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BOUGHT WITH THE INCOMEFROM THE
SAGE ENDOWMENT FUNDTHE GIFT OF
Henrg W. Sage1891
-
ABEL'S THEOEEM AND THE ALLIED THEOKY
INCLUDING THE
THEOKY OF THE THETA FUNCTIONS
-
Honbon: C. J. CLAY and SONS,CAMBBIDGE UNIVEBSITT PRESS
WAREHOUSE,
AVE MARIA LANE.
ffilaagota: 263, AEGYLE STEEET.
lEipjig: F. A. BEOCKHAUS.J^ito gotk: THE MACMILLAN COMPANY.
-
ABEL'S THEOEEMAND THE
ALLIED THEOBYINCLUDING THE THEORY OF THE
THETA FUNCTIONS
BY
H. F. ^AKER, M.A.
FELLOW AiJD LECTURER OF ST JOHN'S COLLEGE,UNIVERSITY LECTURER IN
MATHEMATICS.
CAMBRIDGE:AT THE UNIVERSITY PRESS.
1897
[All Rights TMerved]
-
dDambdDge
:
PRINTED BY J. AND C. F. CLAY,
AT THE UNIVERSITY PRESS.
-
PREFACE.
It may perhaps be fairly stated that no better guide can be
found to theanalytical developments of Pure Mathematics during the
last seventy yearsthan a study of the problems presented by the
subject whereof this volumetreats. This book is published in the
hope that it may be found worthy toform the basis for such study.
It is also hoped that the book may beserviceable to those who use
it for a first introduction to the subject.And an endeavour has
been made to point out what are conceived to be themost artistic
ways of formally developing the theory regarded as complete.
The matter is arranged primarily with a view to obtaining
perfectlygeneral, and not merely illustrative, theorems, in an
order in which they canbe immediately utilised for the subsequent
theory
;particular results, however
interesting, or important iu special applications, which are not
an integralportion of the continuous argument of the book, are
introduced only so faras they appeared necessary to explain the
general results, mainly in theexamples, or are postponed, or are
excluded altogether. The sequence andscope of ideas to which this
has led will be clear from an examination of thetable of
Contents.
The methods of Riemanh, as far as they are explained in books on
thegeneral theory of functions, are provisionally regarded as
fundamental ; butprecise references are given for all results
assumed, and great pains havebeen taken, in the theory of algebraic
functions and their integrals, and inthe analytic theory of theta
functions, to provide for alternative developments
of the theory. If it is desired to dispense with Riemann's
existence theorems,
the theory of algebraic functions may be founded either on the
arithmeticalideas introduced by Kronecker and by Dedekind and Weber
; or on thequasi-geometrical ideas associated with the theory of
adjoint polynomials
;
while in any case it does not appear to be convenient to avoid
reference to
either class of ideas. It is believed that, save for some points
in the
periodicity of Abelian integrals, all that is necessary to the
former ele-
mentary development will be found in Chapters IV. and VII., in
connection
with which the reader may consult the recent paper of Hensel,
Acta
Mathematica, xviii. (1894), and also the papers of Kronecker and
of
B. b
-
vi PREFACE.
Dedekind and Weber, GreUe's Journal, xci., xcii. (1882). Aud it
is hopedthat what is necessary for the development of the theory
from the elemen-
tary geometrical point of view will be understood from Chapter
VI., in
connection with which the reader may consult the Ahel'sche
Functionen of
Clebsch and Gordan (Leipzig, 1866) and the paper of Noether,
Mathematische
Annalen, vii. (1873). In the theory of Riemann's theta
functions, the
formulae which are given relatively to the f and p functions,
and the
general formulae given near the end of Chapter XIV., will
provide sufficient
indications of how the theta functions can be algebraically
defined; the
reader may consult Noether, Mathematische Annalen, XXXVII.
(1890), and
Klein and Burkhardt, ibid, xxxil.XXXVI. In Chapters XV., XVII.,
and
XIX., and in Chapters XVIII. and XX., are given the beginnings
of that
analytical theory of theta functions from which, in conjunction
with thegeneral theory of functions of several independent
variables, so much is tobe hoped ; the latter theory is however
excluded from this volume.
To the reader who does not desire to follow the development of
thisvolume consecutively through, the following course may perhaps
be sug-gested ; Chapters I., II., III. (in part), IV., VI. (to 98),
VIII., IX., X.,
XI. (in part), XVIII. (in part), XII., XV. (in part) ; it is
also possible tobegin with the analytical theory of theta
functions, reading in order Chapters
XV., XVI., XVII.. XIX., XX.
The footnotes throughout the volume are intended to contain
themention of all authorities used in its preparation ;
occasionally the hazardousplan of adding to the lists of references
during the passage of the sheetsthrough the press, has been adopted
; for references omitted, and f
-
PREFACE. VU
on two occasions, by Professor Felix Klein. In the preparation
of the bookI have been largely indebted to his printed
publications; the reader isrecommended to consult also his
lithographed lectures, especially the onedealing with Biemann
surfaces. In the final revision of the sheets intheir passage
through the press, I have received help from several friends.Mr A.
E. H. Love, Fellow and Lecturer of St John's College, has readthe
proofs of the volume ; in the removal of obscurities of
expressionand in the correction of press, his untiring assistance
has been of greatvalue to me. Mr J. Harkness, Professor of
Mathematics at Bryn MawrCollege, Pennsylvania, has read the proofs
from Chapter XV. onwards ; manyfaults, undetected by Mr Love or
myself, have yielded to his perusal; andI have been greatly helped
by his sympathy in the subject-matter of thevolume. To both these
friends I am under obligations not easy to discharge.
My gratitude is also due to Professor Forsyth for the generous
interest hehas taken in the book from its commencement. While, it
should be added,the task carried through by the Staff of the
University Press deserves morethan the usual word of
acknowledgment.
This book has a somewhat ambitious aim ; and it has been written
underthe constant pressure of other work. It cannot but be that
many defects
will be found in it. But the author hopes it will be sufficient
to shew that
the subject offers for exploration a country of which the
vastness is equalledby the fascination.
St John's College, Cambkidge.
April 26, 1897.
1)2
-
CONTENTS.
CHAPTER I.
The subject of investigation.I Fundamental algebraic
irrationality
2, 3 The places and infinitesimal on a Riemann surface4, 5 The
theory unaltered by rational transformation6 The invariance of the
deficiency in rational transformation ; if a
rational function exists of order 1, the surface is of
zerodeficiency
7, 8 The greatest number of irremoveable parameters is 3p - 3
.9, 10 The geometrical statement of the theoryII Generality of
Kiemann's methods ...
PAGES
-
CONTENTS.
PAGES
28, 29 Cases when the poles coalesce ; the p critical integers .
. 34, 35
30 Simple anticipatory geometrical illustration 36, 37
31 33 The {p-\)p{p + \) places which are the poles of rational
functionsof order less than p+\ 3840
34_36 There are at least 2p+ 2 such places which are distinct .
. 414437 Statement of the Riemann-Boch theorem, with examples . .
4446
CHAPTER IV.
Specification of a general form of Riemann's integrals.
38 Explanations in regard to Integral Rational Fimctions . . 47,
48
39 Definition of dimension ; fundamental set of functions for
the
expression of rational functions 4852
40 Illustrative example for a surface of four sheets . . . . 53,
54
41 The sum of the dimensions of the fundamental set of
functionsis p+n-\ 54, 55
42 Fundamental set for the expression of integral functions . .
55, 5643 Principal properties of the fundamental set of integral
functions . 5760
44 Definition of derived set of special functions Q, 0i, ...,
n-\ 616445 Algebraical form of elementary integral of the third
kind, whose
infinities are ordinary places ; and of integrals of the
firstkind 6568
46 Algebraical form of elementary integral of the third kind in
general 687047 Algebraical form of integral of the second kind,
independently
deduced . 717348 The discriminant of the fundamental set of
integral functions . 7449 Deduction of the expression of a certain
fundamental rational
function in the general case ....... 75^7750 The algebraical
results of this chapter are sufficient to replace
Riemann's existence theorem .... . . 78, 79
CHAPTER V.
Certain forms of the fundamental equation of the Riemann
surface.
51 Contents of the chapter 8052 When p>l, existence of
rational function of the second order
involves a (1, 1) correspondence 815355 Existence of rational
function of the second order involves the
hypereUiptic equation . . 8184
56, 57 Fundamental integral functions and integrals of the first
kind 858658 Examples 8759 Number of irremoveable parameters in the
hypereUiptic equation ;
transformation to the canonical form .... 88896063 Weierstrass's
canonical equation for any deficiency . 90
92
-
CONTENTS. XI
646667, 68
69717279
Actual formation of the equationIllustrations of the theory of
integral functions for Weierstrasa's
canonical equationThe method can be considerably generalised
....Hensel's determination of the fundamental integral
functions
PAGES
9398
99101102104105112
CHAPTER VI.
Geometrical investigations.
80 Comparison of the theory of rational functions with the
theoryof intersections of curves .... . . 113
8183 Introductory indications of elementary form of theory .
11311684 The method to be followed in this chapter 11785 Treatment
of infinity. Homogeneous variables might be used . 118, 11986 Grade
of an integral polynomial ; number of terms
;generalised
zeros 120, 121
87 Adjoint polynomials ; definition of the index of a singular
place . 12288 Pliicker's equations ; connection with theory of
discriminant . 123, 124
89, 90 Expression of rational functions by adjoint polynomials .
. . 125, 12691 Expression of integral of the first kind 12792
Number of terms in an adjoint polynomial ; determination of
elementary integral of the third kind 12813293 Linear systems of
adjoint polynomials ; reciprocal theorem . . 133, 134
94, 95 Definitions of set, lot, sequent, equivalent sets,
coresidual sets . 135
96, 97 Theorem of coresidual sets ; algebraic basis of tlie
theorem . . 13698 A rational function of order less than ^4-1 is
expressible by
-
Xll CONTENTS.
XX PAGES
126128 The function ylr{x,a; z, c) ; its utility for the
expression ofrational functions 174176
129 The derived prime function ^(^, ); used to express
rationalfunctions....... ^^'
130, 131 Algebraic expression of the functions \jf{x,a; z, c,,
..., Cp),yjf(x,a;z,c) 177, 178
132 Examples of these functions; they determine algebraic
expres-
sions for the elementary integrals 179182
133, 134 Derivation of a canonical integral of the third kind;
for whichinterchange of argument and parameter holds; its
algebraicexpression ; its relation with Riemann's elementary
normalintegral ......... . 182185
135 Algebraic theorem equivalent to interchange of argument
andparameter . 185
136 Elementary canonical integral of the second kind . . . 186,
187
137 Applications. Canonical integral of the third kind deduced
fromthe function i//-(a;, a; z,c^, ...,Cp). Modification for the
func-
tion yfr(x,a; z,c) 188192138 Associated integrals of first and
second kind. Further canonical
integrals. The algebraic theory of the hypereUiptic integi-alsin
one formula 193, 194
139, 140 Deduction of Weierstrass's and Riemann's relations for
periodsof integrals of the first and second kind .... 195197
141 Either form is equivalent to the other . . . . 198142
Alternative proofs of Weierstrass's and Riemann's period relations
199, 200143 Expression of uniform transcendental function by the
function
yjr (x, a ; z, c) . . . . . . . . . 201144, 145 Mittag-Leffler's
theorem 202204
146 Expression of uniform transcendental function in prime
factors 205147 General form of interchange of argument and
parameter, after
Abel . . ... .... 206
CHAPTER VIII.
Abel's Theorem. Abel's differential equations.
148150 Approximative description of Abel's theorem151
Enunciation of the theorem .......152 The general theorem reduced
to two simpler theorems
153, 154 Proof and analytical statement of the theorem .155
Remark ; statement in terms of polynomials156 The disappearance of
the logarithm on the right side of the
equation ......157 Applications of the theorem. Abel's own proof
.
158, 159 The number of algebraically independent equations given
by thetheorem. Inverse of Abel's theorem ....
160, 161 Integration of Abel's differential equations ....162
Abel's theorem proved quite similarly for curves in space
.
207210210
211, 212212214
215
216217222
222224225231231234
-
CONTENTS.
CHAPTER IX.
Jacobi's inversion problem.
163 statement of the problem164 Uniqueness of any solution165
The necessity of using congruences and not equations
166, 167 Avoidance of functions with infinitesimal periods168,
169 Proof of the existence of a solution ....170172 Formation of
functions with which to express the solution
connection with theta functions
PAGES
235236237
238, 239239241
242245
CHAPTER X.
Riemann's theta functions. General theory.
173 Sketch of the history of the introduction of theta functions
. 246174 Convergence. Notation. Introduction of matrices . . . 247,
248
175, 176 Periodicity of the theta functions. Odd and even
functions . 249251177 Number of zeros is p . . . . . . . . . 252178
Position of the zeros in the simple case . 253, 254179 The places
ij, ..., m^ . . . . . . 255180 Position of the zeros in general
256, 257181 Identical vanishing of the theta functions . ... 258,
259
182, 183 Fundamental properties. Geometrical interpretation of
the placesmi, ..., jp 259267
184186 Geometrical developments ; special inversion problem ;
contactcurves .... 268-273
187 Solution of Jacobi's inversion problem by quotients of
thetafunctions 274, 275
188 Theory of the identical vanishing of the theta function.
Ex-pression of (^-polynomials by theta functions . . . 276282
189191 General form of theta function. Fundamental formulae.
Periodicity 283286192 Introduction of the ffimctions. Generahsation
of an elliptic formula 287193 Difference of two f functions
expressed by algebraic integrals and
rational functions 288194196 Development. Expression of single f
function by algebraic integrals 289292197, 198 Introduction of the
f> functions. Expression by rational functions 292295
CHAPTER XI.
The hyperelliptic case of Riemann's theta functions.
199 Hyperelliptic case illustrates the general theory . . .
296200 The places Wi,..., nip. The rule for half periods . . . 297,
298
201, 202 Fundamental set of characteristics defined by branch
places 299301
-
XIV CONTENTS.
203
204, 205206213
214, 215
216
217218220
221
PAGES
Notation. General theorems to be illustrated .... 302
Tables in illustration of the general theory 303309Algebraic
expression of quotients of hyperelliptic theta functions.
Solution of hyperelliptic inversion problem .... 309317
Single f function expressed by algebraical integrals and
rationalfunctions 318323
Rational expression of jl> function. Relation to quotients of
theta
functions. Solution of inversion problem by p function . .
323327Rational expression of f> function 327330Algebraic
deduction of addition equation for theta functions
when p= 2; generaUsation of the equation tr(u+v}a-(u-v)=
(Ai.a^v.{((>v-flhi) 330337
Examples for the case p=2. Qopel's biquadratic relation . .
337342
CHAPTER XII.
A PARTICULAR FORM OF FUNDAMENTAL SURFACE.
222 Chapter introduced as a change of independent variable, and
asintroducing a particular prime function
223225 Definition of a group of substitutions ; fundamental
proi)erties226, 227 Convergence of a series; functions associated
with the group228232 The fundamental functions. Comparison with
foregoing theory
of this volume ...233235 Definition and periodicity of the
Schottky prime function .236, 237 Its connection with the theta
functions
238 A further function connected therewith239 The hyperelliptic
case
343343348349352
353359359364364366367372372, 373
240
-
CONTENTS. XV
PAGES255 Factorial integrals of the primary and associated
systems . . 397, 398256 Factorial integrals which are everywhere
finite, save at the fixed
infinities. Introduction of the numbers 07, 0, there are
-
xyi CONTENTS.
rr PAGES
282, 283 Proof that the 2^p theta functions with half-integer
character-
istics are linearly independent 446 447
284, 285 Definition of general theta function of order r ; its
linear expres-
sion by rP theta functions. Any p+2 theta functions ofsame
order, periods, and characteristic connected by a homo-
geneous polynomial relation 447 455
286 Addition theorem for hyperelliptic theta functions, or for
the
general case when jd
-
CONTENTS. XVU
CHAPTER XVIII.
Transformation of periods, especially linear transformation.
PAGES
318 Bearings of the theory of transformation 528, 529319323 The
general theory of the modification of the period loops on a
Riemann surface 529534324 Analytical theory of transformation of
periods and characteristic
of a theta function 534538325 Convergence of the transformed
fimction ..... 538326 Specialisation of the formulae, for linear
transformation . . 539, 540327 Transformation of theta
characteristics ; of even characteristics ;
of syzygetic characteristics 541, 542
328 Period characteristics and theta characteristics .... 543329
Determination of a linear transformation to transform any even
characteristic into the zero characteristic .... 544, 545
330, 331 Linear transformation of two azygetic systems of theta
charac-teristics into one another ..... . 546550
332 Composition of two transformations of difierent orders ;
supple-mentary transformations ....... 551, 552
333, 334 Formation of p+2 elementary linear transformations by
thecomposition of which every linear transformation can beformed ;
determination of the constant factors for each ofthese 553557
335 The constant factor for any linear transformation . . . 558,
559336 Any linear transformation may be associated with a change
of
the period loops of a Riemann surface 560, 561337, 338 Linear
transformation of the places m,, ..., m, . . . . 562
339 Linear transformation of the characteristics of a radical
function 563, 564
340 Determination of the places to,, ..., m^ upon a Riemann
surfacewhose mode of dissection is assigned 565567
341 Linear transformation of quotients of hyperelliptic theta
functions 568
342 A convenient form of the period loops in a special
hyperellipticcase. Weierstrass's number notation for half-integer
charac-teristics 569, 570
CHAPTER XIX.
On systems of periods and on general Jacobian functions.
343 Scope of this chapter . . 571
344 350 Columns of periods. Exclusion of infinitesimal periods.
Expres-sion of aU period columns by a finite number of columns,with
integer coefficients 571579
351 356 Definition of general Jacobian function, and comparison
withtheta function ... 579588
357 362 Expression of Jacobian function by means of theta
functions.
Any + 2 Jacobian functions of same periods and
parameterconnected by a homogeneous polynomial relation . .
588598
-
xviii CONTENTS.
CHAPTER XX.
Transformation of theta functions.
PAGES
363 Sketch of the results obtained. References to the literature
. 599, 600
364, 365 Elementary theory of transformation of second order . .
600606
366, 367 Investigation of a general formula preliminary to
transformation
of odd order ... 607610368, 369 The general theorem for
transformation of odd order. . . 611616
370 The general treatment of transformation of the second order
. 617619
371 The two steps in the determination of the constant
coefficients 619372 The first step in the determination of the
constant coefficients 619622373 Remarks and examples in regard to
the second step . . . 622624374 Transformation of periods when the
coefficients are not integral 624628375 Reference to the
algebraical applications of the theory . . 628
CHAPTER XXI.
Complex multiplication of theta functions. Correspondenceof
points on a Riemann surface.
376 Scope of the chapter 629377, 378 Necessary conditions for a
complex multiplication, or special
transformation, of theta functions .... 629632379382 Proof, in
one case, that these conditions are sufficient . . 632636
383 Example of the elliptic case 636639384 Meaning of an (r, s)
correspondence on a Riemann surface . 639, 640385 Equations
necessary for the existence of such a correspondence 640642386
Algebraic determination of a correspondence existing on a per-
fectly general Riemann surface 642645387 The coincidences.
Examples of the inflections and bitangents of
a plane curve .......... 645648388 Conditions for a (1,)
correspondence on a special Riemann surface 648, 649389 When p>l
a (1, 1) correspondence is necessarily periodic . . 649, 650390 And
involves a special form of fundamental equation . . 651
391393 When p>l there cannot be an infinite number of (1, 1)
corre-
spondences . . ... ... 652 654394 Example of the case ^= 1 654
656
CHAPTER XXII.
Degenerate Abelian integrals.
395 Example of the property to be considered 657396
Weierstrass's theorem. The property involves a transformation
leading to a theta function which breaks into factors. . 657
658
-
CONTENTS. XIX
PAGES397 Weierstrass's aud Picard's theorem. The property
involves a
linear transformation leading to T^'j= l/r 658, 659
398 Existence of one degenerate integral involves another (jB=
2) . 659399, 400 Connection v?ith theory of special transformation,
when p = 2 . 660, 661401403 Determination of necessary form of
fundamental equation.
References. 661663
APPENDIX I.
On algebraic curves in space.
404 Formal proof that an algebraic curve in space is an
interpreta-tion of the relations connecting three rational
functions ona Biemanu surface (cf. 162) 664, 665
APPENDIX II.
On matrices.
405410 Introductory explanations 666669411415 Decomposition of
an Abelian matrix into simpler ones 669674
416 A particular result .... 674417, 418 Lemmas 675419, 420
Proof of results assumed in 396, 397 .. . 675, 676
Index of authors quoted 677, 678Table of some functional symbols
. 679Subject index 680684
-
ADDENDA. CORRIGENDA.
FAOG LINE
6, 2, for db'^da, read db'^-'^da.8, 22, for deficiency 1, read
deficiency 0.
11, 12, for 2n-2+p, read 2n-2 + 2p.16, 16, 4, for called, read
applied to.
dx . dx18, 25, for , read .
X y37, 31, for in, read is.38, 3, /or BTirfacea, read
surface.43, 20, for w, read u.
56, 22, for {x-a)''-\ read (x-a)''-''+^61, 24, add or 3i(x,
y).66, 22, /or r'-l, read t,'-1.70, 14, /or Tr+i, read t, + 1.
73, 28, for x"'''"'''"^ Sj, j, read x"^''"^ Si, i-81. The
argument of 52 supposes p>l.
104, 72. See also Hensel, Crelle, cxv. (1895).114, 3 from the
bottom, add here.137. To the references, add, Macaulay, Proc. Lon.
Math. Soc, xxvi. p. 495.
157. See also Kraus, Math. Annal. xvi. (1879).166. See also
Zeuthen, Ann. d. Mat. 2* Ser., t. in. (1869).189, 21, for xii, read
xi.
196, 23, for \h, read Wh.
24, for \h, read \h.197, 24, for A, read B.198, 5, for
7(w')~ici), read y{ia')~^u.
18, for fourth minus sign, read sign of equality.206, 4, supply
dz, after third integral sign: the summation is from fc=2, fc'=
0.
5, supply dz, after first integral sign.8, for 4>(X)l
-
CHAPTER I.
The Subject of Investigation.
1. This book is concerned with a particular development of the
theoryof the algebraic irrationality arising when a quantity y is
defined in termsof a quantity x by means of an equation of the
form
Coy" + Oiy"-' + . . . + a_iy + a = 0,
wherein ao, a, a are rational integral polynomials in x. The
equation issupposed to be irreducible ; that is, the left-hand side
cannot be written asthe product of other expressions of the same
rational form.
2. Of the various means by which this dependence may be
represented,that invented by Riemann, the so-called Riemann
surface, is throughoutregarded as fundamental. Of this it is not
necessary to give an accounthere*. But the sense in which we speak
of a place of a Riemann surfacemust be explained. To a value of the
independent variable x there will ingeneral correspond n distinct
values of the dependent variable yrepresentedby as many places,
lying in distinct sheets of the surface. For some valuesof X two of
these n values of y may happen to be equal : in that case
thecorresponding sheets of the surface may behave in one of two
ways. Eitherthey may just touch at one point without having any
further connexion inthe immediate neighbourhood of the point t : in
which case we shall regardthe point where the sheets touch as
constituting two places, one in each
sheet. Or the sheets may wind into one another : in which case
we shallregard this winding point (or branch point) as constituting
one place : this
place belongs then indifferently to either sheet ; the sheets
here merge into
one another. In the first case, if a be the value of x for which
the sheets
just touch, supposed for convenience of statement to be finite,
and x a value
* For references see Chap. U. 12, note.
t Such a point is called by Biemann "ein sich anfhebender
Verzweigungspunkt"
:Gesam-
melte Werke (1876), p. 105.
B.1
-
2 THE PLACES OF A RIEMANN SURFACE. [2
very near to a, and if 6 be the value of y at each of the two
places, also
supposed finite, and y y, be values of y very near to 6,
represented by
points in the two sheets very near to the point of contact of
the two
sheets, each of y,-h. y,-b can be expressed as a power-series in
x-a
with integral exponents. In the second case with a similar
notation each
of yj-b,y,-b can be expressed as a power-series in {x-a)i with
integralexponents.' In the first case a small closed curve can be
drawn on either
of the two sheets considered, to enclose the point at which the
sheets touch :
and the value of the integral ^jd log (x - a) taken round this
closed curvewill be 1 ; hence, adopting a definition given by
Rieraann*. we shall say that
a; -a is an infinitesimal of the first order at each of the
places. In the
second case the attempt to enclose the place by a curve leads to
a curve
lying partly in one sheet and partly in the other; in fact, in
order that
the curve may be closed it must pass twice round the branch
place. In this
case the integral g. id log [(x - a)i] taken round the closed
curve will be 1
:
and we speak of {x - o)* as an infinitesimal of the first order
at the place.
In either case, if t denote the infinitesimal, x and y are
uniform functions
of t in the immediate neighbourhood of the place ; conversely,
to each point
on the surface in the immediate neighbourhood of the place there
corre-
sponds uniformly a certain value of tf. The quantity t effects
therefore aconformal representation of this neighbourhood upon a
small simple area in
the plane of t, surrounding t = 0.
3. This description of a simple case will make the general case
clear.In general for any finite value of x, x=ia, there may be
several, say k, branchpoints^ ; the number of sheets that wind at
these branch points may bedenoted by Wj + 1, Wo-t- 1, ..., wic+ 1
respectively, where
(w,+l) + (w^+l) + ...+(wt+l) = n,so that the case of no branch
point is characterised by a zero value of thecorresponding w. For
instance in the first case above, notwithstanding thattwo of the n
values of y are the same, each of Wj, Wa, ..., w^ is zero and k
isequal to n : and in the second case above, the values are A; = n.
1 , Wi= 1 , Wj = 0,Wj = 0, . .
., Wt = 0. In the general case ea
-
4] TRANSFORMATION OF THE EQUATION. 3
are infinitesimals of the first order. For the infinite value of
x we shallsimilarly have n or a less number of places and as many
infinitesimals, say
t)"'^'-' 0"^^' ^^^^^ K + l)+ ... +(w, + l) = n. And as in the
par-
ticular cases discussed above, the infinitesimal t thus defined
for every placeof the surface has the two characteristics that for
the immediate neighbour-hood of the place a? and y are uniquely
expressible thereby (in series ofintegral powers), and conversely t
is a uniform function of position on thesurface in this
neighbourhood. Both these are expressed by saying thatt effects a
reversible conformal representation of this neighbourhood upon
asimple area enclosing t=0. It is obvious of course that quantities
otherthan t have the same property.
A place of the Riemann surface will generally be denoted by a
singleletter. And in fact a place {x, y) will generally be called
the place x.When we have occasion to speak of the {n or less)
places where the inde-pendent variable x has the same value, a
different notation will be used.
4. We have said that the subject of enquiry in this book is a
certainalgebraic irrationality. We may expect therefore that the
theory is practi-cally unaltered by a rational transformation of
the variables x, y which is ofa reversible character. Without
entering here into the theory of such trans-formations, which comes
more properly later, in connexion with the theoryof correspondence,
it is necessary to give sufficient explanations to make itclear
that the functions to be considered belong to a whole class of
Riemannsurfaces and are not the exclusive outcome of that one which
we adopt initially.
Let f be any one of those uniform functions of position on the
funda-mental (undissected) Riemann surface whose infinities are all
of finite order.Such functions can be expressed rationally by x and
y*- For that reason weshall speak of them shortly as the rational
functions of the surface. Theorder of infinity of such a function
at any place of the surface where thefunction becomes infinite is
the same as that of a certain integral power of
the inverse - of the infinitesimal at that place. The sum of
these orders ofV
infinity for all the infinities of the function is called the
order of the function.
The number of places at which the function f assumes any other
value o isthe same as this order : it being understood that a place
at which | - a iszero in a finite ratio to the rth order of t is
counted as r places at which f is
equal to a^f. Let v be the order of ^. Let rj be another
rational function of
* Forsyth, Theory of Functions, p. 370.
+ For the integral =. /dlog(f-o), taken round an infinity of
log(|-o), is equal to the
order of zero of f - o at the place, or to the negative of the
order of infinity of |, as the case may
be. And the sum of the integrals for all such places is equal to
the value round the boundary of
the surfacewhich is zero. Cf. Forsyth, Theory of Functions, p.
372.
12
-
4 CONDITION OF REVERSIBILITY. [4
order fi. Take a plane whose real points represent all the
possible values of^ in the ordinary way. To any value of ^, say f =
a, will correspond vpositions X,, ..., X,on the original Riemann
surface, those namely where fis equal to a : it is quite possible
that they lie at less than v places of thesurface. The values of t]
at X^, ..., X^ may or may not be different. LetH denote any
definite rational symmetrical function of these v values of 17.Then
to each position of a in the f plane will correspond a perfectly
uniquevalue of H, namely, ^ is a one-valued function of f.
Moreover, since r] andf are rational functions on the original
surface, the character ofH for valuesof f in the immediate
neighbourhood of a value a, for which H is infinite, isclearly the
same as that of a finite power of f o. Hence H ia a.
rationalfunction of ^. Hence, if Hr denote the sum of the products
of the values of7] sX Xi, ..., X,r together, 77 satisfies an
equation
r - ^"-^i^i + ^"-^ir. -... + (-)- = 0,whose coefficients are
rational functions of f.
It is conceivable that the left side of this equation can be
written as theproduct of several factors each rational in f and 17.
If possible let this bedone. Construct over the f plane the Riemann
surfaces corresponding tothese irreducible factors, 77 being the
dependent variable and the varioussurfaces lying above one another
in some order. It is a known fact, alreadyused in defining the
order of a rational function on a Riemann surface, thatthe values
of t? repi-esented by any one of these superimposed surfaces
in-clude all possible valueseach value in fact occurring the same
number oftimes on each surface. To any place of the original
surface, where f 77 havedefinite values, and to the neighbourhood
of this place, will correspond there-fore a definite place (^, 17)
(and its neighbourhood) on each of these super-imposed surfaces.
Let %, ...,77, be the values of 77 belonging, on one ofthese
surfaces, to a value of f : and 77/, ...,77/ the values belonging
to thesame value of
^ on another of these surfaces. Since for each of these
surfacesthere are only a finite number of values of ^ at which the
values of 77 arenot all different, we may suppose that all these r
values on the onesurface are different from one another, and
likewise the s values on the othersurface. Since each of the pairs
of values (f, ,7.), ..., (f, 77,) must arise onboth these surfaces,
it follows that the values 77, ,7, are included among771',
..., V- Similarly the values 77/ ,7; are included among 77,,
..., ,,^Hence these two sets are the same and r = s. Since this is
true for aninfinite number of values of f, it follows that these
two surfaces are merelyrepetitions of one another. The same is true
for every such two surfaces.Hence ris a divisor of v and the
equation
V" - H.r,"--^ + ...-^{-y H, = 0,
when reducible, is the vjrih power of a rational equation of
order r in ,7. Itwill be sufficient to confine our attention to one
of the factors and the (f, ,,)
-
5] CORRESPONDENCE OF TWO SURFACES. 5
surface represented thereby. Let now X^ Z be the places on the
originalsurface where f has a certain value. Then the values of rj
at Xi, . . , X willconsist of vjr repetitions of r vahies, these r
values being different from oneanother except for a finite number
of values of ^. Thus to any place (f, rj) onone of the v/r derived
surfaces will correspond vjr places on the originalsurface, those
namely where the pair (f, 17) take the supposed values. Denotethese
by Pi,P Let Y be any rational symmetrical function of the vfrpairs
of values {x^, y^, (x^, y^), ..., which the fundamental variables
x, y of theoriginal surface assume at Pj, P^, Then to any pair of
values (f, t)) willcorrespond only one value of Ynamely, Y is a
one-valued function on the(^, 17) surface. It has clearly also only
finite orders of infinity. Hence Y isa rational function of f, 17.
In particular x^, x^, ... are the roots of an
equation whose coefficients are rational in f, 17as also are
j/i, y^,There exists therefore a correspondence between the (^, 17)
and {x, y)
surfacesof the kind which we call a (1, - j correspondence: to
every place
of the (x, y) surface corresponds one place of the (f, 17)
surface; to every
place of this surface correspond - places of the {x, y)
surface.
The case which most commonly arises is that in which the
rationalirreducible equation satisfied by 17 is of the I'th degree
in 77 : then only oneplace of the original surface is associated
with any place of the new surface.
In that case, as will appear, the new surface is as general as
the original
surface. Many advantages may be expected to accrue from the
utilization ofthat fact. We may compare the case of the reduction
of the general equationof a conic to an equation referred to the
principal axes of the conic.
5. The following method* is theoretically effective for the
expression of x, y in terms
of I, r,.
Let the rational expression of ^, 7 in terms of x, y be given
by
ix, y) - t\f {x, y)=0, ^ {x, y) -r)x{x,y)= 0,
and let the rational result of eliminating x, y between these
equations and the initial
equation connecting ar, y be denoted by F{^, ij) = 0, each
of
-
6 ALGEBRAICAL FORMULATION. [5
Then we obtain the equation
This is an equation of the form above referred to, by which x is
determinate from | andI). And y is similarly determinate.
It will be noticed that the rational expression of x, y by |,
ij, when it is possible
from the equations
*(-i>2/)-^'K-^.^')=o. ^(^,3')-w(^'.y)=. /(^.y)=o.
will not be possible, iu general, from the first two eqiiations
: it is only the places x, ysatisfying the equation f{x, y) = which
are rationally obtainable from the places |, i;
satisfying the equation F{^, i) = 0. There do exist
transformations, rationally reversible,
subject to no such restriction. They are those known as
Cremona-transformations*.They can be compounded by reapplication of
the transformation x : y : l = t] : ^ : ^
AVe may give an example of both of these transformations
For the surface
y'^^ny^x'^+x+l)+ 5y{x^+ x+ lf-2x{x^+x+lf=the function
^=y''/{x''+x+\) is of order 2, being infinite at the places where
x^+x+l=0,in each case like (.?; a)~, and the function Ti = xjy is
of order 4, being infinite at theplaces x-+x-{-l=0, in each case
like (.r-a)"^, a being the value of x at the place.
From the given equation we immediately find, as the relation
connecting ^ and t/,
2,,-f + 5^-5=0,and infer, since the equation formed as in the
general statement above should be oforder 2 in i;, that this
general equation will be
Thence in accordance with that general statement we infer that
to each place (^, ij) onthe new surface should correspond two
places of the original surface : and in fact these areobviously
given by the equations
,,^=xy{x''+x+l), y=x/,,.
If however we take
i=yV{x^+x+l), ,=y/(^-6,2),where a> is an imaginary cube root
of unity, so that ij is a fimction of order 3, theseequations are
reversible independently of the original equation, giving in
fact
^= (| - u,Y)l{i - r,% y= (,o- 0,2) f,/(^ - ,2),
and we obtain the surface
having a (1, I) correspondence with the original one.
It ought however to be remarked that it is generally jHjssible
to obtain reversibletransformations which are not
Cremona-transformations.
6. When a surface (x, y) is (1, 1) related to a (f, t)) surface,
the defi-ciencies of the surfaces, as defined by Riemann by means
of the connectivity,must clearly be the same.
* See Salmon, Higher Plane Curves (1879), 362, p. 322.
-
6] RELATION OF DEFICIENCES. 7
It is instructive to verify this from another point of
view*.Consider at
how many places on the original surface the function -^ is zero.
It is infiniteax
at the places where f is infinite: suppose for simplicity that
these areseparated places on the original surface or in other words
are infinities ofthe first order, and are not at the branch points
of the original surface. At
a pole of ?> ^ is infinite twice. It is infinite like t^ at a
branch place (a)
where a; - a = '*+': namely it is infinite Sw = 2m + 2p - 2
timesf at the branchplaces of the original surface. It is zero In
times at the infinite places of theoriginal surface. There remain
therefore 2i>+ 2n + 2p-2 2n = 2v+2p -2places where j- is zero.
If a branch place of the original sufface be a pole
1 /Vt 1of f, and f be there infinite like -, -^^ is infinite
like , namely 2+w
t ax V . t "^
times: the total number of infinities of -^ will therefore be
the same asax
before. Now at a finite place of the original surface where t^ =
0, there are
two consecutive places for which f has the same value. Since - =
1 they can
only arise fi-om consecutive places of the new surface for which
f has thesame value. The only consecutive places of a surface for
which this is thecase are the branch places. Hence
-f- there are 2v+2p 2 branch places ofthe new surface. This
shews that the new surface is of deficiency p.
When v/r is not equal to 1, the case is different. The
consecutive placesof the old surface, for which ^ has the same
value, may either be those arisingfrom consecutive places of the
new surfaceor may be what we may callaccidental coincidences among
the v/r places which correspond to one placeof the new surface.
Conversely, to a branch place of the new surface,characterised by
the same value for f for consecutive placesj, will
correspondi>/r places on the old surface where ^ has the same
value for consecutiveplaces. In fact to two very near places of the
new surface will correspond
v/r pairs each of very near places on the old surface. If then G
denote thenumber of places on the old surface at which two of the
v/r places corre-sponding to a place on the new surface happen to
coincide, and w' the numberof branch points of the new surface, we
have the equation
w'- + C=2v + 2p-2,r
* Compare the interesting geometrical account, Salmon, Higher
Plane Curves (1879), p. 326,
364, and the references there given.t Forsyth, Theory of
Functions, p. 34a.
X Namely, near such a branch place f to, { - a is zero of higher
order than the first.
-
8 PARAMETERS NOT REMOVED [6
and if p' be the deficiency of the new surface (of r sheets),
this leads to the
equation
(2r+2p'-2)-^+G = 2v + 2p-2,
from whichV
C = 2p-2-(2p-2)-.r
Corollary*- If p =p', then C = {2p- 2) (l -^)
Thus^
>> 1, so that
(7 = 0, and the correspondence is reversible.
We have, herein, excluded the case when some of the poles of ^
are ofhigher than the first order. In that case the new surface has
branch places
at infinity. The number of finite branch places is
correspondingly less. The
reader can verify that the general result is unaffected.
Ex. In the example previously given ( 5) shew that the function
| takes any givenvalue at two points of the original surface (other
than the branch places where it is
infinite), i; having the same value for these two points, and
that there are six places at
which these two places coincide. (These are the place {x=Q, y=
0) and the five placeswhere x= 2.)
There is one remark of considerable importance which follows
from the
theory here given. We have shewn that the number of places of
the {x, y)
surface which correspond to one place of the (f, tj) surface is
- , where v is the
order of f and r is not greater than v, being the number of
sheets of the (f, r))surface ; hence, if there were a function f of
order 1 the correspondence wouldbe reversible and therefore the
original surface would be of deficiency 1.
7. This notion of the transformation of a Riemann surface
suggests aninference of a fundamental character.
The original equation contains only a finite number of terms :
the originalsurface depends therefore upon a finite number of
constants, namely, thecoefficients in the equation. But conversely
it is not necessary, in order thatthe equation be reversibly
transformable into another given one, that theequation of the new
surface contain as many constants as that of the originalsurface.
For we may hope to be able to choose a transformation
whosecoefficients so depend on the coefficients of the original
equation as to reducethis number. If we speak of all surfaces of
which any two are connected bya rational reversible transformation
as belonging to the same class
"f, it becomesa question whether there is any limit to the
reduction obtainable, by rationalreversible transformation, in the
number of constants in the equation of asurface of the class.
* See Weber, Crelh, 76, 345.t So that surfaces of the same class
will be of the same deficiency.
-
'7] BY TRANSFORMATION. 9
It will appear in the course of the book* that there is a limit,
and thatthe various classes of surfaces of given deficiency are of
essentially differentcharacter according to the least number of
constants upon which they depend.Further it will appear, that the
most general class of deficiency p ischaracterised by 3p 3
constants when p > 1the number tor p = 1 beingone, and for ^ =
none.
For the explanatory purposes of the present Chapter we shall
contentourselves with the proof of the following statementWhen a
surface isreversibly transformed as explained in this Chapter, we
cannot, even thoughwe choose the new independent variable f to
contain a very large number ofdisposeable constants, prescribe the
position of all the branch points of thenew surface ; there will be
dp 3 of them whose position is settled by theposition of the
others. Since the correspondence is reversible we may regardthe new
surface as fundamental, equally with the original surface. Weinfer
therefore that the original surface depends on 3^ 3 parameters
or on less, for the 3p 3 undetermined branch points of the new
surface mayhave mutually dependent positions.
In order to prove this statement we recall the fact that a
function
of order Q contains^ Qp+l linearly entering constants when its
polesare prescribed: it may contain more for values of Q2p 2and
large enough. Also the Q infinities are at our disposal. We can
thenpresumably dispose of 2Q p + 1 of the branch points of the new
surface.But these are, in number, 2Q + 2p 2 when the correspondence
is reversible.Hence we can dispose of all but 2Q+2p 2 {2Q p + l) =
3p 3 of thebranch points of the new surface J.
Ex. 1. The surface associated with the equation
f=x(l-x) (1-k^x) (1 -yx) {l-fi'x) (1 -A') (1 -p^x)is of
deficiency 3. It depends on 5=2p- 1 parameters, k% X^ /j,', i^,
p\
Ex. 2. The surface associated with the equation
wherein the coefficients are integral polynomials of the orders
specified by the suffixes, isof deficiency 3. Shew that it can be
transformed to a form containing only 6= 2/> -
1
parametric constants.
* See the Chapters on the geometrical theoiy and on the
inversion of Abelian Integrals. The
reason for the exception in case p=0 or 1 will appear most
clearly in the Chapter on the self-correspondence of a Biemann
surface. But it is a familiar fact that the elliptic functions
whichcan be constructed for a surface of deficiency 1 depend upon
one parameter, commonly called
the modulus : and the trigonometrical functions involve no such
parameter.
+ Forsyth, p. 459. The theorems here quoted are considered in
detail in Chapter III. of the
present book.
X CI. Biemann, Gei. Werke (1876), p. 113. Klein, Ueber Riemann's
Theorie (Leipzig,
Teubner, 1882), p. 65.
-
JO SELF-COERESPONDENCE. [8
8. But there is a case in which this argument fails. If it
bepossible to
transform the original surface into itself by a rational
reversible transforma-
tion involving parameters, any r places on the surfaceare
efifectively
equivalent with, as being transformable into, any other r
places.Then the
Q poles of the function | do not effectively supply Q but only
Q-r dispose-able constants with which to fix the new surface. So
that there are 3/) - 3 +
r
branch points of the new surface which remain beyond our
control. Inthis
case we may say that all the surfaces of the class contain 3p -3
disposeableparameters bm.de r parameters which remain indeterminate
and serve to
represent the possibility of the self-transformation of the
surface.It will be
shewn in the chapter on self-transformation that the possibility
only arises
for p = or i) = 1, and that the values of r are, in thesecases,
respectively
3 and 1. We remark as to the case p = that when the
fundamentalsurface has only one sheet it can clearly be transformed
into itself by
a transformation involving three constants x = ^tl^g ' ^^^ ^
regardtop = l,
the case of elliptic functions, that effectively a point
represented by the
elliptic argument h is equivalent to any other point represented
by an
argument u + 7. For instance a function of two poles is
and clearly i^.,p has the same value at u as has ^a+y.ff+r at i6
-(- 7 : so that the
poles (a, /3) are not, so far as absolute determinations are
concerned, effective
for the determination of more than one point.
9. The fundamental equation
a2/"4-ai2/"-'-|-...-l-a = 0,so far considered as associated with
a Riemann surface, may also be regardedas the equation of a plane
curve : and it is possible to base our theory on thegeometrical
notions thus suggested. Without doing this we shall in thefollowing
pages make frequent use of them for purposes of illustration. It
istherefore proper to remind the reader of some fundamental
properties*.
The branch points of the surface correspond to those points of
the curvewhere a line x = constant meets the curve in two or more
consecutive points
:
as for instance when it touches the curve, or passes through a
cusp. On theother hand a double point of the curve corresponds to a
point on the surfacewhere two sheets just touch without further
connexion. Thus the branchplace of the surface which corresponds to
a cusp is really a different singu-larity to that which corresponds
to a place where the curve is touched by a
' Cf. Forsyth, Theory of Functions, p. 355 etc. HarkneBs and
Morley, Theory of Functiorts,p. 273 etc.
-
9] GEOMETRICAL VIEW. 11
line X = constant, being obtained by the coincidence of an
ordinary branchplace with such a place of the Riemann surface as
corresponds to a doublepoint of the curve.
Properties of either the Riemann surface or a plane curve are,
in thesimpler cases, immediately transformed. For instance, by
Pliicker's formulaefor a curve, since the number of tangents from
any point is
= (i-l)n-28-3,where n is the aggregate order in x and y, it
follows that the number ofbranch places of the corresponding
surface is
w = < 4- K = (n - 1) n _ 2 (8 + k)= 2?i- 2 + 2 {^71 - 1) (ft-
2) - S- /cj.
Thus since w = 2n 2-^p, the deficiency of the surface is
\{n-V,{n-2)-h-K,namely the number which is ordinarily called the
deficiency of the curve.
To the theory of the birational transformation of the surface
correspondsa theory of the birational transformation of plane
curves. For example, thebranch places of the new surface obtained
from the surface f{x, y) = bymeans of equations of the form
{x, y) fi/r {x, y) = 0, S {x, y) i;;^ (*, y) =
will arise for those values of f for which the curve
-
12 GENERALITY [9
at the point, and k be the number of cusps of the curveand if v
be the number of
points other than the point itself in which the curve is
intersected by an arbitrary line
through the pointthen t+K-Zv is independent of the position of
the point. If the
equation of the variable lines through the point be written
u-^v=0, interpret the result
by regarding the curve as giving rise to a Riemann surface whose
independent variable
isg*.
10. The geometrical considerations here referred to may however
be
stated with advantage in a very general manner.
In space of any (k) dimensions let there be a curve(a
one-dimension-
ality). Let points on this curve be given by the ratios of the
k+1 homo-geneous variables x^, ... , .Vk+i- Let u, v be any two
rational integral homo-
geneous functions of these variables of the same order. The
locus u-^ =will intersect the curve in a certain number, say v,
points
we assume the
curve to be such that this is the same for all values of f, and
is finite. Let all
the possible values of f be represented by the real points of an
infinite plane
in the ordinary way. Let w, t be any two other integral
functions of the
coordinates of the same order. The values of ; = at the points
where
u ^v cuts the curve for any specified value of f will be v in
number.
As before it follows thence that rj satisfies an algebraic
equation of order v
whose coefficients are one-valued functions of f. Since tj can
only be infinite
to a finite order it follows that these coefficients are
rational functions of f.Thence we can construct a Riemann surface,
associated with this algebraicequation connecting f and >;, such
that every point of the curve gives rise toa place of the surface.
In all cases in which the converse is true we mayregard the curve
as a representation of the surface, or conversely.
Thus such curves in space are divisible into sets according to
theirdeficiency. And in connexion with such curves we can construct
all thefunctions with which we deal upon a Riemann surface.
Of these principles sufficient account will be given below
(Chapter VI.)
:
familiar examples are the space cubic, of deficiency zero, and
the most generalspace quartic of deficiency 1 which is
representable by elliptic functions.
11. In this chapter we have spoken primarily of the algebraic
equationand of the curve or the Riemann surface as determined
thereby. But thisis by no means the necessary order. If the Riemann
surface be given, thealgebraic equation can be determined from
itand in many forms, accordingto the function selected as dependent
variable {y). It is necessary to keepthis in view in order fully to
appreciate the generality of Riemann's methods.For instance, we may
start with a surface in space whose shape is that of an
* The reader who desires to study the geometrical theory
referred to may consult:Cayley, Quart. Journal, vii. ; H. J. S.
Smith, Proc. Land. Math. Soc. vi.; Noether, Math. Annul.9 ; BriU,
Math. Annal. 16 ; Brill u. Noether, Math. Annal. 7.
-
11] OF THE THEORY. 13
anchor ring*, and construct upon this surface a set of elliptic
functions. Orwe may start with the surface on a plane which is
exterior to two circlesdrawn upon the plane, and construct for this
surface a set of elliptic functions.Much light is thrown upon the
functions occurring in the theory by thusconsidering them in terms
of what are in fact diiferent independent variables.And further
gain arises by going a step further. The infinite plane uponwhich
uniform functions of a single variable are represented may be
regardedas an infinite sphere ; and such surfaces as that of which
the anchor ringabove is an example may be regarded as
generalizations of that simple case.Now we can treat of branches of
a multiform function without the use of aRiemann surface, by
supposing the branch points of the function marked ona single
infinite plane and suitably connected by barriers, or cuts, across
whichthe independent variable is supposed not to pass. In the same
way, for anygeneral Riemann surface, we may consider branches of
functions which arenot uniform upon that surface, the branches
being separated by drawingbarriers upon the surface. The properties
obtained will obviously generalizethe properties of the functions
which are uniform upon the surface.
Forsyth, p. 318 ; Eiemann, Ges. Werke (1876), pp. 89, 415.
-
[12
CHAPTER II.
The Fundamental Functions on a Riemann Surface.
12. In the present chapter the theory of the fundamental
functions is
based upon certain a priori existence theorems*, originally
given by
Riemann. At least two other methods might be followed : in
Chapters IV.
and VI. sufficient indications are given to enable the reader to
establish
the theory independently upon purely algebraical considerations:
from
Chapter VI. it will be seen that still another basis is found in
a preliminary
theory of plane curves. In both these cases the ideas primarily
involved are
of a very elementary character. Nevertheless it appears that
Riemann's
descriptive theory is of more than equal power with any other ;
and that
it offers a generality of conception to which no other theory
can lay claim.
It is therefore regarded as fundamental throughout the book.
It is assumed that the Theory of Functions of Forsyth will be
accessible
to readers of the present book ; the aim in the present chapter
has been toexclude all matter already contained there. References
are given also to
the treatise of Harkness and Morley*.
13. Let t be the infinitesimali" at any place of a Riemann
surface : if it isa finite place, namely, a place at which the
independent variable a; is finite,the values of x for all points in
the immediate neighbourhood of the placeare expressible in the form
a; = a + t'"+^ : if an infinite place, x = (""'"+''.
There exists a function which save for certain additive moduli
is one-valuedon the whole surface and everywhere finite and
continuous, save at theplace in question, in the neighbourhood of
which it can be expressed in theform
* See for instaDce : Forsyth, Theory of Functions of a Complex
Variable, 1893 ; Harkness andMorley, Treatise on the Theory of
Functions, 1893 ; Schwarz, Gesam. math. Abhandlungen, 1890.The best
of the early systematic expositions of many of the ideas involved
is fonnd inC. Neumann, Vorlesungen iiber Eiemann's Theorie, 1884,
which the reader is recommended tostudy. See also Picard, Traite
d'Analyse, Tom. ii. pp. 273, 42 and 77.
t For the notation see Chapter I. 2, 3.
-
14] ELEMENTARY NORMAL INTEGRALS. 15
Herein, as throughout, P (t) denotes a series of positive
integral powei-s of tvanishing when t = 0, C, A, ..., Ar-i, are
constants whose values can bearbitrarily assigned beforehand, and r
is a positive integer whose value can beassigned beforehand.
We shall speak of all such functions as integrals of the second
kind
:
but the name will be generally restricted to that * particular
function whosebehaviour near the place is that of
-l + G + P(t).
This function is not entirely unique. We suppose the surface
dissectedby 2p cuts+, which we shall call period loops; they
subserve the purpose ofrendering the function one-valued over the
whole of the dissected surface.We impose the further condition that
the periods of the function for transitacross the p loops of the
first kind J shall be zero ; then the function is uniquesave for an
additive constant. It can therefore be made to vanish at
anarbitrary place. The special function so obtained whose infinity
is that
of - is then denoted by Fo^' \ c denoting the place where the
function
vanishes and a; the current place. When the infinity is an
ordinary place,
at which either x = a or a; = oo , the function is infinite
either likeX a
or X. The periods of F/'" for transit of the period loops of the
secondkind will be denoted by fl,, ..., ilp.
14. Let (Xiyi), (x^y^) be any two places of the surface : and
let the
infinitesimals be respectively denoted by ti, t^, so that in the
neighbourhood
of these places we have the equations a; a;i =
-
16 ELEMENTARY NORMAL INTEGRALS. [14
path by which the variable is supposed to pass from c. It will
be called* the
integral of the third kind whose infinity is like that of log
(ti/ti).
15. Beside these functions there exist also certain integrals of
the first
kindin number p. They are everywhere continuous and finite and
one-
valued on the dissected surface. For transit of the period loops
of the
first kind, one of them, say Vi, has no periods except for
transit of the i"" loop,
Qi. This period is here taken to be 1. The periods of Vi for
transit of the
period loops of the second kind are here denoted by t^.^, ...,
np. We maytherefore form the scheme of periods
-
16] VARYING PARAMETER NEARLY EQUAL TO ARGUMENT. 17
A more general integral of the third kind having the same
property is
i.J
wherein the arbitrary coefficients satisfy the equations Afj =
Aji. The pro-perty is usually referred to as the theorem of the
interchange of argument(x) and parameter (xi).
The property allows the consideration of
as a function of Xi for fixed positions of x, c, x. In this
regard a remarkshould be made
:
For an ordinary position of x, the function
KL - ig (< - ^) = n:;'r^ - log (x,' - x)is a finite
continuous function of x^ when x^ is in the neighbourhood of x.But
if x-i be a bi-anch place where w + 1 sheets wind, and a;,', x be
twopositions in its neighbourhood, the functions of x
KU - ^"& (*'' - *)' n^'x, - ^i^i ig (* - ^)are respectively
finite as x approaches x^' and Xi, so that
n;-':''-iog(^.'-^)
is not a finite and continuous function of x/ for positions of
ar/ up to andincluding the branch place x^.
In this case, let the neighbourhood of the branch place be
conformallyrepresented upon a simple plane closed area and let |i,
^j', f be the represent-atives thereon of the places Xi, a;/, x.
Then the correct statement is that
is a continuous function of a;/ or f/ up to and including the
branch place Xi.
This is in fact the form in which the fimction n^'|. arises in
the proofof its existence upon which our account is based*.
In a similar way the function
^Vregarded as a function of jt,', is such that
1r'-' +
is a finite continuous function of f/ in the immediate
neighbourhood of x.
* The reader may consult Neumann, p. 220.
B. 2
-
r18 ONE INFINITY AT A BRANCH PLACE. [17
17. It may be desirable to give some simple examples of these
integrals.
(a) For the surface represented by
y2=a;(;r_ai)...(a:-a2p + i),
wherein Oj, ..., a^p+j are all finite and different from zero
and each other, consider the
integral
/ y \^-k ^-U'(I. v): (fi> Vi) being places of the surface
other than the branch places, which are
(0, 0),(ai, 0), ..., (ajp + i, 0).
It is clearly infinite at these places respectively like log {x
- 1), - log (x - fj).
It is not infinite at (|, -,), (|,, -); for (y+7)/(-^-^)>
(y+liM^-fi) are finite atthese places respectively.
At a place a;= oo , where ;;:=r', y= er? " (1 +A (0). f being 1
, and Pj (J) a series of
positive integral powers of t vanishing for
-
17] ONE INFINITY AT A DOUBLE POINT. 19
the integral becomes
which is equal toI.
f^ v+{^-$)'i'+H^-$f')"+-(^-|)=''7+(^-f)V+i(^-f)S"+..."
/
-
20 EXAMPLES. [17
is a constant multiple of an integral of the third kind,
provided A, B, C be so chosen thaty-m^x+AaP'+Bxy + Cy'^ vanishes at
one of the two places other than (0, 0) at whichLx-^My is zero. Its
infinities are at (i) the uncompensated zero of Lx-k-My which is
notat (0, 0), (ii) the place (0, 0) at which the expression of y in
terms of x is of the form
y= m-fS+ P.'t'+ a,-3+ . .
.
In fact, at a branch place of the surface where x=a->tt^,
f'{y) is zero of the first order,[ dx
and dx=2t dt; thus I^tt. is finite at the branch places. At each
of the places (0, 0),
/'(j/) is zero of the first order, i^+ il/y is zero of the first
order andy OTia;+ j4a7^+.ry+ Cy*is zero at these places to the
first and second order respectively. These statements areeasy to
verify ; they lead immediately to the proof that the integrals have
the characterenunciated.
The condition given for the choice oi A, B, C will not determine
them uniquelytheintegral will be determined save for an additive
term of the form
/'""*'/^'where F, Q are undetermined constants. The reader may
prove that this is a generalintegral of the first kind. The
constants P, Q may be determined so that the integral ofthe third
kind has no periods at the period loops of the first kind, whose
number in thiscase is two. The reasons that suggest the general
form written down will appear in theexplanation of the geometrical
theory.
(y) The reader may verify that for the respective cases
y
-
18] PERIODS OF INTEGRAL OF SECOND KIND. 21
It is supposed that the number of places where negative powers
of t occur in theexpansion of F is finite, but it is not necessary
that the number of negative powers befinite. The theorem may be
obtained by contour integration of JFdx, and clearly
generalizes a property of the integral of the third kind.
18. The value of the integral* IT"'' dv".'" taken round the p
closed curves
formed by the two sides of the pairs of period loops (a,, 6,), .
.. ,
(a^, bp), in such
a direction that the interior of the surface is always on the
left hand, is equal
to the value taken round the sole infinity, namely the place a,
in a counter-clockwise direction. Round the pair a^, br the value
obtained is
fir j dv''" ,
taken once positively in the direction of the arrow head round
what in thefigure is the outer side of br. This value is fl, (
&),>), where toir denotes theperiod of Vi for transit of a^,
namely, from what in the figure is the inside of
the oval a, to the outside.
The relations indicated by the figure for the signs adopted for
cair, t,> and
the periods of F^' will be preserved throughout the book.
Since aur is zero except when r = i, the sum of these p contour
integrals
is-ai,ifi. Taken in a counter-clockwise direction, round the
pole of V^'',
where
t'-'' = -- + A+Bt + Gf + ..., t
the integral gives
where D denotes -y, . Hence, as Wj, i = 1,at
Dv''"' + tD
-
22 ALL INTEGRALS AND RATIONAL FUNCTIONS [18
This is true whether a be a branch place or a place at infinity
(for which,
if not a branch place, x = tr') or an ordinary finite place. In
the latter case
. d ^ x,c\
Similai-ly the reader may prove that the periods of J^JJ^^
are
0, 0,2-,ri/"'\ 27n-/"'^.
In this case it is necessary to enclose x^ and x^ in a curve
winding Wi + 1
times at iCi, Wj + 1 times at x^, in order that this curve may
be closed.
19. From these results we can shew that the integral of the
second kind
is derivable by differentiation from the integral of the third
kind. Apart
from the simplicity thus obtained, the fact is interesting
because, as will
appear, the analytical expression of an integral of the third
kind is of the
same general form whether its infinities be branch places or not
; this is not
the case for integrals of the second kind.
We can in fact prove the equation
namely, if, to take the most general case, x^ be a winding place
and a;/ a place
in its neighbourhood such that Xi = Xi + t , the equation,
lim. ,-rn"v -n"'"^^l = r-'fx=0 tx, I ^*' ^"^J '
For, let the neighbourhood of the branch place x^ be conformally
representedupon a simple closed area without branch place, by means
of the infinitesimalof X, as explained in the previous chapter. Let
f,', fi be the representativesof the places x^', a;,, and f the
representative of a place x which is very nearto Xi, but is so
situate that we may regard Xi as ultimately infinitely closerto Xi
than x is.
Then a; - a;, = (| - f,)"'+',^-^/ = (?-?.')[c+P{f-f')].
where C does not vanish for x^' = x,
^d n^;,;^= log (x - xo + ^' = log (f - ?/) + 4,'.
where'
is finite for the specified positions of the places and reviains
finitewhen fi' is taken infinitely near to f, ( 16).
^'^ K^,x, =^1 log {>-X,) + ^ = log (? - fO + 4>,
-
19] DERIVABLE FROM INTEGRAL OF THIRD KIND. 23
where
is also finite. Therefore
X,', X, xx, ^y ^_^^y^T' r
and thus
lira.
where yjr is finite.
Now as fi' moves up to fi, for a fixed position of ^, we
have
J-/, X, X Xj_
-^
, ,.
-
24 PROOF FOB RATIONAL FUNCTIONS. [19
are respectively infinite like
_1 _1 _1fl, I'll "X,
We shall generally write D^,, D|,, ... instead of A^,, X)?^,
When a;,
is an ordinary place D^, will therefore mean t , etc.
Corollary iii.
By means of the example (S) of 17 it can now be shewn that the
infinite
parts of the integral
[Fdx,I'
in which F is any uniform function of position on the
undissected surfacehaving only iirfinities of finite order, are
those of a sum of terms consisting of
proper constant multiples of integrals of the third kind and
differential
coefficients of these in regard to the parametric place.
20. One particular case of Cor. iii. of the last Article should
be stated.
A function which is everywhere one-valued on the undissected
surface mustbe somewhere infinite. As in the case of uniform
functions on a single
infinite plane (which is the particular case of a Riemann
surface for whichthe deficiency is zero), such functions can be
divided into rational andtranscendental, according as all their
infinities are of finite order and of finitenumber or not.
Transcendental functions which are uniform on the surfacewill be
more particularly considered later. A rational uniform function
canbe expressed rationally in terms of w and y*. But since the
function can beexpressed in the neighbourhood of any of its poles
in the form
we can, by subtracting from the function a series of terms of
the form
- r^.r^ ' + A,D^r^-''+ ... +-^ 2)^-> r^ "1
,
obtain a function nowhere infinite on the surface and having no
periods at thefirst p period loops. Such a function is a constantf.
Hence F can also beexpressed by means of normal integrals of the
second kind only. Since Fhas no periods at the period loops of the
second kind there are for all rationalfunctions certain necessary
relations among the coefficients Ai,...,Am.These are considered in
the next Chapter.
* Forsyth, p. 369. Harkness and Morley, p. 262.t Forsyth, p.
439.
-
^IJ SPECIAL RATIONAL FUNCTIONS. 25
21. Of all rational functions there are p whose importance
justifies aspecial mention here; namely, the functions
dvi dVi dvpdx ' dx ' " dx
In the first place, these cannot be all zero for any ordinary
finite place a ofthe surface. For they are, save for a factor 2in,
the periods of the normalintegral T^ ^ If the periods of this
integral were zero, it would be a rationaluniform function of the
first order ; in that case the surface would be repre-sentable
conformally upon another surface of one sheet*, f = ra"^-" being
thenew independent variable; and the transformation would be
reversible(Chap. I. 6). Hence the original surface would be of
deficiency zero
;
in which case the only integral of the first kind is a constant.
The functionsare all infinite at a branch place a. But it can be
shewn as here that thequantities to which they are there
proportional, namely DaV^, ..., DaVp, cannotbe all zero. The
functions are all zero at infinity, but similarly it can beshewn
that the quantities, Dvi, ... , Dvp, cannot be all zero there.
Thus p linearly independent linear aggregates of these
quantities cannot all vanish atthe same place. We remark, in
connexion with this property, that surfaces exist of
alldeficiencies such that p-l linearly independent linear
aggregates of these quantitiesvanish in an infinite number of sets
of two places. Such surfaces are however special, andtheir equation
can be putf into the form
We have seen that the statement of the property requires
modificationat the branch places, and at infinity ; this
particularity is however due to thebehaviour of the independent
variable x. We shall therefore state the pro-perty by saying: there
is no place at which all the differentials dvi, ..., dvpvanish. A
similar phraseology will be adopted in similar cases. For
instance,we shall say that each of dvi, dv^, ... , dvp has| 2p 2
zeros, some of whichmay occur at infinity.
In the next place, since any general integral of the first
kind
\^Vi'+... -\-\pVp'
must necessarily be finite all over any other surface upon which
the originalsurface is conformally and reversibly represented and
therefore must be auintegral of the first kind thereon, it follows
that the rational function
* I owe this argument to Prof. Klein. + See below, Chap. V.
X See Forsyth, p. 461. Harkness and Morley, p. 450.
-
26 INVARIANCE OF THEIR RATIOS. [21
is necessarily transformed with the surface into
M {\,^^+...+\^^
/
where Fj = Vi is an integral of the first kind, not necessarily
normal, on thedP
new surface, f being the new independent variable, and M = -^
.
Thus, the ratios of the integrands of the first kind are
transformedinto ratios of integrands of the first kind ; they may
be said to be invariantfor birational transformation.
This point may be made clearer by an example. The general
integralof the first kind for the surface
r = (a;, 1)8can be shewn to be
^{A+Bx + Ga?),y
A, B, G being arbitrary constants.
If then , denote the ratios of any three linearly
independentintegrands of the first kind for this surface, we
have
1 : X : x' = ai
-
24] 27
CHAPTER III.
The Infinities of Rational Uniform Functions.
23. In this chapter and in general we shall use the term
rational functionto denote a uniform function of position on the
surface of which all theinfinities are of finite order, their
number being finite. We deal first of allwith the case in which
these infinities are all of the first order.
If k places of the surface, say tt,, aj ... a^, be arbitrarily
assigned we canalways specify a function with p periods having
these places as poles, of thefirst order, and otherwise continuous
and uniform ; namely, the function is ofthe form
where the coefficients fi,,, fJ'i ft^k are constants, the zeros
of the functions Fbeing left undetermined. Conversely, as remarked
in the previous chapter
( 20), a rational function having a,, ..., aj; as its poles must
be of this form.In order that the expression may represent a
rational function the periodsmust all be zero. Writing the periods
of FJ in the form ni(a), ..., f2p(a),this requires the
equations
^^ Qi (a,) + fi,ni (fl.) + . . . + /iifii (at) = 0,
for all the p values, i=l,2,...,p, of i. In what follows we
shall for the sake
of brevity say that a place c depends upon r places Ci, Cj, ...,
c^ when for allvalues of i, the equations
n.- (C) =f,ni (C,) + . . . +frili (Cr)
hold for finite values of the coefficients /i,...,/r, these
coefficients being
independent ofi. Hence we may also say
:
In order that a rational function should exist having k assigned
places as
its poles, each simple, me at least of these places must depend
upon the others.
24. Taking the k places d, a,, . . . , a* in the order of their
suffixes, it may
of course happen that several of them depend upon the others,
say a,+i, . . . , a*
-
28 DEPENDENCE OF POLES OF A KATIONAL FUNCTION [24
upon tti, ...,athe latter set Oj, ..., a being independent: then
we have
equations of the form
D,i (a,+i) = n,+,, , Hi (a,) + . . . + n,+i, s ^i (a.)
^i (a*) = i, 1 ^t (oi) ++*,. fii (a).
the coefficients in any of the rows here being the same for all
the p values ofi. In particular, if s be as great as p and aj , . .
. , a, be independent, equations
of this form will hold for all positions of a,+i, ...,*. For
then we have
enough disposeable coefficients to satisfy the necessary p
equations.
When it does so happen, that as+i,...,aie depend upon a^-.-ag,
thereexist rational functions, of the form
Rk =
-
25] DETERMINES EXPRESSION OF FUNCTION. 29
which has all its poles among Oj,...,ai be reckoned a particular
case of a
function having each of these as poles ; for it is clear that,
for instance, iZ^ isonly infinite at Oj, ..., o a^. The proposition
with a slightly altered enuncia-tion, given below in 27 and more
particularly dealt with in 37, is calledthe Riemann-Roch Theorem,
having been first enunciated by Riemann*,and afterwards
particularized by Rochf.
25. Take now other places at+i, at+j, ... upon the surface in a
definiteorder, and consider the possibility of forming a rational
function, which besidesimple infinities at Oj, ..., at has other
simple poles at, say, ajc+i, a^+t, ,*By the first Article of the
present chapter it follows that the least valueof h for which this
will be possible will be that for which a^ dependson Oj ... tttOi+i
... oa-i, that is, depends on Oi . . . a, at+i . . . a^-i. This
willcertainly arise at latest when the number of these places a,
... a, Ot+i ... ai,_iis as great as p, namely h l=k + ps, and if
none of the places at+i . . . a/,_idepend upon the preceding places
a, ... a,, it will not arise before: in thatcase there will be no
rational function having for poles the places
i a* Ot+i *+j
for any value oij from 1 to p s.
But in order to state the general case, suppose there is a value
of j less
than or equal to p s, such that each of the places
depends upon the places! ttj at+i ajc+j,
the smallest value of j for which this occurs being taken, so
that no one of
tti+i at-Lj depends on the places which precede it in the
series
ffll fflg t+i O'k+j-
Then there exists no rational function with its poles at aj-.-at
at+i . . . at+j,but there exist functions
Hk+j+i = O'k+j+i + ^t+j+i Lran-> + 1 ~ "*+J+. > ^ a.~
nic+j+1,8 ^a, Jlt+j+i.t+i rojt + i~
~ "t+j+i.i+j I ai + ,J
.
^k-\-j+i =
-
30 STATEMENT OF COMPLETE RESULT. [25
Then the most general rational function with poles at
! aag+i ajcttk+i ajt+j^t+j+i *+>+
is in fact
j/ + 1/,+, Rs+i + +vtRi + vjc+j+iBk+j+i + + ^t+j+iHk+j+i
and involves k-s + i+1 arbitrary constants, namely the same
number asthat of the places of the set
Oi asOn+i *ti+i at+iflifc+j+i "t+j+i
which depend upon the places that precede them.
For such a function must have the form
/i+/i,r^_ + +/islt, + Mmlt,+i+ + /it r^i + f^i+i 1^1+1+
namely,
k-s+ 2 /ig+r
r=lhr + "+r,l "^ a, T T
which is of the form
I'o + J'ir^, + +Vs^a,+ Vs+iJis+l+ + VtRk
+ vk+i Iti^, + + "t+jlt^^,. + vk+i+iR^^i^i + + Vk+j+iR^^j^r,
and the p periods of this, each of the form
i/,n(a,)+ + v,D, (a^) + vk+iil (ak+i) + +'t+jQ(ai+_,),
cannot be zero unless each of v^ ... VgVk+i Vk+j be zero, for it
is part of
the hypothesis that none of at+i . . . ak+j depend upon
preceding places.
26. Proceeding in this way we shall clearly be able to state the
followingresult
Let there be taken upon the surface, in a definite order, an
unlimitednumber of places a,, a, Suppose that each of !..._ is
inde-pendent of those preceding it, but each of a
_
,. ... depends on
i...a_ . Suppose that each of o-q +\^q +i,--- ^o - ^^
independent ofthose that precede it in the series aj . . . o a
.
, . . . o but each of* Vl-9l Vl+ l V9~V2>
a, a depends upon Oi ... a a ^^ ... a This requires
that&-,,+! ", "-r" "f ' "e,-?, e.+i,-,,
Q.-9. + [Q.-9.-Q.]>p.
-
26] EXPRESSION OF FUNCTION OF ASSIGNED POLES. 31
Suppose that each of a^^^j . . . a.Q^.,^ is independent of those
that precede itin the series ai...OQ_aQa ao,, ...a , but each of a
^,...adepends upon the places of this series. This requires
that
Let this enumeration be continued. We shall eventually come to
places"o,_i+i' ", 1+2' % -q ' ^^^^ independent of the places
preceding, for whichthe total number of independent places
included, that is, of places whichdo not depend upon those of our
series which precede them, is pso thatthe equation
P = {Qh-qh- Qk-i}+ + (Q, - q,-Q,) + (Q, - q,)= Qh-qi-q2-
-qhwill hold. Then every additional place of our series, those,
namely, chosenin order ffom a^
^^^,a^
,... will depend on the preceding places of the
whole series.
This being the case, it follows, using R/ as a notation for a
rationalfunction having its poles among a, . . . a/, that rational
functions
do not exist.
The number of these non-existent functions is p.
For all other values off, a rational function Rf eadsts.
To exhibit the general form of these existing rational functions
in the
present notation, let m be one of the numbers l,2,...,h; i be
one of thenumbers 1, 2, ... q^, and let the dependence of a _ .
upon the preceding
places arise by p equations of the form
then, denoting F* by r^, there is a rational function
which has its poles at
and the general rational function having its poles at
a, ... a^,a5,+i - aQ,%,+i - V-9+'-
-
32 THERE ARE p GAPS. [26
is of the form
and involves 5, + g'j + . . . + gm-i + ^ + 1 arbitrary
coefficients.
The result may be summarised by putting down the line of
symbols
1, 2, ... (Qa -?.), (Q>-?. + l), -!. Qi + 1. (Q.-?2),
(Q.,-9, + l),...,QQ.+ l,...,Qft-i + l, ..(QA-?^).(Qft-?^+l).
with a bar drawn above the indices corresponding to the places
which depend
upon those preceding them in the series. The bar beginning over
Qa - g/i + 1
is then continuous to any length. The total number of indices
over which
no bar is drawn is p. There exists a rational function Rf, in
the notation
above, for every index which is beneath a bar.
The proposition here obtained is of a very fundamental
character. Sup-
pose that for our initial algebraic equation or our initial
surface, we were able
only to shew, algebraically or otherwise, that for an arbitrary
place a there
exists a function Kl, discontinuous at a only and there infinite
to the firstorder, this function being one valued save for additive
multiples of k periods,and these periods finite and uniquely
dependent upon a, then, taking arbitraryplaces Oi, 02, ... upon the
surface, in a definite order, and considering func-tions of the
form
that is, functions having simple poles at a^, ..., ajv, we could
prove, just asabove, that there are k values of N for which such
functions cannot be onevalued ; and obtain the number of arbitrary
coefficients in uniform functionsof given poles. Namely, the
proposition would furnish a definition of thecharacteristic number
kwhich is the deficiency, here denoted by pbasedupon the properties
of the uniform rational functions.
We shall sometimes refer to the proposition as Weierstrass's
gaptheorem*.
27. When a place a is, in the sense here described, dependent
upon places61, 62. , 6r, it is clear that of the equations
* "Luckensatz." The proposition has been used by Weierstrass, I
believe primarily underthe form considered below, in which the
places a,, a^, ... are consecative at one place of thesurface, as
the definition of p. Weierstrass's theory of algebraic functions,
preliminary to a theoryof Abelian functions, is not considered in
the present volume. His lectures are in coarse ofpublication. The
theorem here referred to is published by Schottky : Conforme
Abbildungmehrfach zusammenbiingender ebener Flachen, Crelle Bd. 83.
A proof, with full reference toSchottky, is given by Noether,
Crelle Bd. 97, p. 224.
-
27] TRANSPOSITION OF THE LINEAR CONDITIONS. 33
A,a,(b,)+...+Apnp(b,)=o
A,a, (6,) + . . . + Apilj, (K) =A^iliia) +... + Apnp(a)=0
the last is a consequence of those precedingand conversely that
when thelast equation is a consequence of the preceding equations
the place a dependsupon the places by, 62, ..., 6^-
Hence the conditions that the linear aggregate
il (x) = A,n,(x) + ...+ Apilp (x)
should vanish at the places
wherein i^ qm, are equivalent to only{Q.-qd +
(Q2-q^-Qi)+---+{Qm-qm-Qm-,)
or
Qm-qi- -qmlinearly independent equations.
If then T + 1 be the number of linearly independent linear
aggregates ofthe form il {x), which vanish in the Qm qm + i
specified places, we have
T + l=p-{Qm-qi- -qm)-Denoting Qm qm+ i by Q, and the number of
constants in the generalrational function with poles at the Q
specified places, of which constants oneis merely additive, by 5 +
1,
q + l=qi + qi+ ...+ g^^i + i+l.
We therefore haveQ-g=;)-(T + l).
Recalling the values of n,(a;)... fij,(a;) and the fact (Chapter
II. 21)that every linear aggregate of them vanishes in just 2p 2
places, we see
that when Q is greater than 2p - 2, t + I is necessarily
zero.
In the case under consideration in the preceding article the
number
T + 1 for the function i2 , namely the number of linearly
independent
linear aggregates il {x) which vanish in the places
Oi'*!! ,%+! ",.!'
is given, by taking 7n = h-l and i = qh-i in the formula of the
presentarticle, by the equation
T + 1 = p - (Qh-i - 9i- - ^A-i)
= Qh-qh- Qh-i-B.
3
-
34, POLES AT ONE PLACE. [27
Hence one such linear aggregate vanishes in the places
and thereforeQA-gA-i>>2p-2
or, the index associated with the last place a^^^_^^ of our
series, correspondingto
which a rational function Rq^^,^ does not exist, is not greater
than 2p-l. A
case in which this limit is reached, which also furnishes an
example of the
theory, is given below 37, Ex. 2.
28. A limiting case of the problem just discussed is that in
which theseries of points a a^, ... are all consecutive at one
place of the surface.
A rational function which becomes infinite only at a place, a,
of thesurface, and there like
t"^ f"^ '*'
r'
where any of the constants C O^, . . . 0^_ but not Cr, may be
zero, t being the
infinitesimal, is said to be there infinite to the rth order. If
X; = Ci/(i 1) !,
such a function can be expressed in a form
\ + X, rs + x^DaFs + . . . + vDj-'
n
where, in order that the function be one valued on the
undissected surface,
the p equations
Xifii (a) + \Dani ()+...+ KD'a-'^i (a) =must be satisfied : and
conversely these equations give sufficient conditionsfor the
coefficients Xi, X,, ... , X^.
In other words, since Xr cannot be zero because the function is
infinite to
the rth order, the p differential coefficients Z)^~' fi,- (a),
each of the r 1thorder, must be expressible linearly in terms of
those of lower order,
a,(a), Z)fli(a), ...,D'-^ni(a),
with coefficients which are independent of i. We imagine the p
quantitiesD^~^D,i(a), for i = l, 2, ...,p, written in a column,
which we call the rthcolumn ; and for the moment we say that the
nec
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