A Generalised Spheroidal Wave Equation

A Generalised Spheroidal Wave EquationAuthor(s): A. H. WilsonSource: Proceedings of the Royal Society of London. Series A, Containing Papers of aMathematical and Physical Character, Vol. 118, No. 780 (Apr. 2, 1928), pp. 617-635Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/94926 .

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617

A Generalised Spheroidal Wave Equation. By A. H. WILSON, B.A., Emmanuel College, Cambridge.

(Communicated by R. H. Fowler, F.R.S.-Received December 19, 1927.)

1. Introduction. 1.1. The differential equation considered in this paper may be written in the

form d

(1 2) d}+ {X2 _ -1 _ + t}X = ? (1)

When p = 0 this equation becomes the equation giving the solution of V2X - X2X = 0 in spheroidal co-ordinates. The equation may therefore be called a generalised spheroidal wave equation. In Part I of this paper we shall consider equation (1) when p 0, and in Part II, which consists of sections 6 to 10, we shall consider the equation with p = 0. The transformation X = (-2 - 1)"3/2f reduces the equation to one with polynomial coefficients

(1 - 2)2fd- 2 (n3 + 1) iS f + W2_ -p2 - -- n3 (ns3 + )} f = 0. (2)

The further transformations - 1 = x andf = e-Ay bring the equation to the following form,

x (x + 2) 2 + 2 {n3 + 1 + (n + - 2X) x - x}y dx2 dx

+{(n3-2X) (ns+1) - [ - 2+2p -2X(n + 1 -p) x y 0, (3) or as we shall write it,

x (z +2) d + 2{n3 +1 + (n3 + 1- 2X) x-- X2}di

+ {t + 2pX - 2X (n3 + 1 --p) } y -= 0. (3A) We shall take (3A) as our standard form.

1.2. The equation (1) occurs in the important physical problem of deter- mining the possible energies of the ion of molecular hydrogen by means of Schrodinger's wave theory. In this problem n3 is a positive integer or zero, and in cases where this fact makes any difference to the results we shall assume it. We shall also assume X to be real. In the physical problem solutions are

required which are bounded in the real interval 1 < < oo . When (p - n3) is a positive integer n, there are n values of ,i for which equation (3A)

VOL. CXVIII.-A. 2

S



618 A. H. Wilson.

admits of a polynomial solution of degree (n -1). There are also other types of solutions.

1.3. The equation is a confluent form of the general second order linear differential equation with four regular singularities. Such an equation may be

represented schematically as

0 b c oo

y-=P 0 0 0 n3+l1-p x ,

-n - n 3 q n3 +l +p-q

the notation being an obvious generalisation of Riemann's P-function. Equation

(3A) is obtained by taking b = 2, n3 = n3' and by making c -> oo while qc -2 X.

The point at infinity is therefore an irregular singular point of the equation. The solution of an equation of this type is probably expressible as a homo-

geneous integral equation, but we shall obtain our results by means of a solution

in series.

2. The Solutions in a Finite Form.

We put p -n3 = n a positive integer and assume a solution

n-1 y = a, x.

0

The coefficients are determined by the following relations

2 (n3 + l) a] + (~ + 2pX) ao =

2 (m + 1) (m + q3 + 1- l)am+i

+ {m(m + 2n3+1-4X)+ +-2p X} a - 2X(m-n)aml=0 . (1)

for m 1, 2, 3,..., n - 2

{(n - 1) (n + 2n - 4X) + ,u + 2pX}a_i +- 2Xa_2 0

These linear equations for the coefficients will have a non-zero solution if and

only if the determinant of the system vanishes.

[4+2pA 2 (n + 1) 0 0 ... 0 0

2X(n-l1) [+2pX+2n3+-2-4X 4 (3 -+ 2) 0 ... 00

0 2 (n - 2) - +2pX-f 2(2ns33--4X) 6(n3--2)... 0 0 = 0 0 0 ..........................................)(+2).

.............. ..............oo.........

0 0 O 0 ... 2X [+2pX+(n-l)(n+-2n--4X). (2:

. .. .



Generalised Spheroidal' Wave Equation. 6i19

This relation between p- and X is of the .th degree in ~, and is the necessary and sufficient condition for the existence of a solution of the type assumed.

,L can easily be determined for any given values of n. The most interesting values of n are n = 1, 2, or 3. We denote the determinant by Dn (,u, X).

2.2. Particular Cases. 2.21. n 1. The determinant reduces to the single term u +- 2pX, and the corresponding

solution is y = ao. Therefore

y (x) = a0

- 2X (- 1) (3) or

'= 3= (n3 + 1)- X2_

2.22. n= 2.

D2 (OA;) -(V + 2pX)2 + 2 (i +- 2pX) (n3 + 1 - 2X) - 4 (n3 - 1) - 0.

Therefore

: = 2X(n3 - 1)- (n3 + 1) 1) + 2 + 4X2 (

PL == (n3 + 1)2 _ 2 /(n3 + 1)2 + 4X2 j To each value of X there are two polynomial solutions y (x) of the first degree. 2.23. n==3. The expression for a now becomes more complicated. For simplicity we

take ns =0, and put , +- '6X = ,i.

D3 (uA ) = 32$X2 (E + 2) - 4 (3,2 + 16l + 12) + _13 + 8[12 + 121L = 0.

2.3. All the solutions obtained so far give an expression for X (i) which is finite in any interval along the real axis not containing - oo. We shall show that these are the only solutions having this property. There are other solutions expressible in a finite form (when n3 : 0), which are bounded in the interval 1 < i < oo but which have a pole of order n3/2 at ~ = -1.

We proceed to obtain the most important types of solutions.

3. Solutions which are Finite in the Range (- 2 < x < 0).

3.1. If n is a positive integer, the solutions obtained in ? 2 satisfy the condition. When n is unrestricted such solutions will not be possible; but in either event a on-tnrminating solution can be foind.

2 s2



A. H. Wilson.

We assume a solution of equation (3A)? 1.1 to be of the form

y = aMxM. o

The recurrence relations are

2 (n3 + 1)a1 + (x + 2pX)ao = 01 2 (m + 1) (m - n3 + 1) a+(1)

+{m (m +2n +1 - 4X)+ + , 2p} a, -2 (m- n) a - 0

for m = 1, 2, 3,... J

By eliminating the coefficients we obtain an infinite determinantal equation connecting t, X and n. This will give a solution whether or not n is an integer. It is necessary to find the radius of convergence of this solution.

Writing (1) in the form

am+l =_ m (m + 2n3- 1 - 4X) + p + 2pX J 2X (m -n) a_ml am 2(m + 1) (m + n3 - 1) 2 (m + 1) (m + n3 +l)j

' a,,

we see that L am-1/%a is either finite or infinite. m->oo

If the limit is finite the radius of convergence is 2 and it is easily seen that the series is divergent at x = - 2. Choosing the second possibility, the solution

becomes an integral function of x.

Let Nm = am+,1/a,, and write the recurrence relation as

Nm = - t + v1N;1-lT l

The condition that N - 0 can be expressed in the form of the following infinite

continued fraction

- u = No == ___ - - ...O since Nm -0. 0 ul+N1 u1+-J-u2 --u+ 3 +

This continued fraction is convergent* and is the expansion of the infinite

determinant obtained by eliminating the coefficients from (1). The series (1),

subject to the condition expressed either by the determinant or by the continued

fraction, is therefore an integral function of x, and is of course finite in the range (- 2 < x < 0). It is not, however, finite in (0 < x < oc ). In fact to make

2X Nm-> 0 we must have N, - - and y (x) ~ e2A for large x. The correspond-

ing function X (i) behaves like eaX at +- co, and so differs essentially from the

solutions discussed in ? 2 which tend to zero as X ->o .

3.2. The other solutions associated with this expansion are of no interest. * Perron, " Die Lehre von den Kettenbriichen," p. 285.

620



Generalised Spheroidal Wave Equation.

Unless the relation between ,u, X and n is satisfied the solution will only be valid in the circle I x = 2 and the solution is unbounded near x = - 2. The second root of the indicial equation (- n3) gives a solution unbounded near the origin, and this type of solution is without interest.

4. Solutions in Descending Powers of the Variable.

4.1. The chief physical interest lies in the class of solutions X (i) which are

finite in the range (1 < i < o ). We have already obtained such solutions in

? 2, but it does not follow that these are the only ones satisfying the condition. If X (i) is finite in (1 < ? < oo ) then X (~) has either no finite singularities or else

a singularity at ~ = - 1. The functions of ? 2 are those of the first type, and

it remains to determine those of the second type. The obvious expansion is in

powers of (1 -- i). If we expand y (i) round = - 1 the solution will be of the form y ( A) = Au log ( +- 1) + B ( I -- 1)-n3 v, if n3 is an integer, where u and v are infinite series. The solution then has the desired singularity at - = - 1, but the series are not of a simple form, and the conditions at i = 1 and

- = oo will not be easy to satisfy. A more logical method is to expand the solu-

tion in descending powers of (1 + i). The expansion will then be valid outside

the circle 11 -+ =- 2; its infinity condition is satisfied and it only remains

to make the solution finite at =- 1.

4.2. Making the transformation t 1/(1 -+ ), the equation for y becomes

,t (1-2t) dY + 2 {(n. - 1)t2 - (n3 + 2X) t + X}t dt

+ [{(na +- 2X) (3 +- 1) - ' - 2kp - 2t - 2 (n3 + -p)] y . (1)

The important range of values is 0 < t < -2. We expand y as an ascending series in t

o y= tP Y amtm. 0

The indicial equation is p =- n-- + - p; and so only one integral is regular in this expansion; and even the regular integral is in general divergent.

The recurrence relations giving the coefficients are

2Xa1 + {-(n (3 -) (n3 + 4 + p) + (n3 + 2X) (n3 + 1) - '- 2Xp - 2)}a 0

2Xmam + {(m + n3 - p) (m- - p - - - p)

+'(n3 + 2X) (n3 + 1) - - 2 - X2- }am_

- 2 a_2 (m + n3 - 1 -- p) (m - 1 - ) = 0

for m >2

621



622 A. H. Wilson.

These equations give two possibilities. L a,,,/ai, is either infinite or

finite. In the first case the series is divergent everywhere, and is the asymptotic expansion of some solution. We must therefore choose the latter alternative

L am/am_ = 2 M-->o0 '

The radius of convergence is ? and we must examine more closely the case t = -. We can obtain an asymptotic expression for the coefficients by putting aml/an - =2 (1 - /n . ..) and equating powers of m in (2). It is found that

= 1 - n3. Therefore for large values of m the coefficients approximate to those in the expansion of (1 - 2t)-3.

Therefore

y ,t+l- (i - 2t)- n = (1 + [)P-1(1 -_ )- as t o ,

therefore X () - e- (1 -+ )"si/2+P-1 (1 -- )-n/2 near - 1.

If n3 = 0 then y - tl-" log (1 - 2t). Therefore X ([) has either a pole or a

logarithmic singularity at i = 1, and it is impossible to make X (i) bounded as i- 1 by imposing one relation between ,z', X and p, since X (i) is uniformly

infinite for all V', X, and p provided the series does not terminate. It is therefore necessary, and possible, to make the series terminate. It is

found that there are two different types of terminating series.

4.31. The Solutions in a Finite Form. If the series y = tP amtm is to terminate, then it is necessary from (2) that

(m + n3 - 1 - p) (m - 1 - p) = 0 for some m.

The first factor gives p - n3 = n a positive integer > 1. In this case, provided a certain condition between ,u', X and n is satisfied, the series will be

n-1 n- 1

y = tP a amt = tl-n E a,tm n > 1. 0 0

y is then a polynomial of degree (n - 1) in i, and must be the solution found in

? 2. The relation between [, X and n is given by ? 2.1 equation (2). This solution therefore gives us nothing new.

The second factor gives p to be a positive integer > 1. The solution then is

y = tn+l-P amttm (p> 1). 0

This solution is the product of (1 + )~: and a polynomial in [ of degree



Generalised Spheroidal Wave Equation. 623

(p- 1), and is not included in the previous expansions. It is given by the relations (2) for m = 2, 3, ... (p- 1), together with

{2 (1 - hn) - 2 - p2Xp - X2)}a: + 2 (%a -1) a2- 0

Eliminating the coefficients we obtain the following relation between t, X and p

p2p-_ ,+2X (p-n3-1)- X 2X 0 ...

-2(n3+l--p) (-p) (2+n3-p) (1-ns-4X-p) 4X 0 ...

+ (ln+2 X) (n +l) --pg-- 2Xp-- 2

o -2 (ng+2-p) (2-p)

o 0 ...

4.32. Specal Cases. We can discuss the special cases on exactly similar lines to those employed in

?2.2. 1. p =1.

t '= -2X - X2 (3) 2. p =2,

pt 1 -- 2-3 - X' :k 5/1 + 4X + 4X (4) 3. If n3 = 1 we can find an explicit impression for t'. I is ' 2pX - )2.

4.4. We have yet to prove that there are no further solutions in the range

Let the equation for y be written

4+ pay + y=o t2 dt

and denote the solution in series given by (2) by y. A second solution is given by

Y =Y 3. - e J

r 2^ e2j,! y

t"'3+1-j2' (1 - 2 2x)"n- Yand X" (2 illbe fl.;'-

2 '

and X (). will be finite.,.



A. H. Wilson.

Near t = 0

y ~ tn3+l-p e2AI dx y- it 3 x2-2p dx,

and since X ( 0) = O (e-t y), X (e) will be unbounded near t = 0. There are therefore no solutions which are not included in the terminating series.

5.1. Relations between the Solutions. We have obtained various types of solutions of the equation and we now

summarise our results and state various relations between the solutions. The fundamental solution is the integral function obtained in ? 3.1, subject to

a condition expressed by a continued fraction. This solution for y (x) is finite at x = 0, x = - 2, but behaves like e2x at infinity. The corresponding function X ([) is of the form

XI. () = Ae^ (a, + al- + ...) + Be-^t (b, + bl +.- ... ), where the infinite series are integral functions of [. Neither A nor B can be zero.

From this solution we can obtain a second solution in the usual manner. This second solution X12 (w) will have the same form at infinity as Xn1 (e) but will have singularities at i - i 1. These two associated sets of solutions form a denumerable infinity corresponding to the roots of the infinite continued fraction. There is a third set of solutions which are linear combinations of X11 ([) and X12 (i) for the same root of the continued fraction. This type of solution is that obtained in ? 4.2 which converges for i > 1. This last type gives a solution finite along the real axis as 0 o .

X([) = e-^ - n1+P (o +1 ...).

By changing the sign of X we obtain another solution

X (i) = et [-"-1- (do + ~ + +..)

These last two solutions are normal integrals and are convergent for l [ > 1. Here again, of course, there is a condition expressed by means of a continued fraction.

None of these solutions is finite in the range 1 < [ 6 oo, but by imposing an extra condition we can obtain polynomial solutions for y (x), and hence finite solutions for X ([) of the form

n-1 X () = e-At (2 _ )n3l/2 a m.

This completes the specification of the solutions arising from X11 (e).

624




The other type of solution arises from the solution mentioned in ? 3.2. This solution, which we shall denote by 1X1 (~ - 1), is in ascending powers of I - 1, has a singularity at - = - 1, and so is only valid inside the circle I - 11 = 2. This solution exists for all values of ,u except those for which X1, (E) exists. Also the coefficients in the expansions of lX1 ( - 1) and Xn1 (E) are the same inside the circle, except that each solution has its appropriate value of .i.

There is a second solution 2X1 ( - 1) which is finite at - 1 and has a singularity at ~ = 1. The expansion is therefore valid in the annulus 0< - 11 < o . There are corresponding expansions at - 1; 1X2 ( + 1) which is finite at ~ = 1 and infinite at - 1; and 2X2 (G + 1) which is infinite at -=--1 and finite at = - 1. Only two of these functions are

independent. In fact we have

LX (~- 1)= 1X2( + 1)

2X1(- )= 2X2 (+ 1).

The only interest we have in these functions is to prove that they are not finite in the real interval 1 < e < o. This is a matter of some difficulty as we cannot obtain explicit expressions for the functions which are valid' in the whole of the range in question. We proceed to give the form of the solutions, and then to give a proof of their unsuitability for the physical problem.

With these functions is associated an expansion at infinity. This expansion is the asymptotic one obtained in ? 4.2, and is divergent everywhere. There is also another expansion obtained by changing the sign of X.

The solutions are of the following form

aX (- 1) = Alog (~- 1)u (u - 1) + B (- 1)-"s vI(-1)

1X2 (E + 1) = Clog (G + 1) U2 (~ + 1) + D ([ +1'2 ( + 1).

When I | is large these two functions must be linear combinations of the asymptotic expansions.

2X ( -1) e- A r-1+ (Co + + ...)+ el 0n (d +i +

(i + l) yeAt70 t-X( + L ) + + 3e ~-n31P (o++...).

What we have to prove is that none of oc , y, 8 can be zero. We shall only consider the function 1X1 (E - 1) (or 1X2 (E + 1)) as this is the only function which could give a solution finite in (1 6 < o? )*

The form at infinity of Xi ( - 1) cannot be altered by a change in the value

625



A H. . Wilson.

of -,. Now the two functions XX1 ( - 1) and Xln (~) have the same expansions in the circle | - 1 t 2 except that the values of ,i are different. By choosing

- suitably lX1 (~ - 1) becomes Xl (~) which always involves both exponentials

eAt and e-". We therefore see that 1X1 ( 1 - 1) is not finite along the real axis as e -+ oo.

We can obtain a solution of the required form by imposing two conditions. This type of solution is obtained in ? 4.31 from the asymptotic expansion; it has a pole at i = - 1. In a similar manner we could obtain a solution in a finite form, having a pole at = 1.

It may seem at first sight that the solutions in a finite form are incompatible with the fact that jX1 ([ - 1) and X1 (1) are unbounded at infinity. This is not so. For definiteness consider Xn (). We may write the solution as-

Xn () = e-^ Z amm = e-^t y () if n3 = 0.

Our argument shows that if this function is to have no singularities in the finite

part of the plane, then y ( e) e2a. If, however, the series for y ([) terminates

am- values of m > 0.

More precisely, what we have proved is that neither X11 () nor 1X1 (, - 1) is finite in 1 < X < o unless it consists of the product of e-^t and a terminating series. To make the series terminate requires two conditions.

5.2. Asymptotic Expressions when X is Large. We can obtain a formal solution of equation (2) ? 1 which corresponds to

the solution in a finite form found in ? 2. Since we have proved that these solutions are finite in the range - 1 < < 1 we shall confine ourselves to this

range. For the purpose of obtaining solutions in descending powers of X, the follow-

ing transformations are convenient.

f() -=e^- y(e) and e = cos x.

The equation for y then is

? + {(2n3+ 1) cot x+ X sin X} dy+ {i + 2 (n+ 1 -p) X cos x}y = (1) dx2 dx

where tx l= +X2 - n(n3-+).

Owing to the occurrence of the term cot x this equation is not of Hill's type, and therefore there is no advantage in reducing it to the normal form. The

626




boundary condition is now replaced by the condition of the :solution being periodic.

X is large and positive and we assume solutions of the form

Y -- Yo+ Xyl+ -2y2+... - X (o0? + 1() X,1 + 2(2) X-2 + ..)

If we substitute these expressions in (1) and equate powers of X we obtain the following set of equations

2 sin x dq + {tO? + 2 (n3 + 1-p)cos x yO = O, (2) dx

2 sin x d + {Io + 2 (n3 + 1 -p) cos x} y + 2c sin x C

- - +;A2na+ l)ootxdJ + Fl(2)y -o (3

2sinx 2 +{0io+2 (n3 + 1 p) COSx$}Y2 dx&2

+ (2n3 + 1) cot x + c(L) + (2)Y= -0 (4)

The solution of equation (2) is

Yo = (sin ?x) p-ns-l -1 (cos X)?-l.-1 +1Ao.

Periodic solutions are then given by

P? = - 2 (p - n- 1 - r), Yo = (sin Jx) (cos X)2 (p-",-)-r

where r O0, 2, 4, ... 2 (p - n3 - 1), since if r is an even integer the expansiong in g will be terminating series.

Equation (3) then takes the form

2 sin x Y - {2? + 2 (3 + 1 -p) cos 3x}y

+ ir (r + 2n3) (sin x)r-2 (cos )()2-"-l)-r

+- {(1) - ( -- 3 -- 1) (p + n3)} (sin ix) t(COS ~)2(cos -n)--

+ (2p 2) (2p - 2n3 - 2 - r) (sin 2Z) (cos a)a ("-X--)2 0 O.

If yl is purely periodic we must have

1a) = (p + 3) (P -- n - 1). Then

i 2 (?osec2 x+2 log tan )x) (p-1) (2p-2n3-2-r)

^^ ^(se8: 2- log taUrt )-t( ()2n(COS 1)2 L - (sect ix+2 log ta.i: ja)r (r;+-2n) J

627



A. H. Wilson.

To this approximation we have,

t l 2 (p -n3 -1 -r) X + (p + n3) (p - n3-1) or

.' =p(p- 1) +2(p-n- l-r) X-- X2 +0{ (/X (5)

If we compare this expression with the explicit values found for particular cases in ? 2.2, we see that the constant term in [' is only correct when p = n3 + 1. The other terms agree in all cases. It is therefore useless to, calculate the

higher coefficients t(2, etc.

PART II.

6. The Spheroidal Wave Function.

If p 0= the equation takes a much simpler form. Many methods are now

available for the discussion of the equation which are inapplicable when p ? 0.

There will, however, be no solutions in a finite form, and so no solutions which are finite in 1 < <6 o. All the results of the previous sections which do not

depend on (p - n) being a positive integer are also true when p = 0, but they can usually be obtained more simply. We shall write the equations as

di { _( 2) } { 22 _ X= (IA)

and X = (1 - [z)"/2f

(] -- 2)22 (zn3 + 1)1 + {X22+ - s(ns + 1)}f= . (1B)

The equation is similar to Mathieu's equation, and occurs in many physical

problems connected with wave motion in spheroids. Solutions are required which are finite for - 1 < o < o 1 or for 1 < < where a is some finite con-

stant. The equation is well known and has been considered by various writers.*

The account which follows is simpler than those referred to, and obtains the

results in a more manageable form for physical applications. The methods used

subsequently are inapplicable when p $ 0 as they depend on the fact that the

expansion proceeds in powers of [2. The solutions may be either odd or even

functions, and can be expanded in powers of i or as series of associated Legendre functions. The expansion in Legendre functions has the advantage of rapid

convergence when X is small, but gives rise to complicated expressions. We shall

*Niven, ' PhiL, Trans.,' A, vol. 171; Maclaurin, 'Trans. Camb. Phil. Soc.,' vol. 17; Abraham,' Math. Ann.,' vol. 52; Poole, 'Quarterly J. Math.,' vol 49.

628



Generalised Spheroidal Wave Equation. 629

be mainly concerned with solutions which are finite in the range - 1 < < 1, and this condition determines A as a function of X. Unless X is small this relation is best determined from the series in i. Finally we can obtain expansions valid for I 1 large, and also for X large. These expansions have not been

given before.

7. Expansions in Powers of i.

There are two different types of series. One an odd series; the other an even series. They refer, however, to different values of J.

7.1. The Even Series.

0 If we assume a solution f-= a,i;2B, then we have the recurrence relation

(2m + 2) (2m + 1) a2,+2 = {2m (2m - 1) - 4m (n3 + 1) - ( + n3 (n3 + 1)} a2m --2 az2m-2.

Putting N == a,2M+2/a2-, this becomes

N.~ _ ({2m + n3 + 1) (2m + n3)- _ 21 (1)

(2m + 2) (2m +1) (2m + 2) (2m + 1) N,i'

Two cases arise. 1. If N;' is finite then L N = 1.

in -> , .'

The next approximation to Nm is obtained by putting Nm =1 - in (1).

We find that o = 1 - n. Therefore for large values of m the coefficients approximate to those in the expansion of (1 - 2)-n if n3 ? 0, and of

log (1 --2) if n3 = 0. In this case X (E) is infinite at i = : 1. 2. If N;1 is unbounded, then the series represents an integral function, and

is therefore finite in the range --1 1. Also N~ k2/4m2 and f(E) -'cosh X.

The condition that L N = 0 gives a transcendental equation between m -> m'x

[ and A2. We shall obtain this relation in several equivalent forms. The recurrence relation for the coefficients can be written

V2m+-2 a2m+2 + a2., + U a2-2 a2,, 2 0 where

(2m + 2) (2m + 1) 2m"+2 -(2m + n3 +-1) (2m+ n3) - -

U2 = - ( 82m-2 -(2m + n3 + 1) (2m + n3) -



A. . Wilson.

Therefore

Va4? -2+ a% = O

v46 + a2 + Uao0 = 0

V6a6 a4 + t2'2 a= 0

and so on. The condition N - 0 is equivalent to the vanishing of the following infinite

determinant,

1 V2 0 0 0

v4 0 0

1 v6 0

U4 1 Vs

0 u6 1

= 0. (2)

This determinant is of von Koch's type and is absolutely convergent. The

rapidity of convergence is not great enough to enable ( to pe calculated easily from the determinant. A more suitable form is obtained by expanding the determinant as a continued fraction, but we can obtain the expansion directly from equation (1).

We have

m~l (2m + n + 1) (2m + n3) - - - (2m +-) (2m + 1) N.'

therefore

N - 3.432 5 6X: 0 (n3+3) (n3+2)-2.- (n3+5) (n3+4)-V.- (n3+7) (n,+6)-ji -'

since N. - 0 as m - 00o, and since the continued fraction is convergent* for all values of X and p. Also No = a2/aO = {n3 (n3 +- 1) - P}/2, and so the equation for pV becomes

n3 (n3+ 1)-

1.2X2 3.4X2 6X2

(n.3+3) (3+2),t-- (n3+5) (n+"4)-- (n3+7) (n+6)---' (3)

* Perron, lo. cit., p. 468.

uo 1

0 u

O O

O0

6.aoD;'




This is the most symmetrical form of the equation, but we can exhibit its

convergence better by dividing by suitable factors

n3 (n3 + - 1)

_ 1. 22/{ (n3 + 2) (n + 3) } 3. B2/{ ( ( rn + 3)2 ( + 4)( + 5) *

,^. '

. ^ / '

- ~

i 1

(n%3 + 2) (n. + 3) '< 3+4)3 + 5) (4)

This is the best form for the relation between p and A. The expression which has usually been used for i is an expansion in asoeding

powers of X2. The method used by Niven and Maclaurin for obtaining the series is very laborious. The series can be easily obtained from equation (4). The continued fraction has an infinite number of roots and we can approximate to

any of these. For simplicity we take n3 = 0. The first root . for p is that one for which the zero approximation is l = 0.

If we substitute this value for p on the right-hand side of (4) we obtain the first approximation = - -A2. This can be again substituted on the right and a second approximation obtained. In this way t, can be evaluated to any desired power of X2 with very little labour. The result is

,, _ 1 v 2 4 16 3 _ 135 35 . 5 7 37 . 53 72

To find the second root s3 we invert the continued fraction into the following form, 1 . 2X21{ (n + 2) (n3 + 3)}

%3 (n3 + 1)-i

~=1-. E^ -3 4X2/{(s -2) (n3 + 3) (n3 + 4) (a +5)}. (n3 +2) (n + 3)- 1-

(n3 + 4) (n3 + 6) We can then evaluate the root by the same method.

When n3 = 0 211 +94 9

6-21 (21)3

The roots can be evaluated similarly for any given value of n3.

7.2. The Odd Series. 0 If we assume a solution f = S b&nzl 12OT1 the recurrence relation for the

coefficients is

(2m + 2)(2m - 3) b2m+3 = {(2m + n3 + 1)(2m + n3 + 2) - X bb2+ - -r (5)

631,



A. H. Wilson.

The work is exactly parallel to that in ? 7.1, and we shall only give the expression for t, as a continued fraction,

(n3+1)(n3+2) - ~

2 . 3X2/{(n +3) (n3+4)} 4. 5X2/{(n+3) (3+4) (n,+5) (+6)} (6)

1-- - 1--. -. (n3+3)(n3+4) (n3+5) (n3+6)

By expanding the continued fraction we can obtain the roots (2' , JL6, * * .

as power series in X2.

8. Expansions in Series of Associated Legendre Functions.

In this type of expansion we deal directly with X (i). The form of the equation suggests the expansion

X (i) = aS ().

We have the following identity for 2p3 (a).

02p () - (m - n3 + 1) (m - f3 + 2) pn m ( -(2m + 1) (2m + 3)

+

+ 2m (m + 1) - 232 -1 p (e) + (m + 3) (m + n3 -) p ( (2m-1 ) (2m + 3) (2m -1) (2m- +1) -

There will therefore be two different types of series according as m is even or odd.

8.1. The Even Series.

X(),-= P ,2m + ().

By substituting this expression in the differential equation, using the relation for 2P%(E) and equating to zero the coefficients of the various harmonics, we obtain the following recurrence relation for the coefficients.

(2m + 2n3 +1) (2m + 23 + 2) (4m + 2n3 + 3) (4m + 2n + 5)a2+2

+ 2~ [(2m_. na) (2mm?n43+_1) + ;2 (4m-2n3) (2m+-2n3+1) --2n32--1 a2 [F_(2m+n.,) ? (4m+-2n-1) (4m{2na,+3)

(2m- 1)2m (4m + 23- 3) (4m + 2% -1) a2m2 = (1)

There are two possibilities. Either a2m+1a2/m tends to zero or it is unbounded. We must choose the former alternative. As above, this condition is best

632




expressed by an infinite continued fraction. The expression is very com-

plicated and makes the result of little value except for small values of X2.

We can obtain the series for V. from the continued fraction by successive

approximations. The first approximation is obtained by equating to zero the coefficient of a2m in (1). Even for this purpose it is preferable to use the expression found in ? 7 as the quantities are less complicated. 8.2. The Odd Series.

X(i)-= b2,+4 P2+n+l (k). m=0

Using the same method as in ? 8.1 we obtain a recurrence relation for the coefficients. This equation is the same as (1) with (2m + 1) written for 2m. The equation for . will be given by the corresponding continued fraction.

9. Expansions for Large Values of the Variable.

In ? 4 we obtained an expansion for f(E) round the irregular singularity at

infinity. When p 0 we can apply the same method. The expansion proceeds in inverse powers of ( +- 1) or (t - 1), the corresponding solutions being continuations of one another. (Maclaurin has determined an expansion in E-1, but this involves four successive coefficients and therefore is not so good as the present one.) We shall only give the expansion in terms of (1 + k).

Put f() - eyI ().

Then as in ? 4

Y )(1 + )-X-(1 + -a a (1 + P)-, wM=O

where

2Xa, + {(%3 + 1) (3 - 4X) + X + X2 - (,3% + 1)(%n - 2X)} ao 0

2XWam + {-(m + 3) (m - - 1 + 4X)

+ + x2 (n3 +1) (n3- 2)} a^_, + 2 (m + n- 1) (m - 1) ama = 0

It is usually stated that such an expansion is asymptotic. This is not quite correct. When , has one of the characteristic values determined in ? 7, this series is convergent for I (I> 1. Othe ie the series is divergent but asymptotic.

By changing the sign of X we can obtain another expansion

f () = e- 2 ( T).

VOL. CXVIII.-A.

633

2 T




10. Asymptotic Expansionsfor Large Values of X.

We can obtain two asymptotic expansions by the method used in ? 5.2. The work is very similar and will not be given at length.

Putting fl ( e) = e^ Yi (,) and i = cos x, and expanding y, and {,j in powers of X-1 we arrive at the following equations.

2 sin xdy + {- 1? + 2 (n3 + 1) cos x} y? = 0, (1) dx

2 sin xdx + {- o + 2 (n + 1) cos x} y _ d2Y dx +k\3,Y 1 - dx2

d 0 - (2n + 1) cot x Yl (1) y0 = 0, (2)

and so on. The solution of equation (1) is obtained by taking ,l? = 2 (r + n3 + 1)

where r is an integer, and is

Y1? = (sin ix)7/(cos ? x)2( n2+l) +r

Since there are terms with negative indices it is not possible to obtain a solution in finite terms. The solution breaks down at x = ?i 7.

Equation (2) now becomes

2 sin x dy + {_-- 1io - 2 (n3 - 1) cos x} y1' dx - Tr (r + 2n3) sin ({x)'-2/(cos 2 (xi+1)+r

-- {(.1) + n3 (n3 + 1)} (sin x)7/(cos 1x)2 (n+ 1) +r

- - (2n3+2-+r)(sin ?x)7/(cos -x)2(n3+1) +r+2 - 0,

and so i(1) = - 13 (3 -+ 1).

yli() will have the same form as in ? 5.2. It is probable that the expansion becomes untrustworthy at this point. For this reason and also because the next approximation is very laborious, we shall not pursue the investigation further.

We have - + 2x ?(r^ + )+ on(l/x) ()

and (3) f1 () = e^ C OS e {(sin cx)2/(cos x)2(as+) +r 0 (1/)} J

By assuming a solutionf2 (E) = e"' Y2 ([) we can obtain another solution for the same value of V. This will be obtained by writing (7 - x) for x in (3).

Therefore

f () 2 e e- coS cos X

( x) /(sin -X)2(n+)+' + (1/X)}. (4) f([ = Y2 t 2

634



lonised Hydrogen Mllolecule. lonised Hydrogen Mllolecule.

These solutions fail altogether near certain points, and it is necessary to use the

expansion with care. The value for , is by no means certain, and it is usually necessary to use the continued fraction (4) ? 7 to check any results obtained from the asymptotic expansion.

Any solution of the equation can be expressed as a linear combination of these two solutions,fl, f2. To obtain the odd and even solutions of ? 7 we must satisfy the following relations,

f(I) =f(r - E) for the even functions, and

f(E) -f( -(-) for the odd functions.

Our approximations do not distinguish between the values for ji corresponding to the even and odd functions, which are

f1i() -t f2(S) an even function, and

fl(.) -f2()) an odd function.

The Ionised Hydrogen Molecule.

By A. H. WILSON, B.A., Emmanuel College, Cambridge.

Communicated by R. H. Fowler, F.R.S.-Received December 19, 1927.)

1. Introduction.

The model which has been proposed for the ion of the hydrogen molecule H2+, consists of one electron and two protons. Since the mass of the electron is negligible compared with that of the protons, we may, to a first approximation, consider the protons as at rest. The system is then a particular case of the problem of three bodies, and can be solved completely classically. This has been done by Pauli,* and more recently by Niessen.t The value obtained by Pauli for the energy of the normal state is not in agreement with the experimental result inferred from the ionisation potential and heat of dissociation of the molecule. Niessen obtains the experimental result by the introduction of half integer quantum numbers.

* 'Ann. d. Physik,' vol. 68, p. 177 (1922), t 'Z. f. Physik,' vol. 43, p. 694 (1927).

2T 2

These solutions fail altogether near certain points, and it is necessary to use the

expansion with care. The value for , is by no means certain, and it is usually necessary to use the continued fraction (4) ? 7 to check any results obtained from the asymptotic expansion.

Any solution of the equation can be expressed as a linear combination of these two solutions,fl, f2. To obtain the odd and even solutions of ? 7 we must satisfy the following relations,

f(I) =f(r - E) for the even functions, and

f(E) -f( -(-) for the odd functions.

Our approximations do not distinguish between the values for ji corresponding to the even and odd functions, which are

f1i() -t f2(S) an even function, and

fl(.) -f2()) an odd function.

The Ionised Hydrogen Molecule.

By A. H. WILSON, B.A., Emmanuel College, Cambridge.

Communicated by R. H. Fowler, F.R.S.-Received December 19, 1927.)

1. Introduction.

The model which has been proposed for the ion of the hydrogen molecule H2+, consists of one electron and two protons. Since the mass of the electron is negligible compared with that of the protons, we may, to a first approximation, consider the protons as at rest. The system is then a particular case of the problem of three bodies, and can be solved completely classically. This has been done by Pauli,* and more recently by Niessen.t The value obtained by Pauli for the energy of the normal state is not in agreement with the experimental result inferred from the ionisation potential and heat of dissociation of the molecule. Niessen obtains the experimental result by the introduction of half integer quantum numbers.

* 'Ann. d. Physik,' vol. 68, p. 177 (1922), t 'Z. f. Physik,' vol. 43, p. 694 (1927).

2T 2

635 635



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