravitational constant, and where, of the three values
of a, satisfying (3), that one is to be selected which is real. Also let the
notation be so selected that a l} . . .,a t are in ascending order of magnitude,
the Pi being so selected that no two of the a, are equal and now, vanishes.
If m were zero that is, if the satellites were "infinitesimal" possible
orbits would be circles about the planet with a lt . . .,a k as radii; from (3)
it follows that the angular velocities would be n^ . . ., n,. It is quite
immaterial whether any of the n t are negative; the results obtained hold
irrespective of retrograde motion of some of the bodies. The configuration
of the infinitesimal system would undergo periodic variations with the
period T; for, it follows from (2) that
2t m i.n t -rit p
or each synodic period is a sub-multiple of T. This condition being satisfied,*
the motion of the infinitesimal system would be periodic with respect to a
line through M, rotating with uniform angular velocity that of Mt , or,
indeed, that of any other bodyf Mj though whether or not the systemever returns to the same position in space depends upon whether q t is rational
or irrational.
In describing the orbits mentioned, the infinitesimal satellites would be
subject to certain initial conditions, the 2k coordinates and their derivatives
with respect to the time having at the instant t = t certain values, sayc w
(t=l,. . .
,
k] j=l,. . .
,4);but if the k
finitesatellites are
subjectedto these same initial conditions, their mutual disturbances in general destroy
periodicity. The first problem is, then, to determine what, if any, incre-
ments Ac {J can be given to the former initial values b, to preserve the
periodicity when all the satellites are finite.
*Poincar6, treating three satellites (Mtthodes Nouvelles de la Micaniqve Cileste, vol. 1, pp. 154-6), states
the condition thus: Integers o, 0, y, mutually prime, exist such that a+^+-y=0and on,+/3n,+7n, = 0.
Evidently, in the case of three satellites, this condition is equivalent to (2), since (n, n,)//3 = (n, n,)/( a);but for a greater number it is not so. Thus, if n,
=7, n, = 5, n,=3 vXn, = V27 the integers 5,-7, 1, 3
satisfy a condition similar to Poincar6's, but periodicity is impossible. For the general case a re-formulationsuch as (2) is necessary.
fThe commensurability of n t -n t (i=
l, . . . , fc-1) evidently involves that of ty-nj (t'=
l, . . .,
j 1, 7+1 k). For from n< nt = p v and nj ni = pjv it follows that n, rij**{p t pj)p.
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(4)
PROBLEM OP k SATELLITES. 427
210. The Differential Equations. Let the common plane of relat ive
motion of the k bodies be selected as the .ztf-plane, the origin being at M,
and MS and MHbeing rectangular axes which rotate in the plane with the
uniform angular velocity N. Let the coordinates of Mt referred to these
axes be & and t?,; then the differential equations of motion are
(a) g -2N^
- N%+>(M+M t )jjj
+ Z'*M,(*=& + ^)=>
(b) ^k + 2N& - N\+K\M+M t )%+ 2' tM, (*-*
-J) -0,
wherer)= Z+-n), -(,-,)+(%-*).
and 2' means 2^J {j^i). Except in proving a certain symmetry theorem,these coordinates are less convenient than polar coordinates referred to rotat-
ing reference lines. Besides (1), (2), and (3), let the following definitions
be made :
atj= a
t /a } , vq t=
n, (i=
1, . . . , k),
vt = T, <t>ji= (.Pj-Pi)T+(\-\),
where the X, are arbitrary constants, later to be taken as the longitudes of
the M, at the origin of time for n = 0.
Let polar coordinates be introduced by the equations
,= r,cosw,, ij,
= r i sinu t , r, = a i x l ,u i= wi +p i T-{-\ i , (6)
and N be taken equal to n t ,so that w, is the longitude of Mt referred to a
line rotating with uniform speed n,. The differential equations become
(a) x t w"+2x' (w' +q t )+nl''
d^sinfojt + Wj-wjf-z -) =0,
\Xj <T t j/
(6) x';-x t (w:+q t y+ q2 {1tf
tfi)
x\
+m2'8
<>
[^ +Xj0O8(il>jt
+w,-w,) (-,-
^-J J =0,
(7)
where a)a-\ J=
a]x\-\-a)x) 2a (a
Jx
lx
j cos(<t>j l -t-Wj wj, and where the accents on
the variables indicate derivatives with respect to r.
211. Symmetry Theorem. If a symmetrical conjunction occurs at anyinstant t = t a ,
then the orbit of each satellite before and after the conjunction is
symmetrical, both with regard to geometric equality of figures and with regard
to intervals of time. A proof will be given only for the case / = 0, which does
not limit the generality since any other case is reduced to this one by the
substitution < =,+^o-
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428 PERIODIC ORBITS.
The differential equations (4) are invariant under the substitution
<=
<, m=-vi, l=~t. (8)
Consequently,every solution of (4) is transformed by (8) into some solution
of (4). Moreover, the initial conditions
are transformed into
*-*' * =0 > S =0 > m= b " (9,)
Therefore, that solution of (4) which satisfies the initial conditions (9) is
transformed by (8) into itself. Hence, if that solution is
6-*,(0. *-*<(<) (i=
l, ...,fc), (10)
then
#,(0 -*,(D -*(-<), *<(0 --*(?) --*(-*),whence also
&(vi, , *?i)=
<( lit )Vt)-
It will be noted that the proof holds, whatever the value of N. It is
also geometrically evident that the symmetry, if present at all, is independent
of the rate of rotation of the reference line.
212. Conditions for Periodic Solutions. Since the differential equations
(7) are unchanged if t is replaced by T+2nx, or t by t+nT (n being an
integer), it follows that if
X, = Xi(t), W, = W,(t) (l=
l, .,.,*) (11)
is a solution, then so is
x, = x t (T+2mr), w t= wt (T+2mr). (12)
These two will be the same solution if the coordinatesand
their derivatives
have the same values at t = t ; that is, if
x l (T +2nw)=x t (T ), w; 1 (r +2n7r)=w; ( (T ), 1
x'l (T +2nT)=x' ( To ), w'(T +2nir)=w' (T ). J
If these conditions are satisfied, then, for all values of r,
x i (T+2mr)=x i (T), w,(r+2n7r) =w,(t);
that is, (13) are sufficient conditions for the periodicity of the solutions. That
they are also necessary is obvious, if the period is to be 2mr.
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PROBLEM OF k SATELLITES. 429
Special case. In the case of a symmetrical conjunction at t = 0, other
sufficient conditions can be formulated. For, if x'(0) =w,(0) =0 (t=
l, ...,*),
and if every \ t is a multiple of ir, it follows from the symmetry theorem that
x,(ir)= +X i (-Tr), w,(ir)=-iv i (-ir),
Xt (Tr)
= -x' (-ir), w'l (Tr)
= +w'(-ir). j
But, by equations (13), if t is put equal to w, the conditions for period-
icity of Xi(r) and w,(t) are
(a) X( (7r)=x i (-7r), (c) w ( (ir)=W t ( ir),
(b) x'M=x' (-T), (d) iv'M=w' {-ir). )
(15)
Of these conditions (a) and (d) are satisfied by virtue of (14), while (6) and
(c) are also satisfied if x'^ir) =w t (Tr) = 0. It may then be stated that sufficient
conditions for the periodicity of x t and w( (with period in t equal to T) are
(a) a<(0)-0, w,(0) = 0, \-0ott, 1
(&) x (V)=0, w,()-0. J
Moreover, after conditions (16a) have been imposed, conditions (6) are
necessary as well as sufficient.
213. Nature of the Periodicity Conditions. For /x= the differential
equations (7)admit the solution with
period2ir
(orT in
t)
x t=
\, Wi = Q ('=1, ..'.,*),
giving the circular orbits rt =a, ,
u t =\+p t T, in which at t=0, x ( =1, x' t =w t =w' t =0.
If these initial values are given increments Ac ( ;(i=l, . . ., k; j=l, 2, 3, 4),
then the solutions of the differential equations (7) for /xt^O are developableas power series in n and the Ac
,which converge throughout a preassigned in-
terval of t for sufficiently small values of those parameters.* Such solutions
are in general non-periodic; in fact, the periodicity conditions (13) or (16)
impose the condition that 4A; power series in these 4&+1 parameters shall
vanish. In the cases to be considered these Ak equationswill
determinethe Actj as unique functions of p, holomorphic in the vicinity of m = and
vanishing with n; so that, for sufficiently small values of n, there exist initial
conditions (depending upon T, g t , the p t , n, and the ft) such that the orbits
described are periodic with the required period.
Evidently for smaller and smaller values of ;u, smaller and smaller devi-
ations from the initial conditions of undisturbed motion are sufficient in
order to get periodic orbits. These orbits for fj. 5^0 may be said to "grow out
of" the undisturbed circular orbits as n grows from zero. Of course, for
any given masses, n and the ft being fixed, the possible orbits of this sort can
See 14-16.
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430 PERIODIC ORBITS.
vary only with T, q k ,or the p, ; but to a range of values of n there corresponds
a class of orbits.
In what follows it will be inquired whether the conditions for periodicity
can besatisfied
bysuch values of the
Ac,, asto
prove the existence of aclass of periodic orbits of each of the following types :
Type I. The finite system has a symmetrical conjunction.
Type II. The infinitesimal system has a symmetrical conjunction, but
the finite system has none.
Type III. Neither system has a symmetrical conjunction.
214. Integration of the Differential Equations as Power Series in
Parameters. It will be necessary to obtain the first few terms of the devel-
opments mentioned in the preceding article. Instead of increments Ac,,
to the initial undisturbed values of the coordinates it will be more convenient,in finding the properties of the solutions, to employ parameters An, , e, , co, , t< ,
defined as follows. At t = let
xt= (l+Ac ti )
= a+An i )-i
(l-e ico S 6
t ), ^=Ac=n,(l+An,)*1 ^^bO ,'
\ a i_i COS0, e ,. , ,.v,-\ = wt
= Ac 13= w,+ cos t
1 _ eeo8 g (*-!,...,*),
^; = n,Ac <4= n,(
l*_
A^ly
e] ~nt , -<Z,(l+An,)r, = 0,-c,sin0,,
(17)
the v t being equal to n,+g t T, the true longitudes from a fixed reference line.
It is evident that the Ac,, are holomorphic functions of the An, , e, , a>, ,
and t, for sufficiently small values of the latter quantities. Consequently,solutions of (7) exist also as power series in the new parameters. Further,
since the real positive values of the radicals and the smallest values of the
inverse cosines are to be taken in (17), the Ac,, are given uniquely in terms
of the An,, e,, a>,, and t,. From these two facts it follows that, if the latter
quantities can be determined as unique power series in p., satisfying the con-
ditions for periodicity, then also there exist for the Ac,, unique power series
in n, satisfying the conditions. Conversely, while the Jacobian of the Ac,,
with respect to the new parameters is zero for An, = e, = w, = r, = 0, yet, in
the only case where discussion will be necessary (viz., for Ac, 2 =Ac,, = 0,
whence a>,=
t,= 0) , the solution for the An, and e
tin terms of the Ac, x and
Ac, 4 is unique; for the Jacobian of the Ac (1 and Ac, 4 with respect to An, and
e, is distinct from zero for An, =e,= 0. Hence, in this case, if the Ac n and
Ac, 4 exist as unique series in n, satisfying the periodicity conditions, so also
must the An, and e, exist as such series.
In the developments of the coordinates as power series in i* and the new
parameters all those terms independent of n may be obtained, together with
a knowledge of their properties, in the following simple manner: The terms
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PROBLEM OF k SATELLITES. 431
in question are those remaining when fj. is put equal to zero, and are there-
fore the solution of the problem of k infinitesimal satellites when the initial
conditions are (17) in other words, the solutions of k two-body problems.
The dynamical meaning of the new parameters is then evident. Theorbits of the infinitesimal system, subjected to the initial conditions (17), are
ellipses in which the mean angular motions, major semi-axes, eccentricities,
longitudes of pericenter, and times of pericenter passage are respectively
n,(l+An,), Ckil+An,)-*, e X,+o<, ^.
If in the development of x t the coefficient of Anf ej u>* t\ be denoted by x iJ(lh i
then, by applying Taylor's theorem to the well-known developments of
the coordinates in elliptic motion as power series in the eccentricity,* it is
found that the followingcoefficients of first
and second degree terms do notvanish :
^i.oooo 1> Ef.iooo o> Pi.oioo COS^T, #(,jooo n >9 1
Z(. o*oo= sin
sg (T , x UBm = -q t s\nq,T, x
i , nm =^cosq iT+q i Tsmq ( T,
w.iooo=
?iT, io <iW0 ,= 2sing,T J
wumt =l, w,, mi =q
5.w.i,aioo=jsm2q t T, wt , mn =-2q,coaq,T, wt , im = 2q t TCOsq t T.
(18)
From simple dynamical considerations the following important properties
can be established. Let x uim be written x]l t to indicate its dependence
upon r. Then
*S?-CL-o -i, ...,*), (19)
where m is any integer; for, the coefficients x't>Bltt , etc., are those of the
terms which do not involve An, , w, , and t, , these terms being obtained byputting Arij = a> (
= r, = in the developments. But for these parameters equalto zero the initial positions are apses and the periods (in t) are 2w/n { .
Hence, at T = mv/q { , M, is at an apse and x't= wt
=0, whatever the value
of e t . Since this is true for a range of values of e t ,it follows that the coeffi-
cient of each power of e, in x\ and in wt is zero at r mir/q t .
It is evident that in the terms independent of n only those parameters
appear whose subscript is the same as that of the coordinate developed; the
terms involving n introduce, however, the other 4(/c 1) parameters.Terms involving /x. The only terms involving n whose coefficients are
needed in the sequel are ix and tie, (j=l, . . ., k). Let the coefficient of
H in the development of x { be a;,(0; t), and that of ne, be x,(j; t); let the
coefficients of the same quantities in wt be respectively w t (0; r) and w t (j; t)
(t'=
l, . . . ,k;j =l, . . . ,k). The process of finding these depends as fol-
lows upon two properties of the solutions:
*Moulton,Introduction to Celestial Mechanics
(new edition), p.171.
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432 PERIODIC OKBITS.
(a) Since the solutions must satisfy the differential equations identi-
cally in the parameters, the equating of coefficients of corresponding powerson both sides furnishes sets of differential equations for the successive co-
efficients in the solutions.
(6) The arbitrary constants which the successive coefficients carry are
determined by the conditions that the solutions shall reduce identically to
equations (17) at r = 0.
For each pair "of coefficients x,(f; r) and wt (f; r) (/=0, . . .
, k),
equations (7) give two simultaneous differential equations of the second order.
The one from (7a) can be integrated once immediately, and its integral
combined with the equation from (76) renders the latter a well-known type,
oo
z<"C/+o?z,(/;t)+ 2 (7 m cos mT+5 m sin mT) + aT = 0. (20)
Its solution, x,(f; t), when substituted into the first integral, permits the
final integration for wt (f; t). The initial conditions are
x t (f;0)=x' (f;0) = wi (f;0)=w' (f;0)=0 (t=
l, . ., ft;/-0 ft), (21)
for the conditions (17) do not involve n at all.
Now the form of the solution varies greatly according as a term cosg<t
or sinOiT is or is not present in (20). In the former case the solutions con-
tain a so-called Poisson term, rcosqa or Tsing,r, and in the latter case they
do not. In all the z<(/;r) and w,(/;t) (/ = 1, . . ., k), a Poisson term is
present; they are present in the x,(0;t) if, and only if, for some pair of the
n ( , say n, and n,there exists an integer J such that
J {n t -n t )=n l. (22)
The meaning and consequences of such a relation will be discussed in 219.
In performing the integrations it is necessary to expand
( 1 - 2e tJ cos <f> it + el) "1 (s=
3, 5)
as a cosine series, where, for the sake of a uniform notation, the followingdefinitions are made:
= a and &&=o, if /<t; e,j= a and %1, if j>i. (23)
Then(JO
(l-2ecos<k,+e? j ,)-,/ *= 2 F^t^cosm^,
n-2ecos</>+?,)-6/s = 2 r; n (e,/)cosw4> J( ,
where the F m and Gm are well-known power series in W , beginning with e".
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PROBLEM OF k SATELLITES. 433
Finally, the desired coefficients of the functions x t and w t are, for
i-\, . . .,
k and for/=0, 1, . . .,k
Xi(Jm
tT)=A t/ +BifCOBq t T+Ci f 8inqiT+D f fT
+E tf TCOsq lT+H l fTsmq i T-i-J t/ TCOii2qtT00
+K lf Tsm2q (T+ 2 (a^'cosmT+c^sinmr),
w,(f ; r) - L lf +N if r - q t Dif r* - 2 |X + ^1 sin q iT
+2\C if-
JLjcosq t T + 2H if TCOsqiT
<1
(25)
-2E if
Tsmqt T
^Jtf
Tsin2qt T
+^K l ,TC0s2q i T+ 2 (&*? sin mr +< /
>
cos(t),^ m-l
where Dto= and J l/ =K if
= for/^t, while if every A, = or x, then
J =0, c i/ =A / =^ (/ =A / =c (
(
7, =C = o (/=0, ...,*); (26)
but if no relation (22) holds, then
E io= H io
= J tt= K = 0, Dlf
= (/ =(), ...,*).
Those constants whose values are needed in the proofs are
+go
( /'7?/Cri+4'7?/(l+a? / )(y, ga^ij^Crj + X,,],
C-2% ze( M)rinm(x J -*^,1 m=l
where
2?
(27)
6-(J- 2 q t (mi
q l -q i
i -q2
i )LZa "' - *--+*-+>
where the A' (/ and F vanish, except for special relations among the q,.
(28)
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434 PERIODIC ORBITS.
215. Existence of Periodic Orbits of Type I. When k finite satellites
are subjected to the initial conditions (17), the solutions of the differential
equations are expressible as power series whose first coefficients have been
tabulated in equations (18) and (25) in /x and the An,, e,, ,, and t,, con-verging throughout an arbitrarily preassigned time interval for sufficiently
small values of these 4&+1 quantities. And, although the orbits are in
general non-periodic (as shown by the non-periodic terms in the tabulated
coefficients), it will now be proved that the conditions for periodicity can
nevertheless be satisfied, provided that every X, = or tt, by assigning to the
co, and t, the value zero (identically as to n) and to the An, and e, certain
values dependent upon n, that is, at t = there is to be a symmetrical con-
junction in which the velocities and distances must be properly chosen
with reference to the masses if periodic motion is to result.
In this case the conditions for periodicity are (16), of which equations(a) are already satisfied. The necessary and sufficient conditions are then
(a) = An,(g ; 7r)f e,(2sint7 ( 7r)+M^,(0;7r)-|- ,
(b) 0=et (q i s\nq,ir)+fi(x' (0;ir))+An i
eix', Am (ir)
+ f i-2e,x'(f]Tr)+ (t=l,. . . ,*).
(29)
The problem of solving these 2k equations has been treated in Chapter I.
Here the functional determinant, taken at An, = e, = ju=
0, is merely the deter-
minant of the linear terms, whose value is
4 = 71-' ITtf sing, 7T, (30)
so that its vanishing depends upon the q, . Since, from (2) and (5),
?+P (i=l, . . ., fc-1),
it follows that At vanishes if, and only if, the arbitrary q k is selected as an
integer.
Case I: q t is not an integer. Since A^O, the An, and etexist* as unique
power series in ju> vanishing with n, converging for n sufficiently small, andsatisfying (29). Thus, for all sufficiently small masses of the k finite satel-
lites there exist initial conditions for which the resulting orbits are periodic ;
and the presence of q t and of the arbitrary integers p tshows that for given
masses there is a A--fold infinitude of orbits of this type. A family of such
orbits exists "growing out of" any set of circles for which the infinitesimal
system would have commensurable synodic periods, whose least commonmultiple is not divisible by the sidereal period of satellite k. In Case I consecu-
tive symmetrical conjunctions do not occur at the same absolute longitude;nor will any later ones occur at the same longitude if q k is irrational. t
See 51, 2. fSee J219.
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PROBLEM OF k SATELLITES. 435
Case II: q t is an integer. Here Aj = 0. Nevertheless the Jacobian of the
ivJtt) with respect to the An,, taken at An, = e, = /x= 0, is
A2 = tt* II o,^0. (31)
(The only possibility for q t= is p, = q t which requires n t
= 0, and such a
selection for p, has been excluded.) Hence (29a) can be solved for the An*as power series in the e
t and fx, converging for sufficiently small values of the
latter quantities. Now, by (19), every term in (29a, b) has either n or some
An, as a factor; hence the solutions have the form
An J =/P(c y ,M) (*-l fc; i-i , Jb). (32)
If the power series (32) are substituted in (296), the resulting series convergefor sufficiently small values of ll and e s and contain n as a factor. (This
merely means that An, = ll = satisfy the periodicity conditions, whatever
may be the values of the e t ). If n is divided out,f relations are obtained
among the e, and ll of the form
Q = x[(0;w)-e tx'
t , im U) wt (0;r)+ JJ^O'JlO+MJ !< + (33)
At this point two questions of importance arise, viz., as to the vanishingof the x',(0; ir) and as to the vanishing of the determinant of the coefficients
of the linear terms in e,. By (25) and (27) every x' t (0; ir) is zero unless the
relation (22) holds for some pair of the fit. When such a relation does
hold, the equations (33) are not satisfied by e, = it = 0, so that solutions forthe e )} vanishing with ll, do not exist. Hence, periodic orbits of Type I,
"growing out of the circular orbits," do not exist, if, for any n, and n,J (n, n) = n, where J is an integer. When no such relation exists,!
the equations (33) are satisfied by ej = Li = 0. It remains to examine the
determinant A, of the first degree terms in the e s . This involves, in each
of its elements, power series in the e,or a t /aj, a,/a t ;and it is unknown
whether there are any sets of values of the ui for which A3= 0. It will, how-
ever, be shown that there is an infinite number of values for which the
determinant is distinct from zero.
Let P (f be any element of A3 , the first subscript indicating the row andthe second the column; then, by equations (33) and (25),
P (/=
a;;(/;7r) =</,(- l)'<7r/y f/ (/*), 1
I\ = x:(i;T r )-q t (-iy>w t (0;Tr)=q ( (-iy.Tr[H u -NJ. J
The e tJ depend upon v, q t ,the /3,, and the p, which were arbitrarily chosen.
It will now be shown that for a fixed selection of the fi, , v, and q t there is an
This case is not essentially different from that treated in 4. For although Ai has no minors of order
less than k distinct from zero, yet, so far as the linear terms alone are concerned, the equations (29) may be
regarded as k independent pairs.
fit is precisely this step which makes the selection of the parameters An, , c, especially advantageous.
{That qt an integer does not involve the existence of such a relation is shown in 219.
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436 PERIODIC ORBITS.
infinite number of selections of the p, (viz., all for which the e 4/ are "suffi-
ciently small") for which A ^ 0. For convenience, let all the e be expressed
in terms of a single parameter a by
o,-6Ja*- a i 0-1,..., *-l), (35)
where the b t , v, and q t (hence also a t and n k ) are constants independent of a.
Every element of A3 is, then, a power series in a; for by (35), (34), and (27)
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PROBLEM OF k SATELLITES. 437
main diagonal element than elsewhere in the same row, except in the first
row, where P a may now carry a ls_M, as does P u . And even this exception
is immaterial, since, when the rows have been factored as before, the
constant term in the element Qa is to be multiplied by a minor whose first
column contains a 8 as a factor, and hence can not destroy the constant
term in the main diagonal product. Therefore, it is true without exceptionthat A3 5* for all values of a sufficiently small.
Therefore, equations (33) can be solved for the e t in terms of /*, vanishingwith ju- The substitution of these solutions in (32) gives the An t , as well
as the e t , as holomorphic functions of n, for /x sufficiently small. Hence there
exist initial conditions giving periodic orbits of Type I even when q t is an
integer, provided that no relation (22) holds and that a is sufficiently small.
In drawing this last conclusion, however, a point of delicacy arises.
The p t are functions of a in the foregoing argument, and obviously the p t
are not integers (as the formulation of the problem requires them to be)for all values of a on any interval. The question arises as to whether there
are, indeed, any values of a, "sufficiently small," for which the p t are integers.
From (36) and p< = <7< ?*, it follows that
ft-ft( ffj =3-l) (i= l,..., k-1). (37)
The present discussion will be confined to exhibiting a selection of the b t
such that there is an infinite number of values of a less than any assigned
quantity,for each of which the
p{ are
integerswithout common divisor.
Let the assigned value be a and let the &< be defined by
and consider (37) for a = a^y/l/n, where n is an integer. Evidently, since
q t is an integer, and since
?<=(?*+ /c-*>t-'-? t (i~i,:..,h-i), (39)
the p t are integers. Consider the possibility of a common factor. If p k _ t
and p t _j have a common factor, their difference has the same factor. Thus,if there is a factor common to
(qt
+l)n qt and
(gt
+2)n2
qk ,
it is also a
factor of n 2 +n(w l)(q t -\-l). Hence if n is prime and greater than q t ,
such a factor must divide n+(n l)(g t +l) and also the difference between
this number and p t _ 1; or (n 1). But, as n and n 1 are mutually prime,there is no factor of n 1 which divides n+(n l)(g t +l), and hence no
factor common to p t _j and p t _, if n is chosen a prime number greater than q t .
There is an infinite number of primes; hence the p t have the stated property.The periodic solutions exist, then, and might be obtained as series
in ix alone (convergent for sufficiently small values) by substituting in the
original series in n and the Aw, and e ( the values of the latter 2k parameters,
as obtained in terms of n from the periodicity conditions. A far more advan-
tageous method is, however, available.
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438 PERIODIC ORBITS.
216. Method of Construction of Solutions. Type I. It has been shown
that, for m sufficiently small, there exist series
x,(t)= 1
+ 2^,b(t)m", w,(t)
=2w (
,,(r) M
"
(i'=1 k), (40)n.l -l
which (a) converge for ^ r ^ 2ir, (b) satisfy the differential equations
(7), (c) satisfy x\ (0) = 0, to, (0)= identically in n, and (rf) satisfy
x, (t + 27r)- x
s (t)= w
( (t+2jt) w,(t) =0 identically in //
The permanent convergence of series (40) follows from (d) and (a).
From (c) and (d) follow respectively
xl ,M=w l .M=0, (41)
^, B (r+2x)-x ( , B (T)=u\ n (T + 27r)-w,,(T) = 0. (42)
These equations (41) and (42) will determine the constants of integration
arising at each step.
First order terms. Since (40) must satisfy (7) identically in m, the
:r,,,(r) and u\ x (t) must satisfy
(a) iv" l +2q tx' ul +2'd IJ sm<t> Jl (l-rf J
-2 F^ejcosmfa) =0,
(6) x,
;A -'2q t w' u -Hx^+<t^ l
+2 /
5jcos4), ( +7j* J (a ( ,-cos(^,.) 2 F ra (e)cosw<^, 1 =0.y L m=o J
(43)
Since every $ is a multiple of r plus a multiple of ir, equations (43) are of the
type
() w"A + 2q t x' tA + 2 Z) sin Wt = 0,
(/>) x",-
2q liv'
t . l
~ W*iA +E?\ + ^ Kl cos mr = 0.
(44)
where the DJ and ^J^i arc linearly related to the Fm , and can be expressedin terms of the latter as soon as the p, are chosen. The solutions are
x ul (r)= - d2q t c+E)+c% Cosq iT +cZsmq l T+ 2 A^cosmr,
Hi "=!
wiA ( T )=(3q i c^+2E)T+tt-2csmq i T+2Ccosq l T+ 2 B^anmr,(45)
where the c |; j ( ,/=
1, . . ., 4; tl, . . . , k) are the constants of integration and
{r -q])A = E% - 2&DJJ, m*B*J = D*J - 2m$ .4 |j. (46)
titi
Poisson terms do not appear in (45); for, since no relation (22) holds, noterm in cosq,r or ednftr is present in (44). Now, by (41) and (42),
i.i c(.i u
)c i.\ 3o ''
so that the k constants c, alone remain to be determined. And here
arise two cases, just as in the existence proof:
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PROBLEM OF k SATELLITES. 439
Case I. q t is not an integer. Here, g, not being an integer, cosg (r does
not have the period 2-nr; consequently, by (42), c = (i=l, . . ., k).
Case II. q t is an integer. Here cos^r has the period 2x, and (42) is
satisfied for the arbitrary c^'^O. These k constants remain undetermineduntil the second-order terms are found, when the cf} are uniquely determined
in destroying Poisson terms.
Terms of any order. Case I. Assume that for n=l, . . .,
h 1,
the x ( , (t) and wt , n (r) have been found, the constants being determined, andhave the form
x,, %(t)= 2 AcosniT, wun (r)= 2 B^s&tmit (-!,..., t). (47)
m=0 m=0
An induction will show that x, ih (j) and wiih (r) have the form (47); that,
moreover,the differential
equationsfor x
tihand
w(iare of the
type (44) ; andthat the constants of integration are determined just as in the preceding case
for w = 1. The differential equations are [see (7)]
(a) 2 x,, t w",+2 2 x' ^w'i, l -{-2q i Xi iht+t=A t + l = h
(48)
+2'5 y 2 x Jik (sm<j> Jf +w 1 -w t ) l [x J
3
-a7j3
} m = 0,1 k+l+m=h-\
(b) x" h- 2 x i .M(w', t w' i ,,+2q i w', t )-q
i
lx
t , >,+qi
l (x7') l ,
+tfA(.Tr2
)*-.-f2%ja, ; 2 x ui {<rZ3
)i
+ 2 Xj, k (cos 07. + Wj - w^lxj 3 - aZ 3] m \ =0,
where an expression in parenthesis, having a subscript I outside, denotes the
sum of all those terms in the expression which involve /j.
1. It is to be shown
(a) that the variables in (48) whose second subscript is h enter in the sameform as the x (il and wliX enter (44), and (/3) that the remaining terms of
(48) (a) and (6) reduce respectively to a sine series and a cosine series in
multiples of r.
Evidently in (48a) the only terms involving the x iih and wiih (i= 1, . . ., h)
are w't [h -\-2q l x' ult ;
and in (486), aside from {x~2
) n , they are x" h 2q i w' Uh q]x Uh .
Now it can be shown easily by induction that
where the N are positive numbers and the v, are positive integers (or zero)
satisfying the conditions
Z.;-*, 2 fr,~h. (49)
Now x Uh enters only through (dhx i /dix
k
)"h, and for this term N=2, v =1,
v h=
l, and vf= 0(f= 1, . . .
,h 1); hence, in (x'
2
),,, x un appears with the
coefficient2.
Therefore,in
(48b),the
terms involvingx
Uhand
wUhare
"2?, W(, %q\x tA ,
and this establishes statement (a) above.
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440 PERIODIC ORBITS.
On using the notation F'(j) and F'(r) to designate respectively a cosine
series and a sine series in multiples of t, it is evident that
F[-F\ = F e
, Fl-Fl-P, F eF' = F\
(FC
)"= F% (F')
i * = F% (Fr)WJrl = F'-
Hence, as every x Un and w'Un (n=l, . . .,h 1) is a F\t), so also are all
sums of products of these quantities, and also all polynomials in the x iM) e. g.
(x,~x) t
and parts of (a~/),. Similarly, the sums of all products x U) w" t andx' Ui w' ut are F'(t).
There remain in (48) only the terms (sin mu+wj w), where mtJ=
<p Jtin
(48a) and mu=
<f>j l +ir/2 in (486). Let, for the moment, zlJ= m
iJ +w i wt
.
Then, since [dr z
ij /dix'] ll ^ = w} , f wun it can be shown by a simple induction
that
[^"]...= ?*.n sm(m-K!) (,,-,.,)"'.
where the Nt are numbers and the v, satisfy (49) after h is replaced by /.
Now since every {w h , wuf ) (/ =1, . . .
,h 1) is a F\r), the product
R't-\( w).r~ wi.tY' i s a ^'( T ) or F"(t) according as 2'zlv f is odd or even. Butwhen this sum is odd, j> is odd; and when even, v is even. The entire
product no is then always a F\t) if mt j=
<t> )t , or a F"(t) if mtJ=
(j> il -\-T/2.
The differential equations (48) are, therefore, of the form
(a) w'[ h + 2q i
x' u1t + 2 D^sinmT=0,m-l
(b) x'i :-2q i
w,
{ ,,-Hxun+Efi+ 2 E%cosmT=0,
(50)
where no term in cos q,r or sin q t r occurs under the summation sign, since g,
is not an integer. Obviously the integration of (50) is the same problem as
that of (44), so that the solutions are
x,M= -^(2q t c^+Eti)+Ccosq l T+c^smq l r+ 2 A^cosmr,HI -l
^,(T) =^(3fl ( c^4-20T+c^-2c^sin 0j r+2c*cos 0< r+ 2 B^mmr
(i = l, ...,*),whore
(51)
(m*-
<ll) A IS= E - 2f Ag , wBj- = D -2mq (
A . (52)m
Also, by (41) and (42),
K=-~E?:, <-<-<&-<) (i = l,...,k), (53)
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PROBLEM OF A; SATELLITES. 441
so that the induction is completely established. Thus the successive x tM
and wUn reduce to
Xt.n(r)=^ ? E+ 2 ^'cosmr, xo Un {r)= 2 Bt'smmr (-l,. ..,*), (51')
and may be obtained without integration by applying (52). It is merely
necessary to compute the D and E from equations (48) at each step.
Terms of any order. Case II. Assume, as in Case I, that for n =1, . . .
,
(h 1) the xt Jj) and wUn (r) have the form (47), all the constants of
integration having been determined except the c*_ x . The differential
equations for the x u1t and wuh are again (50) (a) and (b), where, however,since q, is an integer, cos q, r and sin q t r may occur under the summation sign.
These terms arisefrom two
sources : from the termscJ5_,
cosq t
r andc2_,
sinq t
r
in the x Uh ^ and w,,^ ,and from similar terms in the earlier x tM and wUn , as
well as (usually) from combinations of the <f> n in the coefficients.
When (50a) are integrated and combined with (506), equations for the
x lih are obtained; to avoid Poisson terms in the x Uh , the coefficients of the
terms in cos a, r in these last equations must be made to vanish. These co-
efficients, E^ 2D\^, involve the cj"-! and various known constants; e.g.,
the e (- 1, . . ., h-2); and the vanishing of the E%-2D% will usually
determine the c^_, .
In the first place, the only terms of (48) in which the x uh ^ and wuh -i
appearare*
(O) X uh . l w'l l + XuM[n-x + ^\.H-iWu^2x\ AWuh ^
+2 ,
1
,|w,.,_x -U7, 1
_l cos0 i ,
-2.T,,_ 1 sin</>
-^ao^-isin^i+^M-i-^..*-! cos^ ( +3^(T fJ
6 sin <, ( (aX-i
(&) - 2o 1x (iA _ 1 m;;. 1 - 2q lx
l , l w' t , h - l - 2w' iA w' Uh - l + Gtfx,^^ - 2gfr3 j a;,, ft_ 1
+ 2 ' K \~ (2x-i cosi ,+w /l4
_ l -W.JI-, sin ^,)
+j!i'iiIli(iir , ;i*-i-*M- 1 cos^ l -|-to M-i-w M_ 1 8in^ J )
-3i?J/r f/ i(o
- COS0) (^j^j+a;^-,- a^x^^+x,,^ cos<
; ,
(54)
For h = 2 the first four terms in (a) and in (b) are to be divided by 2.
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442 PERIODIC ORBITS.
where <r iJA =(l2e u cos^+e*,)171
. Whence it is seen that the cf\^ enter the
E% 2D% linearly; and, denoting their coefficients by R t/ (i=l, . . ., k;
/= 1, . . ., k), it is found that
R{-l)"i >(i-l,...,*;/l, ...,*), (55)
where the P are the elements of the determinant A,, discussed in the
existence proof. Hence, when A, 5^0, the determinant of the coefficients of
the <$_! in the equations E$ - 2D?j is zero. The constants cj?_, (/= 1, . . ., k)
are then uniquely determined.
The values of the x iJt {r) and wuh (t) are given by (51) and (52) ; equations
(53) are, however, replaced by
C=-
E, cl= c = 0-1, ... , lb), (56)
the c remaining undetermined at this step. The induction is thus estab-
lished. In this case, too, the successive x Un and wUn have the form (51'), the
c "n appearing as Ajjf, and are obtained without integration. But, besides
computing the Dj' and E^ and applying (52), it is necessary to obtain
also the A(
!n+i and E% +1 and to determine the cj" by solving the equations
Remarks: (1) In constructing the solutions it has been tacitly assumedthat the Fourier series representation of the x Un and wUn and the attendant
manipulations of these series are valid. This is justified by the considera-
tion that the well-known functions encountered in the first step are repre-
sentable, together with their derivatives, by uniformly convergent Fourier
series. But the products of two such series is another of the same type, andalso the integral of a uniformly convergent Fourier series is uniformly con-
vergent. At every step the convergence remains uniform.
(2) The question naturally arises as to whether, in any orbits of Type I,
the smallest period is a multiple of the period of the infinitesimal system,
namely, mT. An examination of the method of constructing the solutions
furnishes the answer. If mq t is not an integer, the constants are determined
precisely as in Case I; if mq k is an integer, then by the method used in Case II.
In any event, for a given value of n and a given set of p, and q k , there is
a unique solution satisfying (41) and (42) (with 2mir replacing 27r). Thesolutions found above, however, satisfy these conditions; hence there are noorbits of this type whose smallest period is a multiple of the period of the
infinitesimal system.
217. Concerning Orbits of Type II. For Type II, as for Type I, all the
X, are multiples of t; but it is proposed to ascertain whether the co, and t,
exist as functions of /x (not identically zero, but vanishing with n) so as to
satisfy the periodicity conditions (13). The question can be studied most
easily by examining the method of constructing the solutions.
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PROBLEM OF k SATELLITES. 443
Let the origin of time be selected as the instant when satellite k has an
apbidal passage, and the origin of longitude at that satellite's apsidal positionso that co
t
= Tt
=0;
and let it be assumed for themoment
that the wt and t,
(i=l, . . .,k 1) exist as functions of n satisfying (13); then there exist
solutions of the type (40) satisfying (42), but not satisfying (41), for all values
of n except for i = k. Because of the absence of conditions (41) some of the
constants in each step are left undetermined until terms of higher orders are
found.
For the first-order terms the differential equations are again (44), andthe solutions are (45), the coefficients being determined by (46). Evidentlythe c
{
( "are determined as in Type I, but for the other c"J two cases arise.
Case I. q t is not an integer. Here, by (42), c,= c* = (i=l, . . .
, k),
but the c"' are atpresent undetermined, except c"J
= 0.
Case II. q t is an integer. Here the terms cos^t and sing,T have the
required period 2ir, and hence the cf\ and c*, together with the c, remain
undetermined, except c^ = c 0.
Terms of any order. Case I. Consider next the terms of order two. Thedifferential equations are still (48), using h = 2; however, because of the ejj
these equations now reduce not to (50), but to the form
(a) w'*+2q ix' u2 +H + 2 (D$MLmr+H%co8mT)**0,
(b) x" t -2q,w' , i -3q2
tx Ui +E + 2 (E^cosmT+J^slnmr) =0,
(57)
where the flJJ and J,'
1
vanish with the c. The first integration apparentlyintroduces non-periodic terms Hf^r; but computation shows that these coeffi-
cients vanish. Thus
//^ = 2'(^-^)5 ( ^[fa,^(2G -r; 2 )-|F 1 ],
and by Le Verrier's relations among the F(e u ), and (7(e f ,),* it is found that
the bracketed expression vanishes identically in the e tJ. Hence the solutions
of (57) have the form
.+ 2 (A^cosmr+P^smmr),o'/i m= l
-cJS + 2 (B% sin mr + QJJ cos mr ) ,
(58)
where the P and Q vanish with the c1
*}, which (together with the c%)remain at present undetermined, the other constants arising in the x u% and
Tisserand: Mtcaniqw Celeste, vol. 1, pp. 276, 278.
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444 PERIODIC ORBITS.
the wIA having been determined as were the corresponding ones in the first-
order terms.
For the terms of order three, the differential equations are of the form
(57), and the H do not vanish identically. For, while the c% drop out just
as the c% did in the preceding step, the c"J are now present (linearly only),
entering both through the P and Q% and directly from the wm (t) . Thedestruction of the non-periodic terms H t, by setting the #Jj = 0, will be
found to require the vanishing of these c"J which have been previously unde-
termined.
The H consist of two sorts of terms: those which arise identically bycombining trigonometric functions of a single <j> it
and its multiples, and those
which may appear because of relations among the various tf> /(. Terms of
the latter sort can not be collected, unless the numerical values of the p,
have been chosen; but their possible presence will be shown to be immaterialso far as the conclusions are concerned.
In equations (48a), using h = 3, the terms which reduce identically to
constants may be selected by fixing upon some one <f> Jt and expressing the
coefficients of cos w4> tiin x ul and wui , and of sin rrup Jt
in wul and x lA . Incident-
ally it may be noted that the series in each x tM and wUn , as given by (47) or
(58), is in reality a rearrangement of several Fourier series in the various
<f> ti. The resulting complicated constant, involving several series in the F n (e v )
and Gn (e tJ ), can be treated advantageously by expressing all the a n e u , q t ,
and5in terms of a single parameter a, just as in (35) and (36). In the
H the coefficient of each cjjj, say MtJ , becomes then a power series in a;
and it is found that Mu = 2' Mw and
Mtl= *+-. N
t3 (j>i), Mu = a"- .N ti (j<i), (59)
where the NtJ
are infinite series in their respective etJ , beginning with con-
stant terms.
Now the vanishing of the H requires that the c% satisfy k linear
equations, which must be homogeneous, since the vanishing of all of the un-
knowns would reduce the H to their values in Type I, viz., zero. That is,
2Jf*-0 =i, ...,*). (60)y-i
But, from the value of Mlt above, it is evident that the determinant of the
MtJis identically zero, so that any equation is a consequence of the others
and may be suppressed. Let the first equation be the one which is dropped.Then the remaining k 1 equations (i
=2, . . .
, k) will determine the
cm (i 1> >k l) uniquely in terms of c"J if their determinant A,, of
order A; 1 in the MtJ , is distinct from zero. From (59) the lowest powers of
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PROBLEM OF k SATELLITES. 445
o in the various elements of any row of AB may be ascertained. Evidentlythe exponent i+2j6k takes its smallest value 4i+2 6k when j=\, andits laigest value 6i 2 6k when j = il; while 6j6k takes its smallest
value 6i+6 6k whenj = i-\-l. As this last is greater than the highest value
of the former exponent, it is clear that the lowest power of a in any of the
Mu (j=l, . . . ,il, i+1, . . . ,k l) is4i+2 6k, which occurs in Mnand also in M. Thus, in the r th row of A6 (where t = r+l), the lowest
power is 4r+6 6k; and this appears for r=l, . . .,& 2 in two columns,
the first and (r-\-l)th
. But, in the last row, where r = k 1, the lowest
power can appear only in the first column. Hence if the factor a 4r+6_w is
removed from the elements of the r th row (r= 1, . . .,k 1), a new deter-
minant A6 is obtained in whose first column the series of each element
begins with a constant term, as does also the series of one other element in
each row except the last row.
Now, if A, is developed by the minors of its last row, it is clear that a
constant term can not be lacking when the development is rearranged as a
single power series in a. For the only element of the last row which can con-
tribute to the constant term is that in the first column; and in the minor of
this element constant terms are present in all the elements of the main
diagonal, and nowhere else.
Thus A 6 7^0 at a = 0; and hence A and likewise A 6 are distinct from zero
for all values of a sufficiently small. Therefore the vanishing of the Hfldetermines the c"} (7
=1, . . .
,k 1) uniquely and homogeneously in terms
of c"J . But this latter constant is to be put equal to zero by reason of the
choice of the origin of time, as noted above in discussing the first-order terms.
Hence every c"J= 0. Thus the wui (t) reduce to the values which they would
have for orbits of Type I. Then also every PJJ = Q = 0; and the x ui (t) andwt.t( T ) reduce to the values in Type I, except for the c% which remain as yetundetermined for j=l, . . .
,k 1.
In the next step these c% will enter the H (by identity) in precisely
the same way as the c'fl entered the Hfl, and must likewise vanish. Simi-
larly, the cl
*l in the terms of order n will remain undetermined until the H +i
are set equal to zero, when they must vanish together with the P +1 an d
Q]"Vi> to which they will in the meantime have given rise. Thus the final
determination of the constants arising at any step reduces the terms of that
order to the values which they would have for Type I.
If various relations among the p t give rise to other terms in the Hthan those which are present identically in some one <fr n , even if these
terms introduce into various elements lower powers of a than have been
treated in A6 , these new terms can not affect the argument in general;
for, if they introduce into the development lower powers of a than were
previously present, with non-vanishing coefficients, then this new developmentis equally as useful as the former value of A, ; while, if they do not furnish
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446 PERIODIC ORBITS.
terms of lower order, neither can they in general destroy the terms treated
in A6 , since they must involve new @'s in their coefficients. Any cancellation
could occur, then, only for a few special relations among the masses.
The conclusion is not yet warranted that no orbits of Type II exist inCase I; but there can be none when a is below some ''sufficiently small"
finite value, unless possibly for a few very special relations among the masses.
Terms of any order. Case II. The differential equations for the second-
order terms x,,,(t), w,;,(t) are again (48); but because of the c and c
the equations reduce to (57), as in Case I above, where, however, the //*' and./' now involve them (linearly) and vanish with these two sets of constants.
In order that x ta and wu2 shall be periodic, it is necessary, just as for
Type I, that E i$ 2D\', the coefficient of cosqj in the final differential equa-tion for x ia {t), shall vanish. It is equally necessary that J?-\-2H? , the
coefficient of sin q,r in the same equation, shall vanish. Further, the "secularterms" Ht must vanish. The H are free from the c%, just as in Case I
above; but they now contain the cjl and ej3, vanishing with the latter set.
Now, from (54) and the fact that no relation (22) holds, it is evident
that the E-2D% do not involve the eg nor the cj; also the f+2H$involve neither the c nor the c
, and vanish with the c . Further, since
the x ui (t) and w>,.,(t) may be written
xl . l {T)=c
i
sin((] lT + ?-)+c?^mq i T+ A^cosmr,
w,, l (T)=c,
;+2c%cos(q lT + ?:)+2ccosq l T+ I Bfifanmr,
(61)
it is found that the c% enter the J$+2H$, which are the coefficients of
sin q,T, in precisely the same way as the cf\ enter the E$2D$, which
are the coefficients of sin(</,T +*, 2).
In the treatment of Case II for Type I it was shown that, when A?^0, the
equations E^ 2D^ = admit a unique solution for the cj\ (.7=
1, . . ., k).
The same conclusion is evidently valid here; and it also follows at once that
the equations
J%+2H ($ = (i=i,...,k) (62)
admit a unique solution for the c =1, , k), namely, c0. In
(62), c, does not appear, being already zero; but this does not affect the
conclusion, since there are obviously no more solutions for k 1 of the
unknowns, after the last one has been determined, than there are for all k.
The H now vanish identically.
The c"J , cfl ,c are now determined as for Type I
;but the c remain
undetermined until the nextstep,
and the solutions are atpresent given by
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PROBLEM OF k SATELLITES. 447
(58), as in Case I of Type II, where, however, the c% and c$, as well as the
c%, remain undetermined.
In getting the third-order terms certain constants will be determined:
the cfl and c% by the vanishing of the E^-2D^ and J*g +2H(
% respec-
tively; and the c^ by the vanishing of the Hfl, which reduce to their values
in Case I when the c%=0.Likewise, the cf n , c of any order are determined in the next order,
(n+1), and the cf n in the order (?i+2). Each set is obtained from linear
equations, whose determinant remains the same for each successive order.
Thus, with the same possible exceptions as in Case I, it is impossible in CaseII to determine the constants otherwise than as for Type I.
Remarks: (1) If either of the determinants A3 or A5 vanishes for somevalue of a, there may still be no new values for the constants of integration
which would satisfy later conditions in subsequent differential equationsand render the solutions periodic in form. Whether the series converge for
this value of a would be unknown; so that the mere vanishing of A3 or
A6 would not warrant the conclusion that orbits of Type II exist.
(2) The foregoing conclusions extend beyond a denial of the possibility
of obtaining equations of periodic orbits by a certain method of analysis,
or solving the differential equations in a certain way: the non-existence of
a class of physical orbits of a certain type is asserted, though possibly there
exist numerous individual orbits of the type. All orbits, periodic or not,
arising from any set of Ac,, and ju [see (17) and (7)], can be represented by
power series in these parameters converging through the interval 5 r 5! 2t,
provided that the parameters are sufficiently small. And from the exist-
ence of a class of periodic orbits of the type sought would follow the existence
of a range of values of the Aci}and n (including zero) satisfying the periodicity
conditions. These equations obviously could not be satisfied for a range of
values of y. by arbitrary values of the Ac,,, and therefore they would define
the Ac,, as functions of n, holomorphic for /x sufficiently small and van-
ishing with fi. The substitution of these values of the Ac,, into the original
developments of the coordinates would render the latter series in /x alone,
having properties (40a, b, d). If these series are impossible save for Type I,
then there does not exist a class of physical orbits of Type II growing out ofthe circles named.
218. Concerning Orbits of Type III. It may be inquired whether there
exists a class of periodic orbits growing out of circular orbits of an infini-
tesimal system which has no "grand conjunctions"; that is, whether there
are periodic solutions of (7) when some of the X, have other values than are
possible in Type I.
Let the initial conditions be (17), let some instant when Mt is at an
apse be selected as the origin of time, and let the origin of longitude be the
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148 PERIODIC ORBITS.
apsidal position of Mt , both for n and for n^O* Then A = 0, and
u> = t=0 identically in k and /xj and the conditions for periodicity are (13).
But equations (7) admit two integrals, an examination of which shows that
two equations of (13), namely z t (27r) =x k (0) and x' t (2T) = x' t (0) , are a conse-
quence of the other 4A 2 equations. Hence equations (13) become
(a)
(&)
(c)
(d)
=e,(l cos2<7,7r) +juX (i0 -!-e (
T( (-
sin2^ ( 7r)
+^Un,X?>+eX,Z+o>,XZ+T / X <
}+. ,
= e,(g,sin 2q i ir)+iJ.X'j ) +c 1T
( (qi
i \-cos2q i ir)
+^\^hxy+e,x:+u>,x:y+r,x' y}+..,
r-i J
= An,(2^x) +e,(2sin2r/,7r) +AiJF,, +e ( r,(2<7,f cos2g,ir)
= et (-
2q iT :-
cos2q tir ) +nW' <lt +e (T
( (2q i sin2g,7r)
+2\to,W$+e,W' +t* /W%+T,W%}+
(-l *-i),,
(63)
where
X, =X,(0;2t)-Z i (0;0), .
X^ = X,(/;2tt)-Z,(/;0), .
pr;, =T7;(o;27r)-i7;(o;o),
Tf,'!2
;=^;(/;27r)-TF;(/ ; o),
and the X (j=l, 3, 4), etc., are constants whose values will not be needed
here.
*In treating orbits of Typo I, the instant of a symmetrical conjunction was regarded as the beginning of
a period and was taken as the origin of time; but this was merely for simplicity, since in periodic motionany other instant could be so regarded. By taking as t = the instant when Mt of the infinitesimal systemis at arbitrarily selected longitude, and choosing that longitude as a new origin of longitude (so that X* = 0),it is clear t ha t t he longitudes which the other infinitesimal satellites have at that instant constitute a set
of X's, not all of which are multiples of ir. Each family of orbits of Type I may thus be said to arise fromany one of an infinitude of sets o f the X, other than multiples of ir, though all the sets have X* = 0. By reasonof the u>i (t), which are in general distinct from zero, the absolute longitude v ( of any finite satellite at thenew t = would vary with n) (but, since families of orbits of Type I exist for every position of the line of
conjunction, one may obtain a family in which the initial longitude of Mt is zero, identically as to it, byselecting orbits for different it a from different families, taking the conjunction line as needed for the it used.)For such an origin of time Mt would not in general be at an apse for r = 0.
Hence if the origins of time and longitude be chosen at an apsidal position of Mt for all values of n, theset s o f X4 other than multiples of x, which can give rise to orbits of Type I, are largely excluded. (Whetherany such sets remain, depends upon whether, in Type I, Mt has any apses other than at the symmetricalconjunctions of the system; and this has not been ascertained.)
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PROBLEM OF k SATELLITES. 449
While the determinant of the linear terms of the An u e t) u>( , and r
(in
(63) is zero, yet, when q t is not an integer, solutions of (63) (c) and (d)
exist for the Aw, and el
as power series in n and the w, and t\. Prop-erties of the series (63), similar to (19), are easily established, which showthat the solutions for the An
tand e
t ,and also the series obtained by sub-
stitution of these solutions into (63) (a) and (b), carry n as a factor. After
this substitution and a division by ju, (63) (a) and (6) become
+r/[+^r i/ ]}+..'.l
=^+2(l-co^)^+|{^[^sin 2q l7r w , m -i
,r Y ,< 4 , ,
sin 2g (7 r W, M i \2(l-cos27 f 7r)
"'' J'tT/ l^'' ,
' t2(l-C0S2q i T)
" *' J J
(6)(64)
T'l-cos2<7,7r
The constant term in (64a) vanishes, since for q t not an integer no relation
(22) holds; the constant term in (646) reduces to 2q tCu0 . If any of the
C,, are distinct from zero, (64) are not satisfied by w(= T
(=
/i= 0; hence
w, and t, do not exist as holomorphic functions of m vanishing with n, or
periodic orbits of the type sought do not exist. The necessary condition
for periodicity, namely, that the C (i0 vanish, is, by (28),
2'(/2 e m(,>imn(X,-X,)=0 (-l, , . . , ifc). (65)
Of these^-equations
in thequantities \
=1,
. . .,k
1),one is evidently
a consequence of the others; for, before Xt is put equal to zero, the Jacobian
of the C,, with respect to the X,(t'=l, . . . k; j=l, . . ., k) is identically
zero. Let the equation for i = 1 be suppressed.If particular values be assigned to all, save one, of the X, ,
the last X can still
be given an infinitude of values for which any one C,, is distinct from zero;
hence equations (65) impose very special conditions upon the X, . Whether
there are any sets of X's, other than those of Type I which satisfy (65), is
unknown.* It will, however, be shown that there are no others "in the
vicinity of" multiples of t, provided a is sufficiently small.
*Th<- limitations upon any such sets seem fully as severe here as in Type I.
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450 PERIODIC ORBITS.
If the C tA are developed as power series in the \ t J lw (J,
= 0, 1), the
coefficients of the linear terms are simply [dC^/dX,] for X,= J y x. Evidently,
from (28),
[i&'L =* l/ Swe.(, / )cosm(,/ ; -;> (./Vt),
ac1*^*1 =8 <f=
~ 2 '^ 2 we( ( ,)cosw(./,-J ( )7r (jVi).L CM/ J X/-J.T > ra-1
(66)
Denoting by S f/the coefficient of X, in the I th
equation of (65), and intro-
ducing a by (35), the S u are obtained as power series in a. It is found
that the lowest exponent of a present in S if is (fi) if f<i and is 5(f i) if
f>i. Hence, of all the S (f (f<i), S n carries the lowest power of a, namely,
the (1th
; and of all the S tf (f>i), S ul+l carries the lowest, namely, thefifth. Consequently, if a
u_,) be removed as a factor from all the S {/
(/=1, . . ., k1; i = 2, . . .
, k), a determinant A, is obtained (equal to
the determinant of the coefficients S t/ multiplied by a power of a), in whoser* A row all elements save those of the first and (r+l)' A columns begin with
a power of a. Therefore A7 is of precisely the same type as A, ,and the dis-
cussion of the latter shows also that A7 (and hence the determinant of the
S (f ) is distinct from zero for all values of a sufficiently small. Therefore
quantities A,(/ =l, . . .
,k 1) exist such that equations (65) are not
satisfied for any values of the X, for which |X r J f ir\ <A, except X7 =0 or ir.
Moreover, the existence of a set of X r satisfying (65) would not prove theexistence of w, and t, as functions of /x satisfying (64). If it is assumed for
the moment that orbits of Type III exist, and the method of construction is
examined, equations related to (65) are encountered. Thus, in finding the
second-order terms, the H must be made to vanish, and each of these involves
Fourier series in the various \ X, . When q k is an integer, somewhat differ-
ent difficulties arise in the existence proof; the same difficulty is, however,encountered in the construction.
A general negative conclusion is not yet warranted; but it is evident
that if any orbits of Type III exist, they must satisfy very special conditions.
In every case, however, periodic orbits of the type sought do not exist if
(22) holds.
219. Concerning Lacunary Spaces. The relation (22) may be expressedin the form J(p f p l )=p l,+q k ;
hence this relation can hold only if q t is an
integer. But the converse holds true only when k = 2. For example, in
the following selections q k is an integer, but no relation (22) holds:
ft-2, P*-. = 3, Pt-i-5, . . ., p,
= 2A:-l,
q t= 2, p*-i = 3, P-t-7, ..., p 1
= 4fc-l, etc.
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PROBLEM OF k SATELLITES. 451
When k = 2, (22) always holds* if q k is an integer; for then p, = l (other-
wise T would not be the smallest synodic period of the infinitesimal system),and hence Jj) i
= q i is satisfied by giving J the integral value q t .
Since n k T = 2q k ir, the case q k an integer is the one where the consecutive
conjunctions of the infinitesimal system occur at the same absolute longitude;
and, denoting the synodic period of the two infinitesimal satellites M, and M,by Tfl ,
since
nj,. = n. '" 'n t na
it follows that n l Trg= 2irJ when (22) holds; or the consecutive conjunc-
tions of the infinitesimal pair M} and Mgoccur all in the same absolute
longitude.
Moreover, since all the w, vanish at the beginning and end of each period,all the "grand conjunctions" of the finite system occur at the same longi-
tudes as those of the infinitesimal system, and intermediate conjunctionsof any finite pair occur at very nearly the same longitudes as those of the
corresponding infinitesimal pair.
These facts suggest a physical reason for the non-existence of periodic
orbits under certain circumstances. The greater part of the mutual dis-
turbances of two bodies occur while they are near conjunction; and, if the
consecutive grand conjunctions occur at exactly the same longitude, the
perturbations of the elements would tend to be cumulative. Nevertheless,
if there are more than two bodies, the mutual disturbances may so balanceeach other as to yield periodic orbits, especially if the bodies are far apart
(i. e., a sufficiently small) unless (22) holds. But if two bodies have con-
junctions between the grand conjunctions, all occurring very near the same
longitude, the other bodies can not counterbalance the large perturbations of
the two.
More exactly, there exists a range of values of the masses and the e
including zero, for which periodic orbits are impossible; so that, unless the
orbits for n = are eccentric rather than circular, there are for small values of
H no periodic plane orbits of k satellites when (22) holds.
Ho far as this result extends, it would indicate that no asteroids havingnearly circular orbits would be found, whose periods compared to Jupiter's
are in the ratios \, f, f ,etc. Those whose periods are nearly in any such
ratio should be found subject to very great perturbations.It is well known that lacunary spaces of the sort just mentioned do
occur among the asteroids. That there are such spaces also when the ratio
of the periods is f , and other such values, is not surprising, as in any case
the slightest deviation from the correct initial values destroys periodicity,
there being Poisson terms in the solutions.
*In Poincarti's discussion of the problem of three bodies, therefore, the case where q k is an integer
without such arelation as
(22) holdingdoes not arise.
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452 PERIODIC ORBITS.
220. Jupiter's Satellites I, II, and III. Of Jupiter's longer known satel-
lites, the innermost three move almost exactly in a plane, apparently in
periodic orbits having symmetrical conjunctions; and their masses with
respect to that of the planet are very small. Since for orbits of Type I theincrease in the longitude of M, during a period is independent of m, being
equal to n x T, the average angular velocity of each finite satellite for a period
may be taken as the corresponding n.
The unit of time being the sidereal day, and the unit of mass being the
mass of Jupiter, the observational data are :*
Xj=
7T, X, = 0, \ 0,
M, = 0.000017, Mt= 0.000023, M3
=. 000088,
,= 3. 551552261, 2= 1.769322711, w,= .878207937.
(67)
Since (w,-n 1 )/3 = .891114775, and n t -n 3=
. 891 114774, then, of (67) satisfy,
far beyond observational accuracy, the equations (2), where v = .891114774.
Then the period T is 7.0509271 days, and
Pl = 3, Pt = l, q 3= .985516077, 1
</>U= T + 2t, </> 13
= 7T + 3t, <As3= r
> J
so that satellite III advances 354?785788 during each period, while satel-
lites I and II advance 1080 and 360 respectively more than this. If nis taken arbitrarily as .0001, then
ft = .17, ft = .23, ft = .88.
It seems desirable, however, to retain the fts in the computations, inasmuchas a new determination of the masses may render it necessary to use other
values than those given above.
The D and E of (46) are obtained by writing equations (43) in the
form
C+2 2< *'W+ 2'5
'#2 Csin>/^, =0,/ m= l
< -2 9X -3?X 1 +9;ft+ 2'5
"<r'2 C'cos//^, = 0,
(69)
where, rearranging according to multiples of r,
];' = 2a u F -I'\,
K)
=^-+F 2 -2F,. V"=\+2a IJF l -(F t +2F 3 ),
^^rO, V^ = 2a u F m~(F m_^F mil ) (m> i ,
Tisserand, Traiti de Mtcanique Celeste, vol. 4, p. 2. The n, given there (203? 48895528, 101?3747239<>,and 50?3 1760833) are here reduced to circular measure.
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PROBLEM OF k SATELLITES. 453
Evidently the U%, Ug, and 0JJ enter respectively the Dff, D* and Dg,etc. From (23), (5), and the relation {ajatf = (%/nO*, the e and S are found
to be
m - . - .6284333, w = M= .3939606, ( n - e 32= .6268932,
^a= 3-1366 ^ hga = 1>2 3875 /S x ,
m = .19132 ft ,
Suv\ 3 = j 2326 ft , h^a= .77464 ft ,
5^ 2 = .30443 ft .
(70)
The F m{t Xj ), and also the Om(eJ), etc., encountered in the higher orders are
readily computed by using the tables of coefficients given by LeVerrier.*
In obtaining the successive A and Bjf from (52), the smallest divisors
introduced are 16-tf , 4-^, and 1 -q\, or .1156616, .05772591, and .02875806
respectively. These divisors decrease materially the effectiveness of the
small value of /x; nevertheless the terms above those of the second order
seem relatively unimportant and will not be computed. The coefficients
of /x in the z,(t) and w,(t) are found to be:f
^i,iW = (-3ft-.lft)-.8ftcos2r-.lftcos3r-203.7ftcos4r
-K.7ft-.2ft)cos6r-.2ftcos8r-hlftcosl0r+ .
,
Wu(r)-
+.3ftsin2r+.4ftsin3r+406.3ftsin4r- (1.1ft-. 3ft) sin 6r+.3ftsin8r-3ftsin9T-.lftsinlOT+ ,
r 2 ,,W = (.5ft + 3ft- .1ft) + . 8ft cost- (58.4ft + 100.1ft) cos2r- .7ft cos 3r
+ (.7ft-.2ft ! )cos4T-.lftcos5T-.2ftcos6r + .lftcos8r+ ,
u\Jt) = -2.9ftsinr + (114.3ft + 199.2ft)sin2r + l.lftsin3r
-(.8ft-.2ft)sin4T + .lftsin5r + .2ftsin6r-.lftsin8r ,
ar,. t (t)=
(.4/3, + .5ft + .3ft) + 28.9ft cos t + .6ft cos 2t -(.4/3,
-.2ft) cos 3f
+ .lftcos4T ,
w3 , 1 (t)= -55.2ftsinr-.8ftsin2T-(.4ft + .2ft)sin3r-.lftsin4r
For the second-order terms the D|' and E are found from (54), where it
is convenient first to rearrange the coefficients of o^? and <r~5
according to
multiples of r. The x ul and wut involve all multiples of r; but, as they
* Annates de I'Observatoire de Paris (Memoires) vol. 2. Supplement.
fThe writer regrets to sta te that the values formerly published {Transactions of the American Mathe-matical Society, vol. 9, pp. 29-33) are practically all erroneous, the factor q t of the last terms of (46) havingbeen overlooked in making the calculation.
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454 PERIODIC ORBITS.
carry the factor ju1
,all save the following terms fall below the limit of accu-
racy in the xt , x and w,,, above. To facilitate comparison, the second-order
terms are shown multiplied by n = .0001 . They are
.0001x u (t)= -.6^+(.1|8 1 j9;+.1^+.1AA)cos2t
-(2.90,0,-12.202
-9.20,03)cos4t-2.O0,!
cos8t+ ,
.0001 wu (t)= .30,0 3 sin r - (.30,0,+ .2#+ .30,0,) sin 2r + (5.80,0,
-24.40,
5
-18.3/3,0,) sin4T+.l&jS 3 sin6T+5.1$sin8T ,
.0001.r,,,(r) - (.202
+.60,0 3 +.50 3 )- (.10:03+- ltf) cost
+ (6.302 + 12.60,0,+ 12.60,0 3 +3.10
2
)cos2t
-(.20I+.10,0,+60,0 3 +.50Dcos4t+ ,
.OOOl^ 2 ,,(r) = (.20,03+.20,0 s +.40Dsinr-(12.505+25.O0,0,+25.10,0 3
+ 1.30,03+6.20Dsin2T+(.10^sin3T+.40?+.10,0,+ 1.40,0,
+ 1.302
)sin4r+ ,
.0001 xm{t)= - (2.30,0,+2.50
2
+3.90,0 3 ) cost- .10,2cos2r
-.10,0,cos7t+ ,
.OOOlw3.,(r) = (4.70,0,+4.90l+7.60,0,)sinr+.20,2
sin2r+- .
Since ?-!
= a< (l+z MM+z,.jM
2 + ) and vt
= \+q iT+w tA n-\-w tA iii + ,
the radius vector and absolute longitude of each satellite are obtained by
computing the atfrom (3) and using the coefficients above; the deviations
from their values in the undisturbed circular orbits are given to five significant
figures, so far as the terms of the first two orders are concerned, How muchthese would be affected by terms of higher orders is unknown; in fact no proof
has been given that the series converge for /x= .0001, although they have been
proved convergent for all m sufficiently small.
To show the general shape of these orbits, the values of the i\ and r, a{
will be given to four decimals, using the values of the 0, tabulated above:
r,/a, = l-.0044cos4T, ,= ir+3.9855T+.0089sin4r,
r,/a,= l-.0093cos2T, v t= 1. 9855 r-. 0003 sin t + . 0185 sin 2 t
+ .0001 sin 3 t+. 0001 sin 4 t,
r,/a,= l + . 0006 cost, v t=
. 9855 t-. 0011 sin r.
Hence if these orbits are thought of as ellipses rotating in the plane,
the major semi-axes would be the respective a the several eccentricities
would be .0044, .0093, .0006, and the axes would rotate forward at rates
whose average values are the nt
. The three satellites are in line with Jupiter
at the beginning and middle of each period, II and III being on the same side
of the planet at t=
0, and I and III on the same side at r=
w. Whenever
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PROBLEM OP k SATELLITES. 455
II is in conjunction with I or III, the inner of the pair is near a perijove andthe outer is near an apojove, which decreases the amount of their mutual
perturbations.
No radius vector or longitude differs very widely at any time from its
value in a circular orbit (a, and X, +q l r, respectively) . The largest departuresare for satellite II, as r 2 /a t reaches a minimum of .9907 at t = anda maximum of 1.0093 at about t = 7t/2 and every half-period thereafter,
v t meanwhile ranging from 64' more to 64' less than the mean longitude of II.
Similarly, satellites I and III get 30' and 4' respectively ahead of and behind
their mean positions, and the r{ /a, at such instants closely approximate their
mean value, unity. For satellite I the maxima of r/a occur at intervals of
a quarter-period, and for satellite III they occur at intervals of a period.
Finally, it may be noted that, for this system of bodies, the increments
Ac tj (see 209) which have been given to the initial values of the coordinatesand their time-derivatives to preserve periodicity when the bodies are finite
are approximately
rt
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456 PERIODIC ORBITS.
This substitution requires the flattening to vanish with the masses
M,,..., Mt ,so that the central body becomes spherical if the others
become infinitesimal; but the amount of flattening corresponding to any
given set of finite masses remains quite arbitrary, even if the p t and q k arespecified; for the values of the a
t merely determine the ratios of the 7,,
and one 7 may be taken at pleasure. In the solutions of (71) satisfying
initial conditions (17), the terms independent of n are the same as formerly.
Hence in Case I where q t is not integral, the An, and e, still exist as convergent
power series in n satisfying (29 a, b), though of course their values in terms
of n are now different because of the 7, . Thus periodic orbits exist.
In case q k is an integer the 7, enter the x,(0; t) and x,(i; t), but do not
appear in the P t)or P (/ of A3 . Thus the argument in Case II is likewise
unaltered, and periodic orbits of Type I exist under the same conditions
as when the bodies are all spherical.The numerical results for Jupiter's satellites given above are affected
but slightly in the first and second orders by including the flattening of
Jupiter. The corrections to be added are in fact
to x u (t) add .17, cos 4t
to .0001 x-lt2 add 1.8 &7, cos 4r
to .0001 wM add -3.6 7, sin At
to x 2fl (r) add .l7 2 cos It
to.0001a- 2i2 add (.2/3 1 +.4/3 3 ) T2 cos 2r
to .0001 u?M add -(.5/3, + . 8/^)7* sin It
tox 3A (r) add .17, cos r
to .OOOLr,,, add .l/3 2 7 3 cos r
to .0001 ir M add .l/3 27 s sin r
And since 7, =.36, 7 2
= 14.3, and 7 3= 5.7, the values of the r,/a t
and t>,
given above are changed as follows:
tor, a, add
.0018 cosAt, to
r2 /a 2 add .0007 cos 2t,
to v { add .0030 sin 4r, to v 2 add .0011 sin 2r;
r, a, = 1 -.0026 cos At, ^ = 7t+3.99855t+.0059 sin At,
r 2 /a 2= 1 - .0086 cos 2t, e t - 1.9855r- .0003 sinr+.0174 sin 2t
+ .0001 sin 3t+.0()01 sin At,
r, a, - l + .0006 cos t, v 3= .9855r - .001 1 sin r.
then
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Chapter XV.
CLOSED ORBITS OF EJECTION AND RELATEDPERIODIC ORBITS.
222. Introduction. In the problem of two bodies there is in no sense
continuity between circular orbits revolving in the forward and retrograde
directions, except where their dimensions shrink to zero or become infinitely
great. But in the restricted problem of three bodies the deviations from
the circular forms of the orbits are such that there is geometrical continuityin some classes between those which revolve in the forward direction andthose which are retrograde; and the limit between the two types is an orbit
passing through one of the finite bodies. If the infinitesimal body leaves
one of the finite bodies, its orbit is called an orbit of ejection; and if it strikes
a finite mass, it is called an orbit of collision.
In certain cases orbits of ejection are also orbits of collision, or closed
orbits of ejection. When the direction of collision is exactly opposite to
that of ejection, they are the limits of two classes of periodic orbits, in one
of which the motion is direct and in the other of which it is retrograde.
The closed orbits of ejection are not themselves periodic orbits, even if the
physical impossibility be disregarded and the problem considered purelyfrom the mathematical point of view; for, if the expressions for the coordi-
nates are followed, in the sense of analytic continuity, beyond the values of
t for which a collision occurs they become complex, and never become real
again for increasing real values of t. Those orbits in which the ejection andcollision are not in opposite directions are not the limits of periodic orbits,
or at least of orbits which re-enter after a single revolution.
The object of the investigations of this chapter is to determine the
limiting types of certain classes of periodic orbits, and thus partially to
prepare the way for the discussion of the evolution of the various classes of
periodic orbits with varying values of the parameters on which they depend,and to show the relations among these various classes. The existence of
the closed orbits of ejection will be established, some of their properties will
be derived, and it will be proved that each one in which the direction of ejec-
tion and collision is opposite is the limit of two series of periodic orbits.
223. Ejectional Orbits in the Two-Body Problem. As preliminary to
the general problem, the special case in which there is only one finite mass
will first be treated. Let the mass of the finite body be 1 n and let the units
be so chosen that the gravitational constant is unity. Then the motion of
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458 PERIODIC ORBITS.
the infinitesimal body projected along the fixed -axis satisfies the differential
equation d? -\~n ,,
de= + -T' (1)
where the sign is or + according as the motion is in the positive or negativedirection from the origin.
Suppose f = f and d/dt = $' = at t = t . Then the first integral of
equation (1) becomes
(|)
2
= r= 2(l^) ? 2(l^)+ ^ =
2(lp) +Ci> (2)
If c, is negative, || has a finite maximum for which '
vanishes; if c, is zero,'
approaches zero as || becomes infinitely large; if c, is positive,'
is finite
for || infinite. It will be assumed that c, is negative in order to get orbits
of ejection which are closed. Then, without loss of generality, it can be
supposed that is the greatest value of for projection in the positive direc-
tion, or the least for projection in the negative direction. Then
-0, *-+'*&=**. (3)6b
With the initial values (3), the integral (2) becomes
vb=H*-i(-i+D - ^^ ('-'.)- G+>H w
where j is an integer.
Now consider as a function of (t Q. Since the right members of
(1)and
(2)are
regularfor all values of t and all values of
except=
0,it follows that is a regular function of t for all values of t except those for
which vanishes. These values of t are easily determined from (4), and are
found to be t x \/
t
where j takes all integral values.
The character of as a function of (t 1 ) in the vicinity of t = tj is easily
determined from (4). The left member is expansible as a power series in
V+ =i7, and the equation can be written in the form
It is found that
#-**, g-* d-=o, ^).-4(-Therefore r? is expansible as a power series in (t tj)
1 ' 3
, starting with a termof the first degree in (t t)
Vi. Since =
*7
2
, it follows that is expansibleas a power series in (t ti)
1
'*, starting with a term of the second degree in
(t tj)l/i
. It is easily seen from (4) that F(ri) is an odd function of ?j. There-fore t] is an odd series in (t tj)
l/3,and is an even series in (t tj)
l/i. Since
the only singularities are given by (5), the radius of the circle of conver-
gence for the series for both i\ and is ^lUJ-^/f^-)'
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(9)
CLOSED ORBITS OF EJECTION AND RELATED PERIODIC ORBITS. 459
The form of the solution in the vicinity of t = tj being known, the co-
efficients of the series can easily be found from (1) by the method of
undetermined coefficients. It is convenient in the computation to let
r-it-w, (6)
after which (1) becomes
The solution of this equation with initial value of equal to zero has the form
Z = a 2 T>+a i Ti + +a u T
in + (8)
By direct substitution and comparison of coefficients, it is found that
|_> 2 1 1 I 2 " 2 4 I"3 3 6 1894 4 a i 3293 5 in
> t = Ct I 1 + clt a t H a t a r H a r .S ctj^i-r
u.i7
u ' '63 4851 ^7007 J'
-(1 /x) ,a = arbitrary constant.
More convenient formulas for use can be developed by eliminating the
term in-2 from (7). After the transformation (6) the integral (2) becomes
=.8(l-)(!-i),.On using this equation to eliminate
2 from (7), the result is found to be
Now it follows from equations (8) and (9) that
=cr[l + ar 2 + 2 a ^ 2y
]'j = 2
g=2cr[l + 2aT 2
+20'+lKT2
'],
P=2c[n-6ar2
+20'+l)(2i+l)a 2^J,
T^ = 2cV["l + 7ar 2 +6aV+ J (2f +3/4-2)a 2/ r
2J +a f* (2;2
-j + 6)a w _ 2 rj;
+S{So-*+i)'(2y-2i!H r i)<i lA ,. It V],00
- 2$^ = -4cV f"l+3aT2 +2aV+ J\ (j+2) a 2 ,r
2;
+*2 (i+^vZ+Sflo'-Hi)^-^]-J=3 ;=i * *=2 J
(gy =2cV[l
+ 4ar 2 +4aV + 2 jg (i+l)a 2;r^
+4a 2K- 2 r 2J + 2{S *+i)c/-*+i)<v*W-j=4 L A =2 J
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4G0 PERIODIC ORBITS.
Hence, on equating to zero the coefficient of tv
after these series have been
substituted in (10), it is found that/-*
i(2i+3)o,,= -a(2f+j+2)aiJ _ i
-
2 0'-*+ l)(2j-k)
a^^, (11)
which gives the coefficients very simply for all values oij greater than unity.
The result of applying (11) and reducing the coefficients to the decimal
form is
+ = cr-[\ + o.T- - 0.42857aV + 0.36508a V - 0.39044aV
+0.46996aV-0.60863aV ! +0.82861aV 4 -1.16832aV 6 + ].
So far as this solution is written the signs of the terms alternate and
a tJ has a' as a factor. This is a general property which will be needed in
establishing the existence of the closed orbits of ejection in the problem of
three bodies (230). It follows at once from (11) that the part of the co-
efficient Ojy which comes from the first term on the right is opposite in sign
to the coefficient of the preceding term. Since the sum of the subscripts of
the product terms in the right member of (11) is 2j, their product has the
same sign as the coefficient of a,,_ 2 . Therefore, a,, and Ojy_, are opposite in
sign for all j. It also follows from (11) that a,, contains o as a factor to one
degree higher than it appears in a w_ 2 .
The expression for has a branch-point at t =t,, where three branches
unite. If t1]= pe
v ~=r w+in * )
,the three distinct branches are
*-* V=, *i W^1^, &^* v=t l ttH *\ (12)
If ttj is real and negative, <p= ir and the second of these expressions alone
is real. Iittj is real and positive, <p= and the first of them alone is real.
Consequently the analytic continuation of the real branch of the expression
for through t = t } leads to a complex value, and this value remains complexas t, remaining real, increases to + . Therefore the motion is not strictly
periodic.
In case the orbit is not one of ejection the branch-points of the functions
which express the coordinates in terms of the time are not on the real axis.
If the major axis is kept fixed, and if the eccentricity is varied from zero tounity, an examination of the equations from which the character of the
functions can be determined shows that the singularities start from both
positive and negative infinity on the lines t]t a=
jiT $v J jT_ \and approach
the real axis as a limit as e approaches unity. At these singular points twobranches of the function permute except when, for e = l, two of them haveunited on the real axis, and then three branches permute. If e increases
beyond unity, each of the branch-points divides into two which move equallyin opposite directions along the real axis, and for e = oo one of each pair unites
with another of anadjacent pair.
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CLOSED ORBITS OF EJECTION AND RELATED PERIODIC ORBITS. 461
224. The Integral. Equation (2) holds for all values of t, and therefore
wHn the series (9) is substituted in it the result is an identity in r. Theconditions that the coefficients of the various powers of r shall be identical
in the left and right members furnish severe tests of the accuracy of the com-putation of (9). The integral also gives the relation between the arbitrarya and the greatest distance . By direct substitution, it is found that
The interval from ejection to collision is found from equations (5) , (9) ,
and (13) to be
P= tKJ-Jq)3 '' 2 .
y w) (14)"
V2(1-m) \/2(l~-V)
For a = 0, which corresponds to a parabolic orbit, P is infinite.
225. Orbits of Ejection in Rotating Axes. In the problem of three
bodies the motion of the infinitesimal body will be referred to rotating axes.
In the demonstration of the existence of closed orbits of ejection in the prob-lem of three bodies it will be necessary to use some of the properties of the
orbits of ejection in that of two bodies. For this reason the orbits now under
consideration will be referred to axes rotating uniformly with the period 2r.
Suppose the ejection takes place at t = ^ and along the x-axis, where the
rectangular coordinates are denoted by x and y. Then x and y are given
by the equations
x = +cos( Q, y= sin (t t,). (15)
Those orbits which re-enter in the direction opposite to that of ejection
are of greatest interest in the present connection. The condition that theyshall have this property is that their period in t from ejection to collision
shall be a multiple of 2w. This condition becomes, by virtue of (14),
(y3/2 =
2iV2TT=^), (16)
where j is a positive integer.
Figs. 15, 16, and 17 show the curves for j equal to 1, 2, and 3, at least as
to general form, in full lines for ejection along the x-axis in the positive direc-
tion, and in dotted lines for ejection in the negative direction. These curves
for the three values of j are not drawn to the same scale, for it follows from (16)
that their linear dimensions are proportional to j2/
*. One of the important
properties of all these curves is that they intersect the x-axis perpendicularly
at their mid-points.
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4f>2 PERIODIC ORBITS.
226. Ejectional Orbits in the Problem of Three Bodies. The differential
equations of motion for the infinitesimal body when the finite masses describe
circular orbits are, in canonical units,
<Fx 2 dy = dUdf dt dx'
V=\{x>+y>)+]- (t + -
r 2
d*y , ndy = dUdt2 ^ dt dy'
n = (x+fx)2
+tf, r|= (x-l+M) 2
+2/2
(17)
These equations have the integral
'dxV.(dl\> u _,+<D--* (18)
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CLOSED ORBITS OF EJECTION AND RELATED PERIODIC ORBITS. 463
x+(j.+x^iy = (p+V^Tq)2
,
r
x'+y-vr=T(x+n+y') = uV^iv
2(p+V-Iq)In these variables equations (17) become
du = dHda dp
dt=(p2 +q 2
)da = p2da.
(20)
dp . dHda du
dq' = dHda dv
dv
dadHdq
// =i|(
W+2p 2
5)2
+(?;-2p2
p)2
|-|l- M-6V+^P6 +MP 2
F}:
V = I -p 2
+9=- M,r 2 =l-2{p*-q>) + (p*+q%
'2
(21)
It follows from (20) that p = q = \ix+n = y = 0, and that u and v are finite
for r,= 0. Therefore the form of H shows that the differential equations (21)
are regular in the vicinity of p = q = 0. Consequently, p and q are developable
v
Fig. 16. Fig. 17.
as power series in a, vanishing with a. It follows from (20) that x, y, and
tti, considered as functions of a, have the form (19); and therefore x and y
expressed in terms of r, defined in (6), have the form
x+n = c[r*+ay+ ]. y = c[b 2 T*+b 3 T+ ] (22)
227. Construction of the Solutions of Ejection. The character of the
solutions in the vicinity of (t t 1 )'/ '= r = having been found, they can be
obtained without difficulty from (17). Upon using t as the independentvariable and expanding the expression for r 2 ,
these equations become
drx _ dxTdr*
Zdr UT [(x+nY+yT 2
+9 MT5
[2(x+M)+30r+ M)2
-fi/2 +
Tdr*
Zdr +bT dr TV [(x+tf+yT* WT VL 1 + 6 W+U+
]'|
(23)
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Mil PERIODIC ORBIT&
It will now be supposed that the line of ejection is along the x-axis.
Therefore the initial conditions are
x(0)+/x =2/(0)
=rA;r => F
g:t " M
rCT
1=ca = arb. const. (24)
or CLt L r Jr=o
With the initial conditions (24) the solutions have an important prop-
erty of symmetry. Let them be written in the form
x+ m = t7(t)," ?/=
t<<7(t), ^=Mt), jjjU(r)'. (25)
Now make the transformation of variables
xMumX v=- v t--t <k__dxi dy_ dy.
Equations (23) are not changed in form by this substitution. Therefore
the solution of the transformed equations with the initial conditions
*,(0) - yM =^P=o, p^f| = a
are
Xi-tJ/W, fc-tfcfr,), ^-^W fr^W,where /, <?, ^, and 4> are identical with the functions represented by the
same symbols in (25). Therefore
T*f(T) = +T-J(T l )= +T-f(-T), T<p(T)= -T 1 <p(t 1 )
= +T<p(-t),\
TA
9(T)=-r\g(T l )=-T<g(- T ), tV(t) = +t3
1 +(t 1 )= -tV(-t).J(26)
It follows that x and dy/dr are even functions of t and that dx/dr and j/ are
odd functions of t. The first equation of (22) contains only even powers of
r, and the second contains only odd powers, starting with a term of the fifth
degree as the lowest.
With the initial conditions (24), the solution of (23) is found to be
+ (x +M)=c[r
2 +ar 4
-faV+[-i+|a3 + | M]r
H
+---}'
y= -c|r
s -ar 7 -f aV+[|a
3 -/*>" + j-
T9(l u)l l/3
c= 9' a = arbitrary constant,
where the positive or negative signs are to be used in the left members
according as the initial projection is in the positive or negative direction.
The constant of the integral (18) is given by the equation
c~ [()+ $)>(*+*>
+l(-t- + M)
2
+ Z/
2
]
l/i +[l-2(x + M) + (x + M)
2
+ rJ1
(28)
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CLOSED ORBITS OF EJECTION AND RELATED PERIODIC ORBITS. 465
It follows from (27) that the right member of this equation can be developedas a series of the form
c= c->+c +c 2 T*+cy+T
Since this equation must be an identity in r it follows that C = CmC_ 2
= Ci= C l
= C 6= =0. Those expressions which are zero constitute
a check on the computation of the coefficients of (27) . By direct substitution
of (27) in (28), it is found that
C = C =-fc2a+ M (2+M). (29)
The force function is sometimes used in the symmetrical form
F=(l-M)(r1+^)+M(r1+^)=t/+M(2-M),instead of in the form given in (17). Then the constant C becomes
C=-?c 2 a+3 M. (30)
228. Recursion Formulas for Solutions. The second terms in the
right members of (23) give rise to a large part of the labor of constructing
the solutions. They can be eliminated by use of the integral, and relatively
simple recursion formulas can be developed for the construction of the solu-
tion after the terms of lowest order have been found.
The integral (18) becomes in the notation of (23)
(31)
+ (z+m)2
-|2/2
+(x+m)3
-|(z+m)2/2
}-9rC.
On multiplying the first equation of (23) by x+n and the second by yand adding the results; and then multiplying the first by y and the second
by (x+m) and adding the results, it is found that
T( , +M) ^ +T ,^_ 2(x+M) g_ 2 ,g_ 64 (x+M) g-,g]=
9r-[(,+M)2
+^]- [( ^ff r+ 9Mt
5
J2(x+m)2
-2/2
+3(x+m)3
-!(z+m)2/2
"}'[
= 9Mt5
J+3(x-+m)2/+6(x+/x)
2
2/
]
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(33)
(34)
466 PERIODIC ORBITS.
On eliminating the second term of the first of these equations by means of
the integral (31), the simplified equations become
=fr'[(
I + (I )'+,
]-|[(^)!
+(g)']+9 f,r>{3(a
; + > ,)!
-f!+4(x+M)"
= 27 Mt5
{(x+m)2/+2(z+m)
j
2/+ }The solution (27) may be written in the form
oo oo
z+M = cr s
[l+ 2a 2 ,r2
']> y = cA-l + 2 6^]'
c =2 , a, = b t
= a = arbitrary constant.
The next step is to form general expressions for the terms involved in (33).
Nearly all of the terms written are of the second degree in x+M and y; those
which are of degree higher than the second are all multiplied by the factor
H and will be in general of little importance. They will not be includedin the general formula and must be added to it when it is used. Since these
terms contain t11 as a factor, they will not contribute much to the solution
unless it is carried very far. The general expressions for the terms of the
second degree in x+ix and y are found from (34) to be
r(x+M);p =2cv{l+ ^(2f+3j+2)a v T2J + jj ^(k+l^k+Da^^A,
rt/^=2cv{lOr
6 -^(2f-3j+U)b 2J _y 1 + J ^(k+2)(2k+5)b u b
2i - 2t - 6 A'
- 2(x+ M)# =-4cv{l+ 20'+2)a 2J
r2/ + JJ JJ (&+l)a 2t a 2 ,_ 2t r4>
-2yf= -2cV
J5r-2 jg +2 )6 2; _ c r^+ %^{2k+b)b 2t h v -n-A<oo
- 6r 3
(x+ M)g= -6c s
r'[-5T
6
-2[5a 2 ,- 6 -(2y-l)6 2J _ 6 ]r^GO -.J-4
>-5 t-1
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CLOSED ORBITS OF EJECTION AND RELATED PERIODIC ORBITS. 467
C J 4
Gr'yf= 12c 2A - r 6 - 2 [0'-2K_ 6 -& 2 ,_ 6 ]r
2
'+ 2 2 (k+l)a u b21 -*-*A>
* J=i J=i t=l *
-f (1+2m)t
6
(x+m)2 = -
f C2
(l+2 M)r3
{r6
+22^-eTy + J IX^-e^'}'
-f (l-jtffV- -
f c2
(1-m)t3
{tI2 -22Vi 2 ^+ S S^V-a^}'
| (f-Y -2cv{l+2 J 0+l)a 2^+ 2 2 (*+DO'-*+i)^-T*'i" i=l j=2 Jt=l J
I (g)2
=|c2
r'{25r6 -102(2i-DV 6 r w
+ 22 ( 2 ^+ 5 ) (2i-2A- i)&A--eT2 '
[;
y$*=2cv{-l- 2 [0 + l)(2i+l)a w -6 2; ]r"
2S(Hl)(2Hl)a !t VB Tu}
)
r(x+ M)? =-2cV|-10- 2 [l0a 2 ,-(i+2) (2j+5)^]t*
* =1> j-i ..
+ 22 ( fc + 2 ) (2A;+5)a 2; _ M6 2 ,r^ k;=2 t=i -
-2y^ - -4cv{-l- 2 rO'+lK-wV+ 2 2 ^+1)VHT L /-IL J
J=2 t-1 J
_ Q^M . _6c 2r
6
{5r6 -2 2 0'+2)&2,-o^+ 2 2 (2^+5)6 2t 6w_ 2t _ 6 r4.
2(x+ M)^=2cV|-5- 2 [5a2 ,-(2i+5)6 2Jr
2H 2 2( 2 *+ 5K- 6 T*}'
-6r 3
(x+ M) ^ = -12cv{l+ 2 0'+2) 2;
r2J + 2 2 (k+\)a ik a v _ HA,
-27 Mr5
(x+ M)2/= -27cV 6
{-t6 - 2[^-*-^-sV+ 22^-- 6 t4'
L i=4L J
J-I -l J
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468 PERIODIC ORBITS.
On substituting these expressions in (33) and equating the coefficients of
t v+s and t w+'
respectively, it is found that
)-\
2j(2i+3)a w=-22J(*+l)(A;+j)a u a w_ tt +3[4i-9(l+2M)]a t/ .,
+ [4j2
+12;-9]& s ,_,-27(l- M)& l ;- ls
-^[(2k+5)(2j+2k+3)b tt +12(k+l)a n ]b v _ ili _
1-i 1-4
+62(2fc+5)& 2t a 2v _, t _ 6+f (l+2M)2 a **,--.1=4 *-l
-\-(ln)b ik b t j_ u _ n + quantities coming from*=i
terms in (33) of the third and higher degrees,i-\
-2U+2)(2j+Z)b v =2(j+2)(2j+Z)a v -22(k+l)(2k-l)a u b2j_ ik
*=ii-i /-i
+ 122(*+Dua w-u+22J(*+l)(2*+5)a w_ lt 6 kt
,_4M
- 12(i+2)& 2 ,_ 6+ 62J(2*+5)6A--.;-4
-27/i[a, J _,-6,_,_J+27M2/ a 2* &2j-st-6+ quantities
coming from terms in (33) of the third and
higher degrees.
These formulas are to be used when j is 3 or greater, and care must be taken
in adding terms coming from the higher powers of (x+m) and y in the
right members of equations (33) .
The results of applying (35) are
x+ii = rcosT*+x l n+x i ni + ,
-y=rsin T*-y 1 n-y i fx'
l + ,
r = cAl + aT 2 --a?T i +-a zT*- a*T*+ aVL 7 ^63 4851- '7007
_ 2,418,092 6_12_l 55,964,945 , M_ 38,481,084,886 . ,
"|
3,972,769
~T67,540,473 32,937,237,333
*"" '
J'
-(]%*-&*- &- fry,, ^nT l
_l_ 2 2 /4392 2 380 27 Vv , "1
(36)
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CLOSED ORBITS OF EJECTION AND RELATED PERIODIC ORBITS. 469
229. The Conditions for Existence of Closed Orbits of Ejection. TheStries (34) converge for all |p|^Sp and |o a [^p provided |r|^fl, where Ris a positive constant depending on p ,
a,and p. The coefficients of the
various powers of t are polynomials in a and p, so far as p occurs explicitly,
and they also involve p implicitly through c. These coefficients are expansibleas power series in a a and p which converge for all finite values of |a o
|
and for |p|<l. Therefore the expressions for z+p and y are expansible as
power series in a a =/3 and p, and if
|/3| ^p, |p| ^p the series converge for
all |t| ^.R. They may be written
dx dvx = Pi(P,h',t), ^=p,G8,m;t), y=*p t (P,v;T), *g -p(0,j! r), (37)
where p lt . . ., p t are power series in /3 and p.
Now a will be determined so that when p is zero the period from
ejection to collision shall be 2jir. From (5) and (13) it is found that a
satisfying this condition is
a =-^(2j)-2/s
. (38)
Suppose T<R and p = and that for t = T the coordinates of the
infinitesimal body are x,
x', y 0) y' ,
where the accents denote derivatives
with respect to t. Suppose now < p < p and let the values of the coordinates
atr = Tbe
x=
x +p l , x'=
x' +p t , y=
y +p 3 , y'=
yl+P>. (39)
The conditions that these values of the coordinates shall belong to an orbi t
of ejection are
sb+ft =P,(/3, m; T), y +&=pM p; T),
\
(40)arJ+A = P.08,M; T), y'o+P< = p t tf, n; T). j
Since the right members of these equations are expansible as converging
power series in /3 and p, it follows from the definitions of x,x'
, y a ,and y' that
0i=
g,(/3,/i), A = 9i08,m), ft=
g,(i8, m), A-g 4 G8,M), (41)
where ?, ,. . .
, g 4 are power series in /3 and p, vanishing with /3 and p.
If the infinitesimal body crosses the x-axis perpendicularly at any time
the orbit is symmetrical with respect to the x-axis. It follows from the
definitions of x, x'^ , y , and y' that, when p = 0,
x' (P/2) = 0, 2/ (P/2) =0, (42)
where P is the period in t from ejection to collision. It will be shown that
analogous conditions can be satisfied when p is distinct from zero, and there-
fore that closed orbits of ejection exist in the restricted problem of three
bodies.
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470 PERIODIC ORBITS.
Suppose n is distinct from zero and consider the solution with the
initial conditions x +/3, , Zo+ft , 2/ +ft , 2/+ft In order to leave the period
arbitrary an undetermined parameter b is introduced by the transformation
t = t1 (1 + 8), (43)
where t, is the new independent variable. Now if the orbit for n = does
not pass through the position of n the solutions can be expanded as powerseries in ft, . . .
, ft", B, and n; and if Q is arbitrarily chosen in advance,
the moduli of ft ,. . .
, ft , 5, and m can be taken so small that the solutions
converge for all T^t,(1 + 5)^Q. The quantity Q will be taken equal to
(l + 5)P/2 where, as before, P is the period from ejection to collision when
M-6.In order to complete the discussion in regard to the convergence it is
necessary to show that none of the orbits in question for /x = passes throughthe position of /x. Suppose r = t is the time at which the infinitesimal bodycrosses the positive x-axis on which, for it = 0, the body it lies at the distance
unity from the origin. It is necessary to show that for none of the values of
defined in (16) is equation (4) satisfied by = 1 and t t^mr, where ^ is
the time of ejection.
It follows from (4) that the larger | |
is the shorter is the time requiredfor the infinitesimal body to pass from ejection to the distance unity.
Therefore, if for the smallest value of belonging to the problem, viz.,
$ = 2, it reaches the distance unity in less than t, then it will always be
at a distance greater than unity at t ^ = t and all multiples of w until it
reaches the greatest distance . Consequently, it can not pass through the
point occupied by n while receding from 1 n; and since the path referred
to rotating axes is symmetrical with respect to the z-axis, it can not pass
through the position of m on its return to 1 m- In making the computa-tion it is convenient to let = p. Then, transferring the origin to t
t , equa-tion (4) becomes
<-< 1 =^[T-4V t7=7+2sin-,
(-l+2p)](^ )3s
.
It is found from thisequation
that if i- = 2 the value of t txfor = 1 is
0.187T,which is less than t. Therefore none of the orbits in question for /x
= passes
through the position of m-
If the infinitesimal body is moving in an orbit of ejection and crosses
the a;-axis perpendicularly at any time, then it follows from the symmetryof its motion that its orbit is also an orbit of collision. Therefore sufficient
conditions for a closed orbit of ejection are
dx n , P
If the initial conditions are x +ft , ^o+ft , 2/ +ft , 2/o+ft and the parameter5 has been introduced by (43), these equations become
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CLOSED ORBITS OF EJECTION AND RELATED PERIODIC ORBITS. 471
|^=P 1 03 1 ,...,/3 4 ,5,m;/Y2) =O, y-P i (ft,...,A,,>;P/2)-0i (44)
where P t and P 2 are converging power series in ft ,. . .
, ft , 5, and /x. It
follows from (42) that Pj and P 2 vanish for ft = =ft= 5 = ^ = 0.
If ft, . . ., ft are determined by (41) the orbit is an orbit of ejection.
Therefore, upon substituting the series for these constants in (44), sufficient
conditions for the existence of closed orbits of ejection become
^ =Q 1 (/3, 5, M ; P/2) =0, y = Q2 (ft 8, m; P/2) = 0, (45)
where Q t and Q2 are power series in ft 5, and m, which vanish with # = 5 = n = 0.
The coordinates can, therefore, be developed as power series in ft 5, and y.
and the moduli of these parameters can be taken so small that the series
convergefor
\t\ <P,where P is the
periodfrom
ejectionto collision for /z
= 0.
230. Proof of the Existence of Closed Orbits of Ejection. The proofof the existence of closed orbits of ejection resolves itself into the demon-stration that equations (45) have solutions when /* is distinct from zero.
These equations are not satisfied by /x= unless and 5 are both also zero,
because, when n = 0, the problem reduces to that of two bodies in which the
period in r from ejection to greatest distance depends upon ft and in whichthe distance depends upon 8. Therefore equations (45) have one or moresolutions for /3 and 8 as power series in n, vanishing with n, according as the
functional determinant is distinct from zero or is zero for /3= 6 = /x
= 0.
Since the functional determinant involves derivatives only with respect
to /3 and 8, the n may be put equal to zero before forming it. Then the
determinant in question is
A =
dx dxda' d8
dy dyda' d8 0_Oo =
8 = 5=0
where x is the derivative of x with respect to r t. Before forming the elements
of the first column 5 may be put equal to zero, and before forming the
elements of the second column 0=a a o
maybe put equal to zero.
It follows from (15) that when 5 =/x= and t t 1= 2jw the value of y is
zero whatever a may be. Therefore dy/da is zero and the determinant
becomes simply A= (dx/da)(dy/d8).In the case under consideration the value of t t x for which A is formed is
t-t i= T
3 = T3
l (l + 8)3 =- P
^(l + 8)3
=j*(l + 8y. (46)
The second factor of A will be computed first. Upon putting a a = 8 = 0,
it is found from the second equation of (15) that
||=[-||sini7r(l+5)
3
-3i7r(l+5)^cosi7r(l+5)3
]^=(-l)^3i7rf^0.
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472 PERIODIC ORBITS.
Before computing the first factor of A the parameter 5 may be put equal to
zero. Hence it follows, from (46), (15), and (9), that
da da L 7 aT ^2i aT 4851 OT ^ J?rOfP/2
It was proved in 223 that the signs in this series alternate and that a is
negative for those orbits which lie entirely in the finite part of the plane.
Therefore dx/da is distinct from zero for all values of a under consideration.
It follows from this discussion that A is distinct from zero for
/3=o-a=5=0, t=jtt, and consequently that the sufficient conditions for the
existence of closed orbits of ejection can be uniquely satisfied for |/x|suffi-
ciently small. There is a closed orbit of the type in question for ejection
in both the positive and the negative direction for all integral values of j
upon which the , or a , of (16) depends.In the special case in which the finite masses are equal, a closed orbit
of ejection for j = 2, with ejection in the positive direction,* was discovered
from numerical experiments by Burrau in two interesting memoirs. f Since
in his problem y. had the large value 0.5, it is not to be expected that the
results of this analysis would agree very closely with the results of his
computations. Hence the comparison will be made only for the constant
of the Jacobian integral. Upon taking into account the difference in his
units and those employed here, it is found that his Jacobian constant CB ,
equation (5) loc. cit., is related to C of (29) by the equation
_2C fl= C=-fca+ M(2+M) = -^F^+M(2+M).
So
Burrau's computation gave-2C B =2.2528; and for /z= 0.5, t?i?=2V2(l-n)
it is found that C = 2.38, and the agreement is fully as close as would be
expected. It follows from these numbers that a larger value of the con-
stant a, corresponding to a smaller value of, belongs to the undisturbed
orbit having the period 2ir than to that computed by Burrau. In the undis-
turbed orbit the greatest distance to which the infinitesimal body recedes is,
by (16), 2; it has this value at t t x= t, and it is then on the negative
half of the z-axis. The greatest distance found by Burrau in his computationwas 1.9972, or a little less than that in the undisturbed motion.
If the infinitesimal body is ejected toward or from the body n with a
small value of| |,
it will be disturbed so that on its return it will revolve
around 1 /* in the positive direction. This can be seen when the motion
is considered in fixed axes, for under the conditions postulated the disturb-
ance is positive all the time that the infinitesimal body is going out and
returning. If it is ejected farther, it will be accelerated by n in the negative
Burrau's orbit of ejection was from the body called m here, but permuting 1 n and p and changingthe positive directions of the axes, the statements are correct.
fRecherches numcriques concernant des solutions periodiques d'un cas special du probleme destrois corps, Astronomische Nachrichlen, vol. 135 (1894), No. 3230; and ibid, vol. 136 (1894), No. 3251.
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CLOSED ORBITS OF EJECTION AND RELATED PERIODIC ORBITS. 473
direction part of the time. While in general the body will not collide with
l-*/i on its return, it may possibly do so under special conditions. Indeed,Sir George Darwin has discovered one such orbit by numerical experiment*
having the period ir. The ejection was from /x in the direction of 1 n, andthe body collided with /* going in the same direction. f This orbit is one of a
pair which together are the limit of certain periodic orbits, though they are
not periodic themselves, either physically or mathematically. The constant
C belonging to this orbit in the units employed here is 20/11 = 1.818. Thevalues of the masses used by Darwin were 1 ju= 10/11, /x
= 1/11. It follows
from (16) that =2 l/3 for this period, and from (30) that C= 1.716. In this
case the belonging to the undisturbed orbit is larger than that belonging to
Darwin's orbit. The value of is 2 1/3 = 1 .26; the greatest distance in Darwin's
orbit, according to his diagram, is 1.3.
231. Conditions at an Arbitrary Point for an Orbit of Ejection. Since
the motion of the infinitesimal body is regular for all finite values of t andall finite values of the coordinates except those for which it collides withone of the finite masses, { it becomes a matter of interest to determine in
any special case whether the trajectory is one of ejection or collision for a
finite value of t. It is sufficient, as Painleve conjectured and as Levi-Civita
proved,! that the coordinates and velocities shall satisfy one analytic con-
dition in order that the orbit shall pass through one of the finite masses for
a finite value of t. This conclusion will be established here in a different way.Suppose n is zero and consider the problem of defining the initial con-
ditions for an orbit of ejection so that it shall pass through the point in
question, and so that the components of velocity at the point shall satisfy as
many conditions as possible. The velocity in rotating axes at any distance
from the finite mass 1 n is the resultant of the velocity with respect to
fixed axes and that due to the rotation of the axes. The velocity with
respect to fixed axes at any finite distance can be made any finite quantity
by a suitable determination of the constant a, or the equivalent constant .
Consequently, an arbitrary speed, or one of the components of velocity,
with respect to rotating axes at any distance can be secured.
Suppose the speed at a given distance has been assigned; then it is
possible to determine the initial direction of ejection so that the orbit of
the body will pass through any point having the given distance, for it is
possible to do it in fixed axes and the rotation simply changes the direction
of ejection by an angle which is proportional to the time required for the
body to reach the distance in question. It is clear from this that when
jt= the conditions of ejection can be so determined that the infinitesimal
*On certain families of periodic orbits, Monthly Notices of the Royal Astronomical Society, vol. 70
(1909), p. 134.
fSee further remarks on this orbit at the close of 234.
tPainlev6, Lecons sur la Theorie Analytique des Equations Difffrentielles, p. 583.
Acta Malhematica, vol. 30 (1906), pp. 306-327.
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474 PERIODIC ORBITS.
body shall pass through any assigned point with any assigned speed. Of
the four quantities required to define an orbit, viz., two coordinates and the
speed and direction of motion, three can be taken arbitrarily and the fourth
is determined by the condition that the orbit shall be one of ejection. Thedetermination of the fourth quantity is double because the body has the
same speed twice, once when it is receding from 1 n, and once when it is
returning toward 1fi.
Suppose that, for m= 0, an orbit of ejection passes through the point
x T , y T with the speed v T = Vx't+v't at (tI/3 = T. If
'
r represents the speedwith respect to fixed axes, then, since the component of velocity due to the
rotation of the axes equals numerically the distance of the point from the
origin, the relation between v T and'
T is
4-1+& (47)
Equation (2) determines ,the greatest distance to which the body recedes,
and (13) gives the constant a,,. Equation (4) gives the value of T, and
the direction of ejection is T degrees in the negative direction from the line
joining 1 n and the point (x T , y T ). Let the angle of ejection be O .
Now suppose that n is distinct from zero, but small. Let the initial
values of a and 6 be a +/3 and o +7- Let a new independent variable t x
and a parameter 8 be introduced by (43). Then the solution can be written
in the form
z=Pi(/S, y, s, m r,), y=p,(P, y, s, m; n),
^ = Pi(0>7, 5, m; rj, ^=p 4 (/3, t, 5, m; t,),
where p, , . . ., p 4 are power series in /3, y, 8, and m- The moduli of these
parameters can be taken so small that the series converge for <! r t ^ T.
The conditions that the body shall pass through the point (x T , y T ) with
the velocity v T at Tj= T are
Pl (l3,y,8,n;T)-x r = 0, p 3 (/3, y, 5, M; T)-y r= 0, V^+f-v T
= Q. (48)
Since these equations are satisfied by = y = 8 = n=O, they can be written
as power series in /3, y, 8, and fi, vanishing with j3 = y = 8 = n = 0, of the formP t (A%l,ff T) =
0, P 2 (0,y,8,;T) = O, P 3 (/3, 7 ,5,m; T) = 0. (49)
Equations (49) are not satisfied by /x= unless also 13
= y = 8 = 0. There-
fore they have solutions for j8, y, and 8 in terms of n which vanish for n = 0.
If the determinant of the linear terms in 0, y, and 8 is distinct from zero
the solution is unique. In treating the problem it is convenient to use
equations derived from (49) rather than these equations themselves. Let
<p represent the angle between the positive end of the x-axis and the line
from the origin to the point (x T ,y T ). Then let Ql; Qiy and Q, be defined by
Q^P.cos^+Pjsin^, Q,= -P,sin(p+P t cos^, Q, = P 3- (50)
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CLOSED ORBITS OF EJECTION AND RELATED PERIODIC ORBITS. 475
This transformation is equivalent to rotating the axes so that (x T , y T ) lies
on the positive half of the new z-axis. The solutions of
Q1 (0,y,S, (x;T)=O, Qt (J3,y, B tft ;T) = 0, Q,(0, y, 5, M ; T) = (51)
are identical with those of (49), for the two sets of functions are linearly
related with non-vanishing determinant.
The determinant of the terms of the Q t which are linear in 0, y, and b is
A =
3Q.
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476 PERIODIC OIIBITS.
Now suppose 7 = n = 0. Then <p= 9 T 3 and Q, becomes
Q, = (o,+0, 5; n-(a,+0, 0; 7"). (55)
Therefore dQ/d/3 = d/d/3, dQ,/d5 = d/dS, the first of which is positive bythe properties of which were derived in 223. The second one of these
partial derivatives is positive or negative according as, for n = (i= 8 = 0, the
infinitesimal body is receding from, or approaching toward, the origin at
t <,= T 3. If has its greatest value for tt l =T 3
,then dQJdb, which is the
derivative of with respect to t, is zero.
It follows from (47), (48), and (50) that, for y = m= 0,
Q, = Vl+K /
(o + i8,;r)], -Vl+K'(o +0,0; T)Y (56)
Therefore the partial derivatives of Q3 which appear in (52) are
aS'(q.+iB > 0;!r) dt'(a +0,8;T)dQ 3 = dp dQ 3 _ as (57)
dp"
Vl+te'K+O, 0; T)Y' dS Vl + te'K+0/0; T)f
From equation (9) it is found that
S'= 2c(l + 5)V 1 [(l+2a(l + 5)M-fa
2
(l + 5)V;+ ].
where the signs of the coefficients alternate. Therefore the partial deriva-
tives in question are
||= +4rJ- f a rj+ '
, ff -4^+100.1?-^ 0^+ , (58)
Since a is negative, the first of these expressions is positive; since the
velocity decreases or increases with increasing time (according as the infini-
tesimal body is receding from or approaching toward the origin) the second
of these expressions is negative if the body is receding from, and positive
if it is approaching toward, the origin. If'
is zero for t tl =T 3
,is an
even function of 5 because the motion is symmetrical with respect to T 3as
initial time. In this case the second of (58) is zero. Therefore if theinfinitesimal body for n = P = 5 = is not at its greatest distance at t t^T 3
,
the form of A is
++ +
A = (59)
where the upper sign is to be used if it is receding from, and the lower if it
is approaching toward, the origin. Since dQ t /dy is distinct from zero, Ais distinct from zero. Therefore in this case equations (48) are uniquelysolvable for p, y, and 5 as power series in n, vanishing with n. This means
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CLOSED ORBITS OF EJECTION AND RELATED PERIODIC ORBITS. 477
that if an arbitrary point in the xy-plane, and a velocity greater than thatof this point with respect to fixed axes, be selected, then there exist twoorbits of ejection passing through this point such that, for (x sufficiently
small, the infinitesimal body will pass the point with the given velocity.The direction with which the body passes the point depends upon the initial
conditions, of which it is a regular analytic function. This is Painleve's
theorem for the restricted problem of three bodies.
If the velocity chosen equals that of the arbitrary point with respectto fixed axes, so that, for n = 0, the point in question is at the greatest dis-
tance to which the infinitesimal body recedes, then the determinant A is
zero and the solution is multiple. The reason for it is that the two solu-
tions which were distinct in the other case have united, and the solution
has become double.
232. Closed Orbits of Ejection for Large Values of li. It was provedin 230 that closed orbits of ejection exist provided |m| is sufficiently small.
The question of their existence for large values of li will now be considered.
If t= ^(l-r-5), the solutions of (23) may be written
x=f 1 (a +P, 5, M5 r,), 2/=/ 2 (a 3 +/3 5, M ; rj, (60)
and the conditions for a closed orbit of ejection with the period 2t in r, are
^-fi(%+fi, *,l;*)-0, *()*/(+&*, *)-0. (61)
It has been shown that, for |pj sufficiently small, equations (61) can be solved
for /S and 5 as converging power series in ll, vanishing with li, and that whenthese results are substituted in (60) the latter become power series in nwhich converge for 0^t^7t provided \p\
is sufficiently small.
Suppose the series for the closed orbit converge for li = p, ,and that the
values of a o +0 and 5 for this value of m are c, and 5 t . The solutions of (23)
are expansible as power series in t for values of a, 8, and n in the vicinity
of a,, 5,, and ft- If |o o t |<r 1; \8 8i\<r 3 |/x /*,|<r, the series converge if
Tt <T,
where T is
any arbitrary quantityless than the
periodfrom
ejectionto collision. The result will have the form of (27) where r is replaced byT t (l + 5). Each term of (27) can be expanded as a power series in a a lt
8 Si , m Mi which will converge provided \p /*, | < 1 p, . The solution
may be written in the form
x- ^(a-a,, 5-6j, /x-MiJ n)* 2/= Ps(a-a 1 ,
5 5 X , /x~Mi ; r,), (62)
where y x and p 2 are power series in a ,, 5 ^, and li m^. Suppose the
values of x, y, and their derivatives at t x =T for a c^ = 5 = 5 1=
/x_
Mi =are x T , y T ,
x' T ,and y'r . Let their values for an arbitrary set of values of
a aj , 8 5i , and m Mi > satisfying the inequalities which insure convergency ,
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478 PERIODIC ORBITS.
be x T + 0, , y T + 0, , x' r + 3 , and 2/^ 4- /3 4. Then 0, ,
. . . , 4 are expansibleas power series in a a, ,
5 5, , m Mi of the form
A-(-a r *- ; 4iJ'-J%;^ (<-l, 4), (63)
where the g, vanish f or a a!= 5 5,
= m Mi = 0.
Now consider a solution with the initial conditions x T + 0, , y r +P t ,
z'r+A, y'r+ft- Suppose for 0, = 2=
3=
4= O the infinitesimal body does
not pass through the position of n for I5t 1 ^t. Therefore the solutions
can be expanded as power series in 0, ,. . .
, 4 which will converge for
!r^Tj(l + 6)^7r provided the moduli of 0, ,. . .
, 4 are sufficiently small.
At t,= 7t the expressions for the coordinates become, making use of (63),
Z = -Pi(0i, . . . ,0 4 ; r) = Qi(a-a lt 5-5,,m-Mi),
y = F 2 (0i, . . . ,0 4 ; 7r) = Q2 (a-a, , 5-5,, n~Mi),
x' = P,(Pi, ... ,04 ; -)= Q,(o-o,, 6-5,, m-Mi),
y' = P*(Pi, ,0i! 7r)=Q 4 (a-a x , 5-5,, /x-Mi).
(64)
Conditions that the orbit shall be closed with the period 27r/(l + 5) are
Q2 (a-a,, 5-5,, m-Mi)=0, Q3 (a-a,, 5-5, , /x-/i,) = 0. (65)
It has been seen that in general the solution of these equations for a a,
and 5 5, in terms of n /x, , vanishing with n /*, ,is unique. This is always
true unless the solution becomes multiple. If the multiplicity is odd,there is one real solution for both positive and negative values of n p,,
There are three solutions altogether for \n\ sufficiently small because ,
defined in (16), has three values for which the conditions of a closed orbit
of ejection can be satisfied, but only one of them is real. Consequentlythe real solution can not disappear by uniting with one of the others unless
they first unite and become real. Then, if two of the real solutions should
unite and become complex, there would be one real one left. That is, there
is one real closed orbit of ejection from 1 m for all values of m from to 1,
excluding the value unity. The argument has been made for the period
2ir, but it is entirely similar for any multiple of 2ir.
It has been tacitly assumed in the argument that none of the orbits
of ejection under consideration passes through the position of n for anyvalue of n; for it was only under this condition that the convergence of
the series was assured. It has been proved that the closed orbits of ejection
do not pass through m for n sufficiently small. Since the coordinates in
these orbits are regular analytic functions of n, it follows that if any one of
them passes through the position of m for any value of n, then for values
near this one it will pass near n. The motion of the infinitesimal body in
the vicinity of a finite body when referred to rotating axes is always in the
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CLOSED ORBITS OF EJECTION AND RELATED PERIODIC ORBITS. 479
retrograde direction, and the orbits in question are always symmetrical with
respect to the x-axis.
Consider the motion with respect to it in a closed orbit of ejection from
1 fx. Whether the ejection is toward or from it the motion with respect to it
in those parts of the orbit which are near to it is direct instead of retrograde.
Therefore, the orbit can not pass near it without first developing folds so
that a line from it in certain directions will intersect it three times. It
is extremely improbable, though not absolutely certain, that this sort of
development could take place. It is probable that the real orbits of ejection
which exist for it sufficiently small continue in the analytic sense for all
values of it from to unity, and that they do not pass near it.
For it = the orbits in which the ejection is toward it are symmetrically
opposite to those in which the ejection is away from it, and these conditions
\
t
l-JJ'C. v-l-l
S-l
/
Fig. 18. Fig. 19.
are approximately realized when it is small. When it increases, the loopswhich surround 1 it diminish in size and preserve their approximate forms,while those which surround it approach circles whose radii are f'
3, where
2jir is the period, and whose centers are at it. For it 1 e, where e is verysmall, the orbits have the form shown in Figs. 18 and 19. There are, of
course, closed orbits of ejection from both of the finite bodies.
233. Periodic Orbits Related to Closed Orbits of Ejection. There are
periodic orbits passing near one of the finite bodies of which the closed
orbits of ejection are the limits. Suppose it = and consider the motion of
the infinitesimal body with respect to the finite body 1 it. Let the infini-
tesimal body cross the x-axis perpendicularly at < = near the body 1 it,
and let the initial velocity be determined so that the period is 2w, or a mul-
tiple of 2t. Then the motion with respect to the rotating axes is periodic,
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ISO PERIODIC OKBITS.
the orbit is symmetrical with respect to the rc-axis, and crosses it perpen-
dicularly at the half period. The limits of these orbits, as the nearest
approach to 1 m becomes zero, are the closed orbits of ejection.
The orbits under consideration exist for initial motion near the finite
body in both the positive and the retrograde directions, but in both cases
the motion in the remote parts of the orbits when referred to rotating axes
is in the retrograde direction. Therefore, those in which the motion in the
vicinity of the finite body is direct have loops, while the others do not. Thecharacter of the two classes of orbits for periods 2x and 4?r are shown in Figs.
20 and 21. There are, of course, orbits of a similar character which are sym-metrically opposite with respect to the y-axis.
Suppose the initial conditions for one of the periodic orbits in questionwhen n = 0, are x(0) =
a, z'(0)=0, y(0) = Q ! y'(0) =b, and represent the
coordinates for this solutionby
x,
x',
y, and
y'. The distance a is small
and the orbit does not pass through the point (1, 0) occupied by ju-
Fig. 20. Fio. 21.
Now suppose fx is distinct from zero and that the initial conditions are
x(0) = a+a, x'(0) = 0, 2/(0) = 0, t/'(0)= 6+|8. Let the variable t be intro-
duced by t = r(l + 8), Where 8 is an undetermined parameter. Then the
solutions of the differential equations of motion, which are regular functions
of the coordinates and 8 in the vicinity of x = x,x' = x
, y = y , y' = y[, 5 =for ^ t ^ jit (j an integer), can be written in the form
x = x + =a; u (T)4-p 1 (a,0, 5,m; r), y = y +V =y {r)-\-p 3 (a, ^, 8, fi; t),)
x' = x o +? = x Q (T)+p 2 (a,0, 5, M ; t), y'=
y'<,+r)'= y'o(T)+p i (a, (3, 8,n; t),\
where p t ,. . .
, p 4 are power series in a, P, and 8, vanishing with a, /3, and 8.
Moreover, the moduli of o, 0, and 5 can be taken so small that the series
converge for ^ r ^jV, where,;'
is any integer.
Sufficient conditions that the orbit for /x distinct from zero shall be
periodic with the period 2jir are
p s (o, 0, 8, M ; jV)=
0, p,(a, p,8,; jir)= 0.
(67)
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CLOSED ORBITS OF EJECTION AND RELATED PERIODIC ORBITS. 481
The problem is to show that a, /3, and 8 can be determined so that these
equations shall be satisfied for an arbitrary /x sufficiently small. In the
problem for m = 0, either a or 6 can be taken arbitrarily when the other is
determined in terms of j except for sign ;one sign belongs to the direct and
the other to the retrograde orbit. It will be supposed that a is the arbitrary.Therefore it may be supposed that it absorbs the undetermined a and leaves
only two parameters in (67) besides the arbitrary /x.
Equations (67) can be solved uniquely for /3 and 8 as power series in n,
vanishing with /x, provided
A =
dp? djpi
3/T ~d8
djh djh
dp' d8
(68)
/3-5 = M=
is distinct from zero. Before this determinant is formed ju can be put equalto zero, and therefore A depends only upon the two-body problem. Before
the second column is formed /8 can be put equal to zero. When n = 3 =the period in t is 2jv; hence at the half period the infinitesimal body is
on the x-axis and the value of x is an even function of tjw. Now the
parameter 8 serves only to vary the period in t (keeping it fixed in t) ,and is
therefore equivalent to varying t from the half period. When t is near jw,
8 can be determined so that tjir=JT(l+8)jir consistently with the
definition of r. Therefore p 2 (0, 0, 8, /*; jir) is an even function of 5, and
consequently dp 2/ 68 = Ofor/3 = 5 = /x= 0. Hence the determinant 5 becomes
A = ^^i.(69)
The second factor is distinct from zero because, except for a constant
factor, it is the derivative of y with respect to t at t = jw, and this derivative
is distinct from zero.
In considering the first factor of (69) the parameter 8 can be put equalto zero. If and r\ represent the coordinates referred to fixed axes, the
expression for x becomes
x = % cos^ + j; sinf.
Therefore the value of x' at t =jir is
The problem is reduced to finding whether or not the expression
r^Mjl + dl(70 )v ;
a/3 d/3^d/3{<U)
is zero under the assumed conditions.
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482 PERIODIC ORBITS.
The expressions for , ', and rj, as given by the two-body problem, are
dE wdsmE= a[cos#-e], ? = ~ asinE
-dt= ~
1-ecosE' V = ^l-e i smE,
wliere E is the eccentric anomaly, a is the major semi-axis of the orbit, and
w is the mean angular motion in the orbit. Since sin' = and cosE= 1
for t =jw and /3= 0, it follows that
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CLOSED ORBITS OF EJECTION AND RELATED PERIODIC ORBITS. 483
234. Periodic Orbits having Many Near Approaches. -The orbits con-
sidered in the preceding article are characterized by the fact that, at least
for small values of ll, after the infinitesimal body leaves the point nearest
1 fj. itcontinually
recedes until themid-period,
which is amultiple
of ir,
and then returns symmetrically. Orbits will now be considered in which
the infinitesimal body recedes from and returns toward ix many times
before they re-enter.
Suppose fx is zero and that the infinitesimal body is started near 1 ll
on, and perpendicularly to, the line joining 1 m and ll; and suppose the
initial conditions are such that its period is commensurable with 2ir without
being a multiple of 2x. Let the period be
P = 2ttP
(72)
where p and q are relatively prime integers. Then the motion with respect
to the rotating axes is periodic with the period
T = Pq =2irp. (73)
In this period the infinitesimal body runs through q of its periods with
respect to fixed axes and the movable axes make p rotations.
Now suppose that ll is distinct from zero and that the initial conditions
are given slight variations, but of such a character that the infinitesimal
body is started at right angles to the line joining the finite bodies. A newindependent variable t and a parameter 5 are
introduced by the relation t = r(l+5). Thesolutions can be developed as power series in
ix, 8, and the increments to the initial condi-
tions, and the moduli of these quantities can
be taken so small that these series convergefor t<T/2. Then the conditions
that the solution shall be periodic
are that the orbit shall cross the
x-axisperpendicularly at r
=T/2. These condi-
tions have the form (67) and all of the propertiesof (67) which were used in proving the existence
and character of their solut ion. Therefore, the
periodic orbits which are in question exist.
Now suppose that the initial distance from1 fx, which was arbitrary, is made to approach
zero as a limit. During this approach to zero the distances to the other
near apses vary, but there is no apparent reason why all these apsidal dis-
tances should vanish at the same time. In fact, from the lack of symmetryit is doubtful whether
anytwo of them arc
simultaneouslyzero.
Jig. 22.
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IS 4 PERIODIC ORBITS.
The simplest orbits of the type under consideration are those for which
p=l, q = 2. Their general form for retrograde motion in the vicinity of
the finite body 1 // is shown in Fig. 22. If the distance from 1 n to a
becomes zero, the orbit of ejection discovered by Darwin is obtained. The
question whether the distance from 1 m to b becomes zero is one that is
hard to answer. Certainly it can not be answered affirmatively with com-
plete rigor by numerical experiments, though the existence of certain classes
of periodic orbits can be proved in this way. If, for perpendicular projectionfrom a given point on the z-axis with a certain speed, the next crossing of
the x-axis is at an angle which is greater than ir/2; and if, for a perpendicular
projection from the same point with a different speed, the next crossing is
at an angle less than ir/2, then, from the analytic continuity, it can beinferred that there is an intermediate speed at which the crossing will be
perpendicular. But in the present case these conditions are not present, andall that can be said is that when the distance from 1 n to a vanishes, the
distance from 1 fi to b is small, and the approach to 1 n is almost exactly
along the z-axis. This is, of course, to be expected from the nature of
these orbits.
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^ Chapter XVI.
SYNTHESIS OF PERIODIC ORBITS IN THE RESTRICTEDPROBLEM OF THREE BODIES
235. Statement of Problem. In the problem of two bodies there are
circular orbits whose dimensions range from infinitely great to infinitely
small. They form a continuous series geometrically and their coordinates
are continuous functions of the various parameters by which they may be
defined. There are orbits in which the direction of motion is forward, andothers in which it is retrograde. The two series are identical only when the
orbits are infinitely great and when they are infinitely small.
In the restricted problem of three bodies* the orbits which are analogousto the circular orbits in the problem of two bodies are those which revolve
around one or both of the finite bodies and which re-enter, when referred to
rotating axes, after one synodical revolution. Those inclosing but a single
finite body were treated in Chapter XII, and it was shown there that the
deviations from uniform circular motion are due to the attraction of the
second finite mass. Those orbits which revolve around both finite masses,
and which are analogous to circular orbits, were treated in Chapter XIII,and it was shown there that the deviations from uniform circular motion
are due to the fact that the finite masses are separated by a finite distance.
The problem of the present chapter is to trace, so far as possible, a con-tinuous series of orbits from those inclosing both finite masses and having
infinitely great dimensions to those revolving around the two finite masses
separately in orbits of infinitesimal dimensions. There are no difficulties
for very great or for very small orbits, but since in some way the infinitely
great are eventually divided into two series which become infinitely small,
it is clear that there is a region in which the resemblance to the two-body
problem is very remote. The difficulties arise in following the orbits throughthese critical forms.
The method of treatment is that of analytic continuation of the solu-
tions with respect to the parameters upon which they depend. The differ-
ential equations which define the motion are, when referred to rotating axes
and in canonical units,
d 2 x 9 dy_ (I -aQQe+m) (x-l+it)dt 2 dt n s M
r 23 '
dt 2+Z dt V u MV Vr? = {x+n)-+y\ r 2
2 =(x-l+ M)2
+2/2
-
(1)
* The restricted problem of three bodies is that in which there are two finite bodies and one infinitesimal,
the finite bodies revolving in circles.485
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486 PERIODIC ORBITS.
These equations admit Jacobi's integral
(fKi) w+! ' ,+2 -%r }+ |- c - (2)
The generalsolutions of
(1)can be written in the form
*-/i(i, . . .,a 4 , m; t), y=f 2 (oi, . . .
,a 4 , m; 0, (3)
where ai, . . .,a 4 are the initial values of x, y and their first derivatives.
The conditions that the solutions (3) shall be periodic eliminate three of the
four a,, and the periodic solutions have the form
x = F, (a, m; rd), y = F 2 (a, M ; nt), (4)
where a is one of the a,, or a function of them (for example, the Jacobian
constant C), and n is a constant depending on a and n and so associated with
i that the period is 2ir/n. The problem under consideration is to follow
the solutions as a, n, or n varies through its possible range of values. It will
be found convenient in the discussion to use sometimes one and sometimes
another of these parameters as independent. The correspondence between
them is not one-to-one, so that a series of orbits may branch at a certain
point with respect to one of them and not with respect to another. Thefunctions in question are highly transcendental and it has not been found
possible to follow them with complete rigor through branch-points andinfinities by direct processes. But the fact that the orbits may have a branch-
ing with respect to one parameter and not with respect to another makes it
possible sometimes to establish the existence of critical forms indirectly.
There are orbits whose coordinates are complex and whose periods are
real. With varying values of the parameter a they may become real. This
does not arise in the problem of two bodies. It makes it necessary to include
in the discussion certain orbits which are complex for the values of the
parameters in terms of which they are most conveniently expressed.Sir George Darwin discovered several classes of periodic orbits by numer-
ical experiments.* Burrau found the limiting form of the orbits of one
family of oscillating satellites by the same process. f Many other families
have been discovered by computations carried out in connection with this
work. All these examples illustrate the theories to be set forth here and
place them on a solid basis at points where they fall short of complete
rigor because of the difficulty of following orbits by analytical processes
through critical forms.
Before taking up the direct synthesis, some of the properties of the
periodic orbits which have been previously given will be enumerated andsome additional ones will be derived.
236. Periodic Satellite and Planetary Orbits. It was shown in ChaptersXII and XIII that there are periodic orbits encircling each of the finite bodies
separately, and others encircling both of them together, which are closed
* Acta Mathematica, vol. 21 (1897).t Astronomische Nachrichtcn, vol. 186 (1894), No. 3230; and
ML,vol. 136 (1894), No. 8251.
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SYNTHESIS OF PERIODIC ORBITS. 487
after one synodical revolution. The direction of motion of the finite bodies
is considered as being forward, and the opposite is considered as being retro-
grade. For small values of the parameter in terms of which the solutions
are developed, corresponding to very small and very large orbits respectively,there are three classes in which the motion is direct and three in which it
is retrograde. Only one class of the direct and one of the retrograde orbits
is real for small values of the parameters. Darwin's computations showthat at least in some cases the complex orbits may become real.
One of the most important properties of the orbits in question is that
they all cross the line joining the finite bodies perpendicularly at every half
period. If the parameters upon which the periodic orbits depend are varied,
these properties persist, for the solutions are expansible in integral or frac-
tional powers of the parameters, and the property in question is a consequence
of the character of their coefficients. Therefore, the whole series of orbits
from infinite to infinitesimal dimensions will possess this property, unless
indeed they branch into two series which are symmetrical with respect to
the line joining the finite bodies.
If the coordinates of a real periodic orbit are analytic in a parameter,then the orbit can not disappear without becoming identical with another
periodic orbit;* and a complex orbit can not become real without becomingidentical with another complex orbit, the two becoming real as they becomeidentical. Real orbits disappear and appear in pairs with the variation of
the parameters in terms of which their coordinates are analytic functions.
237. The Non-Existence of Isolated Periodic Orbits. Appeal will be
made to the numerical computations in establishing the existence of periodic
orbits near certain critical forms. In order that this procedure may be
justified, it is necessary to prove that the orbits which they have shown to
exist are not isolated examples which exist only for special values of the
masses and the other parameter on which they depend.
Suppose that for ju= Mo equations (1) admit the periodic solution
x = Fi (no; not), y = F 2 ( l x ; riot), (5)
where the period is 2w/no. The initial conditions are
x(0)=x , x'(0)=x' , 2/(0) =2/0, y'(0)=y' . (6)
The initial conditions determine the value of the constant of the Jacobian
integral
ft W+yV)+a*+f* ^ly^ +lft ( 7 )"l '2
The orbit in question will not pass through one of the finite bodies, for then
it would not be strictly periodic, as was explained in 222. It will not have
any infinite branches, for then it could not have a finite period.
*
Let M&hodcs Nouvelles dela
M6canique C61este,vol.
I, p. 83.
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(9)
INN PERIODIC ORBITS.
There are two problems to be considered: (A) To determine whether
or not periodic orbits exist for /x= mo+X, C = C , which reduce to (5) for X = 0;
and (B) to determine whether or not periodic orbits exist for /u= Mo, C = C +y,
which reduce to (5) for 7 = 0. If periodic orbits exist when the parameters
are varied separately, they also exist when they are varied simultaneously.
(A) Let n = no+\ and take as initial conditions
x(0)=x + a, x'(0)=x' + a', y(0)=y +P, y'(0) =y'o+P'. (8)
Since the periodic orbit (5) does not pass through a singular point of the
differential equations, the solutions can be developed as series of the form
x =Fi (moJ WoO + a +pi(a, a', P, P' , X; 0,
x' = *YGuo; lht) + a'+p 2 (a, a', p, j8', X; t),
y = F 2 ( Mo ;n t)+p +p 3 (a, a', j8, P', X; t),
^ = F 2 '( Mo ; Wo0+/3' +P4(a, a', 0, P', X; 0,
where p t . . ., p 4 are power series in a, a', p, P' ,
and X which vanish identi-
cally with these parameters and are zero for t = 0. Moreover, if any finite Tis taken in advance the moduli of these parameters can be taken so small
that p\, . . ., Pi converge for all 0<*^7\
The integral (2) can be written
F(p u . . ., Pi , a, a',P, P', \)-F(0, . . .
, 0, a, *', 0, #, X)-0, (10)
where F is a power series in p u . . ., p A , a, a', P, P', and X. Equation (10) is
identically satisfied by p x=
. . .= p t
= 0.
Sufficient conditions that the solution (9) shall be periodic with the
period P = 2r/n are
p,(a, a', p, P', X; P) =0, p 3 (a, a', p, P', X; P) =0,'
p 2 (a, a', /3, 18', X; P) =0, 7^(a, a', /S, P', X; P) =0.J(ID
It follows from the form of (2) that unless x 'is zero, equation (10) can
be solved for p 2 as a power series in p lf p 3 , p 4 , a, a', p, p', and X, vanishing
identically for pi = p 3= p 4
= 0. If x ' were zero and y' were not zero the
solution would be made for p 4 instead of for p 2 . If x' and y
' were both zero
the origin of tune would be shifted so that at least one of them would be dis-
tinct from zero. This is always possible unless they are identically zero.
But they are identically zero only when the infinitesimal body is at an equi-
librium point, and it has been shown in Chapters V and IX that these
solutions are not isolated. Consequently, it may be supposed that (10) is
solved for p 4 in the form
p< = <p(pu p 2 , p.J a, a', p, j8', X), *>(0, 0, 0; a, a', j8, P', X)0. (12)
Therefore, if p lf p 2 ,and p 3 are periodic with the period P, then by virtue of
this relation p 4 is also periodic with the period P. Hence the fourth equa-tion of (11) is redundant and may be suppressed.
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SYNTHESIS OF PERIODIC ORBITS. 489
Now consider the solution of the first three equations of (11) for a, a',
and 8 in terms of X. The parameter 8' is superfluous and may be taken equalto zero. This amounts to a definite determination of the initial time. Since
(5) is a real periodic solution the coefficients of pi, p 2 , and p 3 are real. Equa-tions (11) are not satisfied by X = with a, a', and 8 arbitrary, for then all
orbits would be periodic with the same period when n = /jl. Similarly, they
are not satisfied by X = unless all three of the parameters a, a', and 8 are
zero. They are not satisfied by a = a' = 8 = and X arbitrary, for then fixed
initial conditions would give a periodic orbit for all distribution of massbetween the finite bodies. This is certainly not true for fi near zero. There-
fore, the three equations can be solved for a, a', and 8 as power series in
integral or fractional powers of X. If the powers are integral the solution
will be unique and real for both positive and negative values of X. If the
powers are odd fractional, there will be a single real solution for both positiveand negative values of X. If the powers are even fractional, there will be
two real solutions for X either positive or negative, depending on the signsof the coefficients of certain terms, and all the solutions will be complex for
X negative or positive, depending upon the signs of the same coefficients.
Hence in all cases there are real periodic solutions for small values of X
which reduce to (5) for X = 0.
(B) Let C = C +y and take the initial conditions (8). Also let
-(1+8)t, (13)
where 8 is an arbitrary parameter and t a new independent variable. Thesolutions in this case have the form
X =Fi (moJ n r) + a +pi(o, a', 8, 8', y, 8; r),
x' = Fi'(fi ; noT)+a'+p 2 (a, a', 8, 8', y, 8; t),
y =F 2 (n ; rhT)+8 +p 3 (a, a', 8, 8', y, 8; t),
2/'= F 2 '0uo; n r)+/3'+p 4 (a, a', 8, 8', y, 8; r),
where p u . . ., p 4 are power series in a, a', 8, 8', y, and 8. The parameter y
is determined in terms of a, a', 8, 8', and 8 by equation (2) at t = 0. There-
fore it may be omitted from p u . . . ,p A . Or, if y'
is not zero, equation (2)
determines 8' as a power series in a, a', 8, y, and 5, vanishing with these
quantities. It will be supposed that 8' is eliminated. It follows from theintegral that the last equation corresponding to (11) is redundant, and it will
be suppressed. The constant a' is superfluous and may be taken equal to
zero. Then the conditions that the solution (14) shall be periodic with the
period P in r are
Pl (a, 8, y, 8; P)=0, p 2 (a, 8, y, 8; P)=0, p 3 (a, 8, y, 8; P) = 0. (15)
Consider the solution of equations (15) for a, 8, and 8 in terms of y.
They are not satisfied unless all four of these parameters are zero, and con-
sequently solutions for a, 8, and 5 as power series in integral or fractional
powers of y exist, and the circumstances under which they are real are strictly
analogous to those of Case (A).
(14)
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490 PERIODIC ORBITS.
238. The Persistence of Double Orbits with Changing Mass-Ratio of
the Finite Bodies. In Chapter XI, p. 359, it was stated that Darwin's com-
putations show that the two orbits which are complex for small values of
///, or for large values of the Jacobian constant C, unite and become real for
a certain value of C. In this computation the ratio of the masses of the finite
bodies was 10 to 1. The question naturally arises whether a corresponding
double periodic orbit exists for other rat ios of the finite masses.
The conditions for a double periodic orbit will first be developed.
Suppose
x-fito, x'=h'{t), y=f 2 (t), y'=V(t), (16)
is a periodic solution of equations (1) for m Mo, having the period P. All
orbits except those around the equilateral triangular points, which will be
given special consideration in 242, are symmetrical with respect to the
x-axis. In them the origin of time can be so chosen that the initial con-
ditions are
x(0)=x , x'(0)=0, 2/(0) =0, y\0) =/, (17)
and from (2), C = C .
Now consider a solution with the initial conditions
x(0)=x + a, x'(0)=0, y(0)=0, y'(0)=|/'o+/9, C = C +y. (18)
The constant can be expressed in terms of a and y by means of (2), andit will be supposed that |8 is eliminated by this relation. Let a new inde-
pendent variable t be defined byt = (l+8)r, (19)
where 5 is a parameter as yet undetermined. Then the solution of (1) can
be developed in the form
*-Ji (, V, I; t), x' = p 2 (a, y, 8] t), ,_y = p 3 (a, y, 8; r), J/'
= p 4 (a, y, 8; r),
where p u . . ., p 4 are power series in a, y, and 8.
The conditions that (20) shall be a periodic solution with the period Pin t are
p 2 (a, y, 5; P/2)=0, p 3 (a, y, 8; P/2) = 0. (21)
It will now be shown that 8 can be eliminated by means of the second of these
equations. Suppose equations (1) with t as the independent variable andinitial conditions (17) are integrated as power series in 8. The terms inde-
pendent of 5 will be periodic with the period P. Non-periodic terms will
enter only when the terms in the right members at some stage of the inte-
gration contain terms with the period P. Then t times periodic terms will
appear in the solution, and terms multiplied by t2 and higher powers of I
will not enter until later stages of the integration. Terms of this character
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SYNTHESIS OF PERIODIC ORBITS. 491
will actually arise, for otherwise the solution would be periodic for all 5;
that is, the coordinates would be constants with respect to I. The expres-sion for y = p 3 is an odd function of I with the initial conditions (18) . There-fore the
termin t is
multiplied by an even function, that is, a cosine termwhich does not vanish at t = P/2. Therefore p 3 carries 5 to the first degreeand this parameter can be eliminated, giving
P(o, 7 )=0, (22)
where P is a power series in a and y vanishing for a = y = 0:
Suppose 7 is taken as the independent parameter and that (22) is
solved for a in terms of y. A necessary and sufficient condition that (16)be a double periodic solution with respect to the Jacobian constant C is that
AP = P l (a,y)=0 (23)
for a = 7 = 0. Suppose this condition is satisfied.
Now suppose n = fi -\-\ and consider the question of the existence of a
double periodic solution for this value of fi. The double periodic solution
will exist provided the equation corresponding to (23) is satisfied. Supposethe initial value of x is
x(0)=x +a + a, (24)
where <x is a parameter which remains to be determined. The steps cor-
responding to those leading up to (23) can be taken in this case, and the
equation corresponding to (23) becomesP 2 (<r, X)=0 (25)
for a = 7 = 0, where P 2 is a power series in a and X vanishing for a = X = 0.
The series P 2 contains terms in <r alone; otherwise, not only would everyorbit whose initial conditions were
x(0)=x +a, *'(0)-0, 2/(0) =0, y'(0)=y' , (26)
where x(0) is arbitrary, be a periodic orbit, but it would be a double periodic
orbit. Therefore (25) can be solved for a as a power series in X, or in somefractional power of X, vanishing with X, the multiplicity of the solution being
equal to the degree of the term of lowest degree in <r alone. Hence, in the
analytic sense, if there is a double periodic orbit for X = (that is, for
m = mo), then there is a double periodic orbit for every value of n = n +\.If the solution of (25) has the form
<r = Xp(X), (27)
where p(X) is a power series in X, then there is a single double solution of the
series for every |X| sufficiently small. But if the solution is in X }
,then there
are two real double solutions when X has one sign and none when it has the
other. That is, for X = two real double periodic orbits unite and disappear
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492 PERIODIC ORBITS.
by becoming complex. And in general, double periodic solutions appear or
disappear in pairs, which become identical for certain values of n, a result analo-
gous to Poincare^s theorem respecting the appearance and disappearance of
real periodic orbits.
Now consider the real periodic satellite orbits which are re-entrant after
a single synodical revolution. Darwin's computation shows that for the
mass ratio of 10 to 1 there is one, and but one, real double periodic orbit
in which the motion is direct. Hence, there is no other double periodic orbit
with which it could unite to disappear for any value of n.
The same result also is a consequence of the analysis of Chapter XI,where it was shown that for n = 0, and therefore for all n, there are but three
real and complex orbits of the type in question. From the fact that there
are only three periodic orbits of the type in question it follows that there
can not be more than one direct double periodic orbit; and from its existence
for n = l /u, 1 M= 10/ii) it follows that there is one direct double periodic
orbit for all values of n from zero to unity. The orbits of inferior planets
differ from those of satellites only in the ratio of the masses. Therefore,
there are also double periodic orbits of inferior planets in which the motion
is in the forward direction.
When one of the masses which revolve in circles becomes zero, those
orbits around the other which are complex for small periods are complex for
all periods from zero to infinity. In this case there are no double orbits
except those of infinite and infinitesimal dimensions. The question arises
as to the character of the double orbits for very small values of the second
finite mass. Consider the totality of real circular orbits around one of the
finite bodies when the mass of the other one is zero. As the second mass
becomes finite, the periodic orbits are continuous deformations of the cir-
cular orbits with the exception of that circular orbit whose period is 2t.
It passes through the point where the second mass becomes finite, and the
force function for this orbit has a discontinuity. This means that the orbit
itself has a discontinuity. It is conjectured that the complex orbits have
corresponding discontinuities; that when the second finite mass is verysmall there are three real orbits about the larger mass in which the motion
is in the forward direction and which have a mean distance near unity and a
period near 2tt; that there are three corresponding real orbits of small dimen-
sions about the smaller mass; and, finally, that there are three similar orbits
of the nature of superior planets. For increasing values of the Jacobian
constant two of the three in each case unite and form the double orbits.
Consequently, if this conjecture is correct, and if the three types of double
orbits are followed as one of the finite bodies approaches zero as a limit, the
one around the larger finite mass and the one around both finite masses
approach the unit circle, and the one around the smaller finite mass approacheszero dimensions.
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SYNTHESIS OF PERIODIC ORBITS. 493
The question of direct double periodic orbits was put to numerical
test for /x=
. In all, 53 orbits were computed for various values of C, start-
ing with (7 = 3.58 and ending with (7 = 3.086. For (7 = 3.58 there were twodirect
periodicorbits which were
geometricallymuch alike, and which intersected the x-axis near
.785. For smaller values of (7 the correspond-ing periodic orbits were more nearly identical.
For (7 = 3.086 they were sensibly identical. The
computation for this value of C gave the results
set forth in the following table (Fig. 23) : wn. 23.
I i = y , C = 3.086, Double periodic orbit.
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494 PERIODIC ORBITS.
M= Ji
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SYNTHESIS OF PERIODIC ORBITS. 495
The orbit defined by m=2> C = 2.50, x .719 is nearly periodic.
Its coordinates for various values of t are given in the following table
(Fig: 26, page 496) :
l^xA, C = 2.95, Retrograde periodic orbit.
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496 PERIODIC ORBITS.
The solution of equations (29) as power series mt-ti with the initial
conditions =?j= ' = ?/
= * s
i-4?(t-t l y+%(t-t l y+ [ -^ +i'f
l+B *Bk t-uy+
Bo, A ,
9-fa(-tO ,
-f(*-<i)H
24
\-^Bq-\-AoB 1 +BqB 2
{t-uy+(30)
Fig. 26.
(31)
24
The direction cosines of the normal to the
curve of zero relative velocity at the point (x u yi)
are proportional to A and B . Therefore the
tangent to the cusp is perpendicular to the curve
of zero relative velocity at the point (x u yi).
Now take a new set of axes (u, v) with origin
at (x u j/i), with u having the direction of the tan-
gentand v
perpendicularto it. If the positive ends
of the new axes are chosen so that the cosine and sine of the angle from the
positive end of the -axis counted counter-clockwise to the positive end of
the w-axis are proportional to A and B,and if the positive end of the t;-axis
is 90 forward counter-clockwise from the positive end of the w-axis, then the
equations of transformation are
u = AoZ+Bov = UAo2+B<?](t-uy +
\-MA<?+B 2 )+2A BoA2+A<?A 1 +BoB2]( t f j 4+
v = Bo + Aov= -i[A2 +B 2
](t-t 1y +\A BQ (B 2 -A l ) + (Ao 2 -Bo i )A 2 ]
^ t t ^ +
The value of u is positive for both positive and negative values of
(t-ti), because the coefficient of {t-Uy can vanish only at an equilibrium
point. Therefore the positive end of the w-axis extends from the point
(xi, yi) into the region of real velocities. The value of v is positive for
small negative values of t t u and negative for small
positive values of ttu Therefore, if motion along the
curves of zero relative velocity is taken as positive when the
region of real velocity is on the left, the motion in the cusp
orbits, in the neighborhood of the cusps, is in the positive
direction; that is, the infinitesimal body crosses the tangent
to the cusp in the positive direction. In Fig. 27, C is a
curve of zero velocity, P is a cusp, is the orbit near
the cusp, and T is the tangent to the orbit at the cusp.
Now suppose the orbit having the cusp is a periodic orbit. If it has no
other double points than at the cusps, and if it is inside of the curves of zero
relative velocity, then it revolves around one of the finite bodies in the
positive direction. If it is outside of the curves of zero relative velocity, it
revolves with respect to the rotating axes in the retrograde direction. In-
Fig. 27.
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SYNTHESIS OF PERIODIC ORBITS. 497
deed, all periodic orbits of superior planets revolve in the retrograde direction
with respect to the rotating axes. But the orbits with cusps, if they haveno other double points, revolve in the forward direction with respect to fixed
axes, because at the cusps they have precisely the forward motion of therotating axes.
240. Periodic Orbits Having Loops Which Are Related to Cusps.Suppose the orbit defined by the initial conditions
z(0)=x , x'(0)=0, t/(0)=0, t/'(0)= 2 /', C = Co, (32)
is periodic with the period P. Therefore
x'(P/2)=0, l/(P/2)=0. (33)
Suppose it has a cusp at t = t u or
x{t l ) = x l , x'(t 1 ) = y{k) = yi, *'(ft)0. (34)
If the initial conditions are varied in such a way that the orbit remains
periodic, its character in the vicinity of the cusp will be changed. Thenature of these changes will now be considered. Suppose the initial con-
ditions are
x(0)=x + a, x'(0)=0, 2/(0) =0, y'(0)=y '+p, C = C +y; (35)
and that
t=(l+6)r, (36)
where r is a new independent variable and 5 is an undetermined parameter.
The solutions can be expanded as power series in a, 0, and 5. Theconditions that they shall be periodic are
x'[(l+8)P/2] = Pi (a, 0, 5)=0, y[(l+8)P/2] = p 2 (a, 0, 5)=0, (37)
where p x and p 2 are power series in a, 0, and 5, vanishing identically with a,
0, and 5. Now consider the solution of (37) for and 8 in terms of a, vanish-
ing with a. The solution is always possible either in integral or fractional
powers unless the equations are identically satisfied by = 5 = 0. But this
means that all orbits in which the infinitesimal body crosses the x-axis near
x with fixed velocity y'
are periodic, and that they all have the same
period. Since the results are analytic in a, the orbits may be continued withrespect to a, in the analytic sense, until the place of crossing, x
,is small.
But then the methods of Chapter XIV apply and it is known that the
period depends upon x . Therefore equations (37) are not identically satis-
fied by = 5 = 0, and they can be solved for and 8 in terms of a. The solu-
tions will have the form
= a ,/"P,(a
1
"'), 8 = a v "P 2 (al/
"), (38)
where p is unity if the Jacobian of pi and p 2 with respect to /3 and 8 is dis-
tinct from zero for a = = 5 = 0. If p is not unity, it is some other positive
integer. In general it will be unity.
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I '.is PERIODIC ORBITS.
In the computations of Darwin 7 was taken as the parameter which
defines the orbits. The change can be made here because (2) is uniquely
solvable for a as a power series in /3, 7, and 5 unless
. (1-ju)Qcq+m) /i(xo-l+M) _n 2/o = 0.
;}(40)
But these equalities are satisfied only at one of the collinear solution points.
The orbits in the vicinity of these points will be omitted from this discus-
sion because they belong to quite another category.
If a is eliminated by means of (2), and /J and 5 by means of (38), the
solutions will be expressed as power series in y Up, and the values of the coor-
dinates at r = ti are
x(r) T . h= *, + 7
1/p0i, x'{r) r ^ - +7 l/ "6 2
,\
y{rU= yi +y^e 3 , i/'(r) r=(l= 0+7 ,/ ^J
where d u . . .,d 4 are power series in y y ".
The expressions for the coordinates in the vicinity of t = t u satisfying
the relations (39), can be expanded as power series int ti. The solution is
found from equations (29) to be
x-x 1= t = a 1 {T-U)+at{T-Uy+a 3 {r-U)
3 +y-y 1
= v = b 1 (T-t l )+b 2 (T-t 1 y-+b 3 (T-t i y +where
ai = T 1/p2 , &i = 7
1/p04,
a 2 = (l+5)&i+Kl+5)24 ,
&2 =-(l+5)a 1 +Kl+5)2
fio,
a 3 = f(l+5)6 2 +|(l+5)2[4ia 1 +4 2 6 1 ],
&s - -f(l +5)a 2 +i(l +5)
2[JB 1 a 1 +B 2 b,],
l+5 = l+7 1/p iV /p),
where A,A u A 2 ,
B,B x ,
B 2 are given in (29) and P is a power series in y Vv
The transformation (31) gives
u = 7 1/p (Mo +e*B 6 ) (r-
J,) + (a 2A +b 2B ) (t-
h)2
+(a 3A +b iB )(T-t 1y +v - 7 1/p (Mo - B 2BQ) (t-U) + (b 2A - a 2B Q) (r
- *02
+ (b zA -a 3B)(T-t
1
y+....
It follows from (41) that for 7 =
W= iGV+J5o 2)(T-* 1 )
2 + . . ., V= _I(^ o2+5o 2
)(t_
<i)3 +
The equation r = determines the points at which the periodic orbit
crosses the w-axis. It follows from the second of (43) that (t ti)= is a
triple but not a quadruple solution for 7 = 0. Therefore there are three
solutions of v = for t U as power series in integral or fractional powers of
71/p
, vanishing with 7. One of them is simply
t-<! = (44)
(41)
(42)
(43)
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SYNTHESIS OF PERIODIC ORBITS. 499
The others depend upon the values of the coefficients of the right memberof the second equation of (42). Unless d 4A e 2Bo = K = for 7 = the two
remaining solutions have the form
V3Kh =
s=,71/2P
tx = J/2P(45)VaJ+b} vaj+bs
'
If K = 0, which will be exceptionally if at all, the corresponding solutions
exist but may be in integral powers of y1/p
.
It has been remarked that p will in general be unity. When it is oddthe periodic orbit with the cusp at (x u yi) is a multiple orbit. If p is even,two orbits which are real when 7 has one sign unite for 7 = and disappear bybecoming imaginary when 7 has the other sign. When p is odd there is a
single real orbit for 7 both positive and negative. It is clear that only
exceptionally, if at all, will a double periodic orbit have a cusp. If it wereso in any particular case the value of y. could be changed, when it would no
longer be true. Therefore it will be supposed that p is unity.
It follows from (45) that the second and third intersections of the curve
with the w-axis are real for 7 positive or negative according as K is positive
or negative, and that they are not real when 7 has the other sign. Whenthe second and third intersections of the curve are real the curve consists
of a small loop as is indicated in Fig.
28; and when they are complex the
curve has a point near which the
curvature is sharp, as is indicated in
Fig. 29. When there are three inter-
sections of the curve with the w-axis,
one occurs before t\ and one after t t .
It follows from this discussion
that if a periodic orbit for a certain value Co of the Jacobian constant has a
cusp, then for a slightly larger (or smaller) value it has a point, near a curve of
zero relative velocity, in the vicinity of which there is very sharp curvature; for
diminishing (increasing) values of C the point of sharp curvature approaches
the cusp form on the corresponding curve of zero velocity, which it reaches for
C = C ; and for still further diminishing (increasing) values of C it has asmall
loop near a curve of zero velocity. In Darwin's computations examples of
periodic orbits with cusps were found. It follows, of course, from the sym-
metry of the periodic orbits with respect to the x-axis that if there is a cusp
at x = x,, y = y u then there is also a cusp at x = x u y= Jfc.
241. The Persistence of Cusps with Changing Mass-Ratio of the
Finite Bodies. Suppose for m = Mo equations (1) have a periodic solution
satisfying the initial conditions
x(0)=x, x'(0)=0, ?y(0)=0, y'(0)=2/'o, (46)
and the cusp conditions att =
ti
Fig. 28. Fig. 29.
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500 PERIODIC ORBITS.
*(*i)=z., x'{t l )=0, y{h)=Vx, IfW-. (47)
If P represents the period of the solution, the expressions for x' and y
satisfy the equations
x'(P/2)=0, y(P/2)=0. (48)
Now suppose m= Mo+X and consider the question of the existence of a
ixiriodic orbit having a cusp for this value of n. Let
I-(144K x(0)=x + a, x'(0)=0, y(0)=0, y'(O)=y'o+0. (49)
The solution can be expanded as power series in a, )3, 5, and X. The con-
ditions that it shall be periodic in t with the period P are
x'(t) t=i ,/2= Pi(o, j8, 8, X) = 0, 2/(t) t=/V2
= p 2 (o, j3, 5, X) = 0, (50)
wherepi
andp
2 arepower
series in a, 0, 6, and X,vanishing
with a, 0, 5, and X.
Unless the initial conditions (46) define a double periodic orbit these equa-
tions can be solved uniquely for /3 and 5 as power series in a and X, vanishing
with a and X. The results will have the form
/S= g 1 (o,X) J *-(*, X), (51)
where q x and g 2 are power series in a and X, vanishing for a = X = 0.
Suppose /3 and 5 are eliminated from the solutions by means of equa-tions (51). The results will be expanded as power series in a and X. Thevalues of the coordinates at t = U will be
Xto^mxt+Pfa X), x'(r),= = 0+2V(a, \),\, .
</(r),= = i/i+P 2 (a, X), *'(T)-*-0+P/(a J X)J
where P h P/, P 2 , P 2'
are power series in a and X which vanish with a and X.
The values of the coordinates near r = U can be expanded as powerseries in t t x satisfying equations (52). The results are
x(t)=Xi+Pi(o, X)+ a(r-Uy+ . . .
x'(t)=0 +P 1
/
(o,X)+2o(t-i) + . . .
l/(r)=y 1 +P,(e, X)+ b(r-t i y+ . . .
y'(r)=0 +P/(o, X)+26(t-i) + . . .
The conditions that the orbit shall have a cusp at t = U_ arc
= P,'(a,X)+2a(< 2 -< 1 )+ . . .,
= P./(a,\)+2b(t 2 -t l )+ .... (54)
These equations are not satisfied by X = 0, because a and b contain terms
which depend upon x x and y x alone. Neither are they satisfied by o = 0,
t 2 ti = 0, unless orbits crossing the x-axis at x = x are periodic for all values
of m and have a cusp for the same t = t x . But it is known that the points at
which the periodic orbits cross the x-axis depend upon the value of ju- There-
fore equations (54) can be solved for t 2 t y and a in integral or fractional
powers of X. In general the solution will be in integral powers of X. If the
(53)
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SYNTHESIS OF PERIODIC ORBITS. 501
solution is in integral or odd fractional powers of X, it is real for both positiveand negative values of X. If the solution is in even fractional powers of X,
there are two real solutions when X has one sign and only complex solutions
when it has the other.
It follows from this discussion that if a real cusp exists for any value of
H, it will exist for all other values of n unless two real cusps become identical
and disappear by becoming complex. Since an orbit is uniquely defined
by the conditions for a cusp, as well as by any other initial conditions, cusps
disappear by becoming complex only when two orbits become identical.
Darwin's computations showed that in the case of one of the orbits
which was complex for large values of the Jacobian constant ("satellites of
Class C") there were periodic orbits without loops near the cusp form, andothers for smaller values of the Jacobian constant having loops. It follows
from the results of 240 that between the two orbits there exists one havingtwo cusps which are symmetrically situated with respect to the x-axis; andit follows from the discussion of this article that the orbits with cusps exist
for all values of n unless a cusp develops on another orbit which later
becomes identical with this.
242. Some Properties of the Periodic Oscillating Satellites near the
Equilateral Triangular Points. In Chapter IX, Dr. Buck has treated the
periodic oscillating satellites which are near the equilateral triangular points,
using in a general way the methods of Chapter V. I t wil l be necessary for the
purposesof the latter
partof this
chapterto
developa few additional
prop-erties of these orbits; and the most important of them can not be established
by the methods of Chapter V, but follow from the methods of Chapter VI.
The differential equations will be transformed by letting m = Mo+X.For motion in the vicinity of the equilateral triangular points they are
W 2dt A
X~4~ (1 2lM,)y A
'
d 2y ,o dx _ 3V3 n _ n a \ r _ 9 mY
(55)
where X and Y are of the second and higher degrees in x, y, and X.
In this article the character of the small oscillations will be discussed.
In treating them X and Y may be provisionally put equal to zero. The
characteristic equation on which the nature of the solutions depends is
S 2
-j, -2-^(l-2 Mo)4 4
2S-3^3(1-2mo), S*~l
^-bS 2
+^ Mo(l-Mo)=0. (56)
The roots of this equation are all purely imaginary or complex in conjugate
pairs according as 1-27/z (1-mo) is or is not greater than zero. When
1 -27mo(1 -Mo) is zero there are two pairsof
equal purely imaginarysolutions
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502 PERIODIC ORBITS.
of (56). It will be supposed for the present that p has such a value that the
roots of (56) are pure imaginaries and distinct, and that p has such a value
that X = p-p is very small.
Let the roots of(56)
be+crV^l, -*V^L +pV^l,
~pV-1, where
a and p are real and a<p. There are two periodic solutions of the linear
terms of (55), one with the period 2w/a and the other with the period 2ir/p.
If x = 0, y = ci>0 at t = 0, the solution having the period 2ir/a is
x = ^~ sin at, 2/= nr-
1 sm vt+Ci cos at, (57)Oi 01
where
h- 2(r n - ~ 3 V/ 3 (l-2/lo) n ,PR
x
0l_ ^p' ai4 ^qrs
-
i
The curve described by the infinitesimal body is an ellipse whose equation is
Oi 2 +i 2 2 2d , 2/2
,
, x 2
^xy + , = 1C\ Ci C\
and if 6 represents the angle between the positive end of the x-axis and the
major axis of the ellipse, it is easily found to be defined, except as to quad-
rant, by the equation
tan 2e =i-a/-bS
=2k 4+k 2 +H-?Kl-2^o) 2
]
<0 - (59)
The direction of motion in the orbit is found to be retrograde from
x'(0)=^>0, 2/'(0)=^<0.Oi Oi
There are equations for the period 2ir/p which differ from (57), (58),
and (59) only in that a is replaced by p, and the subscript 1 on a, b, andc is replaced by 2. The motion in these orbits is also in the retrograde
direction.
The linear terms of equations (55) admit the integral
*'2 +U n = !*
2
+^(1-
2mo)**/ +W- C (60)
Let the value of C for the orbits having the period 2w/a and 2x/p be Ca andCp .
Theyare found from equations (57) to have the values
a = c 1
2
[:-^(l+a1
2
)]= |
2
[9-(<T2 + 4
9)2
-^(l-2po)2
])
^ =c/[- 2 (l+a 2
2
)]=^[9-(p2
+|)2
-^(l-2p )2
].
(61)
It follows from (56) that the values of <r and p2 are
, l-Vl-27po(l-po) a _ l + Vl-27/io(l-/io) ((. s
Since p is small a"1
is small and the limit of <r as mo approaches zero is zero.
On the other hand p2
is near unity, and its limit as p approaches zero is 1.
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SYNTHESIS OF PERIODIC ORBITS. 503
Therefore C is positive and C, is negative, at least for small values of
Mo. For the Lagrangian equilateral triangular solution x=y=0 the valueof Ca
= C is zero. Hence the value of the Jacobian constant is greater for
the orbits whose period is 2ir/<r, and less for those whose period is 2x/p, thanit is for the Lagrangian equilateral triangular point solution.
Now consider the curves of zero relative velocity. They are knownto be real only if the value of C is greater than that which belongs to the
equilateral triangular point solution. Therefore they are real only for the
solution with the period 2-ir/a. Their equation is
3V3\a=fx 2>-(l-2 )xy+-ly\ (63)
This is the equation of an ellipse, the direction of whose major axis is given by
tan 2? - - V3(l - 2mo) . (64)
The limit of the right member of this expression for p = is \/3-
Since a"- is small when p is small, the approximate value of the right
member of equation (59) is V3(l 2p ) Therefore the orbit whose periodis 2ir/a has its axes, for small /x , nearly coincident with the axes of the
corrresponding curves of zero relative velocity. The z-axis is in the line
joining the finite body m with the equilateral triangular point, and the other
is of course at right angles to it.
Let the coordinates in the orbit whose period is 2t/<t referred to its
axes be and y ;its equation is then
1Ai =
2 [(ai cos sin 0)2 +&! 2 cos 2
0],
Bi = \[(ai sin 0+cos ey+bS sin2
0].
(65)
The corresponding equations for the curves of zero relative velocity are
A?+Brf =l,
A-Mt cos 2<P + 3
-|^(l-2/ io) sin <p cos <p + l sinV],
B i-[fsin2 3V3
(1 2/i ) sin (p cos <p+l cosV].
(66)
It follows from (58) that when no is small the approximate values
of a! and 6i are y/l/z and zero respectively. Since the limit of for
Mo = is -30, it Is found from (65) that
lim A# Hm cos 2 0+2 V3 sin 6 cos 0+3 sin 2 _ QMo = Ofi 1 ;uo
= O3cos 2 0-2\/3sin0cos0+sin2 (67)
The limit of the ellipse for mo = is a straight line through the origin and
the position of n, and for small values of mo the eccentricity is near unity.
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504 PERIODIC ORBITS.
The approximate value of <p is also 30 when mo is small. Hence it is
found from (66) that the ratio A/B also has the same value as A x /B x .
Therefore at the limit the orbit with period 2ir/r and the corresponding
curve of zero relative velocity have not only the same orientation, but they
have also the same eccentricity.
It follows from (61), (65), and (66) that at the limit mo =
A _ 9 C X
A l
~ rCt
= 4.
The ratio of the dimensions of the periodic orbit whose period is 2w/a to
that of the curve of zero relative velocity corresponding to the same value
of C is equal to the square root of this number, or 2. This is actually the
limit of the ratio of the linear dimensions of the orbits to the curves of zero
relative velocity as mo approaches zero.
The discussion so far has pertained to the linear terms alone of the
differential equations. The results are the first terms of the series for the
periodic solutions which can be shown to exist by the methods of ChapterVI. Consequently, for small values of the parameter X they give close
approximations to the periodic orbits and the corresponding curves of zero
relative velocity. The period 2-k/o is very long for small values of ju .
Now consider the periodic orbits whose period is 2ir/p. The approxi-
mate value of p for small values of juo is unity, and from the equations corre-
sponding to (58) it is found that b 2= ^, Q 2= 3-^f. Therefore, for these
orbits alsotan 20 = -V3
when no has the limit zero. It is found from the equations corresponding to
(65) thatlim Aj _ i
Mo = 2"*'
Therefore in these orbits the length of one axis is twice that of the other.
The limit, for mo = 0, of the eccentricity of the orbits whose period is 2v/ais unity and the limit of the periods is infinity; the corresponding l imits for
the orbits whoseperiod
is2ir/p
areV3/2
and 2t.
243. The Analytic Continuity of the Orbits about the Equilateral
Triangular Points. The periodic solutions as developed by the methods of
Chapter VI are power series in X*, and they involve ju . The coefficients
of the power series are continuous functions of mo- The orbits are real whenX has one sign and complex when it has the other. As X passes through zero
from one sign to the other, two real solutions for X J unite and disappear bybecoming complex. They do not belong to the physical problem exceptwhen \ = n po, but since the orbits exist for every value of fx distinct fromzero it is easy to get an understanding of the situation from the behavior of
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SYNTHESIS OF PERIODIC ORBITS. 505
the more general solutions when X does not equal m~ Mo- Or, since the coef-
ficients are continuous functions of fi , this parameter can be considered as
varying with X so that their sum is p. That is, the solutions and the period
can be expressed in terms of n and X by replacing ju by p X.
The two real orbits which unite and disappear for X = are not geo-
metrically distinct. This appears to be an exception to the theorem that
real orbits appear or disappear only in pairs. It arises because in the
analysis adopted the conditions that the orbit shall be periodic give a double
determination of the same orbit. The two determinations coincide when the
orbits shrink to zero dimensions for X = 0. Such a situation can arise only at
the five equilibrium points. Moreover, the matter is quite different whenthe solutions are developed by the method of Chapter V. When the para-meter t passes through zero the orbits do not disappear, but the same series
is obtained for both positive and negative values of e, the origin of time
belonging to a different place on the orbit. In the symmetrical orbits around
the equilibrium points which are on the z-axis the origin of time is displaced
by half a period. In the non-symmetrical orbits about the equilateral
triangular points the origin is shifted from one point where the arbitrary
initial condition, e. g., x'(0)=0, is satisfied to the other point in the orbit
where it is also satisfied.
The Jacobian integral exists when the right members of the differential
equations are not limited to their linear terms. Hence, in place of (60),
the right member is an infinite series in x and y. When the expressions for
x and y as series in X* are substituted the constant C becomes a power series
in X*, the term of the lowest degree in X being of the first degree. Conse-
quently the series can be solved for X 5 as a power series in ^C*, and the
result substituted for the solution in powers of X 5 will give x and y expressedas power series in C 5
, which converge for \C\ sufficiently small. As C goes
through zero the orbits whose period is 2ir/a change from real to complex,
and those whose period is 2ir/p change from complex to real. There is a
branch on each of the series at C =0, but the two series are distinct.
When ju satisfies the equation
1-27*,(1 -mo) -0
the values of a and p are equal to y/2/2. In this case four solutions branch
at X = 0.
244. The Existence of Periodic Orbits about the Equilateral Trian-
gular Points for Large Values of /x. The orbits heretofore discussed have
been for values of /x such that 1 -27it(l -/*) is positive. If it is zero, there is
a double solution of zero dimensions. Suppose now that n has such a value
that this function of n is very little less than zero, and take a*o so that
l-27/x (l-Mo) is a little greater than zero. Then there are real periodic
orbits with theperiods 2t/<t
and2t/p
for X sufficiently small. When X
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506 PERIODIC ORBITS.
is less than ju Mo these orbits do not belong to the physical problem. The
analytic continuation of the solutions with respect to the parameter X can be
made until X = /x Mo unless they have some singularity for a real positive
value of X. It is very improbable that there is an infinity in the solutions
for a real positive value of X because an infinity implies either an infinite
branch of the orbit or one passing through one of the finite bodies. Theorbit could not acquire an infinite branch without winding infinitely manytimes, in the rotating plane, about the finite bodies. Even if it should pass
through one of the finite bodies its continuity would be maintained, as was
seen in the case of other orbits of ejection in Chapter XV. A branch-point
would imply the existence of other real orbits which could become identical
with the ones under consideration. It also seems very improbable that
there is such a branch-point for small variations in X. Therefore it will be
assumed,as
being probable,that the periodic solutions can be continued to
the ones belonging to the physical problem for X = n mo- This applies both
to those whose periods are 2t/<t and to those whose periods arc 2w/p. The
possibility of their having acquired loops about one or both of the finite
bodies by having passed through ejectional forms must, however, be
admitted. This circumstance makes numerical verification difficult.
Now suppose the value of /z approaches 0.0385 . . . as a limit, the value
of /x which satisfies 1 27juo(l Mo) =0, and that the analytic continuation
can be made with respect to X for all /x . The expressions for the coordinates
in the two classes of orbits are the same except that a and p are interchanged.
As no approaches0.0385 . . . a and
p approach equality,and the
correspond-ing orbits approach identity for X = /i /*<> A difficulty in attempting com-
plete rigor arises from the fact that a certain determinant which is distinct
from zero in the proof of the existence of the solutions approaches zero as ju
approaches 0.0385 .... But if it is admitted that the analytic continua-
tion with respect to X can be made starting with any p ,it follows that even
if h is a little larger than 0.0385 . . . there is a double periodic orbit, and it
surrounds a small real curve of zero relative velocity in the vicinity of one of
the equilateral triangular solution points. As it is decreased toward the
limit 0.0385 . . .,the dimensions of this double periodic orbit diminish
toward zero as a limit. There is in this analysis a double determination ofa double periodic orbit, just as of a single periodic orbit, and the two deter-
minations coincide when it has zero dimensions. Consequently it can disap-
pear at zero dimensions without uniting with another double periodic orbit.
If h increases the double periodic orbit persists, according to the prin-
ciples of 238, unless it becomes identical with another double periodic orbit.
If there were another double periodic orbit with which it could unite it would
envelop neither of the finite masses and would have two distinct branches
which are symmetrical with respect to the z-axis. It is improbable in the
extreme that there is another such double periodic orbit, which would mean
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SYNTHESIS OF PERIODIC ORBITS. 507
the existence of four single periodic orbi ts of the type under consideration.
There are only two periodic orbits which shrink on the equilateral triangular
points, and others of the type could arise only from orbits, which originally
had loops about a finite body, passing through an ejectional form.The existence of a double periodic orbit for all values of n implies the
existence or two single periodic orbits which branch from it for values of the
parameters which define the orbit, for example the Jacobian constant C, its
linear dimension, or its period. It should be added, of , course, that the
two series of orbits may branch at the double orbit when considered with
respect to one parameter, and form a continuous series when considered
with respect to another.
245. Numerical Periodic Orbits about Equilateral Triangular Points.
In accordance with the principles of 244, two periodic orbits about the
equilateral triangular points should exist for all values of n from to \,
and for all values of C near that belonging to the equilateral triangular
equilibrium points. The only way they could cease to exist for at least somevalue of C would be for all of them to pass through an ejectional form for
every C. These orbits have an axis of symmetry only when p*. It is
very difficult to establish by numerical processes the existence of a periodic
orbit when it has no axis of symmetry because, for a given initial point,
there are two arbitrary components of velocity, and interpolations must be
made from a two-parameter family. Therefore the computations were
restricted to the casem=
2- It follows from the differentialequations
that
in this case the orbits in question have the fine x = as a line of symmetry.Since n = \ is far from the values (0 <; n ^ 0.0385 . . .) for which the existence
of the orbits in question was established by direct processes, those found bycomputation can not be expected to have much geometrical resemblance to
those found by analysis.
Since the surfaces of zero relative velocity expand with increasing Cand unite on the rr-axis, it follows that either the periodic orbits about the
equilateral triangular points unite in pairs and disappear with increasing
values of C, or they pass through the collinear equilibrium points with
infiniteperiods.
Therefore it seemed best to startcomputations
for values
of C not much greater than that belonging to the equilibrium point, viz, 3.
In attempting to discover periodic orbits about the equilateral triangular
points 40 orbits were computed. In 17 of these C was taken equal to 3.03;
in 16 it was taken equal to 3.20; in the remaining 7 it was taken equal to
3.284. The initial values of the coordinates and components of velocity
were x =0, y arbitrarily chosen, y
' =0, x ' determined so as to give the
adopted value of C. The computation was continued until x became again
equal to zero, and the approach to periodicity was determined by the approxi-
mation of yd to zero.
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508 PERIODIC ORBITS.
C = 3.03, Period =2X4.388 =8.776 (Fig. 30).
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SYNTHESIS OF PERIODIC ORBITS. 509
It was found that for C = 3.03 there are two periodic orbits aboutthe equilateral triangular points differing considerably in dimensions andperiods; for C = 3.20 there are also two periodic orbits which differ less
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510 PERIODIC OKBITS.
in dimensions and periods; and for C = 3.3284 there is a very close approach
to a periodic orbit, though one was not actually found. The computations
indicate that the two series of periodic orbits unite and disappear for some
value of C slightly smaller than 3.3284. But there is an orbit so nearly
periodic for C = 3.3284 that it is included as being very nearly that double
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SYNTHESIS OF PERIODIC ORBITS. 511
In all cases the infinitesimal body was ejected from the finite body 1 nin the positive or negative z-direction.
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512 PERIODIC ORBITS.
Any orbit which intersects the x-axis perpendicularly is symmetrical
with respect to the x-axis. Hence, it follows that if one of these orbits of
ejection intersects the x-axis perpendicularly, then it is a closed orbit of ejec-
tion of the type treated in Chapter XV.
Computations were first made for m=z to discover orbits of the type
characterized by j= 1 with ejection toward n and shown in Fig. 15. It was
proved in Chapter XV that such an orbit exists for small values of m and that
its period is approximately 2r. Such an orbit was found for m = i, but its
period was about 8. Another orbit, also of a similar type, was discovered
whose period was about 14. One of these orbits is undoubtedly the limit,
for decreasing values of C, of the oscillating satellite about the collinear
equilibrium point, as Burrau's calculations have indicated. The value of
C corresponding to the equilibrium point for n = \ is about 3.46, and the
values of C for these orbits are 2.24 and 2.84. The question arises regarding
the origin of the other orbit of this type. It is probably the limiting form
of a periodic orbit about 1 n consisting of a double loop and having a
double point on the x-axis. Such orbits were treated by Poincare in Les
Methodes Nouvelles de la Mecanique Celeste, Chapter XXXI. The ana-
lytic continuation of the former beyond the ejectional form for decreasing
values of C is also a periodic orbit with two loops. For greater or smaller
values of C the latter will have also the character of an oscillating satellite,
but it can not reduce to the equilibrium point because there is only one orbit
of this type. The results for n = \ are given in the tables of page 511.
For /i=
|similar results were found. Since orbits of this type have
not been computed heretofore, the results for the four orbits will be given
for enough values of t to exhibit their properties.
The corresponding results for m= I> with ejections from 1 n = l, are
given in the following tables :
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SYNTHESIS OF PERIODIC ORBITS. 513
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:.n PERIODIC ORBITS.
If n were zero and the infinitesimal body were ejected from 1 n either
toward or from m in such a way that its period would be ir, the orbits described
in rotating axes would consist of two parts symmetrical with respect to the
z-axis, as shown in Figs. 40, a, and 41, a. These curves are the limits sepa-
rating two types of periodic orbits (for m = 0) in the rotating plane, as is
shown in Figs. 40, b, and 41, c. As n increases a dissymmetry developswith respect to the line through 1 fi perpendicular to the z-axis. Supposethe orbits are followed as n increases in such a way that they shall remain
orbits of ejection in one way or the other. Then orbits of the type Fig.
40, a, will go into types having some of the characteristics of both types a
and c of Fig. 41. That they partake of the characteristics of type c instead
of those of type b was proved by computations for both n = \ and n = 4 .
The two following tables give the results for ejection from 1 m toward
n, with n having the values \ and |. The first orbit lacks a little of being
closed, but an exactly closed orbit exists between this one and the one whichwas computed for C = 3.478.
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SYNTHESIS OF PERIODIC ORBITS. 515
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516 PERIODIC OKBITS.
certain cases by making use of properties of symmetry. Any orbit which,for n = \, intersects the y-axis perpendicularly is symmetrical with respect
to the t/-axis. Hence it follows that if, for /*=
>an orbit of ejection from
1 fx intersects the y-axis perpendicularly, then it has a symmetrical col-
lision with the second finite mass.After 14 computations had been made, an orbit of ejection from 1 n
and collision with m (m = 5) was discovered in which the ejection was toward
n and in which the collision took place without the infinitesimal having
encircled, in the rotating plane, either 1 it or /x. The following table gives
the results at a considerable number of intervals from the time of ejection of
the infinitesimal body from 1 ju until it crossed the line x = 0:
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SYNTHESIS OF PERIODIC ORBITS. 517
This orbit has a loop about the equilateral triangular point. It follows
that there are two families of periodic orbits of the types shown in Fig. 46.
len orbits were computed in an attempt to find one of them. The diffi-
culties ofmaking the calculations when the
infinitesimalbody was near one
of the finite bodies were so great that wide departures from the orbits of
ejection had to be attempted. Indications of such periodic orbits were
obtained, but none was actually found.Axis
X-Axis
Fig. 45. Fig. 46.
248. Proof of the Existence of an Infinite Number of Closed Orbits of
Ejection and of Orbits of Ejection and Collision when n = %. It was provedin Chapter XV that when n is sufficiently small there are infinitely manyclosed orbits of ejection, and reasons were given for believing that these
orbits persist for all values of n- The question of the existence of orbits of
ejection and collision was not considered.
Fig. 47. Fig. 48.
It will now be shown that there are infinitely many closed orbits of ejec-
tion, and of ejection and collision, for M= i- The differential equations of
motion in fixed rectangular axes with the origin at the center of gravity of
the system are
d?x_ i(x-xi ) i(x-Xj) d?y_ h(y-Vi) h(yy*)dl 2
jy ry dt 2ry ry
(68)
where, if the finite bodies are on the z-axis at t = 0,
Xi=-|cos, z 2 =+icosJ, Vi= h s in *> Vi= +h sin I. (69)
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518 PERIODIC ORBITS.
Now let x = r cos 0, and y = r sin 0, after which equations (68) become
(70)
IS
where
r* = :rJ+2/
2, n^^+i+r cos (6-t), r 2
2 = r 2 +\-r cos (0-t). (71)
Suppose it is known that, at t = T,
r*2, |'>0. (72)
Since ^+^}<^ and i[^-^]<^r ?and
r(gf
always positive, it follows from the first of (70) that
d P> ( r _i).vV6
\
The integral of this inequality gives
(ty>^+^iF +K=F(r) - (74>
Suppose K>0 and that -57 >0 at < = T. Then F(r) will always exceed K in
value and r will become infinite with t.
Computations were made in which the infinitesimal body was ejectedfrom 1
nboth toward and from
nwith initial conditions
correspondingto
(It
K>0, and they were both followed until ?>2 with -r- positive. Hence in
both cases the infinitesimal body would recede to infinity. Moreover, it
follows from the second equation of (70) that, when referred to rotating
axes, the infinitesimal body revolves infinitely many times about the finite
bodies, and its distance from the origin continually increases.
Now consider, for example, a closed orbit of ejection from 1 n, for
H = \, in which the infinitesimal body makes at least one circuit about the
finite body m- Its orbit therefore crosses the z-axis perpendicularly exactly
once, but it does not cross the ?/-axisperpendicularly. Moreover,
it follows
from the symmetry of the orbit with respect to the z-axis, that if it crosses
the ?/-axis in the first half of the orbit at an angle tt/2 + a, then in the secondhalf it crosses the y-axis at the angle ir/2 a. One or the other of these anglesis less than ir/2. Suppose it is the latter. Now suppose the initial conditions
of ejection are changed so as to increase K. The orbit will cease to be aclosed orbit of ejection and will tend toward one which winds out to infinitywith continually increasing r. The angles at which the orbit crosses the
axes are continuous functions of the parameter defining the initial conditions,as K for example. Hence the intersection with the y-axis which was at the
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SYNTHESIS OF PERIODIC ORBITS. 519
angle r/2 a and less than t/2 for the closed orbit of ejection will be exactly
perpendicular for a certain value of K. The orbit will be therefore an orbit
of ejection from 1 /x and collision with n.
Now consider an orbit of ejection from 1 n and collision with /x, cross-
ing the x-axis at least once. From the symmetry of these orbits with respectto the y-axis it follows that they cross the x-axis an even number of times
and that if such an orbit crosses the x-axis once at an angle ir/2 +/3, whereis a positive quantity, then it also crosses it at an angle, of 7r/2 /3. If the
initial conditions are so changed as to increase the constant K, the orbit
ceases to be an orbit of ejection and collision, the angle corresponding to
7r/2 /3 eventually becomes greater than ar/2, and therefore, since it is a
continuous function of K, there is at least one value of K for which it is
exactly w/2. Such an orbit is a closed orbit of ejection.
It follows from this discussion that if, for any value of K, there is a closed
orbit of ejection, then for some larger value of K, corresponding to a smaller
value of the Jacobian constant C, there is an orbit of ejection and collision;
and that if, for any value of K, there is an orbit of ejection and collision,
then for some larger value of K, corresponding to some smaller value of the
Jacobian constant C, there is a closed orbit of ejection. Hence, for 1 fj.=
\,
there are infinitely many closed orbits of ejection and collision. They are
all distinct because they have distinct values of K. And since it has been
shown that, when K>0 for an orbit of ejection, r increases continuously to
infinity, it follows that the infinite sets of values of K corresponding to these
classes of orbits are bounded. The orbits may be characterized by the num-ber of times they cross the y-axis. For ejections in both the positive and the
negative direction there are closed orbits of ejection from each of the
finite bodies, and also orbits of ejection from one and collision with the
other, which cross the y-axis 2(2/+l) times, j = 0, 1, 2, . . .
249. On the Evolution of Periodic Orbits about Equilibrium Points.
The evolution of the periodic orbits about the equilibrium points (a) and
(c) which are on the x-axis and not between 1 n and m was traced, for
decreasing values of C, by Burrau's computations from small ovals to the
ejectionalform. For
/j.
=% they
are shown inFig. 35, and
form=
1in
Fig. 37. Beyond these forms they have loops about 1 fx.
The periodic orbits about the equilibrium point (&) between 1 /x
and n undergo corresponding evolutions. In the case m = 2 as the orbit,
for decreasing values of C, becomes an orbit of ejection from one bodyit becomes an orbit of collision with the other. This limiting form is
shown in Fig. 45. It intersects the y-axis six times, twice perpendicularly.
Beyond this form it has loops about the finite bodies, and the motion in
these loops is in the retrograde direction. With decreasing values of Cthese loops probably enlarge, the loop about each body eventually becom-
ing anorbit of collision with the other
body.In this case the orbit of
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520 PERIODIC ORBITS.
ejection and collision intersects the y-axis six times, two of the intersec-
tions being perpendicular.
As C decreases, the ejectional and collisional form passes into a loopabout the second body, which in turn expands and becomes an ejectional
and collisional form with respect to the first body. In this manner the
loops of the periodic orbit, with decreasing values of C, pass through ejec-
tional and collisional forms, first with one finite body and then with the other,
in a never-ending series, the ejectional and collisional forms being those shownto exist, for n = \, in 248, in which the ejections from each body are in the
direction away from the other. They are characterized by the fact that theycross the t/-axis 2(2j+l) times, j = 0, 1, 2, . . .
If the finite masses are unequal the evolution of the periodic orbits is in
a general way similar, except that the ejectional forms for the two masses donot occur for the same values of C.
Periodic orbits about the equilateral triangle equilibrium points have
been shown in Figs. 30 and 34. With decreasing values of C they probablyincrease in size and pass through ejectional forms, but ejectional forms in
which the direction of ejection is not along the x-axis. Consequently theycan not be discovered by numerical processes. If this conjecture is cor-
rect, for still smaller values of C they possess loops about the finite bodies,
and for still smaller values of C the loop about each of the finite bodies
may pass through an ejectional form with the other. There is, however,no evidence to guide conjectures.
250. On the Evolution of Direct Periodic Satellite Orbits. There arethree direct periodic satellite orbits, two of which are complex for large
values of the Jacobian constant, but all of which are real for smaller values
of C. Suppose the orbits about the finite body n are under consideration.
With decreasing value of C they can pass through ejectional forms. In fact,
Darwin's computations showed that two of them were approaching such
forms, one by approaching ju from the positive direction and the other byapproaching it from the negative direction.
The motion of the infinitesimal body when it is near collision is nearlythe same as it would be if the mass of the second body were zero. Conse-
quently its properties can be inferred from a consideration of the motion inthe neighborhood of an ejection in the problem of two bodies referred to
rotating axes. It is clear that if the ejection is in the positive direction, the
curve near the point of ejection lies on the negative side of the rr-axis, while
if the ejection is in the negative direction the curve near the point of ejection
lies on the positive side of the z-axis. When the ejection is in the positive
direction, the two families of periodic orbits which are near the ejectional
orbit both intersect the z-axis in the negative direction from the point of
ejection, and the small complete loop about the point of ejection is thendescribed in the retrograde direction, while the partial loop is described in the
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SYNTHESIS OF PERIODIC ORBITS. 521
positive direction; while if the ejection is in the negative direction, the twofamilies of periodic orbits which are near the ejectional orbit both intersect
the x-axis in the positive direction from the point of ejection, but in this case
the small loops are both describedin
the same directions as in the other case.The x-axis consists of three parts, viz, that extending from oo to the
position of 1 ju, that extending from 1 n to n, and that extending from yto +00 . If a periodic orbit intersects the x-axis perpendicularly in any oneof these three parts before it goes through an ejectional form, then it will
also intersect the x-axis perpendicularly in the same part after it passes
through the ejectional form. Moreover, the branches of a closed orbit of
ejection extend from the finite body with which there is collision in the
direction opposite to that in which the neighboring periodic orbit intersects
the x-axis perpendicularly.
In the case of the direct periodic orbit about 1 m which enlarges in thepositive direction and approaches 1 n from the negative direction, the
ejection is in the positive direction, the collision is in the negative direction,
and the orbit has the form shown in Fig. 42. The computation shows that
it has two loops and, therefore, that it had two cusps symmetrical with
respect to the x-axis before it arrived at the ejectional form. After this
orbit passes beyond the ejectional form, with decreasing values of C, it
acquires a loop about 1 m> which intersects the x-axis perpendicularly in
the negative direction from n and which has a double point on the x-axis
between 1 n and p..
Consider the further evolution of the periodic orbit. If the small loopabout 1 m should again pass to the ejectional form, the ejectional orbit would
be exactly of the type of that from which the loop developed. It is improba-ble that such an additional ejectional orbit exists for another value of C.
Now consider the possibility of that part of the orbit which crosses
the x-axis perpendicularly in the positive direction between 1 n and n
passing through an ejectional form. It can not pass to an ejectional form
with n because, in accordance with the general conclusions respecting the
motion near a point of ejection, the branches of the curve near n would lie
in the positive direction from it, and the partial loop about n just before the
ejectional form was reached would be described in the retrograde direction.But this branch of the curve could evolve to an ejectional form with 1 /x,
when the orbit would have the form shown in Fig. 47. With decreasing
values of C this orbit acquires an additional loop about 1 n, which is
described in the retrograde direction and which intersects the x-axis per-
pendicularly in the negative direction between 1 y. and y.. This loop can
expand and take an ejectional form with n, then acquire a loop about n,
which can become an ejectional form with 1 n, and so on, being an ejectional
form first with one of the finite masses and then with the other in a never-
ending sequence. The ejections from 1 n are all in the negative direction,
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522 PERIODIC ORBITS.
and from n they are all in the positive direction. It is probable, though not
certain, that this is qualitatively the course of evolution of the direct satellite
orbit from which the start was made.
Now consider the direct satellite orbit about 1 /x which enlarges in
the negative direction and which approaches the ejectional form from the
positive direction. The ejectional form was found by computation and is
shown in Fig. 44. With decreasing C this orbit acquires a loop about 1 p,
which may pass to the ejectional form with n, as shown in Fig. 48. The other
branch which crosses the z-axis perpendicularly may pass to the ejectional
form with 1 it. With decreasing values of C this orbit acquires a small
loop about 1 m which never again passes through the ejectional form.
But the loop about n enlarges and becomes an ejectional form with 1 nwith the ejection in the negative direction. Then follows a loop which
becomes an ejectional form with n, followed by a loop about n which be-
comes an ejectional form with 1 n, and so on, first with one finite body andthen with the other in a never-ending sequence. The ejections from 1 ft
are in the negative direction, and from ju they are in the positive direction.
There is a third direct satellite orbit whose evolution has not been
traced. Only a conjecture can be made in regard to it, and that conjectureis that it acquires cusps and then loops, probably about the region of the
equilateral triangular points.
251. On the Evolution of Retrograde Periodic Satellite Orbits. Con-sider the retrograde periodic satellites about 1 n. There are three such
orbits, only one of which is real for large values of C. The numerical experi-ments which were made, 238, indicate that only one of them is real for anyvalue of C.
For large values of C the retrograde periodic orbit about 1 n is small
and nearly circular in form. As C diminishes the orbit increases in size
and departs widely from a circle. Consider the question of its passing
through an ejectional form with 1 fi. If the periodic orbit should approachthe ejectional form by shrinking upon 1 it from the positive or the negative
direction, just before arriving at the ejectional form it would make a partial
loop about 1 m in the retrograde direction, and just after passage through
the ejectional form it would make a complete loop about 1 ju in the positivedirection. But it was seen in 250, in connection with a consideration of
an ejectional orbit in the problem of two bodies referred to rotating axes,that this is impossible. Hence the retrograde satellite orbit about 1 a*
can not become an ejectional orbit with 1 n, at least until after it has
passed through an ejectional form with m-
Now consider the possibility of the retrograde satellite orbit about1 /x passing through a collisional form with n. Since it intersects the x-axis
between 1 it and it such an orbit must be one in which the collision is in the
negative direction, in which the ejection is in the positive direction, in which
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SYNTHESIS OF PERIODIC ORBITS. 523
the partial loop just before collision is described in the positive direction, andin which the complete loop just after collision is described in the retrogradedirection. This is precisely the way in which such a limiting form can be
passed,and the
periodicorbit
passes throughthis form.
An orbit of thistype, with the roles of 1 ll and m interchanged, was computed and is shownin Fig. 39.
After the retrograde periodic orbit about 1 ll passes through an ejec-tional form with ll, it acquires a retrograde loop about ll which crosses thex-axis in the positive direction between 1 ll and ll. This loop enlarges andpasses through an ejectional form with 1 m, after which it acquires a retro-
grade loop about 1 ll, which, in turn, enlarges and passes through an ejec-tional form with fi. This process continues, the form becoming ejectionalfirst with one finite mass and then with the other in a never-ending sequence.In all of these orbits the
partsnear the
ejection points are on the negativeside of 1 ll or the positive side of /x, and never between 1 ll and ll. Theorbits of these series which are closed orbits of ejection with 1 it are a partof those which were shown to exist in Chapter XV for sufficiently small
values of ll; and those which are closed orbits of ejection with ll are the
corresponding orbits for the other finite mass.
Consider first the orbits of the type under consideration which are orbits
of collision and ejection with 1 ll. All these orbits are orbits of ejection
in the negative direction; they have double points on the x-axis in the posi-
tive direction from ll, and intersect the x-axis perpendicularly Only in the
negative direction from1 ll.
They are therefore only those orbits of 226which are characterized by ejection in the negative direction and by even
values of j; those characterized by odd values of j have a different origin.
The orbits of the type under consideration which are orbits of ejection
and' collision with lx also intersect the x-axis perpendicularly only on the
negative side of 1 ll. On interchanging the roles of 1 it and ll in 226,
orbits of ejection from ll in the positive direction were proved to exist for
1 ll sufficiently small. Those which are characterized by odd values of jintersect the x-axis perpendicularly on the negative side of 1 ll. Theyare of the type of the part of those under consideration which are orbits
of collision and ejection withll.
To summarize: The retrograde periodic satellite orbits about 1 ll,
with decreasing values of C, go through an infinite series of ejectional forms
with 1 ll, the ejections all being in the negative direction, and these orbits
are those of the orbits treated in 226 which are ejected from 1 ll in the
negative direction and which are characterized by even values of j. The
retrograde periodic satellite orbits also go through an infinite series of ejec-
tional forms with ll, the ejections all being in the positive direction, and these
orbits are those which can be shown to exist by the methods of 226, andwhich are characterized by ejection in the positive direction from ll and by
odd values of j. There are similar retrograde periodic satellite orbits about
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524 PERIODIC ORBITS.
m. and they go through. a similar series of critical ejectional forms with both
1 n and m- The ejections from 1 n and n are also respectively in the nega-tive and positive directions, but those which are ejectional forms with 1 n
are characterized by odd values of,;', while those which are ejectional forms
with ju are characterized by even values of j. Therefore, the retrograde
periodic satellites about the two finite bodies together pass through all the
ejectional forms from both finite bodies, of the type treated in 226, in
which the ejection from one body is in the opposite direction from the other.
252. On the Evolution of Periodic Orbits of Superior Planets. It wasshown in Chapter XII that for large values of C there ai^ two periodic
orbits in which the infinitesimal body makes simple circuits alrout both of the
finite bodies in the retrograde direction. When the system is referred to
fixed axes, one of the orbits is direct and one is retrograde. There are also
fourorbits in
whichthe coordinates are
complex, twoof
them beingdirect
when referred to fixed axes and two being retrograde. It is not knownwhether or not either pair of the complex orbits becomes real with decreasingvalues of C. Since none of these orbits was computed, very little is positively
known about their geometrical characteristics or about their evolution.
It was shown in 248 that there are two infinite sets of orbits which are
orbits of ejection from one finite body and of collision with the other. Oneset is characterized by the fact that the ejection from each finite body is in
the direction away from the other. Reasons were given in 249 for believingthat they are limiting forms of the analytic continuations of the oscillating
satellites about the equilibrium point b. The other set is characterized bythe fact that the ejection from each finite body is toward the other finite
body. These orbits are probably limiting forms of the analytic continua-
tions of retrograde periodic planetary orbits. The probable series of changesto the first limiting form is shown qualitatively in Figs. 49 and 50. Beyondthe limiting form the orbits acquire loops about each of the finite bodies.
There are two retrograde periodic orbits of the types of superior planets.
Probably they both undergo evolutions to limiting forms of the types de-
scribed. This conjecture is supported by the fact that two closed orbits of
ejection were found b)*- computation, 247, for a related type of orbits.
i ... i". Fig. r>0.
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