giisniNcuLm;£srac[UBiiAR)i :>^ /S/ 4/ 1^.» V, Digitized by the Internet Archive in 2010 with funding from Boston Library Consortium IVIember Libraries http://www.archive.org/details/lecturesintroducOOfors LECTURES INTRODUCTORY TO THE THEORY OF FUNCTIONS OF TWO COMPLEX VARIABLES CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, Manager SonHon: FETTER LANE, E.C. ^Dmiurgt) : loo, PRINCES STREET ^ i.i i i i S^i Pi i i ^ iSerltn: A. ASHER AND CO. IlEipjtg: F. A. BROCKHAUS Porft: G. P. PUTNAM'S SONS Bombag ani Calcutta: MACMILLAN AND CO., Ltd. SToronto : J. M. DENT AND SONS, Ltd. a:oftso: THE MARUZEN-KABUSHIKI-KAISHA ^/i rights reserved LECTURES INTRODUCTORY TO THE THEORY OF FUNCTIONS OF TWO COMPLEX VARIABLES DELIVERED TO THE UNIVERSITY OF CALCUTTA DURING JANUARY AND FEBRUARY 1913 BY A. R. FORSYTH, Sc.D., LL.D., Math.D., F.R.S. CHIEF PROFESSOR OF MATHEMATICS IN THE IMPERIAL COLLEGE OF SCIENCE AND TECHNOLOGY, LONDON BOSTON COLLEGE LIBRARY CHESTNUT HILL, MASS. M ATH. OEPT. Cambridge : at the University Press 1914 ©ambrilrge : PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS. 15 r:: O a O 9 ) — ^tT'-T^:? '.N Q«eAT gP'^VN PKEFACE rilHE present volume consists substantially of a course of -'- lectures which, by special invitation of the authorities, I delivered in the University of Calcutta during parts of January and February, 1913, The invitation was accompanied by a stipulation that the lectures should be published. As regards choice of subject for the course, I was allowed complete freedom. It was intimated that the class would be mainly or entirely of a post-graduate standing. What was desired, above all, was an exposition of some subject that, later on, might suggest openings to those who had the will and the skill to pursue research. Accordingly I selected a subject, which may be regarded as being still in not very advanced stages of development, and into the exposition of which I could incorporate some results of my own which had been in my possession for some time. Owing to the limitations of the period over which the course should extend, it was not practicable to make the lectures a systematic discussion of the whole subject; and I therefore had to choose portions, in order to discuss a variety of topics and to . indicate some paths along which further progress might be possible. Thus, instead of concentrating upon one particular issue, I preferred to. deal with several distinct lines of investigation, even though their treatment had to be relatively brief. VI PREFACE Wherever it was possible to refer to books or to memoirs, I duly referred my students to the authorities. In particular, I urged them to prepare themselves so that they could proceed to the study of algebraic functions of two variables; because happily, in that region, there is the treatise by Picard and Simart, Fonctions algehriques de deux variables independantes, which includes an account of the researches made by Picard and others in the last thirty years. As this treatise is so full, I made no attempt to give to my students what could only have been a truncated account of the elements of that theory; but, as will be seen, what I did was to restate some of its problems from a different (and, as I think, a more general) point of view. At several stages in my lectures, I deviated from the almost usual practice of dealing with only a single uniform function of two complex variables. I thought it preferable to deal with two dependent variables as functions of two independent variables. Characteristic properties of the variation of uniform analytic functions of two variables are brought into fuller discussion, when two such functions are regarded simultaneously. The combination of at least two such functions is necessary when the general theory of quadruply-periodic functions is under review. The same combination of two functions seems to me desirable in the general discussion of the theory of algebraic functions of two variables whether these occur, or do not occur, in connection with quadruply-periodic functions; the considera- tion of relations between independent variables and dependent variables is thereby made more complete, and illustrations will be found in the course of the book. Even in the simplest case that has any significance, when these algebraic relations are nothing more than the expression of the lineo-linear substitutions, it is of course necessary to have two new variables expressible in terms of the variables already adopted. PREFACE Vll The following is a summary outline of the whole course of lectures. The first Chapter deals with the various suggestions that have been made for the geometrical representation of two complex variables. The intuitive usefulness of the Argand representation, when we are concerned with functions of a single independent complex variable, is universally recognised; but there seems to be a deficiency in the usefulness of each of the geometrical representations when more than a single independent complex variable occurs. The second Chapter is devoted to the consideration of the analytical properties of the lineo-linear substitution, defining two variables in terms of two others, each uniquely by means of the others. It is a generalisation of the homographic substitution for a single variable; some of the properties of the latter are extended to the case when there are two variables. In particular, iasistence is laid upon certain invariantive properties of such substitutions. The third Chapter is concerned with the expressibility of uniform analytic functions in power-series. The limitation of the range of convergence of such series leads to the notion of the various kinds of singularity which, under the classification made by Weierstrass, uniform analytic functions can possess. The fourth Chapter is devoted to the consideration of the form of a uniform analytic function in the immediate vicinity of any assigned place in the field of variation. The central theorem is due to Weierstrass, and was established by him for functions of n variables ; I have developed it in some detail when there are only two variables ; and it is applied to the description of the behaviour of a function in the vicinity of any one of its various classes of places, whether ordinary or singular. The fifth Chapter is occupied with two constructive theorems, both of them originally enunciated (without proof) by Weierstrass, VIU PREFACE as to the character of functions either entirely devoid or almost devoid of essential singularities. A function, entirely devoid of essential singularities, is expressible as a rational function of the variables ; the proof given is a modification of the proof first given by Hurwitz. A function, which has essential singularities only in the infinite parts of the field of variation, is expressible as the quotient of two functions which are regular in all finite parts of the field; the proof, which is given, follows Cousin's investigations for the general case of n variables. The next Chapter is devoted to integrals. The earlier paragraphs are concerned with double integrals of quantities which are uniform functions of two variables; after an exposition of Poincare's extension of Cauchy's main integral theorem, these paragraphs are mainly occupied with simple examples of a subject which awaits further development. The later paragraphs are concerned with integrals, whether single or double, of algebraic functions, a theory to which Picard's investigations have made substantial contributions. In restating the problems for the sake of students, I took the line of introducing a couple of algebraic functions, instead of only a single algebraic function, of two variables, so that there may be complete liberty of selection of two independent variables. The geometry of surfaces has led to valuable results connected with integrals of algebraic functions of two variables, just as the geometry of curves led to valuable results connected with integrals of algebraic functions of one variable. But my own view is that the development of the theory, however much it has been helped by the geometry, must (under present methods) ultimately be made to depend completely upon analysis. This will be more complicated when two alge- braic equations are propounded than when there is only a single equation; but its character will be unaltered. And so I have stated the problem for what seems to me the more general case. PREFACE IX In Chapter VII I have discussed the behaviour of two uniform analytic functions considered simultaneously. In particular, when the functions are independent and free (in the sense that they have no common factor), it is shewn that their level places are isolated; and the investigations in Chapter IV are used to obtain an expression for the multiplicity of occurrence of such a level place, when it is not simple. The last Chapter is devoted to the foundations of the theory of uniform periodic functions of two variables. In the early part of the chapter, I have worked out the various kinds of cases that can occur. The method may be deemed tedious ; it certainly could not be used for the functions of n variables with not more than 2n sets of periods ; but it brings into relief the discrimination between the cases which, stated initially only from the point of view of periodicity, are degenerate or resoluble or impossible or actual. The theta-fu net ions are then introduced on the basis of a result in Chapter V ; and the discrimination between functions with three period-pairs and those with four period-pairs is indicated. Later, some theorems enunciated (but not proved) by Weierstrass are established for functions of two variables, together with some extensions, all these being concerned with algebraic relations between homoperiodic uniform functions devoid of essential sin- gularities in the finite part of the field of variation. The Chapter concludes with some simple examples belonging to the simplest class of hyperelliptic functions. But I have not attempted, in these lectures, to expound the details of the theory of quadruply- periodic functions of two variables; it can be found in specific treatises to which references are given in the text. My whole purpose, in the Calcutta course, was to deal with a selection of principles and of generalities that belong to the initial stages of the theory of functions of two complex variables. PREFACE Often before, I have had to thank the Staff of the Cambridge University Press for their efficient help during the progress of proof-sheets of my books. This volume has made special demands upon their patience; throughout, as is their custom within mv experience, they have met my wishes with readiness and skill. To all of them, once again, I tender my grateftd thanks. A. R FOESYTH. Imperial College of Science AND Technology, London, S.W. FebrvAiry, 1914. >!S TABLE OF CONTENTS CHAPTER I. GEOMETKICAL REPRESENTATION OF THE VARIABLES. PAGE 1. General introductory remarks 1 2. Functions of two variables ; reason for occasionally considering two such functions, independent of one another ..... 2 3, 4. Geometrical representation of the variables ; three methods ... 4 5. Representation in four-dimensional space 5 6. Representation by lines in ordinary space ...... 7 7-9. Limitations upon the use of the Line ....... 7 10-13. Other methods of using lines in space for the geometrical representation 12 14 Representation by points in two planes, or by two independent points in the same plane .......... 13 15. Inferences from the two-plane representation of the variables . . 14 16. Extension of Riemann's definition of a function of a single variable to functions of two variables ........ 16 17. Extension of the property of conformal representation, when there are two independent variables ........ 18 18. But it belongs to any place in the field, and does not extend to loci or to areas. 19 19, 20. Analytical expression of frontiers of a doubly infinite region in the field, with examples .......... 20 CHAPTER II. LINEO-LINEAR TRAN'SFOR^MATIONS ; INVARIANTS AND CO VARIANTS. 21. Lineo-hnear transformations in two variables ..... 2-5 22, 23. Canonical form of the transformation, in the alternatives fi'om the characteristic equation, with expressions for its powers ... 26 24. Invariant-centres of transformations 29 25. Curves conserved, in character, under homographic substitutions in one variable ........... 32 26. Frontiers conserved, in character, under lineo-Unear transformations in two variables ........... 32 27. Simplest conserved equations : quadratic frontiers 34 xu CONTENTS 28. When the axes of real quantities are conserved 29. Another method of constructing the equations of some conserved frontiers 30. Invariants and covariants of quadratic frontiers 31. Introduction of homogeneous variables and umbral forms; use of Lie's theory of continuous groups . 32. Simple examples of invariants and covariants 33. The infinitesimal transformations 34. The partial differential equations of the first order characteristic of the infinitesimal transformations 35. Number of algebraically independent integrals 36, 37. Method of determining the integrals, in general 38. Determination of the four invariants 39. Contragredient variables .......... 40. Suggested canonical form of equations for a quadratic frontier 41. Periodic lineo-linear transformations 42. Equation for the multipliers ; conditions for periodicity ; with examples FAGB 35 35 39 39 41 42 44 46 46 48 ■ 49 50 52 52 CHAPTER HI. UNIFORM ANALYTIC FUNCTIONS. 43. 44. 45. 46. 47, 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. Preliminary definitions ; field^ domain, vicinity, for the variables . . 57 Uniform, tmiltiform, for functions, with an example .... 58 Continuous, analytic, regular, integral, transcendental, algebraic, mero- morphic, for functions ......... 59 Property of function establishing its regularity ..... 61 Upper limits for the moduli of derivatives of a regular function; some double integrals 64 A theorem expressing, by means of a double integral, any number of terms in the expansion of a regular function ..... 67 Dominant functions, associated with a regular function ... 70 Absolute convergence of a double power-series ..... 72 A regular function must acquire infinite values somewhere in the whole z, / field ; identity of two regular functions under a condition ; when the regular function reduces to a polynomial ... 72 A regular function must acquire a zero value somewhere in the whole z, z' field , . . 75 The investigations of Picard, Borel, and others, in regard to the same property for a regular function of one variable . . . . 77 Extension of Picard's theorem concerning functions of one variable . 78 Weierstrass's process of analytical continuation of a function ; the region of continuity ............ 79 Singidarities, unessential, essential, of uniform functions ... 82 Two kinds of unessential singularity for a uniform function of two variables; discriminated, in name, by pole and (the other type of) unessential singularity ......... 84 Extension, to functions of two variables, of Laurent's theorem for functions of one variable 86 CONTENTS Xlll CHAPTER IV. UNIFORM FUNCTIONS IN RESTRICTED DOMAINS. §§ PAGE 60. Expression of a regular function / (z, /) — / (0, 0) in the immediate vicinity of 0, 0, the function / {z, z') being regular .... 92 61-63. Weierstrass's theorem for the case when/(2, 0) — /(0, 0) is not an identical zero ; likewise for the case when / (0, z') —f (0, 0) is not an identical zero, together with a corollary from the theorem as to an expression for / (z, z') -^f (0, 0) in these cases 93 64. New expression (distinct from Weierstrass's expression) for/(s, s') — /(0, 0) when either / (^, 0) -/ (0, 0), or / (0, /) - / (0, 0), or both expres- sions, may be an identical zero, with summary of results, and a general example ...... .... 98 65. Weierstrass's method of proceeding, adapted to two variables, for the cases of § 64, with examples ........ 105 66, 67. On the level values of functions in the immediate vicinity of the value / (0, 0) at 0, 0, with examples 108 68. The order of the zero-value of f {z, z') — f {a, a!) Ill 69. Divisibility of one function by another . . . . . . . 112 70. Analytical tests of divisibility, when both /(O, 0) and g (0, 0) are zero 113 71. Analytical tests that the' two functions f {z, z') and g (z, z') in the latter case should possess a common factor h {z, z') such that h (0, 0) is zero; reducibility of functions . . . . . . . . 115 72. Expression of a uniform function in the immediate vicinity of a pole ; it has an infinitude of poles near a pole 119 73. Expression of a uniform function near an unessential singularity ; when the expression is irreducible, the unessential singularity is isolated . 121 74. Expression of a uniform function near an essential singularity ; references to authorities 122 CHAPTER V. FUNCTION'S WITHOUT ESSENTIAL SINGULARITIES IN THE FINITE PART OF THE FIELD OF VARIATION. 75. Two theorems on the expression of a uniform function of two variables . 124 76. Properties of a polynomial function of z and / as regards singularities . 124 77. Properties of a rational ftmction of z and s' as regards singularities . . 125 78. Proof (based on Hurwitz's proof) of Weierstrass's theorem that a uniform function, which has no essential singularity anywhere in the whole field of variation, is rational 125 79. A possible expression for a function which has no essential singularity in an assigned finite part of the field . . . . . . 129 80. Weierstrass's theorem (as adumbrated) on functions having essential . singularities only in the infinite part of the field .... 130 XIV CONTENTS §§ PAGE 81. Cousin's proof of the theorem; his preliminary lemmas • . . 131 82. Division of a domain into regions with which functions are associated ; functions equivalent in a region or at a point . . . . 133 83, 84. Construction of a function F {z, z'), equivalent to the assigned functions associated with each region . . . . . . 135 85. Cousin's extended theorem, so as to lead to an ultimate product- theorem ............ 138 86. The general theorem as to the expression of a function of two variables having no essential singularities in the finite part of the field . 141 87-90. Establishment of the theorem that such a function is expressible as the quotient of two regular functions . . . . . . . 143 91, 92. Appell's sum-theorem, with an example . .... . 147 93. Example of a product-theorem for a special class of functions . . 150 CHAPTER VI. . INTEGRALS ; IN PARTICULAR, DOUBLE INTEGRALS. 94. Two kinds of double integrals ; the class arising from repeated simple integration ........... 152 95. Definition of a double integral with complex variables . . . 153 96. Theorem on double integrals with real variables . . . . 156 97-99. Application to the double integral jjf{z,z')dzdz', taken through a limited four-dimensional space, when f{z,z') is regular; Poincare's extension of Cauchy's Theorem 158 100. Several examples of double integrals when/(z, z') is not regular within the region, deduced by means of the inferences in §§ 97-99 . 161 101. Some remarks on algebraic functions of two variables; the most general form of function . . • . . . . . 170 102. Similarly as to the form of the most general function involving two algebraic functions . . . . . . . . . . 172 103. Single integrals of algebraic Ructions; preliminary condition if they are to have no infinities . . . . . . . . 178 104, 105. Equivalent forms of a single integral . . . . . . . 180 106, 107. For two quite general algebraic equations in two dependent variables, integrals of the first kind do not exist 185 108. Double integrals, in equivalent forms 187 109, 110. Some conditions that a double integral should be of the first kind . 190 111. Partial extension of Abel's theorem . 193 CHAPTER VII. LEVEL PLACES OF TWO SIMULTANEOUS FUNCTIONS. 112. Theorem as to the possession, somewhere in the whole field of variation, of common zeros by two functions which are regular for finite values ............ 198 113. Lemma as to the march of the gradual diminution of | / (2, 2') | and ■ \9{^,^)\ . 199 CONTENTS XV SS PAGE 114. Application of the lemma to establish the theorem enunciated in § 112 . 202 115. Extension of the theorem to a couple of functions devoid of essential singularities in the finite part of the plane 203 116. Order of the conamon zero,, when isolated: the condition for isolation 204 117. A common zero may happen not to be isolated; conditions . , 205 118. Summary of results as to the possession of common zeros . . 206 119. But the two functions must be independent, and must be free, if the common zero is isolated 208 120. Determination of the order of a common isolated zero . . . 209 121, 122. Common level-places of two functions, independent and free . . 210 CHAPTER VIII. •UNIFORM PERIODIC FUNCTIONS. 123. Definition of periodicity, and periods . . . , . . . 213 124. Why infinitesimal periods for functions of two variables are excluded from consideration; Weierstrass's theorem .' . . . . 213 125. A uniform analytic function cannot possess more than four pairs of linearly independent periods ........ 216 126. Preliminary lemma . . . . . . . . . . 217 127-130. Jacobi's theorem : one pair of periods : two pairs of periods . . 222 131-133. Three pairs of periods: canonical form of period-tableau . . . 226 134. Complete field of variation for variables of triply periodic functions 231 135-137. Four pairs of periods 232 138. Representation of variation for variables of quadruply periodic functions 236 139. Uniform triply periodic functions: a general expression . . . 238 140. Triple theta-functions 240 141, 142. Law of coefficients in a double Fourier-series, in order that it may be a triple theta-function ........ 241 143. Periodicity of double Fourier-series in general 243 144. Special forms of coefficients leading to special functions . . . 244 145. Remark on the addition-theorem for theta-functions . . . . 246 146. The sixteen triple theta-functions 246 147, 148. Limitation on coefficients so as to secure oddness or evenness . 248 149. Interchange of triple theta-functions for half-period increments of the variables, (i) in general, (ii) when each of the functions is either even or odd, (iii) when the coefficients are specialised so as to give double theta-functions ........ 250 150. Table of the interchanges in § 149 253 151. Some selected zero places of the theta-fimctions .... 253 152-154. Two uniform quadruply periodic functions considered simultaneously: their common irreducible level places are isolated and are finite in number, when there are no essential singularities for finite values of the variables 255 Xvi CONTENTS PAGE 155. The number of irreducible level places of two uniform quadruply periodic functions is independent of the level values: grade of two functions • • 259 156. Algebraic relations between three imiform quadruply periodic functions, having the same periods 260 157. Any uniform quadruply periodic function satisfies a partial difi'erential equation of the first order 262 158. Expressibility of a quadruply periodic function, (i) in terms of three given homoperiodic functions; (ii) in terms of a homoperiodic function and its first derivatives 262 159. Algebraic equation between two homoperiodic functions and their Jacobian . . . 264 160. Expressibility of a quadruply periodic function in terms of two homo- periodic functions and their Jacobian 265 161, 162. The double theta-functions, and the hyperelliptic functions of order two 266 163-166. Illustrations of the theorems of §§ 156-160 271 Index 278 CHAPTER I Geometrical Representation of the Variables In regard to functions of a single complex variable, reference may generally be made, for statements of results and for quoted theorems, to the author's Theory of Functions. No reference is made to the ultimate foundations of the theory of functions of a single real variable ; a full discussion will be found in Hobson's Functions of a real variable. For a large part of the contents of the fii'st two chapters, reference may be made to two papers by the author*; and particular references to memoirs will be made from time to time as they are quoted. But in addition, reference should be made to a paper t by Poincare, who discusses groups, classes of invariants, and conformation of space, when the representation of the two complex variables is made by means of four-dimensional space. 1. This course of lectures is devoted to the theory of functions of two or more complex variables. It will be assumed that the substantial results of the theory of functions of a single complex variable are known ; so that references to such results may be made briefly or even only indirectly, and suggestions, especially in regard to the extensions of ideas furnished by that theory, can be discussed in their wider aspect without any delay over preliminary explanations. My intention is to deal with some of the principles and the generalities of the selected subject. Special illustrations and developments will be given from time to time; but limitations forbid the possibility of attempting an exposition of the whole range of knowledge already attained. Moreover, my hope is to establish some new results, and suggest some problems; in order to make that hope a reality within this course, some developments must be sacrificed. The sacrifice, however, need only be temporary, in one sense ; because references to the important authorities will be given, and their work can be consulted and studied in amplification of these lectures. * "Simultaneous complex variables and their geometrical representation," Messenger of Math., vol. xl (1910), pp. 113 — 134 ; " Lineo-linear transformations of two complex variables," Quart. Journ. Math., vol. xliii (1912), pp. 178—207. t "Les fonctions analytiques de deux variables et la representation conforme," Rend. Give. Mat. Palermo, t. xxiii (1907), pp. 185—220. F. 1 2 FUNCTIONS OF TWO VARIABLES [CH. I Usually, it will be assumed that the number of independent variables is two. In making this restriction, a double purpose is proposed. Not a few of the propositions for two variables, with appropriate changes, can justly be enunciated for n variables ; and sometimes they will be enunciated explicitly. In such cases, they usually are true for functions of a single variable also ; and they become generalisations of the last- mentioned and simplest form of the corresponding proposition. Results of this type have their importance in the body of the theory. But it is desirable to have other results also, which may be called characteristic of the theory for more than a single variable, in the sense that they have no corresponding counterpart in the theory for a single variable. Again, it is desirable, wherever possible, to obtain results equally character- istic of the theory in another direction, that is to say, results which are not mere specialisations of results for the case of three or more variables. Such a result is provided in the case of the quadruply-periodic functions of two variables and their association with single integrals involving the quadratic radical of a quintic or sextic polynomial. The case might be taken as the appropriate specialisation of 2?i-ply periodic functions of n variables and their proper association with single integrals involving the quadratic radical of a polynomial of order 2?i -|- 1 or 2w -f- 2. These latter functions, however, are notoriously not the most general multiply-periodic functions for values of n from 3, inclusive and upwards. Consequently, it is sufficient to develop the association with quadratic radicals of a quintic or sextic polynomial; the formal generalisations of the results so obtained are only limited and restricted forms of the results belonging to the wider, but not most com- pletely general, theory. These combined considerations constitute my reason for dealing mainly with the theory of functions of tw/) independent complex variables. The two variables will be denoted by z and z'. 2. One illustration of real generalisation from the theory of functions of a single variable arises as follows. In that theory, when a variable w is connected with a variable ^^ by a relation f{w, z) = of any form, we frequently consider that w is defined as a function of z by the relation. But frequently also there is a necessity for regarding ^ as a function of w ; and important results, especially in connection with periodic functions, are obtained by using this dual notion of inversion. A question naturally suggests itself: — what is the general form of this notion of inversion when there are two independent variables ? A function w oi z and z' can be regarded as given by a relation f{w, z, z) = 0, any precision as to the form of/ being irrelevant to the immediate discussion. A limited use of the notion of inversion can be applied at once 2] RECIPROCAL ASPECT OF EQUATIONS 3 to the relation. Just as in the Cartesian equation of a surface in ordinary space it is often a matter of indifference which of the three coordinates is to be regarded as expressed by the equation in terms of the other two, so now we may regard the relation f{w, z, z) = as dejSining any one of the three variables w, z, z' in terms of the other two. Such an interpretation of the . relation does not imply the complete process of inversion in the simpler case, whereby the quantity initially regarded as independent is expressed in terms of the quantity initially regarded as dependent. In the present case, the initially independent variables z and / are not expressible in terms of the single initially dependent variable w. The limitation in the use of the notion, however, disappears when two functionally distinct quantities w and w' occur. This occurrence might arise through the existence of two functional relations f{w, z, z') = 0, g (w', z, z) = 0, or of two apparently more general functional relations F{w, w', z, z') = 0, (w, w', z, z') = 0. We assume that the equations F=0, G — 0, do actually define distinct functions w and w' in the sense that they are independent equations ; that is, we assume that their Jacobian \w, w' does not vanish identically. Moreover, for our purpose, w and w' are not merely to be distinct from one another ; they are to be independent functions of z and z', so that the Jacobian does not vanish identically. Now \W,W J \Z, Z J \z, Z J always ; hence neither of the Jacobians \w, w / \z, can vanish identically. In other words, we can interpret the two relations F=Q and G = in a new way ; they define z and z as two distinct and independent functions of the two independent variables w and w'. Ex. Thus the equations satisfy both conditions ; the quantities w and w' are independent functions of z and z'. And conversely for z and z' as independent functions of w and w'. 1—2 4 GEOMETRICAL [CH. I On the other hand, the equations ww' — z — z' = 0, w^ — ic' — 1 = 0, being independent equations, determine w and w' as distinct functions of the variables, for -^ — , I does not vanish identically. But these distinct functions are not independent w^ w J (w io\ ' , j vanishes identically. As a matter of fact, both w and w' are functions solely of the combination z + ^ of the variables, and therefore w and w' are expressible in terms of each other alone ; the actual relation of expression is the second of the two equations. Thus, by the introduction of a second and independent function w', we are in a position to adopt completely the notion of inversion, as distinct from any precise expression of inversion, for the case of two complex independent variables*. The inversion will be equally possible from any two relations, which are the exact and complete equivalent of F = and (r = in whatever form these relations may be given. In particular, if F and G are algebraical in w and w', they have an exact equivalent in relations of the type /= and g = 0, obtained by eliminating w' and tv in turn between F=0 and G^ = 0. Finally, we could regard any two of the four variables z, /, w, w' as independent and the remaining two as dependent. The necessary and sufficient condition is that no Jacobian of F and G with regard to any two of the variables shall vanish identically. Accordingly, for many purposes, we shall find it desirable to consider simultaneously two independent functions w and w' of the two independent variables 2 and z. Geometrical Representation of the Variables. 3. Next, it proves both convenient and useful in the theory of functions of one variable to associate a geometrical representation of the variables with the analysis. It happens that this representation is simple and complete while full of intuitive suggestions; and thoughf the notion of geometrical interpretation has not been adopted by all investigators and has occasionally been deliberately avoided by the sterner analytical schools, it has acquired importance because of the character of the results to which it has led. The representation, initiated by Argand, is obtained by the customary association of a point upon a plane with one variable, and of a point upon * When there are n independent variables Zi, ..., z^, then n functions wi, ..., /«„ are required for the corresponding complete use of iuversion. t There is a wide diversity of practice, in regard to the extent of the adoption of geometrical notions in the development of the analysis of the theory of functions. As an indication of this variety, it is sufficient to note the different relations to the subject as borne in the work of Cauchy, Hermite, Kronecker, Poincare, Eiemann, and Weierstrass. 5] REPRESENTATION 5 another plane with the other variable ; and the functional relation between the two variables is exhibited as a conformal representation of either plane upon the other. An adequate geometrical representation of two independent complex variables is a more difficult problem than the representation of a single complex variable ; at any rate, there is as yet no unique solution of the problem which has been found quite so satisfactory as the Argand solution of the problem for a single variable. In order to let the full variation appear, we resolve each of the complex variables into its real and its imaginary parts ; so we write 2 = x + iy, z =x + iy'. Here x, y, x , y' are real ; when z and z' are independent in every respect, each of these four real quantities admits of independent variation through the range of reality between — oo and + oo . Thus a four-fold set of variations is required for the purpose ; and such a set cannot be secured simply among the facilities offered by the ordinary space of experience. 4. Several methods have been proposed. No method has been adopted universally. The respective measures of success are attained through some greater or smaller amount of elaboration; but each increase of elaboration causes a decrease of simplicity, and therefore also a decrease of intuitive suggestiveness, in the geometrical representation. Among the methods, there are three which require special mention. In one of them, four-dimensional space is chosen as the field of variation. In the second, a line (straight or curved) is taken as the geometrical entity representing the two variables simultaneously. In the third, each of the variables is represented by a point in a plane (the planes being the same or different), so that two points are taken as the geometrica,l entity repre- senting the two variables simultaneously. 5. Of these methods, the simplest (in a formal analytical bearing) is based upon the use of four-dimensional space; and applications to the theory of functions of two complex variables have been made by Poincare*, Picardf, and others. The four real variables x, y, x', y' are associated with four axes of reference. Sometimes they are taken as the ultimate variables ; sometimes they are made real functions of other ultimate real variables, from one to three in number according to the dimensions of the continuum * "Sur les fonctions de deux variables," Acta Math., t. ii (1883), pp. 97—113; " Sur- les residus des int^grales doubles," Acta Math., t. ix (1887), pp. 321 — 380; " Analysis situs," Journ. de VEcole Polyt., Ser. 2, t. i (1895), pp. 1 — 123; "Analysis situs," Eend. Circ. Mat. Palermo, t. xiii (1899), pp. 285—345, t. xviii (1904), pp. 45—110, and elsewhere. t Traite d' Analyse, t. ii, cb. ix ; Theorie des fonctions algebriques de deux variables in- dependantes, t. i, ch. ii, in the course of which other references are given. 6 SPACE OF FOUR DIMENSIONS [CH. I to be represented. Thus a single relation between x, y, x', y provides a hypersurface (or an ordinary space) in the quadruple space ; and, along the hypersurface, each of the four variables can be conceived as expressible in terms of three variable parameters. Two such relations provide a surface in the quadruple space ; along the surface, each of the variables can be conceived as expressible in terms of two variable parameters. Similarly, three such relations provide a curve along which each of the variables can be conceived as expressible in terms of a single variable parameter. Lastly, four such relations provide a point or a number of points. The intersection of a hypersurface and a surface is made up of a curve or a number of curves. Two surfaces intersect in points; two hypersurfaces intersect in a surface or surfaces. We consider only real surfaces, curves, and points, in such intersections; because what is desired is a representation of the four real variables, from which the complex variables are composed. The representation, by itself, does not seem sufficiently definite and restricted. There is no preferential combination in geometry among the four coordinate axes, which compels a combination of x and y for one of the complex variables, while x' and y' must be combined for the other. But this original lack of restriction is supplied, so far as concerns functions of z and /, by retaining the partial differential equations of the first order, which are satisfied by the real and the imaginary parts of any function w. Writing w = u-\-%v =f{z, z), where a and v are real, we have du dv du _ dv du _ dv du _ dv dx~ dy' dy dx ' dx' dy' ' dy' dx' ' so that u satisfies (as does v also) the equations dx^ dy^ ' dxdx' dydy' ' dx^ dy'^ ' dxdy dydx' From a value of u, satisfying these equations, the value of v to be associated with it in the value of w can be obtained by quadratures. Thus we have a geometry, tempered implicitly by differential equations. The comparative difficulty of dealing with the ideas of four-dimensional geometry tends to prevent this mode of representation from being intuitively useful, at least to those minds who regard the stated results to be analytical relations merely disguised in a geometrical vocabulary. In particular, the method fails to provide (as the other methods equally fail to provide) a representation of quadruple periodicity which serves the same kind of purpose as is served by the plane representation of double periodicity; and a fortiori there is an even graver lack, when divisions of multiple space are required in connection with functions of two variables that are automorphic 7] REPRESENTATION BY A LINE 7 under lineo-linear transformations. Still, it is the fact that certain results have been obtained through the use of this method in the extension of one of Cauchy's integral-theorems, in the formation of the residues of double integrals, in the topology of multiple space, and in the conformation of spaces. 6. The second of the indicated methods of representation of the four variable elements in two complex variables is based upon the fact that four independent coordinates are necessary and sufficient for the complete specification of a straight line in ordinary space. Such a line would be determined uniquely by the two points (and, reciprocally, would uniquely determine the two points) at which it meets a couple of parallel planes ; and therefore, if one variable z is represented by a variable point in one plane and the other variable z' is represented by a variable point in the other plane, we might regard the line joining the points z and / in the respective planes as a geometrical representation of the two variables z and z' con- jointly. (It can also be determined by a point, and a direction through the point ; again, the determination requires four real variables in all.) We must, however, bear in mind that the two points on the line are the ultimate representation of the two variables. When the whole line* (with the assistance of the two invariable parallel planes of reference) is taken to represent the two variables, a question at once arises as to the geometrical relations between a line z, z and a line w, lu , which correspond to two analytical relations between the variables. Does the whole line z, z, under any transforming relation, become the whole line w, w ? 7. It is only a specially restricted set of transforming relations, which admit such a transformation of a whole line. The result can be established as follows. For simplicity, we assume that the planes for z and / are at unit distance apart, and likewise that the planes for w and w' are at unit distance apart ; and we write w = u + iv, w' = u + iv'. The Cartesian coordinates of any point on the z, z' line are o-o; -h (1 — 0-) x, ay +{1 — a) y , 1- a, and those of any point on the w, w' line are pu + {1- p) u', pv -f (1 - p) V, 1-/0, where p and a are real quantities, each parametric along its line. Let. two relations F{w, w', z, z) = 0, G (w, w\ z, z') = 0, be such as to give a birational correspondence between w, w' and z, z . If, * For the following investigation reference may be made to the first of the author's two papers quoted on p. 1. 8 REPRESENTATION BY A [CH. I then, in connection with these relations, the whole z, z' line is transformed uniquely into the whole w, w line, and vice-versa, some birational corre- spondence between the current points upon the lines must exist ; and so the coordinates of the current point upon one line must be connected, by functional relations, with the coordinates of the current point upon the other line. Because of the independent equations F=0, (r = 0, the quantities u, v, u, v' are functions of x, y, x', y' alone ; and these functions do not involve a. Similarly x, y, x, y are functions of u, v, u', v' alone ; and these functions do not involve p. Hence p is a function of a only, such as to take the values and 1 (in either order) when a has the values and 1 ; and, for the current points, we must have pu + {l-p)u =f{^,7),l-a), pv + (l- p)v'=g{^,V, 1-0-), where / and g are appropriate functions of their arguments, and ^ = a-x + (l-^\ The remaining combination, viz. ix = fj!, v = v', under the requirements leads to the expressions w = {Az^ A'z) e'^\ w' = {Cz -\- C'z) e>'\ All these expressions must still satisfy the terminal condition applying to p and 0-, viz. that p must be or 1 when cr is or 1. When these expressions 9] REPRESENTED IN SPACE 11 are inserted for the functions / and g in the earliest equations in § 7, the latter lead to the relations / ja + (l-/j)7 ^ pa! + (1 - /o) ry' (T 1 — cr ' p^+a-p)h _ p^'+{i-p)h' ^ cr 1 — ^\ w = G'z'ei^^, the same as before. In obtaining this result, we neglected temporarily an arbitrary change of origin in each of the planes ; and we assumed that z can be interchanged with z', and w with w'. Thus we have the result : — The only relations which give a birational transformation of the straight line, joining z and z' in two parallel planes, into a straight line, joining w and w' also in two parallel planes, either are w = aze*^^ + 6e^^ w' = a'^e"^ + ce^*, where a, a', b, c, a, ^, y are real constants, or can be changed into this form by interchanging z and z' , or w and w' , or both. These relations, as equations in a general theory, are so trivial as to be negligible; and so we can assert generally that two functional relations F{w, w', z, z') = and G{w, w', z, /) = 0, which transform the variables z 12 REPRESENTATION OF [CH. I and / in their respective parallel planes into the variables w and w' likewise in their respective parallel planes, do not (save in the foregoing trivial cases) admit a birational transformation of the whole straight line joining z and / into the whole straight line joining w and w'. 10. Manifestly; therefore, we need not retain the suggested geometrical representation of two variables by the whole straight line joining the two points z and z', because the only effective part of the representation is provided by the two points in which the line cuts the planes. Nor would any other method of selecting the four real variables for the specification of the straight line be more effective. For example, the line would be uniquely selected by assigning a point where it cuts a given plane and assigning its direction relative to fixed axes in space ; and then we could take z = x-{- iy, z' = e^* tan 6, with the usual significance for x, y, 0, (f). It is easy to see that, when we take a plane at unit distance from the given plane, and we write z" = z + z, the former representation by the straight line arises for z and /'. As before, the whole straight line is not an effective representation of the two complex variables ; the only effective part of the representation is the point in the given plane and the direction relative to fixed axes. 11. Another method of constructing a straight line to represent two complex variables z and / has been propounded by Vivanti*, whereby it is given as the intersection of the two planes ccX + yZ=l, a;'Y+y'Z=l, where X, T, Z are current coordinates in space. The immediate vicinity of a line Zq, Zq is assumed to be the aggregate of all lines such that (x - XoY +{y- yof < r\ {x - x.'f + {y - y^f ^ r\ where r and r are arbitrary small quantities ; and the boundary of the vicinity is made up of the lines {X - XoY + {y- y,f = r\ (x' - x,'y + {y' - y,' )' = r\ It is easy to see that, as before, the Avhole straight line as a single geometrical entity is not an effective representation of the two complex variables z and z ; the only effective part of the representation depends upon the coordinates of the two points in which the line cuts -the planes of reference Y =0, X = (or any two of the coordinate planes). * Rend. Circ. Mat. Palermo, t. ix (1895), pp. 108—124. 14J TWO VARIABLES BY A LINE 13 12. The preceding investigation suggests cognate questions which will only be propounded. Two functional relations, F {lu, w', z, z') = and G (w, w', z, z) = 0, transform a pair of points z and /, in parallel planes, into a pair (or into several pairs) of points w and w , also in parallel planes. Let z and z be connected by any analytical curve ; let a corresponding pair of points w and w also be connected by any analytical curve ; and suppose that the two analytical curves have a birational correspondence with one another. Then (i) How are the equations of this correspondence connected, if at all, with the original functional relations ? and what are these equations when the two analytical curves are assigned ? (ii) What functional relations are possible if, under them, the whole z, z curve is to be transformed into the whole w, w curve ? (iii) When functional relations are given and an analytical z, z curve is assigned, what are the equations of the w, w' curve, if and when the whole curves are transformed into one another ? 13. One warning must be given before we pass away from the con- sideration of a line, straight or curved, as a geometrical representation of a couple of complex variables. The preceding remarks refer to the possibility of this geometrical representation ; they do not refer to functions of two complex variables which are functions of a line. Functions of a real line occur in mathematical physics ; thus the energy of a closed wire, conveying a current in a magnetic field, is a function of the shape of the wire. This notion has been extended by Volterra* on the basis of Poincare's general- isation of one of Cauchy's integral-theorems. In the case of the integral of a uniform function of one complex variable, we know that the value is zero round any contour, which does not enclose a singularity of the function, and that the integral between two assigned points is (subject to the usual proviso as to singularities) independent of the path between the points; that is, the integral can be regarded as a function of the final point. So also (as we shall see) the integral of a function of two complex variables over a closed surface in four-dimensional space is zero if the surface encloses no singularity of the function ; and when the surface is not closed, the integral (subject to a similar proviso as to singularities) depends upon the boundary of the surface ; that is, the integral can be regarded as a function of the boundary-line. This property has nothing in common with the line-representation of two complex variables which has been discussed. 14. The third of the indicated methods of representation of two complex variables is the effective relic of the discarded line-representation. It is the simple, but not very suggestive, method of representing the two variables z * Acta Math., t. xii (1889), pp. 233—286. 14 PROPERTIES OF THE [CH. I and / by two points, either in the same plane or in different planes, the two points always being unrelated. It is the method usually adopted by Picard and others. For quite simple purposes, it proves useful ; thus it is employed by Picard* in dealing with the residues of the double integrals of rational functions, and it is important in his theory of the periods of double integrals of algebraic functions. Let me say at once that the point-representation of z and z' is not completely satisfactory, in the sense that it does not provide a representation which gives a powerful geometrical equivalent for analytical needs. One illustration will suffice for the moment. It is a known theorem f, due originally to Jacobi in a simpler form, that a uniform function of two variables cannot possess more than four pairs of periods. The point- representation of two variables admits of an effective presentation of simple periodicity for either variable or for both variables, of double periodicity for either variable or for both variables separately, of triple periodicity for both variables in combination ; but (as will be seen later in these lectures) it does not lend itself to a presentation of quadruple periodicity for both variables in combination, a presentation which is much needed for functions so fundamental as the quotients of the double theta-functions. An attempt to circumvent the latter difficulty will be made later for one class, of quadruply-periodic functions. But the general difficulty remains. There are other limitations also upon the effectiveness of the method of repre- sentation by points ; they need not be emphasised at this stage. New ideas, or some uniquely effective new idea, can alone supply our needs. In the meanwhile, we possess only two fairly useful methods, viz., the method of four-dimensional space, and the method of two-plane representation. Properties of the fiuo-plane representation. 15. As the principal use of the representation of two variables in four- dimensional space occurs in connection with double integrals, illustrations can be deferred until that subject arises for discussion. We proceed now to make a fe^v simple inferences from the two-plane representation of two variables;!:. We shall use the word place to denote, collectively, the two points in the 2^-plane and the /-plane respectively which represent the values of z and * See the reference to the second treatise by Picard, quoted on p. 5. t The general theorem is that a uniform function of n independent variables cannot possess more than 2ii independent sets of periods. The simplest case, when n=:l, was originally estab- lished by Jacobi, Ges. Werke, t. ii, pp. 27 — 32. For the general theorem, see the author's Theory of Functions, § 110, § 239, where some references are given. J For much of the investigation that follows, reference may be made to the author's paper, quoted on p. 7. 15] TWO-PLANE EEPRESENTATION 15 of /. Let w and lu' be two independent functions of z and /, so that their Jacobian J, where /=/ w, w z, z does not vanish identically ; and let the places z, z' and w, w be associated by functional relations. Any small variation from the former place, repre- sented by dz and dz, determines a small variation from the latter place, which may be represented by dw and dw' ; the analytical relations between these small variations are of the form dw — Adz-\- Bdz, dw' = Gdz + Ddz , where A, B, G, D are free from differential elements, and AD — BG = J. Next, let d-^z and d-^z, d^z and d^z' denote any two small variations from the z, z' place ; and let d-^w and d-^w', d^w and d^^w' denote the consequent small variations from the lu, w place. Then d^w, dxw' doW, d^w' = j AdiZ + BdiZ, I Ad^z + Bd^z', = J Gd^z+Bd^z GdoZ + DdoZ diZ, djZ j . Ct'2 ^j tX'2 ^ I Manifestly, if d^zd.^z' — d^zd-^z vanishes, then d^wd^w' — d^wd-^w' also vanishes ; and the converse holds, because J is not zero. Hence if, at the place z, z' , two similar injfinitesimal triangles are taken in the planes of z and of / respectively, the corresponding infinitesimal triangles at the place w, w in the planes of w and of w' respectively also are similar ; and conversely. This property holds for all pairs of similar infinitesimal triangles ; and therefore, when the 2^-plane and the ^'-plane are put into conformal relation with one another, the w-plane and the w'-plane are also put into conformal relation with one another. This result is the geometrical form of the analytical result that, when the two equations F (lu, w' , z, z) = 0, G {w, w', z, z) = 0, determine w and w' as independent functions of z and z', a relation (f) (z, z') = 0, involving z and / only, leads to some relation i/r {w, w') = 0, involving w and w' only. Another interpretation of the relation di w, di w doW, d^w' J ' diZ, d-^z d^z, d.yz' is as follows :— When w and w' are two independent functions of two independent complex variables z and /, and when d^z, d^z, d^w, d^w' are 16 riemann's definition of a function [CH. I any one set of simultaneous small variations, while d^z, d^^', d^w, d^w' are any other set of simultaneous small variations, the quantity dxW, d\w' d^w, d^w diZ, d^z' is independent of differential elements and depends only upon the places z, z' and w, w'. 16. The converse also is true, viz. : — Let z and z' he two complex variables, such that z = x + iy, a! = cd -\- iy' , where x, y, x' , y are four real independent variables; and let _w and w' be other two complex variables, such that w = u + iv, w = u! + iv\ where u, v, u', v' are four real independent quantities, being functions of x, y, x' , y' ; then, if the magnitude diW, diw' diW, d^w' d.2Z, d^z for all infinitesimal variations, is independent of these variations, w and w are independent functions of z and z alone. This property, which for two independent complex variables corresponds to Riemann's definition -property* for functionality in the case of a single complex variable, can be established as follows. Let dw dx dw' dx By ^' dw dw = 7. dy 7=S, = a, so that dtv' ^, dw' , dw' ^, 8^ = ^' W^"^' dy'^^' Then dw = adx + ^dy -i- "y dx + h dy') dw' = i,x, d^x + /3, ^' d^y, d^y 8, 8' d^y, d^y' Biemann's Ges. Werke, p. 5. a, a 1 d-^x. diV I 8, 8' d^x, d^y' + 7' 7 d^x', d^y' ( N ), 8' d^x, d^y 16] Also EXTENDED TO TWO VARIABLES doZ, doz' 17 d^x + idii/, dix' + id^y' d^x + id2y, d (^> y> ^'. y') = ^> "^ (^' 2/' ^'> y') < ^^ and the range represented by 4> {^> y> ^^ y) < 0, ^ {x, y, x\ y) = 0. These two portions of the three-fold boundary themselves have a common frontier represented by the equations {x, y, x, y') = 0, i/r {x, y, x, y) = 0, which give a two-fold range of variation. This last range is a secondary or subsidiary boundary for the original four-fold field; to distinguish it from the proper boundary, we shall call it the frontier of the field. Accordingly, we may regard the frontier of a field of the suggested kind as given by two equations and yjr are algebraic functions of their arguments, the foregoing relations can be modified into relations of the type d (x, y, x) = 0, 'd {x, y, y') = 0, or into relations of the type X {x, X, y) = ^, X (2/' ^'' y) = 0' which are equivalent to them.) Now this form of the equations of the frontier of the field possesses the analytical advantage that, when the variables are changed fi:om z and z to w and w by equations F (w, w', z, z) = 0, G {w, w', z, z') = 0, 19] FRONTIERS 21 the equations of the frontier of the w, w field are of the same type as before, being of the form ^ {u, V, u, w) = 0, ^ (u, V, u', v') = 0, It is necessary to jfind some analytical expression of the doubly-infinite content of these equations. In the special example arising out of the periodic substitution in § 18, we at once have the expressions u \j1 = a cos Q -k- a cos d\ u' \/2 = a cos 6 — a cos 6', V \J1 = a sin 6 + a' sin 6', v a/2 = a sin 6 — a sin 6', giving the doubly-infinite range of variation for u, v, ii , v' , when 6 and 6' vary independently. But when the equations of the frontier do not lead, by mere inspection, to the needed expressions, we can proceed as follows. Let X, y, X, y' = a, h, a, h' be an ordinary place on the frontier given by the equations = and t/t = 0, in the sense that no one of the first derivatives of and of -v/r vanishes there ; and in its vicinity let x = a + ^, y = b + 7], x' ^ a + ^', y' =b' + rj'. Then we have d(f) 0=f da d(f> ,d(f) /9 . rfc w /-] , da db da' o=r^+^^^+ri5+v'^+[e^,r,v].+ db' there being only a finite number of terms when ^ and ■^ are algebraic in form. Introduce two new parameters s and t, and take S = ^a +r]^ -\-^''y -\-r]'h, t — ^a + 7]^' -H ^Y + rj'h' , where a, /3, 7, S, a, /3', 7', h' are constants such that the determinant 9(^ d(^ d B' does not vanish. Then the four equations can be resolved so as to express ^) V> I'; v' in terms of s and t ; owing to the limitations imposed, the deduced expressions are regular functions of s and t, vanishing with them ; and so we have each of the variables x, y, x, y , expressed as functions of two real variables s and t, regular at least in some non-infinitesimal range. 22 EQUATIONS OF A FRONTIER [CH. I In order to indicate the two-fold variation in the content of the frontier, it noAv is sufficient to consider regions of variation in the plane of the real variables s and t. Thus, corresponding to a region in that plane included within a curve k {s, t) = 0, there are frontier ranges of variation in the z and the z planes, determined respectively by the equations x — a =p' (s, t) y'-b' = q' {s,t)\, O^k (s, t) X — a = p (s, ty y-h^q(s,tn, 0^k{s, t)^ that is, by the interiors of curves f{x -a, y-b) = 0, g (x - a, y' - h') = 0, the current descriptions of these interiors being related. Moreover, the equations J^ = and G = potentially express u, v, u', v' in terms of x, y, x , y' ; and so the frontier range of variation in the w and w planes would be given by substituting the obtained values of x, y, x, y', as regular functions of s and t, in the expressions for u, v, u', v', that is, the frontier range of variation is defined by equations of the form u, V, u, v = functions of two real variables s and t. But, in dealing with the geometrical content of the frontier, whether with the variables z and z' or with the variables w and tu', care must be exercised as to what is justly included. We are not, for instance, to include every point within the curve /(a; — a, y — b) = conjointly with every point within the curve g {x' — a', y' — 6') = 0, even if both curves are closed ; we are to include every point within either curve conjointly with the point within the other curve that is appropriately associable with it through the values of s and t Ex. 1. The method just given for the expression of x, y, x\ y' is general in form ; but there is no necessity to adopt it when simpler processes of expression can be adopted. Thus in the case of the equations _^2^y2^^'2_l^ _^2_y2_y^ a complete representation of the variables is given by .r = sin 5 cos it, y = sin s sin t, x' = cos s, y' = sin^ s cos 2t. A full range of variation in the plane of s and t is ^ S ^ TT, ^ i! < 277-. When we select, as a portion of this range, the area of the triangle bounded by the lines S-t = 0, 8 + 1 = ^77, t = 0, 20] EXAMPLES 23 the limiting curves corresponding to /=0 and g = are a curvilinear figure made up of a straight line and two quarter-circles in the s- plane, and another curvilinear figure in the 2^ -plane made up of a parabola and arcs of the two curves y = (1_^'2) (2^2- 1), y= _ (1 -0/2) (2jr'2- 1). Ex. 2. For the periodic substitution a, z, z' frontier defined by the equations is transformed into a w, iv' frontier defined by the equations that is, the frontier is conserved unchanged. Bx. 3. To shew how a field of variation can be limited, consider the four-fold field represented by the equations x^+f+x'^^l, 2x^ + 3f+i/'^^l. As regards the 2-plane, the first equation allows the whole of the interior of the circle x^ + 1/^= 1. The second equation allows the whole of the interior of the eUipse 2x^ + 83/^ = 1. The region common to these areas is the interior of the ellipse ; hence the content in the 2-plane is the interior of the ellipse 2^2 + 3^2 _i^ go that x^ ranges from to ^, and 3/^ ranges from to J. As regards the s'-plane, we have 30/2-/2 = 2-^2^ 2o;'2-y2 = i+y2_ Because of the range of x% the first of these equations gives the region between the two hyperbolas 3,x''2-y2=,2, 3y2_y2=|. Because of the range of i/% the second of these equations gives the region between the two hyperbolas 2o;'2-y2=A^ 2.«'2-y2=i. The required content in the «' -plane is the area common to these two regions ; that is, it is the interior of two crescent-shaped areas between the hyperbolas 2x'^-i/'^ = l, 3^'2-y2 = 2. The whole field of four-fold variation of the variables z and z' is made by combining any point within or upon the first ellipse with any point within or upon the contour of each of the crescent-shaped areas. Ex. 4. Discuss the four-fold field of variation represented by the equations ^2 ^y2 _|. 2a {xx' +yy') < ^^, ^"i ^y'2 _|_ 2c {xy' — yod) ^ P. 20. The last two examples will give some hint as to the process of estimating the field of variation when it is limited by a couple of frontier equations in the form e {x, y, x) = 0, © {x, y, y') = 0, or in the equivalent form X {00, X, y) = 0, X {y, x , y) = 0. 24 FIELD OF VAEIATION [CH. I We draw the family of curves represented by ^ = for parametric values of x ; for limited forms of 0, there will be a limited range of variation for X and y, bounded by some curve or curves. Similarly, we draw the family of curves represented by @ = for parametric values of y ; as for Q, so for %, there will be a limited range of variation for x and y, bounded by some other curve or other curves. Further, the equations % = and X = may impose restrictions upon the range of x' and the range of y , which are parametric for the preceding curves. In the net result for the 2^- range, when subject to the equations ^ = and @ = 0, we can take the internal region common to all the interiors of these closed curves. The same kind of consideration would be applied to the equations % = and X = 0, so as to obtain the range in the /-plane as dominated by these equations. And the four-fold field of variation for z and ,s' is obtained by combining every point in the admissible region of the ^-plane with every point in the admissible region of the /-plane. Note. In the preceding discussion, a special selection is made of the four-fold fields of variation which are determined by a couple of relations (^*^ 0, x//' < 0. It is of coiu-se possible to have a four-fold field of variation, determined by a single relation 0^0. The boundary of such a field is given by the single equation cf) = ; there is no question of a frontier. It is equally possible to have a four-fold field of variation, determined by more than two relations, say by ^ 0, -v//- ^ 0, ;^ ^ 0. The boundary then consists of three portions, given by ^ = 0, \/^^0, x=^0; 4>^0, i/^ = 0, x^O' <^=^0, ^^0, X=0- The frontier consists of three portions, given by (^^0, •\|/- = 0, x = 0; 0=0, ^//•^0, x=0; cji = 0, -^=0, X ^ 0. And there could arise the consideration of what may be called an edge, defined by the three equations (^=0, yj/ = 0, x = 0- Sufficient illustration of what is desired, for ulterior purposes in these lectures, is provided by the consideration of four-fold fields determined by two relations. CHAPTER II LiNEO-LINEAR TRANSFORMATIONS: INVARIANTS AND Co VARIANTS Lineo-linear transformations. 21. Whatever measure of success may be attained, great or small, with the geometrical representation, the analytical work persists ; the geometry is desired only as ancillary to the analysis. So we shall leave the actual geometrical interpretation at its present stage. The fundamental importance of the lineo-linear transformations of the type az -\-h lu = -. cz + a in the theory of autoraorphic functions of a single variable is well-known. We proceed to a brief, and completely analytical, consideration of lineo- linear transformations of two complex variables*, shewing the type of equations that play in the analytical theory the same kind of invariantive part as does a circle or an arc of a circle in the geometry connected with a single complex variable. These lineo-linear transformations between two sets of non-homogeneous variables have arisen as a subject of investigation in several regions of research. Naturally, their most obvious analytical occurrence is in the theory of groups. When the groups are finite, they have been discussed for real variables by Valentinerf, Gordan|, and others ; they are of special importance for algebraic functions of two variables and for ordinary linear equations of the third order which are algebraically integrable§. Again, and with real variables, they arise in the plane geometry connected with Lie's theory of continuous groups ||. They have been discussed, with complex * For much of the following investigation, as far as the end of this chapter, reference may be made to the second of the author's papers quoted on p. 1. t Vidensk. Selsk. Skr., 6 Ra^kke, naturvid. og math. Afd., v., 2 (1889). + Math. Ann., t. Ixi (1905), pp. 453—526. § See the author's Theory of Differential Equations, vol. iv, ch. v. II Lie-Scheffers, Vorl. il. cont. Gruppen, (1893), pp. 13—82. 26 LINEO-LINEAR TRANSFORMATIONS [CH. II variables, by Picard* in connection with the possible extension, to two in- dependent variables, of the theory of automorphic functions. And a memoir by Poincare has already been mentioned -f*. 22. We take the general lineo-linear transformation (or substitution) between two sets of complex variables in the form w w az + hz' -\-c a'z + Vz' + c a"z t- h"z' + c" ' where all the quantities a, h, c, a', h', c, a", h", c" are constants, real or complex. The first step in the generalisation of the theory for a single variable is the construction of the canonical form ; and this can be achieved simply by using known results^ in the linear transformations of homogeneous variables. For our purpose, these are III = axi + hx.2 + cx.^, 2/3 = a'wi + h"x^ + c"a;3, so that we have '^ _ z _1 w w' _ 1 Xi X2 x^ y^ 2/2 2/3 The quantities w and w' are independent functions of z and z ; and there- fore the determinant a , b , c a, b' , c' denoted by A, is not zero. As a matter of fact, ^w, w'\ A J The equation z, z J {a"z + b"z' + c'y ' a - 6, h , c =0 a , b' — 6, c a" , b" , c"-t is called the characteristic equation of the substitution. This characteristic equation is invariantive when the two sets of variables are subjected to the same transformation ; that is to say, if we take W ^ W _ 1 aw + ^w' + 7 aw + ^'w' -h 7' "~ ol'w + /3"w' -I- 7" ' Z ^ Z' _ 1 a^ -h /3^' + 7 ~ OLZ + ^'z' -h 7' ~ a."z + ^"z + 7" ' * Acta Math., t. i (1882), pp. 297—320; ih., t. ii (1883), pp. 114—135. t See the reference on p. 1. X Jordan, Traite des substitutions, Book ii, ch. ii, § v ; Burnside, Theory of groups, (2nd ed., 1911), ch. xiii. 22] CANONICAL FORM 27 and express W and W in terms of Z and Z', the characteristic equation of the concluding substitution between W, W, Z, Z' is the same as the above characteristic equation of our initial substitution between w, w', z, z . There are three cases to be discussed, according as the characteristic equation, which is of the form ^3 - Ai^2 + Ao6' - A = 0, has three simple roots, or a double root and a simple root, or a triple root. Case I. Let all the roots of the characteristic equation be simple ; and denote them by 6^, 6^, 6^. Then quantities a.^ : /3,. : 7^, determined as to their ratios by the equations aCCf + a'^r + ft'Vr = (^r^r , b0Lr+ b'^r+b"jr = Or0r, Ca,. + C'l3r + c"^r = ^rjr , are such that, if Yr = OLryi + ^ry-2 + Tr^/s, ^r = O^^i + ^r^-i + 7r«?3, we have The canonical form of the homogeneous substitution is Y, = d,X„ Y,= e,x„ Y,= e,X,; and so the canonical form of the lineo-linear transformation is a^w + ^iw' + 7i _ aiZ + jB-jZ + 71 ' ttsW + ^zW' + 73 OizZ + /Sg/ + 73 a^iu + /32w' + 72 _ 02 2 + ^2^' + 72 a3W + /33w' + 73 Oi^z + iS^z ->r 'y^^ where the quantities \ and /a, called the multipliers of the transformation, are ' ^ _ ^1 _ ^2 being the quotients of roots of the characteristic equation. The multipliers are unequal to one another, and neither of them is equal to unity. This canonical form can be expressed by the equations W = \Z, W' = /jiZ'. Case II. Let one root of the characteristic equation be double and the other simple; and denote the roots by 0^, Oi, 63- The canonical form of the homogeneous substitution is Y, = e,X„ Y,= KX, + e,X„ 73 = ^3^3, where the forms of the variables X and Y are the same as in the first case ; and the constant k, in general, is not zero. 28 POWERS OF A TRANSFORMATION ■ [CH. II The canonical form of the lineo-linear transformation is of the type W = \Z, W' = \Z'+'3ri+/33r/ + 73=0, while «iri + /3ir/ + 7i + o. Thus the invariant centres are given by the equations ' «2ri + /S2^i' + 72 = o^ 03^1 + /33r/ + 73 = 0^ «3^2 + /S3r2' + 73 = 0^ «ir2+A^/ + 7, = o, «i^3 + /3:^3' + 7i = 0] «2^3 + /S2r3' + 72 = 0j ' a result which can be inferred also from the canonical form of the trans- formation. In deducing this result, certain tacit assumptions have been made as to the exclusion of critical relations. It will easily be seen that the transformation w s]'^=z-\-^ , w' ^J2 = z-z' , is not an example (for the present purpose) of the general transformation. 30 INVARIANT CENTRES [CH. II Manifestly, we can take w, w , 1 w, w , 1 W, W , 1 w, w', 1 = X 1 ^, ^', 1 2:, /, 1 ^3, fs', 1 = /X ^, z', 1 z, z , 1 as a canonical form of the lineo-linear transformation. This canonical form leads at once to an expression of the relations between the two sets of variables in the immediate vicinity of the invariant centres. Near ^i and ^Z, we have ^ = Ci + ^1^. ^' = ?i' + ^1^'' '^ = ?i + Si'^' '^' = ti' + ^i«^'. where hxW BiW' 1 [ §1-2^ S-^z' ?3-ri ^3'-?/ ^ir3-ri rs'-r/j Near ^o and ^2', we have Z = ^2 + B2^> ^' = ^2 + ^2^^', W = ^2 + ^2^;, W' = ^3' + SgW', where B«w So?/;' X, ( S, > = M S.,/ Near ^3 and ^3', we have S3W 83W;' f StZ S.z' r:-c3 ri'-C3' ^Ui-^3 r/-^' = A. where b 2 b 3 b 2 b 3 ( b:i b 3 b2 b 3 Moreover this new canonical form, involving explicitly the places of the invariant centres in their expressions, shews that the assignment of three invariant centres and two multipliers is generally sufficient for the con- struction of a canonical form of a lineo-linear transformation of the first type. 24] EXAMPLES 31 Ex. 1 . Some very special assignments of invariant centres may lead to equations that do not characterise lineo-linear transformations. The resulting equations, in that event, belong to the range of exceptions. Thus, if we take Ci = l \ C2=« 1 (r3 = a^ 1 Ci'=-lj' C2'=-ai' C3'--a'i' where a is neither zero nor unity, and if we assign arbitrary multipliers X and [i diflferent from unity and different from one another, the canonical equations can be satisfied only by w+w'=0, z+z' = 0, which is not a lineo-linear transformation of the z, z' field into the w, lo' field. Other special examples of this exceptional class can easily be recognised. One inclusive example is given by the relations i% ~ Cs' i% — ^z _ C2C3 —Jzid A B f 3 — f 1 _iz~ f 1 c C3 f 1 ~ Ci C3 C3) f2', =0; A B G ' and then the equations acquire the unsuitable form Aw-Bw'-\-C=Q, Az-Bz' + C=0. Ex. 2. When neither point in any one of the three invariant centres is at infinity, we can (by unessential changes of all the variables that amount to change of origin, rotation of axes, and magnification, in each of the planes independently of one another) give a simplified expression to the canonical form. Suppose that no one of the quantities fi, ^1', ^2) ^2'; Cs? Cs' then is zero ; alternative forms, when this supposition is not justified, are left as an exercise. We then transform the 2-plane and the w-plaue by the congruent relations z-Cx={C2-Ci)Z, ^-Ci=(C2-Ci)^; and we transform the /-plane and the w'-plane by the congruent relations ^ - Ci' = (C2' - Ci') z', w - c' ={a- Ci) w'. All of these are of the type just described ; they require the same chapge of origin, the same magnification, and the same rotation, for the s-plane and the w-plane ; and likewise for the /-plane and the w'-plane. The effect of the transformation is to place, in the Z, Z' field and the W, W field, two of the invariant centres at 0, and 1,1. The third invariant centre then becomes a, a', where Cs~Ci Cz " Ci C2 - Ci ' C2' — C\ The equations, in a canonical form, of the lineo-linear transformations of the Z, Z' field into the W, W field, having 0, ; 1,1; a, d ; for the invariant centres, are W, W, 1 Z, Z', 1 1, 1, 1 1, 1, 1 a, a', 1 = X a, a', 1 W- W Z-Z' Wa'- W'c I i W-Z'a W- w Z-Z' 32 INVARIANTIVE [CH. II where A and jx are diflferent from one another and where (so far as present explanations extend) neither X nor jj. is equal to unity. But it must be remembered, in taking these equations as the canonical form, that definite (if special) identical modifications of the 2-plane and the w-plane have been made simultaneously, and likewise for the /-plane and the w'-plane. The result of these modifications, in so far as they afifect the original lineo-linear transformation, is left for consideration as an exercise. Invariantive Frontiers. 25. In the theory of automorphic functions of a single complex variable, it proves important to have bounded regions of variation of the independent variable which are changed by the homographic substitutions into regions that are similarly bounded. Thus we have the customary period-parallelogram for the doubly-periodic functions ; any parallelogram, under the transformations W = Z -^ Od^, W = Z + CO2, remains a parallelogram and — with an appropriate limitation that the real part of 0)2/6)1 is not zero — the opposite sides of the parallelogram correspond to one another. Similarly a circle or a straight line, under a transformation or a set of transformations of the type (cz + d) w = az + b, remains a circle or sometimes becomes a straight line ; and so we can construct a curvilinear polygon, suited for the discussion of automorphic functions. These boundary curves — straight lines and circles — are the simplest which conserve their general character throughout the trans- formations indicated ; they are the only algebraic curves of order not higher than the second which have this property. They are not the only algebraic curves, which have this property, when we proceed to orders higher than the second ; thus bicircular quartics are homographically transformed into bicircular quartics. For the appropriate division of the plane of the variable, when auto- morphic functions of a single complex variable are under consideration so as to secure an arrangement of polygons in each of which the complete variation of the functions can take place, other limitations— such as relations between constants so as to secure conterminous polygons — are necessary. They need not concern us for the moment. What is of importance is the conservation of general character in the curve or, what is the same thing, conservation of general character in the equation of the curve, under the operation of a homographic transformation. 26. Corresponding questions arise in the theory of functions of two complex variables. We have already seen that, when a z, z field is determined by two relations, its frontier is represented by a couple of equations between the real and the imaginary parts of both variables; and therefore what 26] FRONTIERS 33 is desired, for our immediate illustration, is a determination of the general character of a couple of equations which, giving the frontier of a z, z field, are changed by the lineo-linear transformation into a couple of equations which, giving the frontier of a corresponding w, w' field, are of the same general character for the two fields. The invariance of form of such equations, at any rate for the most simple cases, must therefore be investigated. We shall limit ourselves to the determination of only the simplest of those frontiers of a field of variation which are invariantive in character under a lineo-linear translation. Also, we shall consider only quite general transformations ; special and more obvious forms may occur for special trans- formations, such as those contained in the simplest finite groups. Accordingly, in the equations az + bz' + c a'z -f- b'z' + c a"z + h"z' + c" ' we resolve the variables into their real and imaginary parts, viz. z = X -[- iy, z = X + iy , lu = u + iv, w = u' + iv' ; and we require the simplest equations of the form (f) (x, y, x, y') = 0, -f {x, y, x', ?/') = 0, which, under the foregoing transformation, become ^ (w, V, u', v') = 0, "^ {u, V, u', v') = 0, where and ^ are of the same character, in degree and combinations of the variables, as (f) and yjr. Moreover, the constants in the transformation may be complex ; so we write a = ai + ia^, h =■ h-^ + ih^, c = c-^ + ic^, a' = a/ + ia2 , h' — 6/ + ih^ , c = Cj' + ic^ , a" = a/' + m/', h" = W + ib,", c" = c," + ic,", in order to have the real and imaginary parts. Lastly, let iVi = Or^x + bix' — a^y — h^y' + Ci , iV^2 = ^2^ + ^2^' + ciiV + Ky' + Cs , iVj' = a^x + b-^x — a^y — b^y' + c/, N^ = a^x + b^x + a-[y + b^y + c^' , Nl' = a^'x + 6/V - a^'y - b^'y' + c/', N^' = a^'x + bJ'x' + a^'y + b^'y' + c/', D = i\V"' + N^ ; then the real equations of transformation are Dv = N,N^' - N,N^', F. 3 34 INVARIANTIVE [CH. II Further, we have D (uii +vv') = N^Ni + N^N^, D {uv' - u'v) = N, N,' - N,N,', D (u" + v") = N," + N;\ These equations express each of the quantities u, v, u', v', u^ + v^, uu' + vv , uv — u'v, u'^ + v'^, in the form of a rational fraction that has D for its de- nominator. The denominator D and each of the numerators in the eight fractions are linear combinations (with constant coefficients) of the quantities 1, X, y, X, y, x" + 3/^ xx' + yy' , xy - x'y, x'''^y'\ The same form of result holds when we express x, y, x, y in terms of u, v, u, v' ; any quantity, that is a linear combination of 1, x, y, x, y' , x^ + y\ xx + yy , xy — x'y, x'^ + y'^, comes to be a rational fraction the numerator of which is a linear combination of 1, u, v, u', v', u? + v^, uu +vv', uv' — ^ifv, u'' + v'^; the denominator is a linear combination of the same quantities, and is the same for all the fractions that represent the values of X, y, x , y , x^ + y'^, xx + yy' , xy — yx , x'^ + y''\ Consequently, any equation A (a;2 + 2/2) + C.(xx' + yy') + D (xy' - yx') + B (x" + y'^) + Ex + Fy+ Gx' +Hy' = K ■ is transformed into an equation A' (m^ + v"") + C (uu + vv') + D' {uv - u'v) + B' (u'^ + v"') + E'u + F'v + G'u' + H'v' = K', where all the quantities A, ..., K are constants, as also are A', ..., K', each member of either set being expressible linearly and homogeneously in terms of the members of the other set. 27. Thus the transformed ecjuation is of the same general character, concerning combinations and degree in the variables, as the original equation ; and there is little difficulty in seeing that it is the equation of lowest degree which has this general character of invariance. Further, two such simul- taneous equations are transformed into two such simultaneous equations of the same character. This is the generalisation of the property that the equation of a circle is transformed into the equation of another circle by a homographic sub- stitution in a single complex variable. Accordingly, when a z, z field having a frontier given by two equations of the foregoing character is transformed by a lineo-linear transformation into a w, w field, the frontier of the new field is given by two similar equations. We define such a frontier as quadratic, when it is given by equations of the second degree in the variables; and therefore we can sum up the 29] EQUATIONS 35 whole investigation by declaring that a z, z field, which has a quadratic frontier, is transformed hy a lineo-linear transformation into a w, w' field, which also has a quadratic frontier. 28. One special inference can be made, which has its counterpart in homographic substitutions for a single variable, viz., when all the coefficients in a lineo-linear transformation are real, the axes of real parts of the com- plex variables in their respective planes are conserved. For when all the constants are real, we have vD = (a"b - ah") {xy' - xy) + {ac" - a"c) y + {he" - h"c) y', v'D = (a"b' - ah") {xy' - x'y) + {a'c" - a"c') y + {h'c" - h"c') y' ; and therefore the configuration given by y = and y' = becomes the configuration given by v = and v' = 0. The converse also holds, owing to the lineo-linear character of the transformation. These axes of real quantities in the planes of the complex variables are, of course, an exceedingly special case of the general quadratic frontier, which can be regarded as given by the two equations ^1 (^' + y') + B, (x" + y") + G, {xx + yy) ^ A {ocy - xy) -I- E^x -\- F^y + Q^x + H^y = K^, A^ {x" + 2/') + ^2 (^" + 2/'') + C'a {xx +yy') + A {^y' - oo'y) + E.x + F.y + G^x + H^y = K^. Let z and z' be the conjugates of z and z respectively, so that z = X — iy, z = X — iy ; then the general quadratic fi-ontier can also be regarded as given by the equations A^zz + B^z'z -f- Ci'^i' -I- D^z'z + E^z -t- F^z -J- G^z + H^z = K^, A^zz + B^z'z' -t- G^zz' + Doz'z + Eo^z -{■ F^z -\- G.^z ■{- H.^z = K.^, where A-^, B^, K^, A^, B^, K^ are real constants, while (7/ and A', G^ and A') A' and A', A' and A', G^ and H^, G^ and A', are- pairs of conjugate constants. Manifestly any equation of this latest form is transformable by the lineo-linear substitution into another equation of the same form. 29. Another mode of discussing the frontier of a z, z field, which is represented by two equations that have an invariantive character under a lineo-linear transformation, is provided by the generalisation of a special mode of dealing with the same question for a single complex variable. The general homographic substitution affecting a single complex variable has the canonical form w — a _ z — a w- /3~ z- l3' 3—2 36 INVARIANTIVE [CH. II where a and yS are the double points of the substitution, and K is the multiplier. Let w = u + iv, z = X -{■ iy, a = a + ia, /3 = 6 + ih', K = tce^"", where u, v, x, y, a, a, h, b', k, k are real ; then u — a + i {v - a) _ ^^^ x — a + i (y — a) u — b + i (v — b') X — b + i{y ~b')' and therefore _j {ii —b)(v — a') — (u-a){v — b') tan" {u — a) {u — b) + {v — a) (v - b') — tan , (x-b)(y-a')-(x-a)(y-b') _ = h {x -a){x-b) + {y- a) (y - b') Hence the circle {x -a)(x-b) + (y- a) (y -b') = m [{x -b) {y - a') - {x - a) (y - b')], which passes through the double points (a, a') and (b, b') of the substitution, is transformed into the circle (w -a){u-b) + {v- a') {v -b') = M [{u -b)(v- a') - (u - a) (v - b')], which also passes through those common points. The constants m and M are connected by the relation m - if = (1 + niM) tan k At a common point, the two circles cut at an angle k, which depends only upon the multiplier; thus when an arbitrary circle is taken through the common points, it is transformed by the homographic substitution into another circle through those points cutting it at an angle that depends only upon the constants of the substitution. This process admits of immediate generalisation to the case of two complex variables. Let the lineo-linear transformation in two variables be taken in its canonical form ; and write a^z + I3i2' +yj = l^ + ill', «! w + ySi w' + 7i = Zi' + iL^', a^z + l3^z +r^o^ = l^ + il^', 0L2W -\- ^^w' + ^.^ = L^ + iL^' , ^3 ^ + /Sg^' + 73 = 4' + il^', tts w + /33 w' + 73 = X3' + 1X3", where Z/, l-^', l^, I2", U, h" are real linear functions of x, y, x , y' and L^, X/', X2', L^', Z/3', Z3" are respectively the same real linear functions of u, v, 11, v . The three invariant centres are the places given by the equations 4' = \ 4' = 1 4' = \ 4"=o ^ ^/' = o 4'=o 4' = ' 4'=o 4" = oi ^/' = oJ K = i 29] QUADRATIC FRONTIERS 37 and they are also the same places given by what are effectively the same equations X/ = \ W = X/'=0 x/ = o A" = The canonical form of the lineo-linear transformation now is LJ + iL. Is' + ilz U + il,' and therefore, among other inferences, we have T IT II T 'T " 1 '1 , i/Q J-/0 — -L/o -t-/3 1 ''3 ''2 tan-i^r^T^^; ^^r^, -tan-i LzL^ + L^'Lz n " \7h^ arg/A, Accordingly, the frontier configuration, represented by any two of the three equations ^3 4 ~ 4 4 =p (^3 4 +" ^3 ^2 )) '2 '1 — '1 '2 = ^ ('2 '1 + '2 '1 )) where the three constants -p, q, r are subject to the relation p + q+r=pqr, so that the three equations are really equivalent to only two independent equations, is changed by the transformation into the frontier configuration represented by any two of the three equations L^L^' — h^liz = P'{L^ L2 + -t/3 L2 ), Li'Ls" — L3L1" = Q (L^ L3 + Li lis ), Z/2 Li — Li 1/2 = ^ (-L/2 ll + ^2 -^1 )) where the three constants P, Q, R are subject to the relation P + Q + R=.PQR, so that these three equations are really equivalent to only two independent equations. Moreover, if fji, = Ge('\ X = He''\ 38 EXAMPLES [CH. II where g,h, G, H are real constants while G and H are positive, we have P-p={l-\-Pp)tQ.n.g, Q-q= -(i + Qq) tan h, R-r = (l+Rr) tan (h - g). It is easy to verify that, if either of the relations P + Q + R = PQR, p + qJfr=pqr, is satisfied, the other also is satisfied in virtue of these last equations. The quadratic frontier of the z, z field and the quadratic frontier of the transformed w, w field both pass through the three invariant centres of the lineo-linear transformation. Ex. 1. In connection with the homographic substitution in a single variable to —a_ j^z — a (in the preceding notation), shew that the constant m in the equation of the circle {x - a) {x -h)^-(^ - a') iy -h') = m{{x -h) {y -a') - {x - a) (:y -h')} is the tangent of the angle at which the circle cuts the straight line joining the do.uble points of the substitution. Prove also that, if 2c? is the distance between the double points, r is the radius of the foregoing circle, and R the radius of the circle into which it is transformed, 1 2 cos k 1 _ sin^ k m rBT ■*■ r2 = —^ • Ex. 2. Shew that the circle {x-aY-^{y-hf=n'^{{x-a'f+{y-hy] is transformed, by the homographic substitution, into the circle {u - af + {v - 6)2 = JVmu- a'Y + (v - b'^}, where Interpret the result geometrically. Ex. 3. Construct a lineo-linear transformation which has 0, ; 1, 1 ; i, -i for its invariant centres ; and shew that there are quadratic frontiers of the z, z' field, which pass through these invariant centres and are represented by any two of the three equations x^ + 7/^+x''^+y''''-2 {xx' +1/1/') — 2 (xt/ — x'i/) — 2 (y— /) = a {x^ ■\-y^ — {x"^+y'^) + 'i{x — A'')}, x^ + ?/^ + x'"^ + ?/'2 + 2 {xx^ +yy')-'i ixy' — xfy) — 'i{x+x') = ^ {^2+3/2_ (^.2+y2) _ 2 (3/+y)}, x^-\-y^ — (^'2+?^'2) =r y (xy' — x'y), provided the constants a, /3, y satisfy the relation 7(a + i3) = 2a + 2^--y. 31] INVARIANTS OF FRONTIERS 39 Verify that the lineo-hnear transformation changes these equations into equations in u, V, u', v' of the same form but with different constants a, /3', y satisfying the relation y(a' + /3') = 2a' + 2^'-y. Shew that, at the invariant centre 0, 0, small variations dz and dz' cause small variations dw and d^o' such that dw — dw' = - (dz — dz!)^ A dw + dw' = i^ {dz+dz') ; and obtain the relations between the small variations at each of the other two invariant centres. Invariants and Covariants of quadratic frontiers. 30. Owing to the importance of the quadratic frontier, because it is given by two equations of the second order that are invariantive in general character under any lineo-linear transformation, we shall briefly consider those combinations of the coefficients which are actually invariantive under all such transformations. The proper discussion of the invariants and covariants, which belong to two equations of any order that are invariantive in general character under the transformations, requires an elaboration of analysis that will take us far from the main purpose into what really is the full theory of invariants and covariants. It will be sufficient to give the elements of that theory as connected with the fundamental procedure. Moreover, we shall take a general quadratic frontier and not merely the special class which pass through the invariant centres of an assigned transformation; and we require the quantities which are invariantive under all lineo-linear trans- formations and not merely under one particular transformation. We further shall only deal with such invariantive quantities as are algebraically independent of one another. 31. There are several modes of procedure ; in all of them, it is con- venient to use homogeneous variables, as was done in establishing the canonical form of the lineo-linear substitution. So we take z z' 1 w _w' _\ x-J~ x^ x^' y^ 2/2 2/3" Also, as the variables respectively conjugate to z, z, w, w' have been intro- duced, we shall require variables respectively conjugate to x^, x^, x^, y-^,y^,yzi denoting these by ^, ^2, ^sj 2/i' ^2, ^3, we take z ^'_1 w _w' _1 x^~x^ x.i' y^ % Vs' 40 UMBRAL [CH. II For the present purpose, we take a z, z field determined by two relations Q < 0, Q' ^ 0, where q = Ay,y, + %i^2 + Gy,y, + Dy^y^ + Ey,y., + Fy^% + GyzVi + Sy^y^ + Ky^y%, Q' = A'y.y^ 4- ^'t/^^^ + ^'ViV^ + ^'.Va^i + ^'2/2^2 + F'y.% + G'ysy, + H'y^y.2 + K'y^y^ ; its quadratic frontier is given by the equations Q = 0, Q' = 0, which, on division by the non- vanishing quantity y^y^, acquire the form of our earlier equations. In Q the coefficients A, E, K are real, while B and D, C and G, F and R, are conjugates in the stated pairs; and similarly for the coefficients in Q'. The method of procedure that we shall use is based upon an application of Lie's theory of continuous groups to these quantities Q and Q' ; and the application proves fairly simple in detail when we use umbral forms simultaneously with the expressed forms. Accordingly, we introduce umbral coefficients a^, a^, os, o-/, 0-3', os', with their conjugates a^, a^, 0=3, ffi, a^, a^ ; we take n = o-i^/i + 0-22/2 + 0-32/3 1 n' = 0-1' 2/1 + 0-2' 2/2 + 0-3' 2/3 1 n = CTi ^1 + 0=2^2 + ^3^3 J ' n' = CTi'2/1 + CTa'^a + CTs' Ih J ' and we write _ _ Q = nn, Q' = n'n; We then both define and secure the umbral character of these new coefficients by imposing the customary condition that the only combinations of the umbral constants which have significance are those leading to the expressed coefficients in the form A = (TiOi, D ^= 0"2O'i, G" = O'sCTj, B = 0"iCT2, E = (T.2^2j H ^= Cfz^2y C = cTiCTg, F = (72^3, K = cr^a^', and likewise for the coefficients of Q'. When the lineo-linear transformation, in the form 2/1 = ax^ + hx<;, + cx^ \ 2/2 = a'^i + ^'^2 + 0x3 \ , II I 111 , // 2/3 = a ^1 + X2+ c x-i! and its conjugate, in the form y-i. = aXi +6^2 +c^3 ' ^2 = ol'x-^ + h'x^ + c'x^ \ , y^ = a'% + h"x2 + c-'x^ , 32] NOTATION 41 are applied to Q and Q', these become P and P' respectively, so that we take Q = P, Q' = P', and then P = A-^x^x^ + B1X1X2 4- CiX-^Xs + DiXoXi + E^x^Xj + F-^x^x^ P' = A-^XyX-^ + B 1X1X2 + Ci'xiXs + Dt'xoXi + E-^x.jX.^ + F-^x^x-^ + Gi XsX^ + HiXsX.2 + KiXgXs. We take >Sf = 5i«i + 52^2 + SsOCs , S' = Si'Xi + S2OS2 + S^'Xs, S =SiX-i + S2X2 + SsXs, S' = Si% + S2X2 + s^x^, where Sj, S2, S3, s/, S2, s^' are new umbral coefficients, while Sj, s.2, ^s. ^1', s/, S3' are their conjugates ; and we write P = SS, Q = S'S', regarding IT as transformed into S, IT into S, U' into S', and 11' into S'. Then the laws of relation between the umbral coefficients in IT and S, and in n and S, are 5i = a(Ti + a'cTo 4- a"o-3 ^ Si = So = 60-1 + 6'o-2 + Z)"o-3 V , 5.2 = v.^1 . ^ -2 ' - -3 j S3 = co-i + c'o-2 + c'Vs j 53 = CCTi + c'ffg + c"a3 j and the same laws of relation hold between the umbral coefficients in Q' and S', and in II' and S'. Finally, in connection with our transformation, we write A=a,6,c', A=a, 6,c a' , h' , c' a , , c where A has the same significance as before, A is its conjugate, and neither A nor A vanishes. 32. As an example of an invariant, consider the quantity /= A„ B„ G, A, F„ F, G„ H„ K, To express it in umbral symbols, three sets of these are required because it is of degree three in the non-umbral coefficients. Denoting these by ^1, ^2, S3, ^1, U, ^3, Ui, Uo, Us, with their conjugates, we easily find that / is equal to Si = aai + a' «3 tl, H , t. ii, k, k Ux, U2, Us Ui, Ma, U 42 that is, to that is, to INFINITESIMAL [CH. II a-2, cr, V3 a , h , a , b', c a". b", c" Ol, CT2, ^3 1 ri, T.2, T3 Vi, V2, Vs 6", ^AA cl, 0-2, 0-3 ■^1, T2, T3 fl, 1^2, V3 o"i, o'2, cr3 T^l, T2, T3 and therefore A, E„ F, = AA A, B, C D, E, F 0, H, K a relation which establishes the invariantive property of the quantity / which is a function of the non-umbral coefficients of P alone. The same combination of the coefficients of P' alone is easily seen to be an invariant. The simplest covariants are P and P' ; for we have Q = P, Q' = P'. 33. Passing now to the consideration of invariants and of covariants that belong to the general quadratic frontier, we define any quantity <^{yi, 2/2, Vz, Vi, 2/2, y-i, A, ...,K,A', ..., K') to be such a function if it satisfies a relation O = ApApc/), where O is the same function of Xf, x^, x^, x-^, x^, x^, A-^, ..., K^, J./, ..., K^ as <^ is of its own arguments. We shall deal only with integral (not with fractional) homogeneous combinations of the variables and the coefficients; and we assume that, in the foregoing relation which defines an invariant or a covariant, the index of A is the same as that of A because we are limiting ourselves to the properties of real frontiers as defined by two real equations. And we retain the customary discrimination, by the occur- rence or the non-occurrence of variables, between a covariant and an invariant. By Lie's theory of continuous groups*, it is sufficient to retain the aggregate of the most general infinitesimal transformations of a continuous transformation in order to construct the full effect of the finite continuous * For proofs of this fundamental theorem, see Campbell, Theory of continuous groups, chap. iii. 33] TRANSFORMATIONS 43 transformation. Accordingly, for our immediate purpose, it is sufficient to obtain an algebraically complete aggregate of integrals of the set of partial differential equations which characterise the full tale of the infinitesimal transformations in question. To obtain these, we take a = 1 + ei, 6 = 6.2, a = €4, 6' = 1 + €5 a" = €7, h" = €s, a = 1 + ij, h =€-2, a' = €4, b' =1 + eg ^7, h"=es, c =€s c' =66 C" = 1 + 69 C =63 C' =66 For the most general infinitesimal transformation, all the quantities e and e are small, arbitrary, and independent of one another, subject to the condition that 6,1 and e„, for the nine values of n, are conjugate to one another. The laws of relation among the umbral coefficients now are Si — o-j = eiCTj + 640-2 -1- 670-3 I Si — ffi = ejCTi + 64^2 + ^tO's 82 — 0-2= 620-1 + 65O-2 + 680-3 r , §2 — CT2 = 62^1 + 65^2 + h^3 S3 — 0-3 = egO-i + 660-2 + 69O-3 I S3 — 0^3 = 63^1 + egCTo + 69CT3 Consequently the infinitesimal variations of the coefficients in the equations of the quadratic frontier are given by the equations 8A ==A^-A = e^A+ e,D + e^G + e^A + e^B + e^C ^ SB = B, - B = €,B + e,E + ejl + e.A + e,B + €,C 80 = C\- C = €,G + e,F + e,K+e,A +-e,B + €,C 8D =n,-D = 62^ + e,D+esG + e,D + e,E + e,F , 8E = E,-E = 6^B + €,E + e^H + e^B + e.E + e^F f- ; 8F=F^-F = e2C + e,F+ e,K + e,D + e,E + e.F 8G = G,-G = €,A + €,D + e,G + e,G + e,H + e,K 8H= H,-H=e,B + e,E + e,H + 626^ + e,H + e.K 8K = K,-K=e,G + e,F+e,K + e,G + e,H + e,K with a corresponding set of nine expressions for the infinitesimal variations of the coefficients A', ..., K'. The infinitesimal variations of the variables are given by the relations 2/i — ^1 = €1X1 + €2X2 + 63a;3 \ yi — ^i = ei.«i + 62*2 + ^3^3 ' 2/2 - ^2 = ^4^1 + ^5^2 + ^6^3 [ , ^2 - ^2 = ^4^1 + ^5^2 + ^6^3 yz — X^= 67^1 + e^X2 + 69^3 j y% — ^% = ^7^1 + ^8^2 + ^9^3 , 44 EQUATIONS CHARACTERISTIC OF [CH. II and therefore, so far as small quantities up to the first order are concerned, we have xi-yi = - ei2/i - e.y2 - €^y% «2 - 2/2 = - e4,?/l - ^53/2 - to2/3 X3-y-i = - ^7^/1 - ^8^/2 - eg^/s And, lastly, we have ^1 - ^1 = - ^1^1 - e-^h - e^y-i ^■2-y2 = - e^yi - €,y^ - €6^3 ^3 - ^3 = - €7^1 - 682/2 - eg^B AA = 1+61 + 65+69 + 61 + 65 + 69. 34. Now any covariant or invariant satisfies the equation = (AA)P(^(3/i, y^, y,, y„ y,, y,, A, ..., K, A', ..., K'). Substitute in this defining equation the values, of A^, ..., K^, A(, ...,K^, yi,y2,y3,^^; write T-v9 7-i9 7-r9 -r-v/9 7-r/9 71/ ^V ^• = ^3-5 + ^3^ + ^3? + ^ 32)'+'^ 3¥' + -^3F' ^ ^' = ^H +^34)+«3TJ + ^'33' + ^'3F + <''3l' ^ 5 73^ 7:t9 rr^ Ti' ^ -ni "^ rr ' ^ /I ,<9 TiO /-.3 j/9 -n/S /-*/9 h dG dH dK dG' dH' dK' 0, = A^^+D^+G^ + A'^, + D'^,+ G'^j^, '^-^L^^W^^lG-^'iA'^^'iD'^'^'W \, 34] INVARIANCE 45 a T\ ^ E^^ IP ^ T\' ^ IP' ^ ET/9^ y = ^dA^^dB + ^dU^'^'dA''^^'dB''^^'W I I and expand both sides of the equation in powers of the small quantities e and e. Equating the coefficients of these small quantities on the two sides and denoting our covariantive function (yi, 2/2> 2/3, Vi, y-i, Vs, ^, •••, K, A', ..., K') by (^, we have the partial differential equations 8, which satisfies all these equations, is a covariant (or invariant) of the required type. 35. Having regard to the fact that ultimately we are dealing with quadratic frontiers and with transformations between w, w' and z, z , we shall consider only those integral functions ^, which are homogeneous (say of order rn) in 7/1, 3/2, 2/3 and homogeneous (also then of order m) in ^1, y^, y^. We also shall consider only such functions ^ as are homogeneous (say of degree n) in the coefficients A, ..., K and homogeneous (say of degree n) in the coefficients A', ..., K'. Then, from the first set of equations and by means of Euler's theorem on homogeneous functions, we have n + n' — 171 = 8p. It follows that every integral invariant of a quadratic frontier has its degree in the coefficients of the boundary a multiple of 3. When the index p is taken as equal to the foregoing value, and when we note the equality between the indices of A and A in the relation which defines the covariants, the first six equations can be replaced by the four and we then retain the other twelve equations, so that we have a set of sixteen partial equations of the first order. It is easy to verify that the conditions of co-existence of these sixteen equations are satisfied, either identically or in virtue of the equations in the set. Hence the set of equations constitutes a complete Jacobian system of partial equations of the first order. The possible arguments in any solution <^ are twenty-four in number, viz., the nine coefficients A, ..., K, the nine coefficients A', ..., K', and the six variables yi, yo, yz, y-i, y^, ys', consequently, by the customary theory of such systems*, the number of algebraically independent integrals is eight, the excess of the number of possible arguments over the number of equations in the complete system. 36. After the limitations that have been imposed, every integral <^ of the system is homogeneous of degree m in y^, y^, 3/3, and also homogeneous of degree m in y^, y.2, ys. Let it be represented by ^p,q,p,qyi y-rys^yi " yr ys^ ? * See my Theory of Differential Equations, vol. v, chap. iii. 37] CHARACTERISTIC PARTIAL EQUATIONS 47 then, in order that it may satisfy the equations, we must have the relations (among others) ^4 • Up^q^p'^q' — {p +1) Up+i^q^p'^q' = | ^ "i • Up,q,p',q' ~\P +1) ^p,q,p'+l,q' = "7 • ^ p,q,p',q' ~ W + 1) '-'p,q+i,p',q' = ^ "7- ^p,q,p',q' ~ \9 "^ ^) ^P,q,p',q'+l — ^ \. By the continued use of these equations, all the coefficients Up,q,p',q- can be obtained when once t^o,o,o,o (say U) is known; and therefore, as usual in the theory of homogeneous forms, the whole covariant can be regarded as known when its leading term Uyi'^yi"''^ is known. Again, and just as in the ordinary theory, the leading coefficient C of the covariant satisfies the equations d,u=o, d,u=Q, e,u=o, e,u=o, d,u=o, o,u=o, 0eU=Q, e,u=o, o,u-e,u = o, d,u-d,u=o. These ten equations also are a complete Jacobian system of partial diffe- rential equations of the first order. Each integral can involve the eighteen possible arguments, constituted by the constants in the two equations of the quadratic frontier; and therefore the system of equations possesses eight algebraically independent integrals which are the leading coefficients of the eight covariants constituting the algebraically complete set of integrals of the full system of equations. It follows that, in this method of proceeding, we have to obtain eight algebraically independent integrals of the preceding set of ten equations in the second complete Jacobian system. 37. The actual process of solving the equations is the customary process that applies to complete Jacobian systems that are linear and homogeneous. The algebra required in the manipulation is long and tedious for the present set of equations ; so the results will merely be stated, especially as they can be obtained by another method (or combination of methods) applicable to the equations of the quadratic frontier. The summary of the final integra- tion of the ten equations, which are to possess eight algebraically independent integrals, is as follows : — Every integral of the system is expressible algebraically in terms of the eight independent integrals A, A', I, J, J', I', T, T', where / is the invariant of Q, I' the similar invariant of Q' , (the summation being extended over all the coefficients of Q and Q'), 48 THE FOUR INVARIANTS [CH. II B, G G , D B', C G', U A , C A, G A', C A', G' A, B A, D A', B' A', D' A, B 1 G, A A', B' G', A' A, D 1 , A A', D' c, A' and where T and T' are the coefficients of \ and jx respectively in the expression {\A+fiA') + {\E 4- ixE') + (XK + fjiK') + {XH + fxH') Moreover, A determines a covariant Ayiyi+ ..., that is, Q; A' deter- mines a covariant A'y-^yj^+ ..., that is, Q' ; T determines a covariant Ty-^yi + ..., say R ; T' determines a covariant T'y^y-^ + ■ • • , say R' ; and /, J, J', I' are invariants. Finally, any quantity connected with the quadratic frontier that is invariantive under the lineo-linear trans- formation is expressible in terms of Q, Q', R, R' , I, J, J', I'. 38. Had our quest been for invariants alone, the preceding analysis shews that they must satisfy the equations e,=o, d,=Q, ^4 = 0, e, = o, e,=o, e,=^o, d,= {), ^3 = 0, ^4 = 0, d, = 0, d,= 0, 0, = 0. But always ^1+195-^^9=^1 + ^5+^9, so that, in virtue of the first four we have and therefore 6^ = 65, 6^^ 6^. The two equations e^-e,= ^ and e,-Os = o are therefore satisfied in virtue of and so the system for the invariants contains fourteen independent equations. They are a complete Jacobian system, and involve the eighteen arguments constituted by the coefficients of Q and Q'; hence there are four algebraically independent invariants. They can be obtained simply as follows. We have seen that A, B, C fD, E, F G, H, K 39] CONTRAGREDIENT VARIABLES 49 is an invariant of Q; the same function for aQ + ^Q', where a and yS are arbitrary parameters, also is an invariant of the system. Let aA + ^A', aB + ^B' , aC + /30' = o?I + cl^^J+ a/3' J' + ^T ; aB + /3D' , aE + ^E', aF + /3F' aG+^G', aH+^H', ccK + ^K' then /, J, J', I' are four invariants, independent of one another, and there- fore suitable for the aggregate of the four algebraically independent invariants. They manifestly agree with the four invariants in the earlier aggregate of invariants and covariants. Ex. Prove that the complete system for a single equation §=0 is composed of Q and /. 39. The detailed consideration of the invariant! ve forms will not be con- sidered further. What has actually been done should suffice to shew the march of a general method of proceeding for the particular problem. But one warning must be given if this general method is to be applied to a wider problem, viz. the determination of all the covariantive concomitants of all kinds whatever that are to be associated with any single form or with any couple of forms that are integral and homogeneous in ?/i, y^, y^, and also integral and homogeneous of the same order in yi,y2,ys, where we still assume the lineo-linear transformation for y-^, y^, ys and its conjugate for y^, y^, y^ as the transformations under which the concomitants are to be invariantive. For this problem, it is necessary to introduce variables contragredient to the variables x-^, x^, x^ and y-^, y^, y^, according to the customary law of variation in the theory of forms ; that is, if we denote these further variables by |^i, ^2, ^z, Vi^ Viy Vs> and their conjugates, they are subject to the lineo-linear trans- formations li = avi + (^'V2 + a" Is ] li = <^^i + « ^2 + ^"Vz ] ^2 = 6771 + h'r]^ + 6" 773 L I2 = 6^1 + 1% + h"riz \. ^3 = C97i -I- c'?72 + c"Vz > Is = C^i -I- c'rj^ + c"7js ) It will be noticed (as is to be expected) that the umbral coefficients, used to express a given homogeneous form symbolically, are themselves contragredient to the variables. Manifestly we have yiVi + y2V2 + yzVs = a;i^i + x2^2-\-xs^s, It need hardly be pointed out that, while the complex variables Xi, x^, x^ correspond to the point- variables in the ordinary theory of ternary forms, the complex variables ^1, ^2> ^3 correspond to the line- variables in that theory. In order to obtain the most general concomitant of any kind, we should apply the preceding method to a function of the type (l>(yi, 2/2, yz, Vi, ^2, Vz, Vi> V2, vz, Vi> m, m, ^, ■••), F. 4 50 CANONICAL [CH. II involving all the variables and the coefficients of any or all of the initial given system of forms whose aggregate of concomitants is vs^anted. There is plenty of room and opportunity for research ; but the investigations vi^ould take us into the wider pure algebra of the theory of homogeneous forms, and they will not be pursued in these lectures. Ex. 1. Let C/and F be any two covariants that belong to a form or to a system of homogeneous forms ; and let ^"3.^2 9^3 8j/3 9«/2 a^3F_acraF ac^ar_3^8F ^~dyi dyi dy^ dy^ = _3C_^9F_a£^aF ^~ 8^2 Si's 9^38^2 J _dUdV^ _dlJdV_ ^"3^3 3^1 9^1 9^3 \' y_dudv_dUdv_ ^~9^i 9^2 9^2 9^1 / Prove that Fj, F2, F3 are cogredient with 3/1, 3/2, ys, and that Fj, T^i T3 are cogredient with yi, y2, ^3 ', and shew that C^(Fi, 72, Fs, Fi, F2, F3) and F(Fi, F2, F3, Fj, F2, F3) are covariants of the system. In particular, when U and F are the two initial quantities Q and Q' belonging to a quadratic frontier, determine the two covariants which are thus constructed. Ex. 2. Shew that when a quartic frontier, generally covariantive under a lineo-linear transformation, is given by equations Q = and Q' = 0, where symbolically Q = U.^W and Q' = n'^n.'^, the algebraically complete set of invariants and pure covariants belonging to the system consists, in addition to Q and Q', of sixty functions. 40. One other matter is left for investigation outside the range of these lectures. We have already dealt with the canonical form to which the expression of a lineo-linear transformation can be reduced. Also we have seen that there are quadratic frontiers, represented by the two equations of lowest degree, which keep a general invariantive character under such a trans- formation. It remains to consider what is the simplest canonical form to which two simultaneous equations representing such a quadratic frontier can be reduced, where there no longer is a question of invariance under a single transformation only*. This more general problem has some analogy with the problem of reducing to canonical forms the equations of two conies. * The simplest examples of forms, invariant under a single given transformation, have already been given; they are the equations of the frontier which passes through the three invariant centres of the transformation. 40] FORMS 51 In that solved problem, certain invariants of the system are necessarily conserved ; in this propounded problem, the four invariants of the system of two equations, which already have been obtained, must also be conserved. One appropriate form is suggested almost at once by the known result in the case of two conies referred to their common self-conjugate triangle. It is natural to enquire whether two forms P = AxiXi + Bx-^x^ + Gx^Xs + DX2X1 + EX2X2 + Fx^x^ + Gx^x-^^ + Hx^x^, + Kx^x^ , P'= A'xiXi +B'xiX2 + G'x^Xs+D'x^Xj +E'x2X2 +F'x2Xs + G'x^x^ + H'x^X2 + K'x^x^, can simultaneously, by homogeneous linear transformation of the variables, be changed to forms _ _ _ P = X^Xi-\- X2X2 + XgX-i , P' = A"X,X, + 5"X,Z, + G"X,X„ where no two of the three quantities A", B" , G" are equal to one another, and no one of them is equal to unity. With these last restrictions, we have I + a/ + a^J' + o?r = (1 + OiA") (1 + aB") (1 + aO"), for arbitrary values of a ; consequently, the three invariants ///, J' \I, I' jl (which are absolute invariants) are independent of one another, and no one of them vanishes. Thus the general condition as regards conservation of invariants is satisfied. Now all the quantities A, E, K, A', E' , K' are real ; hence a requirement that they shall respectively acquire the values 1, 1, 1, A", B", G" , where A", B" , G" are real, imposes six conditions. Also B and D, B' and D' , G and G, G' and G' , F and H, F' and E' , are (in each combination) conjugate constants; hence a requirement that all these coefficients shall vanish imposes twelve conditions. In order, therefore, that the suggested canonical forms shall be possible, eighteen conditions of the specified kind must be satisfied. Suppose, then, that the variables are transformed by the relations X2= 6'X^ + (j> X2+ ^r'Xs, Xg = $"Xi + ^"X2 + -^"X^, where the complex constants are at our disposal. Let V = then 4—2 4>^ ^ v = , ^ t ^', ^' d' , W> ^' r, r d", ¥, ^' A" + B" + G" = VV/, B"G" + G"A" + A"B" = VVJ', A"B"G" = vvr, 52 PERIODIC [CH. II SO that the values of A", B" , C" are given bj means of the quantities Jjl, J'jl, I' /I, three real quantities. Also, as each of the nine arbitrary constants 6, ..., ylr" is complex, we have effectively eighteen constants at our disposal, formally sufficient to satisfy the eighteen conditions which take the form of linear equations. It therefore may be inferred that a couple of general forms P and P' can be transformed so that they acquire forms of the suggested type. Periodic transformations. 41. These results, as regards lineo-linear transformations, are general. Simple forms occur when the transformations are periodic, that is, are such that after a finite number of repetitions in succession we return to the initial variables; and these provide the generalisation of finite groups of homo- graphic transformations in a single variable. The requirement of periodicity will impose conditions upon the unequal multipliers X and /a, in the first type (§ 22). The second type cannot be periodic unless cr vanishes. But if j2=z + z\ w' ij2 = z-z!. Ex. 4. Prove that all transformations of the linear type, which have quadratic periodicity, belong either to the form w=—z + c, w' = -z' + c', or to the form , l-a2 , 1+a iv = az + bz +c, w= — T — z-az a~*^' where a, b, c, c' are arbitrary constants. Ex. 5. Prove that all cubic linear transformations have either the form ^ w = 6z + c, i(/ = 6'z' + c' ; or the form w=az + bz' + c, with either w' = -^(a^ + ad^ + 6) z- {a + 6'-) z' -^^{a- 6), or W= -r{a^+a + l)z-{a + l)z' + c', where 6 and 6' are imaginary cube-roots of unity, and a, 6, c, c' are unrestricted constants. Ex. 6. Shew that, if w w then az + bz' + c a'z-^b'z' + c' a"z + b"z! + e'' z ^ 1 Aw+A'w' + A" Bw + B'w' + B" Cw + Cw' + C where A, A\ A", ..., (7, C", C" are the respective minors of a, a\ a", ..., c, c', c" in the non- vanishing determinant A, where and prove that A= a , & , c a' , 6' , c' a", 6", c" {d'z -f 6"2' -f- c )3 J ( -^ ) = A. Prove that the roots of the characteristic equation for this inverse transformation, expressing z and 2' in terms of %o and w\ viz. ^-0, A' , ^" 5 , B'-<^, B" C , (7' , C"-( = 0, are connected with the roots of the characteristic equation of the original transformation by the relation <9(^ = A; and verify that the invariant centres for the inverse transformation are the same as those for the original transformation. Ex. 7. Obtain for a lineo-linear transformation, between two sets of n variables, results corresponding to those in the preceding example. 56 EXAMPLES [CH. II Ex. 8. Prove that the invariant centre ^j and ^i' of the general lineo-linear trans- formation is given by the equations A" + cdi B"+ c'di C" -{a + h') 6^ + 6^^ ' the denominator in the third fraction being distinct from zero. Prove also that, for the quantities ai : ^i : yi, Ex. 9. Shew that, when n is a prime number, all the periodic substitutions 'w = az + hz'+c 1 for s = 2, ..., ?i— 1, are powers of the same periodic substitution for s = \. Shew that all the substitutions where a and a are primitive wth roots of unity, are periodic. Do the two preceding classes contain all the purely linear substitutions which are periodic ? CHAPTER III Uniform Analytic Functions 43. We now proceed to the more immediate and direct consideration of the properties and the characteristics of functions of two independent complex variables, beginning wjth the simplest fundamental propositions. Not a few of these can be considered as well known ; they are included for the sake of completeness, and also for the sake of reference. Some among them are expressed in forms that appear more comprehensive than the customary- enunciations. Others of them appear to be new, such as those which deal with the characteristic relations and the properties of two functions of a couple of variables considered simultaneously ; and these, as being more novel than the others, are expounded at fuller length (Chaps, vii and viii). Though the exposition is restricted to the case when there are only two independent complex variables, it should be noted that many of the theorems belong, mutatis mutandis, also to functiops of n independent variables. For others, however, further ideas are needed before a corresponding extension can similarly be effected. We begin with definitions and explanations of the more frequent terms adopted, many of which are obvious extensions of the corresponding usages for functions of one complex variable. The whole range of the variables z and z' is often called the field of variation. The extent of the field sometimes depends upon the properties of the functions concerned ; otherwise, it implies the four-fold range of variation between — oo and + oo . A restricted portion of a field of variation is called a domain, the range of a domain being usually indicated by analytical relations. Thus we may have the domain of a place a, a', given by relations \z — a\^r, \z' — a \^r' ; we may have a domain given by relations j>{x-a,y -I3,x - a!, y - /3') ^c, ylr{x -a, y - ^, x - a, y - ^')^c', where a=a + i0, a' = a' + i^', the equations being such as to secure a finite 58 DEFINITIONS [CH. Ill range of values of z and a finite range of values of z . When r and r' (or c and c', in the alternative case) are small, the domain of a and a' is sometimes called the vicinity, or the immediate vicinity, of the place a, a'. In these definitions we substitute i— , for \z — a\ when a is at infinity, and —7, for \ z' — a'\ when a' is at infinity. \z \ ' ' '' 44. A function of z and z', say -2^ =f{z, z), is said to be uniform, when every assigned pair of values of z and z gives one (and only one) value of w. Through familiarity with properties subsequently established, the notion that z and z may attain their assigned values in any manner whatever sometimes comes to be associated with the definition ; but the notion is not part of the definition. The function w is said to be multiform, when every assigned pair of values of z and z' gives a finite number of values of w, the finite number being the same for all z, z places where the function exists. Sometimes it is convenient to specify the number in the definition ; when there are on values, and no more than m values, w is sometimes called m-valued. A function w may have an infinite number of values for given values of z and /. Among such functions, each class can be specified by its own general property. Thus one simple class of this kind arises from integrals of functions that have additive periods. Just as with uniform functions, so with multiform and other functions, familiarity with properties subsequently established leads to the notion that a specification of the path or range by which z and / attain their values will lead to the acquisition of some definite one among the m values ; again, the notion is not part of the definition. Even in this matter of the description of the range of z and of z', care must be exercised ; it may become necessary to take account, not merely of the actual range of z and of z', but also of the mode of description of those actual ranges. Consider, for example* the function w = {z^ — z' + l)'^. Take z = and z' = as the initial place; and consider the branch of w which has the value +1 at that place. We make z vary from to +1 by describing (in the direction indicated by the arrow) a simple curve OAB which, when combined with the axis OB of real quantities, encloses the point ^i and does not enclose the point i. * The example was suggested to me by Prof. W. Burnside. Another example, viz. w = (z-z' + 1)^, is given by Sauvage, Ann. de Marseille, t. xiv (1904), section i, a particular path being specified. Obviously any number of special examples of the same type can be constructed. 45] DEFINITIONS 59 We make / vary from to +1 by describing the straight line 0' C in the direction indicated by the arrow; the point D' on that line is given by 2^ = 1. . Consider two different descriptions of these paths. In the first description, keep z' at 0', while s describes the whole path OAB ; and then keep z at B, while z' describes its whole path 0' C. For this description, the final value of w is manifestly + 1. D' +1 C In the second description, keep z at 0, while z' describes the part O'D' of its whole path; then keep z' at D', thus making w = {z^ + ^)i for that value of z', and now make z describe its whole path OAB. When z arrives at B by this path, the value of w is — (£)2, that is, when z is Sit B and / at i)' by this description of paths, the value of {z^— z' + 1)2 has become -(|)2. Now keep z at B, and let z' describe D'C, the remainder of its path ; the final value of w is manifestly — 1. It thus appears in the case of the special function that, even when the range for each variable is perfectly precise, the final value can depend upon the mode of description of the precise ranges. The matter belongs, in its simplest form, to the theory of algebraic functions. 45. A function f{z, z') is said to be continuous if, when the real and imaginary parts of z and of / are substituted and the function is expressed in its real and imaginary parts u + vi, both the functions u and v of cc, y, x', y' are continuous. Let the function f{z, z) be uniform and continuous, everywhere within a field of z, z variation. It is said to be analytic, when it possesses derivatives of all orders with regard to both variables dz ' dz' ' ' which are uniform and continuous everywhere within that field ; or what is equivalent, it is said to be analytic if /(^, z') is an analytic function of z when any arbitrary fixed value is assigned to z and is also an analytic function of z when any arbitrary fixed value is assigned to z. But it need hardly be pointed out that, while f{z, z) is — under this definition — expressible as a power-series of z alone having functions of the parametric / for coefficients and also as a power-series of z' alone having functions of the parametric z for coefficients, an expansion in powers of z and z simultaneously is a matter of proof, to be considered later. 60 DEFINITIONS [CH. Ill It is a known proposition that an absolutely converging double series can be rearranged in any manner and can be summed in any order, the sum being the same in all arrangements and for all orders of summation. Suppose, then, that the double power-series where m and m are positive whole numbers (including zero), and where the coefficients Cm,m' are constants, converges absolutely at every place within some domain of the place a, a'. The series, within the domain, dej&nes a function ; and the function is said to be regular, or to behave regularly, everywhere in the domain of the place a, a'. The domain must not be infinitesimal in extent ; and the place a, a' is said to be an ordinary place for the function. When it is desired to indicate specifically that the double series contains only positive powers oi z — a and / — a in accordance with the definition, we call the series integr'al, or tvhole, or holomorphic \ and sometimes the function is called integral or holomorphic within the domain of the place a, a. When the power-series is finite in both sequences of indices, the function is a polynomial in z and /. When it is infinite in either sequence or in both sequences, the function represented is usually called transcendental, unless it can be represented by algebraic forms. When the function is transcendental, the question arises as to the range of the domain over which the power-series converges. When the domain is limited, a question arises as to whether the power-series, representing the function within the domain, can be continued analytically beyond the limits of the domain. Perhaps the simplest example of a multiform function w of z and / occurs, when the three variables are connected by an algebraic equation A{iu, z, z') = 0, where J. is a polynomial in each of its arguments. As already explained, it sometimes proves desirable in this connection to consider two multiform functions w and w , defined by algebraic equations G (w, w , z, z') =0, D (w, w' , z, z') = 0, where G and D are polynomial in each of their arguments. In this event, the ordinary processes of elimination enable us to substitute equations A {w, z, z) =0, B {lu, z, z') = 0, for the equations C = 0, D = 0; but care must be exercised to secure that the separate roots of ^ = and of i? = must be grouped so as to give the simultaneous roots of C = 0, D = 0, 46] DEFINITIONS 61 For example, we shall have (Chap, vi) to consider an expression R (w, w', z, z') j!0, D 22 where R (w, w', 2, z') denotes an integral polynomial in w and w\ and where the double finite summation extends over the simultaneous roots of (7=0, D—0. In the method adopted for its evaluation, we are led to introduce terms which arise from combinations of the roots of ^ = 0, B = 0, that do not provide simultaneous roots of C=0, D = 0. In the first case, to the function iv : and, in the second case, to the functions w and w' : the epithet algebraic is assigned. Manifestly, among the four variables w, w , z, z, any two can be described as algebraic functions of the other two, unless (in limited cases) elimination should lead to a single relation between two variables alone. In this initiaL stage, it is not necessary to state the definitions of terms pole, accidental (or non-essential) singularity, essential singularity. New and modified definitions are required, because functions of two variables possess properties which have no simple analogue in the properties of functions of a single variable. These definitions will be given later (§§ 57, 58), when the properties are under actual consideration. As will be seen, a dis- crimination between functions of two variables and functions of more than two variables can be made, so as to give a classification proper to functions of two variables. We may, however, mention in passing that, in the vicinity of any non-essential singularity a, a', a uniform analytic function is expressible in a form Q{z— a, z —a!) P {z — a, z — a) ' where Q and P are functions, which are regular in a domain of a and a . Such a function is sometimes called meromorphic in the, vicinity of the place a, a. The simplest example of a meromorphic function occurs when both Q and P are polynomial functions of their arguments ; in that case, the function is called rational. Some properties of regular functions. 46. Consider functions that are regular everywhere in some finite domain of an assigned place a, a. By writing z — a — ^or-r, according as | a | is finite or infinite, and by writing z — a — ^' or -p, , according as \a'\ is finite or is infinite, we can take the assigned place as 0, 0, without any loss of generality. 62 ANALYTIC PROPERTY OF [CH. Ill We then have a theorem* connected with the definition of the analytic property, as follows : — When a function f{z, z),for values of\z\^r and of\z'\^ r, is a regular function of z everywhere within the assigned z-circle for every value of z within its assigned circle, and also is a regular function of z everywhere within the assigned z -circle for every value of z within its assigned circle, it is a regular function of z and z everywhere within the indicated field of z, z variation. Let the function /(^r, /) be represented by a series f{z,z')= i g^{z')z^-, m = as is possible under the first hypothesis. If Mo denote the greatest value of I f{z, z') I for any assigned value V of / within the /-circle, and for all the values of z within its circle, our series gives 00 f{z,Zo)= 2 gmMz"^; m = and then by a well-known theorem f, we have M' \9m{^o)\<-^- Consequently, if M denote the greatest value of \f{z, z') | within the whole z, z field considered, we have m: < M, and therefore I S'^l (^0 ) i < ^ , for all values of m, for any value of zl such that \zl\^ r' . Consequently, for all values of / in question, we have Now f{z, z) is a regular function of z for every value of z for which \z\^r\ hence g^ iz), being the value oif{z, z) when z=0, and ^.. (-) = -, ^f{^,^') for all values oi\m, are regular functions of /. Accordingly, we can write gm \^ ) ^^ ^ ^tn, n ^ > w=0 * The theorem is true unde.^ g^g^ jggg restricted conditions. See two papers by Osgood, Math. Ann., t. lii (1899), pp. 462—454^ ^j_^ ^^ ijij (1900), pp. 461—464 ; and a paper by Hartogs, ?•&., t. Ixii (1906), pp. 1—88. t Theory of Functions, § 22. I f I 46] EEGULAR FUNCTIONS 63 where the series represents a regular function of / ; and as | gm {z') \ throughout the whole range of variation of / is less than M/r'"^, we have, again by the theorem already quoted, On these results, consider the double series 00 00 if it converges absolutely, we can take it in the form 00 1^ 00 n=0 Km=0 that is, Xgm{z')z^, and so we shall have F(z,z')=/{z,z') for the field of variation within which F (z, z') converges absolutely. But we have just proved that M and therefore we have 00 00 t F{z,z')\ = OT = ?j = 1 00 00 mi ^' \n ^ X 1 \c,n,n\ \Z OT = »i = 00 00 J)^ < X X T- \z\'^\z'\'^ 771 = re = ' ' M_ r ) y r for all values of | ^ | = 27r* ^-^dz\ dz T,J{a,z) _ 7tA f f(a,z') the integrals being taken positively round any simple closed curve which lies entirely within the region bounded by \z' —a' \ = r' and encloses the point a'. The variations of z and / are independent of one another, as also are the integrations in the two planes of the variables ; combining the results, we have f(a,a')- ' ^' ^^''^'^ { d^+-f(z, z') {27nyjj (z-a)(z'-a'/^'^''' --l-W-^S^l^l—dzdz' - 47r^jj(^_a)(/-aO ' 'm\n\ dzdz', 47r'^ }} {z- ay^^ {z' - a')'*+^ the integrals being taken round simple closed curves in the ^-plane and the /-plane, the ^-curve lying entirely within the region \z — a\ = r and enclosing the point a, and the ^^'-curve lying entirely within the region \z' — a \= r' and enclosing the point a'. We thus have expressions, in the form of double contour integrals, for the value oif{z, z) and of every derivative off{z, z') at the place a, a. Again, let M denote the greatest value of \f(z, z') \ for places within the whole z, z domain of variation represented by \z — a\-^r, \z' — a'\^r' \ then at every place on the double contour integral we have \f{z,z)\^M. Proceeding exactly as in the case of a single variable, we can shew that — , dzd^ and therefore ^ 477 W, {z — a) {z' — a) \f{a,a')\^M, which is merely a statement that the value of \f{z, z) \ at a particular place in the field is not greater than its greatest value in the field ; and we can also shew that /(^, ^') {z - a)"'+^ (/ - a')**+i dzdz 477^ < T ^i. and therefore which is the former result. F. ^ m ! n ! Milt y 66 SPECIAL CLASS OF [CH. Ill Another method of stating these results is as follows. Let z, z' be any place within the field of variation where fiz, z) is regular; in the ^^-plane, take any simple closed curve lying within the field and enclosing the point z, say a circle of centre z, and let t denote the complex variable of a current point on this curve ; and in the ^'-plane, take any simple closed curve lying within the field and enclosing the point z', say a circle of centre z', and let t' denote the complex variable of a current point on this curve. Then '^ra+nf(^Z, z') _ m\n \ [[ f(t, t') ll<^ dtdt'. Ex. Prove that, for the foregoing function f(z, z') and with the foregoing curves of integration, the value of each of the integrals for all positive integer values (including zero) of m and n, is zero. 48. We shall come later (Chap, vi) to a fuller discussion of double integrals involving complex variables ; meanwhile, it will be sufficient to state that integrals of the foregoing type, in which the integrations with regard to z and to z' are completely independent of one another, belong to a very special and limited class of double integrals. They may even be regarded as merely iterated simple integrals ; and many of their properties can be deduced as mere extensions of corresponding properties for simple integrals. Thus we know that the value of the integral taken positively round the whole boundary of any region within which f{z) is uniform, continuous, and analytic, is zero, even if the region is multiply connected ; and it follows, as a corollary, that the value of the integral taken round any simple closed curve is unaltered if the curve is deformed without crossing any point where f{z) ceases to have any one of the three specified qualities. This result can at once be generalised, merely through a double use of the result, into the following theorems : — I. Let F {z, z') denote a function which, over a limited region in the ^-plane with a complete boundary unaffected by variations of z' , and over a limited region in the /-plane with a complete boundary unaffected by variations of z, is uniform, continuous, and analytic. Then* zero is the value of the integral * The constant - l/iTr^ is inserted here merely for the i^nrpose of formal expression. 49] DOUBLE INTEGRALS 67 taken positively round all parts of the complete boundary* of the 2^-region, and positively over all parts of the complete boundary of the /-region, when these boundaries are entirely unrelated to each other. II. For the same type of function, and with the same type of range of integration, the value of an integral -^jjP(^.^')'\ where p and p' are small. We then expand {(t — z) (t' — z')}~'^ in ascending powers of t/z and t'/z', the expansion being and so the subject of integration becomes The value of the part taken round the contour as indicated, is zero (Ex., § 47), because there are no negative powers of t'. Similarly the value of the part is zero. Again, the value of the integral / dtdf -i^J// ('•*') IS fr+l f's+l \rlsl dVdt'' ji=o,i'=o' for all integers r = 0, 1, . . . , and all integers s = 0, 1, .... When either of the integers r and s is negative, and when both of the integers are negative, the value of the integral is zero. Hence, taken positively along the small contour that encloses the a'-origin in the ^-plane and the ^'-origin in the /-plane, we have I [[ f(t,f) z^i/n+i 47r2 jj (t- z) (f - z) f «+if «+i dtdt' m n = - S 2 ^r/s \d''+'f{t, t') r\s\\ drdf' Ji=o,r=o. 70 DOMINANT [CH. Ill We thus have the full value of the integral 47r2 Jj (t - z) (f - z') 1 1^+^ "•■ t'''+> P+ii''^+i I ^^^^ ' taken positively round our contour in the ^-plane enclosing the ^■-origin and the point z, and our contour in the ^'-plane enclosing the /-origin and the point z'; it is f{z,z')- t 2 r\s\\ dVdt'^ Jj=o,t'=o Consequently our magnitude is equal to the polynomial ^ro 5^0 L^M s ! I Sr8i'^ j j=o,i'=oJ ' and when this polynomial is denoted by P (z, z), we manifestly have \ 'd^^^P{z,z') \ _ f^'fit, i!) \ \ dz^dz' i,=o,/=o 1 c^-ar^ j«=o,r=o' The proposition is thus established. The result, in either form, shews that it is possible to construct an ex- pression the value of which shall be a polynomial approximation to the value of a function /(^, /) in a field where it is a regular function of its arguments. Ex. Evaluate the integral with the same suppositions as to the function / {z, z') and the range of integration. 50. In connection with the function f{z, z), which is regular within the field \z — a\^r and \z' — a \^ r', and for which \f{z, z) \ is never greater than M for places in the field, consider a function <^ {z, z') defined by, the relation (Z, Z ) = ; ; '-. z-a\ / z -a' Evidently <^ (z, z') can be expanded in a double power-series in z —a and z' — a', which converges absolutely for values of z and z' such that \z — a\^p {z, z) is not necessarily unique. In the same circumstances as before, consider a function -^/r {z, z') defined by the relation M ylr {z, z)= ; ; ^ ^ z — a z —a i. -f which also is expressible as a double power-series in z — a and z'—a', con- verging absolutely for the region ^ -\-- ; — ^^k<\. Proceeding as for ^ {z, z), we find, for all integer values of m and n, \ d'^+''y\r{z,z') \ A'^ + n) \^j Now (m + n)\ "^ ml nl ; hence U^ U^ J2=a,z=a I. i^* ^'^ ) z=a, z =a ^ 1 { d^-+^^f(z,z') ^\\ dz^dz'^' )z = a,z=a- SO that -^{z, z') also is a dominant function*. * Poincare uses the term majorante. 72 CONVERGENCE OF SERIES [CH. Ill 51. During the foregoing investigations, particular series in suitable circumstances have been declared to converge; and it will be noted that, in such series as have occurred, the convergence has been absolute. We do not propose to consider, in detail, the general theory of convergence of double series. When convergence is absolute, no other kind of convergence need be considered specially ; and such series, as will be discussed in these lectures, will be discussed with a view to absolute convergence. What is wanted here is a knowledge of some non-infinitesimal region of variation of the variables in which the respective series converge absolutely*. In this regard, one warning must be given. Both in what precedes and in what will follow, a region of variation, in which a double series converges absolutely, is usually defined by a couple of relations of the form \2!\^ p iT^ Rp-'^ R'l M dOdO' Jlp-m Jl'q-n ' By hypothesis, we can increase R and R' without limit ; hence, for all values of p that are greater than m, or for all values of q that are greater than n, and for both sets of values simultaneously, we have and therefore ^p,q -p,q = 0, = 0, for those values. Accordingly, when we remove from the series those terms which have vanishing coefficients, the modified expression for/(^', z') becomes in n p=Oq=0 shewing that/(2^, z') is a polynomial in z and /, of degree m in ^^ alone and of degree n in / alone. 53. It follows, from the first investigation in § 52, that a uniform analytic function must acquire infinite values. In particular, a general polynomial in z and / acquires infinite values, when | ^ | is infinite while \z' \ is not zero, or when \z'\ is infinite while | ^ I is not zero, or when both | z \ and | / ] are infinite, though in the last event conditions may have to be satisfied*. * For example, the function 1 + z + z' does not become infinite when' z | is infinite and \z' [is infinite unless U + a'l also is infinite. 76 INFINITE VALUES AND [CH. Ill The questions then arise : — Must a uniform analytic function of z and z acquire a zero value within the whole field of variation ? And, what is a subsidiary question governed by the answer to this preceding question, must a uniform analytic function of z and / acquire any assigned value within the whole field of variation ? Naturally, in considering the questions, we assume that we are dealing with functions that do not reduce to a mere constant. First, a brief proof will justify the answer that a uniform analytic function of z and z must acquire a zero value somewhere within the whole field of variation. Let / {z, z') be a function of z and /, which is uniform ; con- sequently, if '^^^'^'^ =7(^:7)' the function {z, z) is uniform. Further, ^ (z, z') is continuous, unless f(z, z) has zero values. Let f{z, z') be analytic ; then ^ {z, z) also is analytic. Thus, assuming that f{z, z) is a regular function, that has no zero within the whole field of variation, its reciprocal ^ {z, z) is uniform, continuous, and analytic throughout the domain where f{z, z') is regular. Consequently, ^{z, z') is a function that is regular throughout the whole field. Now we have seen that a uniform analytic function must acquire an infinite value or infinite values somewhere in the field of variation of the variables ; hence our function cf) (z, z') must acquire an infinite value somewhere, that is, the regular function f(z, z') must acquire a zero value somewhere and therefore the hypothesis, that / (z, z') has no zero, is untenable. But as was the case with the place where the function acquires an infinite value, so that the function is not regular there and the place does not belong to the i-egion of continuity of the function, so it may happen that a place where a function acquires a zero value does not belong to the region of continuity of the function. Thus the function e^ + ^' is regular over a domain given by finite values oi \z\ and finite values of 1 2' I ; it is not regular for infinite values oi \z\ alone and of \^\ alone, because it cannot be expanded in powers of - and -, . When z is real, infinite, and negative, while 1 2' I is finite, the function e2+^' = 0; and so for other places. No one of these places belongs to the region of continuity of the regular function e^ + ^'. The corresponding question, as to the acquisition of an assigned value a, would similarly be answered in the affirmative after a consideration of the function f{z, z) — a which, under the foregoing argument, would have to acquire a zer6*^alue ; so/(^, /) would have to acquire an assigned value. The difficulty, that the zero of the function perhaps will not occur in the domain of regularity, may be illustrated by returning to the corresponding question in the theory of functions of a single complex variable ; indeed, it would be raised directly, for example, by taking z = 0, in the case of a regular function. • 54] ZERO VALUES 77 54. It is a result, in Weierstrass's theory of uniform functions of a single variable*, that, in the vicinity z^ of an essential singularity of a uni- form function f{z), there always is at least one point within a circle 1 2; — ^'o I = e, where e is any assigned small quantity, such that !/(^)-a| z' = — - , we have Ao— Jm, 11 m ! ?i ! 1+^ 1^-5) ixm+l/ 1\" + 1 '2\m + re + 2 3/ When we form a series in powers oi z + - and s' + -, so that -- and -5 is the new origin for a new domain, the series converges for values of z and z' such that 1 The series is that is, it is fn 22 2 2 n=0 n=0 \3 ^ + 2 1\™ <'■l such that , 1 ^+2 ^l' <'-, the second series does give a representation of / which is not given by the first series. The first series is the expansion, within a domain round 0, 0, of the function {l-z){l-z') When we sum the second series, we have, as the sum, ^3. (ly that is, -iH)IMG'4 verifying the property that the two series, within their respective domains, are elements of one and the same function. Singularities of uniform functions. 57, Any region of continuity of a function that is uniform, continuous, and analytic has for its boundary a place or an aggregate of places (whether these are given by values of the variables that are continuous in succession or are given by discrete sets of variables) where the function ceases to be regular. Such a place is called singular by Weierstrass*. Let k, k' be a singular place for a uniform function f{z, z') ; then in the immediate vicinity of k, k', the function cannot be expanded as a converging * See the memoir cited (§ 56) above, p. 156. 57] SINGULABITIES 83 series of powers oi z — k and / — k'. Two alternative possibilities present themselves as to the behaviour of functions in the vicinity of such a place. Under the first of these alternatives, it can happen that a power-series Po {z — k, z' — k'), representing some function regular at k, k' and vanishing there, exists such that the product P,{z-k,z'-k')f{z,z) is regular in the immediate vicinity of k and k'. Denote this product by F{z, z). Then F{z, /), being a regular function of z and / in the immediate vicinity of k and k', can be expanded in a double series of powers o^ z — k and z' — k' which converges absolutely within non-infinitesimal regions round k and k'. Denote this new series by P^ {z — k,z'~ k') ; then we have P,{z-k,z'-k') J^'''^-p,{z-k,z'-ky Following Weierstrass *, we call such a place an unessential singularity of the function. Under the second of the alternatives indicated, it can happen that no power-series Po {z — k, z — ^'), representing some function of z and z regular in the immediate vicinity of k, k', exists such that the product P,{z-k,z'-k')f{z,z') is regular in the immediate vicinity of k, k'. Following Weierstrass*, we call such a place k, k' an essential singularity of the function y(2', z'). It is to be noted, in passing, that, for the occurrence of an unessential singularity, it is sufficient to have a single power-series Pq such that the product Po/ is regular in the immediate vicinity of the place. But there is no assumption (and it is not universally the fact) that only a single power- series exists having this property or that all such power-series, as exist having this property, are expressible in terms of Po alone. When two different expressions for the uniform function f{z, /) are obtained in the vicinity of the place k, k', they must be equivalent; and we should then have a relation Q, (z-k, z' - k') ^ Pi (^ - k, z' - k') Qo(z-k, z'-k')~ P,{z-k, Z' -k'Y We shall assume that, while Pi (0, 0) and Pq (0, 0) vanish, the power-series Pi and Po possess -f- no common factor vanishing at k, k' , whether it takes the form of a regular power-series or a mere polynomial which is a special case of a regular power-series. Similarly, we shall assume that Q^ and Q^ possess no common factor vanishing at k, k'. Now * I. c, p. 156. t This matter will be considered la,ter, so as to obtain the conditions necessary and sufficient to justify the assumption. 6~'2 84 SINGULARITIES [CH. Ill Here Qi is regular in the immediate vicinity of k, k', while P^ and Pq have no common factor vanishing at k, k' ; hence Q^ must contain Pq as a factor. Let F denote the quotient of Qq by P^, so that F is regular at k, k' ; then Q, = P,F, Q, = P,F Again, p.(.-M--^')= ^;|::^;:::i?/ .(^-^./-n Here Pi is regular in the immediate vicinity of k, k', while Qi and Qo have no common factor vanishing at k, k' ; hence Pq must contain Q^ as a factor. But and therefore IjF is regular at k, k'. Consequently both F and \\F are regular at k, k' ; and therefore F does not vanish at k, k'. It is not difficult to see that we then may choose a domain round k, k', which may be small but is not infinitesimal, such that F does not vanish in that domain; and then the behaviour of Qq in the immediate vicinity of the place k, k' is effectively the same as the behaviour of Pq in that immediate vicinity. Likewise for Pj and Q^ if they vanish at k, k'. When either does not vanish, the other will not vanish ; they are different from zero at k, k' together. It follows that, in discussing the behaviour of / {z, z') in the immediate vicinity of k, k', any representation oi f{z, z) by a quotient Pi/Po can be used, if Pi and Pq have no common factor*. 58. In the case of functions of a single variable, it is known that there are different types of essential singularities, whether these occur at isolated points, or along lines, or over continuous areas. Special kinds of essential singularities are considered in that theory, and they furnish partial charac- teristics of some classes of functions ; for example, not a few definite results have been achieved when the essential singularities in question can be approached as the limits of groups of particular points of a function ; but the theory is far from easy or complete. A fortiori, it is to be expected that even greater difficulties will arise in the consideration of the types of essential singularities of uniform functions of a couple of variables. But when we deal with unessential singularities of uniform functions, there is a real divergence between the theory of functions of a single variable, and the theory of functions of two variables or more than two variables. In the case of functions of one variable, there is only one type of unessential singularities, the only variation in the type being the variety of the order ; such a point a is said to be an unessential singularity (or a * The relation between two such functions as P^ and Q^ will be considered fully in Chapter iv : in particular, see § 64. 58] TWO TYPES OF UNESSENTIAL SINGULAEITY 85 pole) of a function f{z), and of order n for the function, when there is a positive integer n such that {z-ayf{z) is finite and not zero at the point. In the case of uniform functions of two variables, we arrange the un- essential singularities in two distinct types or classes. After the explanatory- definition we know that, in the immediate vicinity of k, k', the function f{z, z') can be expressed in the form J^^'^->~ p,{z-k,z'-k'y where Po and Pj are converging double series in powers oi z — k and / — k', of which Po vanishes at At, k'. Two different cases then can occur as alternatives, discriminated according to the value acquired by Pj at k, k'. In the one case, leading to one of the two types of unessential singular- ities, it is the fact that Pi does not vanish at k, k'. It then follows that, no matter how z tends to the value k and / to the value k' , the quantity \f{z,z)\ can, for sufficiently small values of \z-k\ and \z' -k'\, be made larger than any assigned magnitude, however large : that is to say, this large magnitude is assigned at will, and the appropriate small values of \z — k\ and \z' -k'\ are determined subsequently to the assignment. We therefore can take infinity as the limit for the assignment ; and the place k, k' then gives a definite and unique value tof{z, z'), this value being infinite. This tj^e of unessential singularity is one of the two kinds of un- essential singularity considered by Weierstrass. It is convenient to use for functions of two variables, the same name as is used, for functions of on variable, when the place gives a definite and unique infinity of the function. Accordingly we shall call this type of unessential singularity the polar type ; and a place k, k', being an unessential singularity of the polar type for the uniform function, will be called a 'pole of the function f{z, z'). In the other case, leading to the other of the two types of unessential singularities, it is the fact that Pi does vanish at k, k'. The place k, k' then does not give a definite and unique infinite value for the function f{z, z). Subsequent explanations may so far be anticipated here as to declare that particular modes of approach oi z io k and of / to k' can be selected, so as to make f{z, z') tend towards any assigned value near k, k' and acquire that assigned value at k, k' ; thus the function f{z, z) does not acquire a definite unique value at the place. This type of unessential singularity is the other of the two kinds of unessential singularity considered by Weierstrass. We have given a definite name to the other type of unessential singularity that can belong 86 UNESSENTIAL SINGULARITY [CH. Ill to uniform functions of two variables; to the type just indicated, we shall give simply the general name unessential singularity and, so far as concerns functions of two variables, there need be no confusion in taking this un- restricted name*. Thus, for the function z+z' z — z the place z = l, z' = l is a, pole ; the place z=0, s' = is an unessential singularity. For the function z + z' l+l, z-z the place 0=1, 2'= - 1 is a zero ; the place 3 = 1, s' = l is a pole ; the place 2=0, 2' = is an essential singularity. For a function where P (2, 2') and Q (2, 2') are polynomials in 2 and 2' having no common factor, all places satisfying the equation $(2,2') = are poles unless they also satisfy the equation P(2,2') = 0; and all places satisfying the two equations . . Q{z,z') = Q, Piz,z')^0, are unessential singularities. As a summary conclusion, we see that there are four kinds of places for a uniform analytic function of two variables, viz. ordinary places, poles, unessential singularities, essential singularities. The first set of these constitute the region of continuity of the function ; the remainder constitute the boundary of the region of continuity of the function. Extension of Laurent's Theorem. 59. As a last theorem for the present, we proceed to an extension of Laurent's theorem on functions of a single variable ; in order to make the establishment simpler, we shall restate Cauchy's theorem concerning the * Corresponding considerations arise for functions of n variables. Weierstrass arranges their unessential singularities in two kinds. One kind includes places that, as in the text, may be called poles ; at such a place, the function definitely and uniquely acquires an infinite value. The other kind includes all unessential singularities which are not poles. Now it is conceivable that an unessential singularity of this second kind for a uniform function of n variables might be ranged in one or other of n-1 classes, according as there are m, 00^, oo 2, ..., 00 "~2 ^ays (where m is finite) in which zi, z^, ... , 2„ could be made to approach the unessential singularity ai, a2, ..., a„ so as to make the function Pi(2i-ai, 22-«2. ••■, z^-a^ ^o('^i-«i, Z2-«2. •••, 2„-a„) acquire an assigned value at the place. The question manifestly does not arise when there are only two independent variables ; hence the adoption of the names pole and unessential singularity in the text. 59] EXTENSION OF LAURENT'S THEOREM 87 expansion of a function in a double series of positive powers. Consider a function f {z, z') within a region where it is continuous, uniform, and analytic. Within that region (assumed to include 0, 0) consider the domain defined by \z\^pR>\z-a\^r>ro, Ro > R' >\z' - a' \^r' > r^. Denote by t and by s current variables {or points) on the circumferences of the outer circle of radius Rq and the inner circle of radius r^ in the z-plane ; and similarly for t' and for s' on the circumferences of the outer circle of radius Rq and the inner circle of radius r^ in the z -plane. Then the function fiz, z') can be expressed as a series of integral powers of z — a and z —a' ; the indices of those powers can range from — oo to + oo for each of the * The analytical work, needed to establish the result, is so similar to the corresponding analysis for functions of a single variable (see my Theory of Functions, § 28) that it need not be set out in detail. 88 EXTENSION OF [CH. Ill variables ; and the double series converges absolutely for values of z and z given by R>\z-a\^r, R'^\z -a'\^ r'. By the generalisation of the first part of Cauchy's theorem, we have f{z, z') = ^^ Wjt^^P-^ dtdt' {ZTTlf J J {t — Z) (t —Z) {2^iyJJ(s - z) {f - z) "^'"^^ (27riy]j{t-z){s'-z') ^^^' I ^ f f ^^''''^ dodo' ^{2'rriy]]{s-z){s'-z'r''^'- Now, for our vahies of a, a, z, z', t, t', we have t-a ^ z — a (z — ay t — a fz-aV''''^^ = 1 + +...+ , + t — z t—a \t —aj t — z \t — a) t'-a' , z'-a fz'-a'Y t' - a /z' - a'y'+' t'-z' " t'-a^'" ' \t'-ay t'-z'\t' -a' and so the integral '{2^f\\{t-z){t'-z)^^'^^' is expressible as a double series of terms X^ Cp^q{z - ay {z - a'Y for p = 0, 1, . . . , m and q = 0, 1, . . . , n, where . _ 1 rr /ao dtdt' ; ^'^ {27riy]}(t-af^^{t'-ay+^ together with a single series of terms 1 f[f{;tJ)(z-aY^'(z'-a'Vi -,,-,. for q = 0, 1, ... , n; and a single series of terms 1 [f f{t, t') ( z - a^ V+^ (z - ay ^,^,. . for p = 0, 1, ..., 771 ; and a term (277^/ j j (t - z) {f - z') \t - a) [f - a') "^^"^^ ' To consider the coefficients in the double series, let 31 denote the greatest value of \f(z, z') | within the whole region considered ; then, as before, M though nothing can be declared as to a relation between Cp^q and the derivative . ,^ ' at a, a, for our function is not defined within the dz^oz '■' domain \z — a\ro>\s-a\, while \t' - z \<\t' -a' \] so we take z—a ^ s-a /s-aY z — a fs — ay-^^ = 1+—+...+ r— : +:r-„ h^-. > s — z z—a \z—aj z — s\z—aj t'-a' , z'~a fz'-a'Y t' - a' f z' - a'y+^ t' -z'~^^t' -a''^'"^ \t' -a) '^t'-z'\t'-a' (^p,,=7^JL .' ^ J.a^. i'-^y-'dsdt'. ^P,1 = 7o57n. I \ ri \1L («' - ^y~' dtds'. 90 EXTENSION OF [CH. Ill We proceed as in the last case. It is possible to increase /u. without limit and n without limit ; and we obtain, as the expression for the integral, where 1 {{ f(s> t') {2irifii{t'-a')i^^ Also \cp,q\R>\z — a\'^r>ro, Rq > R' >\ z' - a' \> r' > i\'. It follows as an immediate corollary that when a f auction j>{z, z') is uniform, continuous, and analytic for all the z, z' region of variation repre- sented hy the relations \z—a\'^r>ro, \z' — a' \'^r' >r^, it is expressible as a double series of negative powers in the form (z,z') = ll c.^, n (z - a)-"^ (z' - a')-^ where \ c^, n \ < Mr^"^ r^'^, M being the greatest value of \ (f> (z, z) j within the foregoing region ; and the series converges absolutely for the specified range of values for z and z . The result is at once derivable from the extension of Laurent's theorem by making R^ and R^ increase without limit ; and it can of course be established independently in the same manner as the general theorem. Ex. 1. The function where P ( 2, - , 2', - J is a polynomial in 2, - , 2', -, , can be expanded in a series — 00 — OC for finite values of 1 2 [ and 1 2' | such that 1 2 I ^ 7- > e, I 2' I ^ 7-' > e', / where e and e' are positive non-zero quantities. Ux. 2. Shew that the coefficient of z'^z''^ (where m and n are positive) in the Laurent expansion of >(-l)4'(^'-a e 5 I ^ I and 1 1} I being finite and independent of 2 and of 2', is Jm (^) "4 (7)) where J„i and Jn are Bessel's functions of order m and n; and obtain the coefficient of ^nt/n JQ ii^Q same expansion (i) when either m or n is negative, (ii) when both m and n are negative. CHAPTEK lY Uniform Functions in Restricted Domains A theorem due to Weierstrass. 60. After these preliminary results relating to expansions of a uniform function, which converge absolutely and are valid over the appropriate domains, it is important to take account of the detailed behaviour of the function in the immediate vicinity of each of its several kinds of places. Accordingly, let a, a' be an ordinary place for a uniform, continuous, analytic function f{z, /) ; the preceding investigations shew that f(z, /), regular in some domain of that place, can be represented within the domain by a double series of positive powers of z — a and z' — a' which there con- verges absolutely. No generality, for our present purpose, is lost by assuming that a = and a' = 0, for the assumption can be secured by taking z — a = Z, z' — a = Z' . Hence we write F {z, z') =f{z, z') -/(O, 0) = t%c^,nz^z\ where the summation is for positive integer values of m and of n save only simultaneous zero values. Also,'|/(0, 0) | is finite and may be zero. The detailed behaviour of the function F (z, z') in the immediate vicinity of the place 0, is governed by an important theorem, originally due to Weierstrass. After the analysis has been given, the principal results will be enunciated in a form that differs from Weierstrass's, because the limitation to two variables renders greater detail possible* than when n is the number of variables. * The theorem is proved by Weierstrass for functions of 7i variables, Ges. Werke, t. ii, pp. 135 — 142. Another proof, due to Simart, is given by Picard, Traite d' Analyse, t. ii, pp. 243—245. The theorem is discussed here for the special case when there are only two variables. For this case, a proof (which follows Weierstrass's proof for the general case) is given in my Theory of Functions, § 297 ; it is modified in the proof given in the text, because the theorem is not regarded from the point of view of estabhshing the existence of implicit functions of a single variable. 61] A THEOREM OF WEIERSTRASS 93 Our function F{z, z'), which is regular in a domain round 0, 0, can be expressed in a form F{z, z') = o(z) + z'<}i,(z) + z''<}>,(z) + .... Two cases arise according as F(z, 0) does not vanish, or does vanish, identically for all values of z within the domain. 61. First, suppose that F (z, 0) does not vanish for all values of z. Denoting F(z, 0) by Fo(z), which is equal to (f>o(z), and introducing a new function F^ (z, z') defined by the equation F(z,z')=F,{z)-F,iz,z'), we have a function F-^ (z, z') which, when / = 0, vanishes for all values of z. Now ^^0 {z) is independent of z' and does not vanish for all values of z \ hence we can choose places z, z in the vicinity of 0, 0, which lie within the region of convergence of F {z, z') and are such that l^o|>|i^i|. It is to be remembered that ^o vanishes when z = 0; and so there may be some lower limit for | ^ | below which this inequality is not satisfied. As | ^ | increases, a zero of Fq may be attained, and then the inequality would not be satisfied. Also as | / 1 increases, the value of | F (z, z') \ may increase ; and so there may be some upper limit for | / 1 above which the inequality is not satisfied. Accordingly, we suppose that, for places satisfying the relations p^<\z\< p, |/|i, the inequality | i'o I > I -^i I holds. For all such places we have, on taking logarithmic derivatives of the equation F = F,{\-^'" the relation 1 a^^j^aPo__a_/| i^' F dz F, dz dz V;,=i \ Fo^. Now i^o (■s') is a regular function of 2^ in a domain round z = 0, and it vanishes when z = 0; hence the lowest exponent in its expansion must be a positive integer greater than zero, say m. Thus Fo(z) = z^-h(z), where h(z) is a regular function of z in the selected domain and has a constant term; consequently Fo dz ~ z^ h (/) where {z) is a converging series of positive powers of z in the selected domain. Similarly ]P\ CO 94 ESTABLISHMENT OF [CH. IV where GK^y.{z'), the coefficients of the powers of ^'j are converging series of positive integral powers of / ; and because F^ {z, z) vanishes when / = for all values of z, each of these coefficients Gk,^ {z) vanishes when / — 0. Take each power of z, and collect all the terms which involve that power of z in the expansion then we have A = l '^ -Po M=-oo while each of the coefficients Gn (z), being a linear combination of the coefficients G\^^{z'), vanishes when z' = 0. Thus and the only term on the right-hand side, which involves the power z~^, is , m the term — . z Now let ^1, ..., ^s denote the zeros of F(z, ^'), regarded as a function of z, when we consider a range of values of z such that \z\ < p, and when we assign to / a parametric value ^' such that \^'\< pi- Repeated zeros of '^(^r, f') are given by repetition in the quantities ^, so that s denotes the tale of zeros of F {z, ^') within the range. Then, as F(z, ^') is regular for all such values of z, the function 1 dFjz, ^') _ I 1 F dz p=i z — tp is finite for those values ; it can therefore be expanded as a converging series of positive powers of z, say P {z), so that F dz p=iz-^p Choose values of z, such that \z] is still less than p and is now greater than the greatest of the quantities \^i\, ..., \^s\- The fractions on the right-hand side of the equation can, for such values , of z, be expanded in descending powers of z ; and the equation, after such expansions, becomes i dFiz,n _p,.. s I where As this result is valid for all values of ^' within the selected ^r'-range, ^' being independent of z, we have m z Z r = l 61] WEIERSTRASS'S THEOREM 95 identically for all values of z ; and therefore, among other results, we have for all values of r. The first result shews that, for any given value of z' such that \z' \< p^, the function F(z, /) has m zeros in the range \ z\< p, where the number m is the index of the lowest exponent in F (z, 0) when expressed as a regular series of positive powers of z. The second result then shews that, for all the positive values of r, the quantity is expressible as a regular function of ^' which vanishes when ^' is zero. Hence all integral symmetric functions of ^i, ..., ^^ are regular functions of ^' which vanish with ^' ; and as ^' is a parametric value of z', we may (within our range) substitute z' for ^'. It therefore follows that, if g{z,z') = (z-^,)...{z-^„^) the coefficients gi, ..., gm are regular functions of z within the selected range, each of them vanishing when z' = 0. Further, from the same equation, we have P{z) = G{z)- X (n + l)z-Gn^d^'\ where all the functions are regular. Thus, if • r (Z, Z') =\'g (z) dz-t Z-+^ Gn^, (/), Jo ?i = where T (z, z') manifestly is a regular function of z and z', and vanishes when z = and z' = 0, we have P{z)=^jr(z,z% Thus^ and therefore F=Ug(z,z')e^^'''\ where U is independent of z. As U is the same for all values of z, and as F and g (z, z') and T (z, z') are regular functions of z and z' for the range considered, it follows that U (if variable) is a regular function of z'. When / = 0, let the first term in the expansion of the regular function Fq, which is all of F (z, z') that then survives. 96 WEIERSTRASS'S THEOREM [CH. IV be Cz^ ; then g (z, z') becomes z^^ ; and F (z, z') is then a regular function of z alone. Thus, when z' = 0, we have U = C ; and U, at the utmost, is a regular function of /; hence U = C (1 + positive powers of z') = Ce'\ where u is a regular function of z which vanishes when z = 0. Let R (z, z') = u+T (z, z'), where again R (z, z') is a regular function of z and z' which vanishes when z = and / = ; and we then have F(z,z')=Cg(z,z')e^^'''\ with the defined significance of g (z, z'), R {z, z), and G. The new expression is valid within the assigned range of z, z in the immediate vicinity of 0, 0. But it must not be assumed — and usually it is not the case in fact — that the new expression is valid over the whole domain where /(^, /) is initially taken as regular. We thus have the result : — I. When a function f{z, z) is regular in some domain of 0, 0, and is such that f{z, 0) — /(O, 0) does not vanish for all values of z in that domain, we have f{z,zf)=f{0,0)+Gg{z,z')e^^^'^\ where g (z, z') = ^"^ + ^1^'^-^ + ... +g,n, the quantities g-^, ..., gm heing functions of z', each of which is regular in the immediate vicinity of z' = and, vanishes when / = ; where Cz''^ is the lowest power in the expansion off(z, 0) — /(O, 0) in positive powers of z ; and where R(z, z') is a function of z and z', which is regular in the immediate vicinity of 0, and vanishes when z = and / = 0, 62. One important corollary can be at once derived from the preceding result. Suppose that 0, is a non-zero place for the function f(z, z'), s6 that /(O, 0) is not zero ; then we have Now R {z, z) is a regular function of z and z\ vanishing when ^ = and z — 0, so that I e*(2'2') | jg finite throughout some definite domain round 0, 0. Also I C//(0, 0) I is finite ; and g {z, z), while polynomial in z and regular in / in the immediate vicinity of / = 0, vanishes at the place 0, 0. It therefore is possible, owing to the regularity of g {z, z') and R (z, z'), to choose a non- infinitesimal domain given by \z\^r, \z'\^r', 63] COROLLARY 97 such that, for all the included values of z and z , ^ '' {z,z)\\e^^^''''^\^M<\, /(O, 0) where if is a real positive quantity. For all such values of z and z' , we have n where R (z, z') is a regular function of z and z', given by the expansion ^ g (z, z') e^<-'^'' - 1 ,.,,^' , g' {z, z) e^^'-.-'* - . . . , that is, R (z, z') is a regular function in a domain of z and / and vanishes when z = and / = 0. This domain does not include any place that is a zero of f{z, z'), because at a zero-place ^, z' of f{z, z') we should have and therefore n I a possibility which is excluded. Hence we must have /(0,0) and therefore f{z,z')=f(0,0)e^^'''\ Our corollary can therefore be stated as follows : — Whenf(z, z') is regular within a finite domain round 0, 0, and f(0, 0) does not vanish, then there is a domain round 0, — usually more limited than the former domain within -which f{z, z') is regidai — such that f{z, z) can he expressed in the form f{z,z')=f{0,0)e^^^'^\ where R (z, z') is a function of z and z , which vanishes when z = and z' = and is regular within the second domain. In particular, this expression is valid in the immediate vicinity of 0, 0, on the supposition adopted. 63. In precisely the same manner and with exactly similar analysis, we can establish the following result which therefore needs only to be stated : — II. When a function f{z, z) is regular in some domain of 0, 0, and is such that /(O, /) — /(O, 0) does not vanish for all values of z in that domain, ive have f {z, z') =/(0, 0) + Kh (z, z) e'S<^' ^'' , where h (z, z) = z"" + h^z'""-^ + . . . + A^, the quantities hi, ..., hn being functions of z, each of which is regular in the imm,ediate vicinity of z = and vanishes luhen z = ; where Kz'"^ is the lowest F. 7 98 SECOND THEOREM [CH. IV power in the expansion off{0, z') -/(O, 0) in positive powers of z ; and where S {z, z) is a function of z and z , which is regidar in the immediate vicinity of 0, and vanishes when z = and z = 0. The postulated circumstances are not the same in these two theorems. If it should be the case that/ (2', 0)-/(0, 0) does not vanish for all values of z within the range, and also the case that /(O, 2f')-/(0, 0) does not vanish for all values of / within the range, then both theorems hold. In that event, we have two different expressions for/(^, z) — /(0, 0) which must be equivalent to one another. This equivalence will be illustrated by an example, that will be given after we have discussed the alternative to the initial hypothesis. 64. Secondly, suppose that the function F{z, 0), where vanishes identically for all values of z. Now F {z, z) is a regular function of z and z , within the range considered ; as before, it can be expressed, by summation of the uniformly converging series which represents it, in the form which itself is a converging series within the range. (As already stated, 00 (^) is the F^{z) of the preceding investigation). If then F {z, 0) vanishes identically for all values of z, then ^^{z) vanishes identically. It may happen that other coefficients <^i {z), (f)^ (z), ..., vanish identically ; let (pt (z) be the first that does not thus vanish, t being a finite integer because F (z, z) is presumably not a constant zero. Consequently F {z, z) = z' {(f>t (z) + z(Pt+i {^)+ ■■ ■], and the series cf)t{z)+z'ci>t+i(z) + ... is a regular function of z and z' ; that is, in the suggested circumstance when the function F {z, 0) vanishes identically for all values of z, our function F (z, z') has some power of z' as a factor. Let this factor be z''^ ; then ^ is a positive integer greater than zero, and it is assumed to be the largest positive integer which allows F (z, z) /"* to be a regular function of z and z' in the vicinity of the place 0, 0. The first of the two preceding theorems does not hold as an expression for/(^, z). But if the function ^(0, z) does not vanish identically for all values of z, the second of the preceding theorems does hold as an expression ior f{z, z). There are, however, limitations upon the forms of the quantities hn, hn-i, ... ; in particular, hn = 0, hn-i = 0, ..., hn-t+i=--0. But the momentarily important result is that f{z, z')-f (0,0) = z'^G(z,z), where G (z, z) is regular in the vicinity of 0, 0, and G {z, 0) does not vanish identically for all values of z. 64] THIRD THEOREM 99 Next, suppose that the function ^(0, /) where (as before) F{z,z')=f{z,z')-f{Q,Q), vanishes identically for all values of /. Then an argument precisely similar to the preceding argument shews that the function F {z, z) has some power of ^ as a factor. Let this factor be z^ ; then s is a positive integer greater than zero, and it is assumed to be the largest positive integer which allows F (z, z') z~^ to be a regular function of z and z' in the vicinity of 0, 0. The second of the two preceding theorems does not now hold as an expression ior f{z, z'). But if the function F (z, 0) does not vanish identically for all values of z, the first of the preceding theorems does hold as an expression for f{z, z). As before, there are limitations upon the forms of the quantities g^, gm-i, ••• i in particular, 9m = 0, gm-i = 0, . . . , gm-s+i = 0. But the momentarily important result is that f(z,z')-f(0,0) = z^H{z,z'), where H (z, z') is regular in the vicinity of 0, 0, and H {0, z) does not vanish identically for all values of /. Next, again taking F{z,z')^f{z,z')-f{0,Q), suppose that the function F (z, 0) vanishes identically for all values of z and that the function F (0, z') vanishes identically for all values of /. As in the preceding cases, F(z, z') has a factor which is now of the form z^z'^, where s and t are positive integers each greater than zero ; and it is assumed that each of them, independently of one another, is the largest positive integer which allows F{z, z') z~^z'~^ to be a regular function of z and / in the vicinity of 0, 0. Neither of the two theorems already proved now holds as an expression for/ (5, z'). The momentarily important result is that f{z, z) -/(O, 0) = z'z'^I (z, z'), where / {z, z') is regular in the vicinity of 0, 0, while / {z, 0) does not vanish identically for all values of z and I (0, z) does not vanish identically for all values of z. Thus in each of the cases contemplated, we have f{z,z')-f{0,0)=z^z''U{z,z'), where s and t are positive integers that are not simultaneous zeros, and U {z, z) is regular in the vicinity of 0, 0, while neither U {z, 0) nor ^7(0, z') vanishes identically for all values of z or of z' respectively. The alternatives are as follows. 7—2 100 THIRD [CH. IV (a) When JJ (0, 0) is not zero, then, within the sufficiently small domain round 0, 0, we have U{z,z')= U (0, 0) e^^'''\ where T(z, z') is a regular function of z and /, vanishing at 0, 0. Then we have f(z, z') =/(0, 0) + Cz'zHT(''^\ where the constant C is the non-zero value of 11(0, 0). (/S) When U (0, 0) is zero, the conditions attaching to U (z, z) require that TI {z, 0) does not vanish identically for all values of z and that ?7(0, /) does not vanish identically for all values of z'.. As VI iz, 0) does not vanish identically for all values of z and as V {z, z') is a regular function, the first of our two earlier theorems applies to U{z, z) ; we have an expression of the form U (z, z) =^ Ag (z, z') e^^'''\ where -4 is a constant ; g {z, /) is a polynomial in z having, as its coefficients, regular functions of / which vanish with / ; and where R (z, z') is a regular function of z and z' which vanishes when z = and / = 0. Then /(5, z') =/(0, 0) + Az'/'g (z, z') e^^'''\ Also U (0, z') does not vanish identically for all values of /, and U(z, z) is a regular function ; hence the second of our two earlier theorems applies to J] (z, /). We have an expression of the form U{z,z')==Bh{z,z')e^^'''\ where 5 is a constant ; h {z, z) is a polynomial in z' having, as its coefficients, regular functions of z which vanish with z; and where S{z,z') is a regular function of z and / which vanishes when z = Q and / = 0. Then f{z, z') =/(0, 0) + Bz'z'h (z, z') e^^'''\ Summarising these results, we have the theorem : — III. When a function f{z, z') is regular in some domain of 0, 0, and is such that either (i) f(z, 0) — /(O, 0) vanishes identically for all values of z while /(O, z') — /(O, 0) does not vanish identically for all values of z , or (ii) /(O, z') — /(O, 0) vanishes identically for all values of z' while f(z, 0) — f(0, 0) does not vanish identically fur all values of z, or (iii) f (^2,0)— f (0,0) vanishes identically for all values of z and f{0,z') —f (0,0) vanishes identically for all values of z', then expressions for f(z, z') in the immediate vicinity of the place 0, are f{z^ /) =/(0, 0) + AzH'^g (z, z) e^'^'^'', f{z, z) =/(0, 0) + BzH'^h (z, z) e'S''^'^'), 64] THEOREM 101 tuhere s and t are positive integers such that s=0, t >Ofor the first hypothesis; s>0,t = for the second hypothesis ; and 5 > 0, i > for the third hypothesis. The quantities A and B are constants ; the functions R {z, z') and S (z, z) are functions of z and z', each of which is regular in the immediate vicinity of 0, and vanishes luhen z=0 and z' = 0; the function g (z, z') is a polynomial in z of the form z'^ + g,z^-^ + ...+gm, luhere the coefiicients g^, , . . , g^ are functions of z which are regidar in the immediate vicinity of z =0 and vanish with z' ; and the function h (z, z) is a polynomial in z of the form z'''+h,Z''-^ + ...+hn, where the coefficients h-^, ..., hn are functions of z which are regular in the immediate vicinity of z and vanish with z. There is a limiting case when both m and n are zero; the expression for f{z,z') in the immediate vicinity of 0,0 is f{z, z) =/(0, 0) + Gz' z''eT(^''\ where G is a constant, luhile T(z, z) is a function of z and z which is regular in the immediate vicinity of 0, and vanishes when z = and z' =0*. Note. We saw before that, in certain circumstances, both Theorem I and Theorem II are valid, thus providing for the regular function f{z, z') two expressions, which are formally distinct from one another, and must be equivalent to one another. In Theorem III it follows that, in certain circumstances, the regular function/ (2^, z') can have two expressions, which are formally distinct from one another and must be equivalent to one another. In the former case, the two expressions for/(^, /) — /(O, 0) are Gg {z, z) e^ '^' ^'» , Kh (z, z') e^^'' ^'' , where g (z, z') is polynomial in z with coefficients that are regular functions of z' vanishing with z', while h (z, z) is polynomial in z with coefficients that are regular functions of z vanishing with z. Thus h(z,z')~G^ ~^^ where Z is a constant and V{z, z') is a regular function of z and / which vanishes when z=0 and z' = 0; hence g{z,z') = Le'^^''''^h{z,z'), h(z,z')=^je-^i^'^'^g(z,z'). Similar relations hold in the latter case. * This theorem is quite distinct from Weierstrass's second preliminary theorem (p. 141 of his memoir already quoted) for the case n = 2 ; the latter will come hereafter (§ 65). 102 GENERAL [CH. IV It follows that, for a regular function /(2;, z'), when it is not expressed as a power-series valid over a domain round 0, 0, but is expressed for con- sideration in the immediate vicinity of 0, 0, we usually can obtain two different expressions according as ^ or ^' is taken as the variable for simplifying the representation. Each of the expressions is unique in its form ; the two expressions are equivalent to one another. Ex. Consider an ordinary place of a regular function /(z, s'), and let it be 0, ; and take the general power-series for /, in that domain, in the form /(.,0')-/(O, 0) = (aioS+aoi2') + («2oS^+aii2s' + «o23'^) !First, assume that neither ajo nor aoi vanishes. It is not difficult to establish the following results*: — where ^02= 2 (<*02^10^~ ^11^10^01 + ^20 "^Ol )) 6o3 = ^ («03 «10^ - «12 «10^ «01 + ^21 «10 «01^ " «30 «01^) 1 («02^10^~%l%0f*01 + <^20<*01^) (2*20 ^01 ~ ^ll^lo)» Z. _^20 »10-— "5 aio .«30_ 1 ^ ■ aio 2 aio^ Z, _"'30_ 20 — ;"" 9 /Y 2 ' ^02 = ;(«12«10^-«21«10«01+«30«01^) «10 i (flt02<^10^ — '^11 ''^10<3f01 +^20^01 )~o I~^ (^n^l0~^20<^0l) ) which is the expression for /(g, 2') under Theorem I. Similarly, as the expression for f{z, z') under Theorem II, we have /(.,2')-/(0,0) = (aoi/-hOio^-f-C20^2 + e3o.3 + ...)e^lO^ + ^01^' + ^20^' + '"^^'+^02^'^+...^ * The expressions suggest that the theory of invariantive forms can be appUed to the expansions, in all the cases stated. 64] EXAMPLE 103 where ^20= ~~^ ('3^02 %0' ■" 0^11«10<^01 + ^20^01 )> C30 = 3 («30 «01^ - «21 «01^ «10 + «12 aoi «10^ " «03 «10^) ' 4(ao2«io^-aii«io«oi + «2o«oi^) (2ao2«io-<^ii«oi) 7 <^02 ?20 = («21 ''^Ol^ ~ <^12 0^01 <3!io + «03 <^10 ) . OOS _ 1 ^^ '«oi 2aoi2' And it is easy to verify that «10^ + «Ol2' + ^02^'H^03g'^+--- _^(/lO - kw) Z + (/oi - h\) z'+ ... ^ ajos 4- aoi •2^' + <^20 2^ + C3o2^ + . . . Secondly, when a^i vanishes but not ajo, the first expression is eflfective for /(2, 0-/(0,0), but the second is ineffective. When aio vanishes but not aoi, the second expression is eflfective but the first is ineffective. Thirdly, when ajo and aoi both vanish, neither of the expressions is eflfective. Then /(S, z')-f{0, 0)=a202H«ll22' + «022'^ + «30S^ + «2l2^«' + «1222'Ha032'^ + --- ; and we find /(^,^')-/(o,o) :{a2oSH2(«ii/ + 6i2S'2+...) + s'2(ao2 + 6o32'+...)}e where fel2 = 2 {^12 '^20^ ~ <^21 «11 «20 + ^30 (Cfll^ ~ <3^02 <3S2o)} j «20 ^03 = 2 {'^03 «20^ ~ (^21 <*02«20 + ^^30^11 <^02}j Ct20 «'10 — — ) CI20 *01 = ~~~2 (^21 0^20 "~ 0^30 <^ll)) «20 104 GENERAL EXAMPLE [CH. IV We also find /(.,2')-/(0,0) where ^21 = ~~2 i^^i^ ^02" ~ 0!i2 ^H 0^02 + ^03 ('^ll' ~ <^02 '^2o)}j ao2 •^SO^ 2 1^30*02 ~<^12^02^20 + <^03^11^20/j «02 ^10 = -— 2 ('^12'^02 — ''^OS'^^ll)? «02 7 (^03 hi- — , ■ «02 The first expression is effective when ^20 do^s not vanish ; but it is ineffective when a2o does vanish. The second expression is effective when ao2 does not vanish ; but it is ineffective when ao2 does vanish. When both a2o ^^'^ <^02 vanish and when an then does not vanish, another expression must be obtained. In that case, we have /(s, z')-f(0, 0) = anZz' + a3oZ^ + a2iS^z' + ai2Zz'^ + ao3z'^ + ..., ■and then we find that f(z,z')-f {0,0) = {a,oZ^+z2(b2iz' + b22z'^+...) + z{bnz' + h2^^ + ...) + bo3z'' + boiZ'i+...}e^^oz + koiz' + ...^ where «^io — — ) (^30 koi = — ^ (a^i a^Q- — a2i a^o a^Q — an Usq a^Q + an a^Q ), <^30 "^20 = ~~~9 (^30 "^SO ~ 1 '*'40 )) «30 ^■30 = -—5 (<^30^ «60 — Of'30 ^40 «50 + 3 '^0^) j <^30 >^11 = i-10-^01 H {«« - «31^10 - «40^01 - «21 (^20 "i-^lO^) " «11 (-^30 " '^'20^10 + J'^IO^)}, <^30 ^11 = ^11) 021 = ^21 ~^U MO) C>i2 ^ 0^12 — ^11 "^01 ) Oo3 = '^03) 022 = 0^22 ~ 0^12^10 ~ '*21'^'01 ~ <^11 ('^^11 ~ '^'10^01)) There is a corresponding expression iorf{z, z') -/(O, 0), in which / is made the dominating variable ; it has the form /(2,.')-/(0,0) = {a032'H2'2(C2l2 + C2222 + ...)+2'(Cll2 + Ci2s2+...)+C30«3 + C4o2*+...}e^lO^ + ''^l^'+'--, 65] ALTERNATIVE METHOD 105 where ?10 = ; (ai3 ao3 ~ ^12 '^O'l <^03 ~ <^11 ^03 <^05 + ''^11 ^^Ol")) «03 7 <^04 ^01 — — > <^03 <^03 ^03 = r~3 ('''03^ ^06 ~ '^los ao4 «05 + 3 '^04^)5 «03 ^11 = ^10^01 H {'^14~<^13^01 ~ Cf04^10- <^12 (^02~"5^01^)~<^11 ('o3 ~ 'o2^01 + 6 ^01 )/) 0^03 <'30 = «30) C2l = «21~^linO) Cl2 = ai2-«ll^01) C22 = «22~<'^2l'oi~<^12'oi~<^ll ('ll ~ 'lo'oi)) The first of these is effective when aao does not vanish. The second is effective when aos does not vanish. The general form of expression for f{z, z')-fiO, 0), when both /(O, z')-f{0, 0) and f{z, 0)-/(0, 0) vanish identically, has been indicated. It then is possible to isolate a factor z^z'\ where _ fiz,z')-f{0,0)=z'z'tf{z,z'), such that both f{z, 0) and /(O, z') do not vanish identically; and expressions, similar to those which precede, can be obtained for f{z, z'). 65. When the function i^(^, 0), =/(^, 0) -/(O, 0), vanishes for all values of z, another method of proceeding was given by Weierstrass*. It was devised for functions of n variables (when n > 2) and some method is needed for them other than the method for functions of two variables, because with n variables it is not generally possible to extract an aggregate factor such as ^*/* from the function corresponding to /(z, z') —f{0, 0). Applied to functions of two variables, the Weierstrass method is as follows. In the double-series expansion of f{z, z) —f{^, 0), valid in a domain round 0, 0, let the terms be gathered together into groups, each group con- taining all the terms of the same order in z and z combined ; and suppose that the group of lowest order is of order fx, so that we have fiz, z') -/(O, 0) = {z, z\ + {z, z'U, + .... Change the variables from z and z to u and u' by relations of the form z = a.u-{- ^u', z' = 'yu + hu, where a, yS, 7, S are constants such that ah — /37 is not zero, so that u and u are new independent variables. Then f{z, z') -/(O, 0) becomes a regular * See p. 140 of his memoir already quoted. 106 METHOD FOR [CH. IV function of u and w', say G (u, u'), the lowest terms in which are of order /x ; and {u, 0) = (a, 7)^ W- + (a, 7V+1 ^/'^+l + . . . , so that, choosing {a, 7)^ to be different from zero, G{u, 0) does not vanish for all values of u. The first of the preceding theorems can therefore be applied to G{u, u')', the result is of the form G {u, u') = («, 7)^ {u>^ + W^-^ g, (!*') + . . . + ^^ ill')] / ("' '''\ where (a, 7)^ is the non- vanishing coefficient, g^, ..., ^^ are regular functions of u' which vanish with ii! , and I{xi, u') is a regular function of u and u which vanishes when u = and u' = 0; moreover, as the lowest terms in G(u, u') are of dimensions /x, the regular series for gr{u') begins with a term in u''^, for ?• = 1, ..., jx. When retransformation to the original variables z and z' is effected, we have f{z,z')-f{0,i)) = G{u,u') = [{z, z%+{z, z']^+, + ...]e^^'^''), where J{z, z) is a regular function of z and z which vanishes when 2^ = and / = 0; and by expanding e'^(^'^) so as to have the complete series for the new expression, we have \z,£]^, = {z,z')^, so that, as is to be expected, the first term in g (z, z'), where /(^,/)-/(0,0) = ^(^,/)/(^'^'), is the aggregate (z, z')^ in the original double series for/(2', z')—f{0, 0). Note 1. It may be pointed out that the preceding method is effective, even if f{z, 0) -/(O, 0) does not vanish. Thus for a function it might happen that, in the regular function /(2', 0) — /(O, 0) when it does not vanish for all values of z identically, the term of lowest order is Az'^, while, in f{z, z) —fiO, 0), the terms of lowest order are of dimensions less than n. (As a matter of fact, each of these terms of lowest order will then contain some positive power of z' as a factor). The application of the method will then lead to an expression of the preceding form. Note 2. In the method, the limitations upon a, ^, 7, S are merely ex- clusive; they are aa-/37=fO, (a,7).=l=0. Thus a certain amount of arbitrary element will appear in the result; by varying these constants a, ^, 7, 8, different expressions will be obtained which are equivalent to one another. 65] MORE THAN TWO VARIABLES 107 Ex. 1. Consider the function* the unexpressed terms being of order higher than 4. We take z = u, z' = u + u', so that fz= u^ + mi' + } (2w3 + Su^u' + Suu"'^ + u'^) + Jj (2u* + 4u^ ?(' + 6lf 2 2<'2 + 4m?('3 + ?4'*) + . . . . This must be equal to where ffi — kiu' + k^iC'^ + k^%i!^ + . . ., Expanding, and equating coefficients, we find ^1 = 1, A'2 = ^-, ^"3=~1T6» •••' ^2 = 0, ^3 = J, ^4=7V, ..•; «2 = T8? ^2 = 0, C2=ifV; and thus the expression for our function becomes g (w, ?(') e ^"' " ', where and When we retransform to the variables z and z' bj the relations rt =■ 2, u' = z' — 2, the terms of the lowest order in g (?« , %') become zz', as is to be expected. But the completely retransformed new expression for / is less effective than the original expression ; and the discussion of / in the vicinity of 0, is more effectively made in connection with the expression in terms of z and z'. Ex. 2. Obtain an expression for the function in the preceding example, when the transformed variables are given by the relations Z = it-{-all', z' = ic + ^u', where the constants a and 13 are unequal ; and prove that, when retransformation takes place, the terms of the first order in /(«, u') become z + z'. This last method of Weierstrass has been outlined, because of its importance when the number of variables is greater than two. When the number of variables is equal to two, the general case for which it was devised falls more simply under the comprehensive results of Theorem III. We may therefore summarise the results of the whole investigation briefly as follows. Whatever be the detailed form of any function f{z, z), regular in a domain round 0, 0, its general characteristic expression in the immediate vicinity of 0, is f{z, z) -f(0, 0) = ^z''P (z, z) /("' ''\ * The expansions under Theorem I and Theorem II arise as special cases of the result given above, p. 104. 108 LEVEL [CH. IV where I {z, z') is a function of z and / which is regular in the immediate vicinity of 0, and vanishes when z = ^ and z' = 0. The quantities s and t are positive integers, which may be zero separately or together. When either of these integers is zero, or when both of them are zero, P(0, 0) can be different from zero for special functions ; for all other functions, P {z, z') is polynomial in one of its variables, the coeflficients of the powers of which are regular functions of the other variable within a limited domain, each such coefficient vanishing when that other variable vanishes. Level values of a regular function. 66. One immediate deduction of substantial importance can be made from the expression for f{z, z') which has just been obtained, viz. F(z, z') =f{z, z') -f(0, 0) = z'z'' A (z, z') e^ (^' ''\ as to the places where f{z, z) acquires the same value as at 0, 0. When /(O, 0) vanishes, we shall call the place a zero for f{z, z'). When /(O, 0) does not vanish, we shall call the value /(O, 0) a level value for all the places z, z' such that f{z, z')=f(0, 0); all these places are therefore zeros of F(z,z'). As B (z, z') is a regular function of z, z' within a limited domain of 0, 0, the quantity e (^' ^ ' cannot vanish at any place in the domain. Consequently the zero-places of F{z, z') within the domain are given by three possible sets. When the positive integer s does not vanish, zero-places of F{z, z') arise when z = 0, z' = any value within the domain. When the positive integer t does not vanish, zero-places of F{z, z) arise when z = any value within the domain, / = 0. When A {z, z') is not merely the constant A (0, 0), all the places in the domain such that A {z, z') = are zero-places for F{z, z'). As regards the first set, we obtain an unlimited number of zero-places of F{z,z') within the domain of 0, 0; they constitute a continuous two- ,,^\v dimensional aggregate, the continuity being associated with the plane of / alone. As regards the second set, we obtain also an unlimited number of zero- places of F{z, z') within the domain of 0, ; they too constitute a continuous two-dimensional aggregate, the continuity now being associated with the plane of z alone. For the third set, there is no additional zero-place for F{z, z'), if A (0, 0) is a non-vanishing constant ; in that event, either s, or t, or both s and t, must be different from zero. When A (0, 0) does vanish, the function 6Q] PLACES 109 A (z, z') either is polynomial in z and (usually) transcendental in /, or is polynomial in / and (usually) transcendental in z ; and these alternatives are not mutually exclusive. In the former case, for any assumed value of z' within the domain, there is a limited number (equal to the polynomial degree of -4) of values of z, which vanish with z' and usually are trans- cendental functions of z! ; hence, taking a succession of continuous values of £ in the domain, we have, with each value of z , a limited number of associated values of z. All these places taken together constitute a continuous two- dimensional aggregate ; the continuity now is associated with both planes, each value of / having a definite value of 2^ or a limited number of definite values of z associated with it, all within the assigned domain of 0, 0. Similarly, in the latter case, as regards A (z, z') ; the same result holds when the appropriate interchange of z and z' is made in the statement ; and the two-dimensional aggregate is unaltered. Ex. 1. Among the simplest examples that occur, are those when A {z, z') can be expressed in a form az + P{z'), where a is a constant and P (2') is a regular function of z' given by P{z') = bz' + cz'^+..., b not being zero. Then A (z, z'\ with an appropriate change in B {z, z') which is the function in the exponential, can also be expressed in the form bz' + R{z), where the regular function R (z) is given by It(z) = az+Cz'^+..., with suitable values of the constants C, .... The zero- values are given by the two- dimensional aggregate ~az = P{z'), -bz'==R{z). The result is the generalisation of the known property whereby, in the vicinity of a real non-singular point ^, rj on an analytical curve /(.r, ?/) = 0, we have the linear term m. Piy-rj) combined with .r-|, and the linear term in R'{x-^) combined with y - T], give the tangent to the curve at the real ordinary point ^, rj on the curve. Ex. 2. In both cases that arise out of the alternative forms of A, the actual determi- nation of the set of values of z in terms of / (or of the set of values of z' in terms of z) can be made as in Puiseux's theory of the algebraical equation f(w, s) = 0, the governing terms being selected by the use of Newton's parallelogram. For example, in the case of the zeros of the function .f (^) 2') - / (0) 0) = an ^2' + «30 ^H «2i ^^^' + ai2Zz'^ + aosz'^ + ... within a small domain round 0, 0, we find three values for z in terms of z', viz. «u\i ,1 1 , V / \ «lA8 ,1 1 , X , I f— _^ -'2 + —1 (^12 «03 - ail «04) 2'H . 11 0^11 110 EXPRESSION NEAR A LEVEL PLACE [CH. IV and there are three corresponding values for z' in terms of z, viz. , / «ii\i 1 1 , , \ z= -— 23 + ^-^(ao4aii-ai2ao3)2 + ... , / a,\\\\ 1 1 , , z = -l - -- 22-j- — ^(ao4aii-ai2ao3)2+... \ «t03/ ^'3!o3 If ^30 is zero, the first two series in the earlier pair are not valid ; if ao3 is zero, the first two series in the later pair are not valid. If all the coefficients a„o vanish so that /(s, 0)— /(O, 0) vanishes for all values of 2, only the third expression in the earlier pair survives. If the first coefficient a„o, which does not vanish, is a,.o, there is a set of ?• - 1 expansions in a cycle corresponding to the above two which exist when a^ does not vanish. And so on, for the respective cases. Ex. 3. Quite generally, it may be stated that the detailed determination of the behaviour of F {z., z') in the vicinity of 0, 0, so as to obtain the nature of its zeros as well as the actual positions of its zero-places, has a close resemblance to the method of proceeding in the consideration of an equation f{w, s)=0, which is algebraical both in w and in z, and in the determination of the associated Riemann surface*. 67. All the results relating to the zeros of F{z, z) can apply, in descriptive range, to a determinate finite level value (say a) of a uniform function f{z, z) in a domain vi^here it is regular. Let a, a' be a place where f acquires the value a ; so that f{a, a') = a. For places a + Z, a' + Z' near a, a' within the domain of a, a', we have Az,z')=f{a + Z,a' + Z') =f(a,a') + ^^CmnZ^Z\ that is, f{z,z')-a=^^^C,nnZ^^Z\ Thus the places within the domain of a, a' where/ acquires the level value a are given by the zeros of the double series which itself vanishes when Z=0, Z' = 0. Hence the level places which give a determinate finite value a to a function /"(2', z') form a continuous aggregate within the domain of any one such level place. Manifestly, as we are dealing with properties of a uniform function of f, which is regular within the domain of an ordinary place, the values of/ must be finite (for poles do not occur within such a domain) and they must be determinate (for singularities, whether unessential or essential, do not occur within such a domain). The behaviour of a function in the vicinity of a pole and in the vicinity of an unessential singularity will be discussed separately. * For this subject, see Chapter viii of my Theory of Functions for the discussion of the algebraical equation and Chapter xv for the construction of the associated Eiemann surface, Reference should also be made to the early chapters of Baker's Abelian Functions. 68] ORDER OF A ZERO 111 68. Not because of any immediate importance for a single function of two variables but mainly because of the need of estimating the multiplicity of a common zero-place or a common level-place of two functions of two variables, it is worth while assigning integers that shall represent the orders, in z and / respectively, of the zero of f{z, z) —f{a, a') at the place {a, a'). By the preceding proposition, for a place z = a + u, z' = a + u' in the im- mediate vicinity of a, a', we have f{z, z') —/{a, a) = uUi* G (u, u'), where G is regular in the domain, and the integers s and t can be chosen so that G (u, 0) does not vanish for all values of u and G (0, u') does not vanish for all values of u'. The positive integers s and t can be zero, either separately or together. As G (ii, 0) does not vanish for all values of u, there exists a series Q (u, u') = u:»' + u^-' q, (u) + . . . -f g,„ (ti), where qi(u'), ..., qm(u') are regular functions of u' vanishing with u, such that G{u, u') = KQ{ii,u')e^^''''''\ where K is a, constant and Q (u, u') is a regular function of u and ii vanishing with u and u'. Thus for any small value of iif, there are m small values of u, making G (it, u') zero. As G (0, u') does not vanish for all values of u, there exists a series R (u, u') = u'« -I- w''»-i r, (u) + ... + rn (ii), where r^ (ii), ..., rn {u) are regular functions of u vanishing with u, such that where X is a constant and R {u, u) is a regular function of u and u' vanishing with u and w'. Thus for any small value of u, there are n small values of u', making G(u, u') zero. In both of these cases, G(u, u) vanishes when u = 0, m' = 0; and then neither of the integers m and n is zero. There remains a third case, when G (0, 0) is not zero ; then where / {u, u') is a regular function of u and u' vanishing when u=0 and li' = 0. Thus no small values of u and u' make G (u, u') vanish ; and then both of the integers m and n are zero. With these explanations, we define the orders of the zero of the function /(^^ ^') -/(a. a') at a, a' as s -I- m for the variable z and as t + n for the variable z . But it must be pointed out that the zero of the function at a, a' is not an isolated 112 DIVISIBILITY OF [CH. IV zero, for it is only a place in a continuous aggregate of zeros; still, a settlement of an order in each variable at a place a, a' is convenient as a preliminary to the settlement of the multiple order (Chap, vii) of such a place when it is a simultaneous and isolated zero of two functions considered together. Relative divisibility of two regular functions near a common zero. 69. Before proceeding to obtain the expression of any uniform analytic function in the vicinity of a singularity, it is important to consider the behaviour of two uniform functions /(^r, z) and g{z, z) simultaneously, both being regular within a common domain which will be taken round 0, 0. First, suppose that g (0, 0) is not zero ; then we have seen that a uniform function S {z, z) exists, which vanishes when z = ^ and ^' = and is regular in a domain in the immediate vicinity of 0, 0, and is such that for that domain. Also, we know that we can take /(^, /) =/(0, 0) + ^<^(^, z)z'z^e^^'^ '''\ where .9 and t are non-negative integers, (/> {z, z') is polynomial in 'z and regular in z, and R {z, z') is a uniform function of z and z' which vanishes when z=0 and / = and is regular in a domain in the immediate vicinity of 0, 0. Consequently f(^, z') 1 g{z,z') ^(0,0) {/(O, 0) + ^ ^/(O, 0) ^^(,, ,') A__ ^'^^^s^'t^Riz, z') -S{z, z') _ ^(0,0)"^ ^^(0,0)'^'^' ^ ^ "^ The right-hand side, whether /(O, 0) vanishes or not, can be expressed as a regular double series U {z, z); that is, f{z,z) , -7 7^ = Uiz, Z ). g(z,z) ^ When a uniform function f{z, z') is expressed as a double series F (z, z'), and another uniform function g {z, z') is expressed also as a double series Q {z, z'), and when a third uniform function U{z, z) exists such that all the functions being regular in a domain round 0, 0, we say, following^ Weierstrass*, that the series P{z, z) is divisible by the series Q{z, z'). * Ges. Werke, t. ii, p. 142. 70] ONE FUNCTION BY ANOTHER 113 It therefore follows that, when g (0, 0) is not zero, the regular function f {z, z) is divisible by the regular function g (z, z), the regularity of both functions extending over a domain round 0, 0; and the result is true whether f(0, 0) is zero or is not zero. 70. Next, suppose that g (0, 0) is zero ; then we know that we can take g{z,z') = Bz-z'^e^('''')x{^,^'), where i5 is a constant; cr and r are non-negative integers; T{z,z') is a function of z and z', regular in the immediate vicinity of 0, and vanishing when z = and / = ; and x {^, ^') is a function which is a polynomial in z having functions of z' for its coefficients, these coefficients being regular in the immediate vicinity of /= and vanishing when z' = 0. The form off{z, z) is the same as before. It at once follows that, when /(O, 0) is not zero, we cannot express g {^, z) in the form of a regular function; in that case, the function /(^^, z') is not divisible by g (z, z'). But when/(0, 0) is zero, as also is g (0, 0) under the present hypothesis, then we have f{z, z) ^ Az'z'' ^') Now B (z, z') — T (z, z) is regular in the immediate vicinity of 0, and vanishes when z = ^ and / = ; hence the exponential factor in the last expression admits the divisibility oi f {z, z') by g {z, z'\ Also this divisibility is admitted, so far as powers of z are concerned, if s ^ o- and, so far as powers of / are concerned, if f^r. There remains therefore the divisibility of (f)(z,z') by x(^> ^')> where (for the present purpose) we shall assume thaf both ^ (z, z') and x (^' ^') ^^^ polynomials in z the coefficients in which are regular functions of / in the immediate vicinity of / = and vanish when z = 0. Manifestly the degree of ^ {z, z') in z cannot be less than that of X (z, z), if divisibility is to be possible ; accordingly, we suppose that (^, /) = ^^ + ^™-i ^Ti + . . . + 5r^, X {z, z') = z^ + z^'-'h, +...+hn, where m^n, and all the coefficients gi,-..,gm, K,-..,hn are regular functions of z in the immediate vicinity of / = and vanish when / = 0. When <^ {z, z') is divisible by x {z, z'), the quotient is manifestly of the form nin—n 1 nnii—n—i h a_ A- h /6 -t z /t-i -f- . . . -(- Hinn,—n> - P. 8 114 DIVISIBILITY [CH. IV where the coefficients ki, ..., km-n are functions of /. Also 9i = K + h, g2 = h2 + hih + h, g,f — hf "T fiy^jKi -+- 1IY—2K2 + . . . , From the first, it follows that the function ki is regular and vanishes when 2=0; from the second, that the function k.2 is regular and vanishes when 2;' — 0; and so on, in succession from the first ni — n of these relations. Also all the relations are to be satisfied, by appropriate values of k^, ..., kjn^n, for all values of z in the immediate vicinity of / = 0. The conditions, necessary and sufficient to satisfy the last requirement, are that, when we form the n independent determinants each of m — n rows and columns from the array 1 , h, , h , ..., , , ..., , , 1 , K , ..., , , ..., , , , , ..., , ..., K , K-i , , , ..., , ..., , K each of these n determinants must vanish identically for all such values of z. Thus there are n conditions. The form of the conditions should, however, be noted. As all the functions g and h are regular functions of z' in the immediate vicinity of z' = and vanish when / = 0, each of the n deter- minants is also a regular function of / in the immediate vicinity of z =0 and vanishes when / = 0. Each determinant is to vanish identically for all values of z' in the range round z' = 0; and therefore every coefficient, in the power-series which is the 'expression of the determinant, must vanish. Thus in practice, when the power-series are infinite, the number of relations among the constants would be infinite for each of the conditions; the arithmetic process could not be carried out in general*. But the n analytical conditions among the functions would still remain, in the form of determinants that are to vanish identically. Thus, in particular, the conditions, that the function should be divisible by the function z^ + zhi-\-h2, are that the two independent determinants from the array- s'! -^H, gi-h, gz 1 , Ai , ^2 * In particular cases, it might be feasible, e.g. when there are known scales of relation governing all the coefficients. 71] RELATIVE REDUCIBILITY 115 shall vanish ideutically. When the two conditions are satisfied, the quotient is The general argument shews that the function 5^3/^2 is to be regular and to vanish with s'; a limit upon the orders of the lowest powers of 2' in h^ and g^ is thereby imposed. Relative reducihility of functions. > 71. Further, it is important to discover whether, even in the case when a function {z, z') is not actually divisible by a function 'x^ {z, z'), both being of the- foregoing type, each of them is actually divisible by a function -^{z, z') also of the same type: that is to say, if '>^{z, z) exists, it is to be a polynomial in z the coefficients of which are regular functions of z in the immediate vicinity oi z' =0 and vanish when z' = 0. A method of determining the fact is as follows. Both ^ {z, z) and 'x^ (z, z') must vanish for all the places where ^|r (z, z') vanishes, if ^jr exists. We therefore regard 4>(z,z')^0, x(^,^') = o, as two simultaneous algebraical equations in z. We eliminate z between these two equations, adopting Sylvester's dialytic process. The resultant is a determinant of m + n rows and columns, every constituent in the deter- minant (other than the zero constituents) being divisible by / ; and therefore this resultant is of the form where /* is a positive integer not less than the smaller of the two integers m and n, and where @ (/) is a regular function of z in the immediate vicinity of / = 0, when it is not an evanescent function. When @ {z') does not become evanescent, the values of z different from 2r' = which make the resultant vanish are given by the equation @ {z') = ; and these values of z form a discrete and not a continuous succession. In that event, for each such value of z and for the specially associated values of z, both ^ and % vanish. But their simultaneous zero values are limited to these isolated places ; there is no function -v/r {z, z) possessing a continuous aggregate of zero-places in the vicinity of 0, 0. When © (/) is evanescent, the functions <^{z,z') and %(2^, ^') become zero together, not merely at the place 0, 0, but at all the continuous aggregate of places where some function -v/r {z, z), as yet unknown, vanishes ; for there is no equation @ (/) = limiting the values of / and requiring associated values of z. 116 RELATIVE REDUCIBILITY OF [CH. IV In the latter case, cf> (z, z') and % (z, z') possess a common factor -v/r (z, z), which' necessarily will be a polynomial in z of degree less than n ; and the polynomial will have functions of / for its coefficients, all of which are regular in the immediate vicinity of 2;' = and vanish when / = 0. Let -^/r (^, /) = ^^ + ^^-Ui + . . . + A;^ ; as i/r is a factor of by hypothesis, and also a factor of % by hypothesis, our earlier analysis shews that (as already stated) k-^, ...,kp are regular functions of / in the immediate vicinity of / = and vanish when z = 0. Accordingly, let (z, z) _ ^ ^ y}r{z, z ) yiz, z') „ „ 7, / = ^"-^ + Z^-P-' B, + ...+ Hn-p , Y{z, z) ^ where all the coefficients Gi, ..., Gm-p, Hi, ...,Hn-p are regular functions of z' in the immediate vicinity of / = and vanish when z = 0. Consequently the relation (^m + ^m-x ^r, + . . . + gn,) (z^-P + Z^'P-^ H, + . . . + H^-p) = (Z"" + Z""-' h^ + ...+K) (Z'^-P + Z^-P-' G,+ ...+ Grr,-p) must be satisfied identically for all values of z and z within the im- mediate vicinity of 0, 0, the common value of the equal expressions being (j) (z, z) 'x^ (z, z') -^ yjr (z, z'). Equating the coefficients of the same powers of z in the expressions, we have m + n — p relations, linear in the (n ~p) + (m—p) unknown functions ifj, ..., Hn-p, Gj, ..., Gm-p- When these are eliminated determinantally, we have m + n — p — (n — p) — {m — p), that is, we have p, equations in / which, being satisfied for all values of z , must become evanescent. The conditions for this evanescence, which are thence derived as existing between the coefficients of <^ and ^, are the conditions necessary and sufficient for the existence of -v/r {z, z'). When these conditions are satisfied, the actual expression of i|r (z, z') can be obtained by constructing the algebraical greatest common measure of <^ {z, z') and 'x, (^' z'), regarded as polynomials in z. We thus have analytical tests determining whether two functions ^ {z, z) and % {z, z'), each polynomial in z and having for the coefficients of powers, of z regular functions of / which vanish when z = 0, are or are not divisible by a common factor of the same type as themselves. To these tests, the same remark applies as in 1 70 ; each condition usually would, in practice with infinite power-series, require an infinite number of arithmetical relations among the constants. Still, the analytical tests remain in the form indicated. When the tests are satisfied, the two functions are said to be relatively reducible ; each of them is said to be reducible by itself. 71] TWO FUNCTIONS 117 Note 1. The processes connected with finding the conditions are those connected with constructing eliminants in algebra. Thus, in order that the functions z" + g^z^ + g^z^ + gzz + g^, z^ + li^z^ + A, should have a common factor linear in z, all the coefficients of powers of z' in the final expansion of the determinant 9i-f^i, 1 , 1, 0, 92 - k, 9i> k, 1 , 9s > 92, h, K 1 9i > 9z, , K k , 9i, , 0, A, must vanish identically. Note 2. In the preceding investigations, we are concerned with the possession by ^ {z, z') and ^ {z, z) of a common factor of the same type as themselves ; that is to say, (/> {z, z'\ y^ (z, z'), and the common factor (if it exists) are polynomial in z. We are not concerned with the comparison of expressions (f>iz,z') and cf>{z, z')e^^'' ''\ where R (z, z') is regular in the immediate vicinity of 0, and vanishes when z = and z' = 0; the latter expression, when expressed in a double series, is no longer polynomial in z. The case, when R (z, z') can be such as to make the second expression polynomial in z alone, has already been discussed (§ 63). Ex. When two functions (ao', ai', a2'Jz, z'f + {bo', b^', b{, b{\z, z'f +..., possess a common factor of the type z + R{z'), where R{z') is regular in the immediate vicinity of z' and vanishes when ^ = 0, we can approximate to its expression as follows. (The algebra will illustrate the distinction between the finite number of analytical tests and the infinite number of arithmetical relations between the constants; the latter, of course, cannot be set out explicitly.) The first function is expressed (§ 64) in the form {ao2' + s(«i2' + a23'2 + ...) + a22'H/33/3 + ...}e^o^ + \i2' + ...^ where 6*0 1 a 02 = — (ao &2 - «2 bo) - -\ (ao 61 - «i 60), «o «0 ag = ^3-— 2(«o&i-ai&o), «0 118 EXAMPLE [CH. IV and so on ; and the second function is expressed in the similar form where V = 3' V=T7^(«o'V-«i'V), •••, 1 a ' 02' = —, {ao'b^' - a^'W) - -\ (ao'W - a^'bo'), /33' = &3' - ;7| («o'&i' - «i'V), and so on. We then must have the condition or conditions that and ao'z'^ + z {ai'z' + a/s's +...) + a^'z'^ + ^s'z'^ + . . . should possess a common factor of the type z + R{z'), say Z + yiz' + y2z'^+.... Let these two expressions, which are quadratic in 0, be denoted by do^^ + zii + ^i, aoZ^ + zr]i + r]2. They both will vanish, if they possess a common factor linear in z and if that factor vanishes. When they vanish, we have aoZ^+z^i + ^2 = 0, ao'z^+zrji + r]2 = 0, simultaneously ; and therefore the relations z^ _ z _ 1 iiri2 — $2Vl l2«o'-'72«0 »?l«0-^l«o' will be satisfied for the value of z, in terms of z', which makes the common factor vanish. Thus we must have (|]'72-^2'?i)('7i«o-|i«o') = (l2«o'-'72«o)^ satisfied identically for all values of z' ; and the value of z, which would make the common factor vanish, is given by l2«o' — T?2«0 j/iao-liao'* Now iinz - ^im=^^ {(«i«2' - «i'«2) + («i^3' - «i'/33 + 02^2' - a2'a2) 2' + .-.}, l2ao' - '72«0 = Z"^ {(«o'«2 - «2'«o) + («o'/33 - ^o/^s') 2' + • • •}) J?! «o - ^1 «o' = ^' {«o «i' - «i «o' + («o 02' - «o'a2) 2' + . . . } ; and therefore, disregarding the factor /*, the expression {ao'oa - a2'ao+ ("o'/Ss - ^o/Ss') 2' + . ■ . }^ - {(aia2' - a/ag) + (ai/33' - aj'^g + a2a2 - a{a^ z' + . . .} {(ao«i' - «i «o') + («oa2' - «o'a2) «' + ...} must vanish identically, for all values of z'. Let the expression be denoted by then we must have as the arithmetical relations between the constants. 72] EXPRESSION NEAR A POLE 119 Also the value of s, which makes the common factor vanish, is s= -f — , '?i<^o-gi«o _ , ao'a2 — «2'o5o + (o^o'i^s — o^o ^3') 2' + . . . ao«i' — (^1 o'o' + (^0 "2' ~ '*o'«2) 2' + . • - ' Consequently, when all the relations between the constants are satisfied, the common factor is 2 + yi2' + 722'^ + ---5 where ^0 ^^2 — ^0^2 ^0 ^1 — (^0^1 (ao'«2 - «2'«o) («o 02' - «o'a2) - («oa/ - «1 «oO («o'^3 - ^ofeO and f30 on. It is clear that, in the absence of general laws giving relations between the coefficients in each of the two functions, we cannot set out the aggregate of relations (7=0 and the aggregate of constants y. Expressions of functions near a pole or an accidental singularity. 72. The non-ordinary places of a uniform function have been sorted into three classes, the poles (or accidental singularities of the first kind), the unessential singularities (or accidental singularities of the second kind), and the essential singularities. The simplest of these, in their analytical character and in their effect upon the function, are the poles. Let p, p' be a pole of a uniform function f{z,2!); then, after the definition, some series of positive powers of z — p, z ~p' exists, say F(z—p, z—p'), which is regular in the immediate vicinity of Pi p' and vanishes when z=p and / =p', and is such that the product f{z,z')F{z-p,z' -p') is regular in the vicinity of p, p and does not vanish when z — p, z' = p\ Thus the function f(z,z') acquires a unique infinite value at a pole; that is, the infinite value is acquired no matter by what laws of variation the variables z and / tend towards, and ultimately reach, the place p, p. Further, the pole-annulling factor F {z — p, z — p') is not unique ; a factor F{z-p,z'-p)e^^'-P'''-P'\ where R{z—p,z'—p') is any regular function of z —p and / —p', would have the same effect. All such factors we shall (for the present purpose) regard as equivalent to one another ; they can be represented by i'' (^ — p, z — p). Moreover, there cannot be more than one such representative factor for f{z, z) at a pole ; if there were two, say F{z —p, z' - p') and G{z—p, zf —p), we should have f{z, z')F{z—p, /— ^')=regular function, not vanishing when z=p and z'=p'\ f{z,^)0{z-p,z'-p)= 120 EXPRESSION NEAR AN [CH. IV and therefore p, p would be an ordinary non-zero place for the quotient F {z-p, z -p') G(2—p, z' -p')' which is impossible unless F is divisible by G, and it would be an ordinary non-zero place for the reciprocal of this function, which is impossible unless G is divisible by F. Hence, denoting the representative factor by F, we have f{z, z')F(z-p, z' -p') = A-oo + h,{z-p) + K{z' -p) + ..., the series on the right-hand side being a regular function in a domain of" p, p' ; and therefore 1 _ F{z-p,z'-p) f{z,z') A;oo + /::io(^-p) + A;oi(/-p')4- ... = a regular function (§ 69) of z^ and z in a domain of p, p', vanishing when z = p, z'=p'. It therefore follows that a pole oi f{z, z) is a zero of -j-. 7-, so that the place p, p' is an ordinary place for the function j- -p.. Hence, in the vicinity of a pole of f{z, z'\ it is convenient to consider the reciprocal function, say and then the behaviour of f{z, z) in the vicinity of the pole p, p' can be described by the behaviour of <^ {z, z') which is regular in the vicinity of its zero there. Moreover, any zero of f{z, z) in a domain of p, p is a pole of ^ {z, z') ; hence the domain of p, p, within which <^ {z, z') is regular, does not extend so far as to include any zero oif{z, z). As ^ {z, z) is regular in this domain of p, p, and as it vanishes at p, p^ it has an unlimited number of zero-values in the immediate vicinity of p, p', and these occur at places forming a continuous two-dimensional aggregate that includes p, p. Hence in the immediate vicinity of any pole of a uniform analytic function, there is an unlimited number of poles form/ing a continuous two-dimensional aggregate that includes the given pole. Further, we have definite integers as the orders of the zero of (f> (z, z') in the two varia];)les at p, p', the integer being derived from the equivalent expressions of (f) (z, z) in the immediate vicinity of p, p ; these integers will be taken as the orders of the pole of f{z, z) in the two variables at p, p . Cor. Manifestly, a pole of f(z, z) of any order is a pole of f{z, z') — a of the same order, where I a I is finite. 73] UNESSENTIAL SINGULARITY 121 73. An unessential singularity (an accidental singularity of the second kind, to use Weierstrass's fuller phrase) of a uniform function f{z, z') at a place s, s is defined by the property that there exists a power-series F{z — s,z' — s'), which is a regular function of z and / in the immediate vicinity of s, s' and vanishes at s, s', and is such that the product f{z,z')F{z-s,z'-s') is a regular function in the immediate vicinity of s, s, and vanishes at s, s. Let this latter regular function be denoted hj H {z — s, z — s'). No generality is lost by assuming that the functions F and H have no common factor vanishing when z = s, z' = s'. We then have a fractional expression for /, viz, . ,^ H{z-s,z' - s) J^'^^^- F{z-s,z'-s')- As in the case of a pole of f(z, z') at p,p', the function F{z—p,z'—p') was representative and unique, so here each of the functions H (z — s, z' — s') and F {z — s, z' — s') is representative and unique, when H and F have no common factor vanishing when z = s, z' = s'. The functions H and F can of course have any number of exponential factors, each exponent being a regular function of z — s, z' — s' ; but no factor of that type affects the characteristic variations of/ in the immediate vicinity of that place. Thus, in our expression for f(z, z'), we can regard the representative functions H and F as unique. To consider the behaviour of / at, and near, the accidental singularity, write z — s = (T, z' — s' = a' ; then we have expressions of the form H(z-s,z- s') = ^o-"*(r''«' (o-^ -V a^-' h W) + . . . + /i; (o"')} e^^*"' '''\ F {z-s,z' - s') = Da^' o-''^' [a^ + a^-^ f^ (cr') + . . . 4-/^ {a')] e^ ^"^ "'l where F and D are constants : m, m', n, n are positive integers, each zero in the simplest cases : I and k are positive integers, each greater than zero in the simplest cases; h^, ...,hi,f^, ...,fk are regular functions of a' in the immediate vicinity of cr' = and vanish with a' ; and H, F are regular functions of a and a in the immediate vicinity of o- = 0, a = and vanish with a and a, so that neither H nor F can acquire a zero value or an infinite value from the factors e^ and e^. Moreover, H and F are devoid of any common factor : so that either m or n (or both) must be zero, and m' or n (or both) must be zero. Also a' + o-^-i h (a') +...+hi (a-'), a^ + cr^-'f, (o"') + . . . +fk ( Kn are themselves polynomials in z. Then we at once have the results : — (i) every finite place is ordinary for p (z, z) ; (ii) with every finite value z' , that is not a zero of t,^, can be associated n finite values of z, such that each of the n places thus constituted is a zero for 'p (z, z'), repetition of values of z causing multiplicity of zero-places for p {z, z') ; (iii) with every finite value z', that is a zero of fo and is such that ^r (r > 0) is the first coefficient of powers of ^ in p (z, z') which does not vanish, can be associated n — r finite values of z, such that each of the n—r places thus constituted is a zero for p (z, z') ; (iv) the poles of p {z, z') are given by infinite values of ] ^ | and finite values of / other than the roots of ^o, and by infinite values of I / i and finite values of z other than the roots of the coefficient * Both theorems were enunciated by Weierstrass for n variables, but without proof ; references will be given later. 78] RATIONAL FUNCTIONS 125 of the highest power of / in p {z, z) arranged in powers of z, and by infinite values oi\z\ and of i / | ; (v) the unessential singularities of p (z, z'\ if any, are given by infinite values of 1 2; I and by the roots of fo) but each such place is an unessential singularity only if other conditions are satisfied ; and similarly for infinite values of \z' \ and by the finite values of z excepted in (iv), but each such place is an unessential singularity only if other conditions are satisfied : so that, in general, p (z, /) has no unessential singularities ; and (vi) there are no essential singularities of p (z, z). 77. In the next place, consider an irreducible rational function of z and z', say where p {z, z) and q {z, z) are polynomials in z and /, P{Z,Z') = ^0^^ +^,Z''-' +...+^n, q {Z, Z) = 97o^"' + ^1^'""' + . . . + ^m, while ^0, •••, ^11, Vo, •••, Vm are polynomials in / alone. Then it is easy to infer the following results : — (i) every finite place, that is not a zero of q(z, z'), is ordinary for B{z,z'); (ii) every zero of p (z, z'), that is not a zero of q (z, z), is a zero of R{z,z'); (iii) every zero of q (z, z'), that is not a zero of p (z, z), is a pole of R(z,z'); (iv) every place, that is a simultaneous zero of p (z, z') and of q (z, z) which have no common factor because our rational function is irreducible, is an unessential singularity of R {z, z) ; (v) the behaviour of R {z, z) for infinite values of \z\ or of \z' \ or of both I z I and | / |, depends upon the degrees of ^ (z, /) and q(z, z') in z and in /, while every such place is either a zero, or ordinary, or a pole, or an unessential singularity ; and (vi) the rational function R {z, z') has no essential singularities. Functions entirely devoid of essential singularities. 78. Now we know that not a few of the important properties of uniform analytic functions of a single variable are deduced from those expressions of the function which arise when special regard is paid to its singularities ; and occasionally some classification of functions can be secured according to the 126 FUNCTIONS DEVOID OF [CH. V number and nature of these points*. In particular, we know that a uniform function, devoid of essential singularities throughout the whole field of variation of the variable z, is a rational function of z. Of this result, there is the generalisation, given by the theorem f : — A uniform analytic function of two complex variables z and z' , having no essential singularity in the whole field of their variation, is a rational function of z and z' . To establish this theorem, we proceed as follows. Let / {z, z') be a uniform function of z and z', entirely devoid of essential singularities; and let any ordinary place (say 0, 0) be chosen which is a non-zero place of the function. In the vicinity of 0, 0, let the expansion of fiz, z') be f{z,z)= i i Cm,nZ'^z'''] m=0 n-O and suppose that this series converges absolutely within a domain | ^ | < r, z'\ (z, z'), which may or may not vanish identically. The quantities Go, ..-, Gr, being lineo-linear in the rational functions A and F, are themselves rational functions of z' ; and not all the functions G can vanish. Then the function (z, z') is a regular function of z and / within the domain \z\^8 and \z' \^ h', because all its components are regular within that domain. The foregoing analysis shews that, for all values of z in the range | ^^ | < S, there is an infinite number of values of z' in the range \z'\^8' for which (z, z') vanishes. If (z, z) does not vanish identically, we take any special value of z within the range \z\^h, say z^c\ then ^ (c, z') is 79] ESSENTIAL SINGULARITIES 129 a regular function of / within the range | / 1 ^ 8', and (after what precedes) there is an infinite number of values of / within that range where (c, /) vanishes. It is a known property* of regular functions of one variable that the number of its zeros, within any finite region where the function is regular, is necessarily finite ; and the preceding result, based immediately upon the h3rpothesis that ^ (2, z') does not vanish identically, does not accord with this requirement. Accordingly, the hypothesis must be abandoned ; the function {z, /) vanishes identically ; and therefore, for all values of z and / within the selected domain, we have {¥, + zF, + ...+z''Fr)f{z,z')^G, + zG,^... +Z^Gr, where Ff^, F^, ..., F^, Go, Gi, ..., Gr are rational functions of z. The function ^0 and the function Gq do not vanish under our initial hypothesis that the ordinary place 0, is not a zero of f{z, z') ; some (but not all) of the other functions F-^, ..., F.y, G^, ..., Gr may vanish. We thus have J., ,. _ G(i + zGi + ... + z'^Gr ^^^'^^~ F, + zF, + ...^Z'^F'r' that is, f{z, z) is a rational function of z and z. The proposition is thus established. 79. One provisional remark will be made at this stage. Let f{z, z') be a uniform function which, within some limited region of its existence, has no essential singularities and, within that region, does possess zeros, and poles, and unessential singularities. Suppose that a uniform function exists, which has those zeros, those poles, and those unessential singularities, all in precisely the same fashion as/(ir,/), and which possesses no others within the region; and suppose that this function has no essential singularity anywhere in the whole field of variation of z and /. The preceding proposition shews that it must ^e a rational function of z and /. (Examples can easily be constructed, in the case of definite simple assignments of such places). We shall, for the moment, assume the possible existence of such a rational function; and then, denoting it by r {z, z'), we write ^^''^^~ r{z,z^)' Within the region, the function g {z, z') has no zeros and it has no singularities of any kind; hence, within the domain of every place in that region, the two functions g^ and g2, where 1 da 1 da y g dz ^^ g dz' can be expressed as absolutely converging power-series, which are elements * See my Theory of Functions, § 37. F. 9 130 SECTION FOR A [CH. V of two regular functions. Moreover, as regards these two power-series for g-^ and g^, we obviously must have dz dz identically; so we denote the common value of these two quantities by d^P {z, z) dzdz' ' where P (z, z) is itself a double series converging absolutely in the domain, and is an element of a single regular function, which may be denoted by Q{z,z'). Then 1 dg_ dP(z,z') 1 dg _ dP{z,z') g dz dz ' g dz' dz and therefore within the domain. Now g (z, z) is regular throughout the region ; and, for each domain within the region, P (z, z) is the element of the regular function Q (z, z'). Consequently, on the assumption that the rational function r (z, z) exists, we have r{z, /)e«'^'^' as a representation oi f{z, z') within the region, Q {z, z) denoting a function that is regular within the region. The definite existence of the function, denoted by r {z, /), has not been established in general. The assumption that has been made raises the question as to whether rational functions exist, defined by the possession solely of assigned zeros, assigned poles, and assigned unessential singularities. Also, that question raises the further question as to what are the limitations (if any) upon the arbitrary assignment of zeros, poles, and unessential singu- larities, in order that it may lead to the existence of a rational function. These questions initiate a subject of separate enquiry which will not be pursued here. Functions having essential singularities only in the infinite part of the field. 80. The other of the theorems already mentioned relates to the expression of a uniform analytic function, of which all the essential singularities arise for infinite values of one or other or both of the variables. It was adumbrated by Weierstrass * ; the following proof is based upon a memoir by Cousin i*. We have to establish the theorem : — A uniform analytic function of two variables, all the essential singu- larities of which arise for infinite values of either of the variables or of * Ges. Werke, t. ii, p. 163. + Acta Math., t. xix (1895), pi?. 1 — 62; it applies to n variables. It may be added that a proof is given by Poincare, Acta Math., t. ii (1883), pp. 97^113 ; 81] SPECIAL INTEGRAL 131 both of the variables, can he expressed as the quotient of two functions which are everywhere regular for finite values of the variables. For this purpose, Cousin uses the Cauchy method of contour integrals. 81. Consider an integral, the variable of integration Z' being taken in the plane of /, as given by B Fig. 1. Fig. 2. ^<^>2i^-j 1 P dZ' 2iri}AZ'-z" where the integration extends along an arc AB from A as the lower limit to B as the upper limit. When we take a closed contour of which AB is a portion, AB is the positive direction of description in figure 1 and is the negative direction of description in figure 2. Now in figure 1, we have '^'^ij A3IB Z' - for all points / within the contour AEBMA, and dZ' ^T^t'J AMB Z' for all points / without the same contour. For all points within the contour, and for all points without the contour, 6 {z) is a regular function of z' . Consequently the line AEB is a section* for the function; the continuation 6 (D), taken from the inside point G to the outside point D across the section AB when the latter is described positively for the area, is — 1 + ^ {G). In the same way for figure 2, the continuation 6 (D), taken from the inside point G to the outside point D across the section AB when the latter is described negatively for the area, is 1 + 6 {G). it is based upon the properties of potential functions. The following memoirs may also be consulted: — • Poincare, Acta Math., t. xxii (1899), pp. 89—178 ; ih., t. xxvi (1902), pp. 43—98. Baker, Camb. Phil. Trans., vol. xviii (1899), p. 431 ; Proc. Lond. Math. Soc, 2nd Ser., vol. i (1903), pp. 14—36. Hartogs, Jahresb. d. deutschen Mathematikervereinigung , t. xvi (1907), pp. 223 — 240 ; and the memoir by Dautheville already (p. 126) quoted. * See my Theory of Functions, § 103 ; the notion is due to Hermite, who called such a line a coupure. 9—2 1 ^^ g{z,Z') 132 PROPERTIES OF [CH. V The general value, of course, is ^(/) = ^.log^, 27^^ ° a — z where a and h' are the variables of A and B. Clearly the quantity «M-~log(6'-.') is regular in the immediate vicinity of B, and the quantity d{z') + ^\og{a:-z') is regular in the immediate vicinity of A. Next, let g {z, z) denote a function of z and /, which is regular for ranges of z and / that have finite values ; and consider an integral A taken precisely as for the preceding integral 6 (/). Then % {z, z) is a regular function of z and z', except when / lies upon the line AEB ; and AEB is a section for the function ;)^ {z, z'). Now let Zj — Z as g{z, z') is a regular function of z and z', it is easy to see* that G(z, z' , Z') is a regular function of z, z', Z'. Hence = E(^z,£),^d{^z')g{z,z'\ where II{z, z) is a regular function of z and / for all the values of ^ and / included, and 6 {z') is the preceding integral already considered. Consequently % {z, /) is a regular function of z and z' for all points z' that do not lie upon the section AEB; and the change in the analytical continuation oi xi^y ^') * If we take g (z, Z') =g, (z) + Z'g, {z) + Z'^g^ (2) + . . . , then G{z,z', Z') = g^{z) + {Z' + z')g2{z) + ..., so that I G (z, z', Z')\^\ g, (z) I +2/ I g2 (z) \ +3r'2 | g^iz) | + ... , for values of z' and Z' such that U' I < r', \Z' \ ^') log (6' - z) is regular in the immediate vicinity of b', and the function % (^> z') + 2^i9 (z> ^') log (a - ^') is regular in the immediate vicinity of a . Next, take in order a finite number of lines A^B, A^B, ... in the plane of /, such that they have a common extremity B, do not meet except at B, and all lie within the z, z' domain considered. Associated with each of the lines ArB, we take a regular function g^ {z, z), occurring precisely as g {z, z') occurred in the preceding discussion of the function xiz, z') over its section ; and write , ,- 1 f^gr(z,Z'),„, the integral being taken from Ar to B. The character of xi^' ■^') is known from the earlier investigation. Let a new function {z, z) be defined by the equation 0(^,/)= S Xr{z^z'\ For all places not lying upon any one of the lines, the function ^ {z, z') is regular. In the immediate vicinity of the place B common to all the lines, the function ^(^>^')- 2^- {log (&'-/)} ^ gr{z,z) is regular ; hence, if {z, z') is regular in the immediate vicinity of B, it is necessary and sufficient that 2 gr (z, z) should vanish at B. Moreover, if 2 gr (z, z') = Ikiri at B, where A; is a constant, then $ {z, z') - k log (6' - /) is regular at B. 82. We are to deal with a uniform analytic function f{z, z'), which has no essential singularity in the finite part of the z, z' field. In this field, take any finite domain. Within the selected domain, / {z, z') deviates from regu- larity at or in the immediate vicinity of poles, and at or in the immediate vicinity of unessential singularities. At a pole and in its vicinity, there is 134 FUNCTIONS AND [CH. V one definite type of representation of / {z, z') which is valid for some region round the pole. At an unessential singularity and in its vicinity, there is another definite type of representation oi f{z,z') which likewise is valid for some region round the unessential singularity. At an ordinary place and within some limited region of the place, f{z, z') is regular ; within that region, there is another definite type of representation of f {z, z') which likewise is valid for the limited region. When any two of these respective regions have any area in common, the respective representations of our uniform function / {z, /) are equivalent to one another over that area. Moreover, we have selected a finite domain in the z, zf field ; so that the total number of these regions in this domain is finite. Now let the whole selected domain in the z, z' field be divided up in different fashion. Let the whole region in one of the two planes (say the /-plane) belonging to this domain in the field be divided into n regions, where n is finite. Each of these regions is to be bounded by a simple contour. With each of these n regions in the /-plane, we combine the whole of the ir-plane that belongs to the selected domain ; so that we now have n domains within the single selected finite domain in the z, z field. At every place in each of these n domains, our function f {z,z') is defined. Let f-i.{z, z') denote the whole representation oif{z, z) in one domain, f^ {z,p^) the whole representation in another domain ; and so on for the n domains, up to fn {z, z). With each region in the /-plane, we associate the function /^ {z, /) giving the representation of f {z, /) for the domain which includes that particular /-region. It may happen that two such regions have a common area, so that the respective functions belonging to the regions coexist over that area; we shall assume that, if deviations from regularity occur within the area, such deviations are the same for the two functions, say /^ {z, /) and fi {z, /), so that f,{z,z')-Mz,z') ■ is a regular function over the area. When two functions are such that their difference over an area is a regular function, they are said* to be equivalent over the area; if their difference is a regular function in the immediate vicinity of a point, they are said to be equivalent at the point. Denote the regions in the /-plane by R^, R2, ..., Rn with which /j (2^, z), /^{z, z), ...,fn{z,z') are respectively associated. Further, denote by ^12 the boundary between R^ and R^, such that when / passes from R^ to R.2 by crossing l-^^, this line is described positively for the boundary of R^; and similarly for the boundary between any two contiguous regions. Lastly, there will be points where three or more boundary lines are concurrent. * Cousin, I. c, p. 10. 83] REGIONS 135 When a point P' lies within the region R^, then f]e{z, z") is the function associated with P'. When a point Q' lies on the boundary between two contiguous regions R]^ and Ri, then either of the functions fie{z, z') and/^ {z, z') is the function associated with Q'. When a point 8' is a point of concurrence of more than two boundary lines of regions Rj, R^, Ri, ..., then any one of the functions fj (z, z'), fk (z, /), fi (z, z), ..., is the function associated with 8\ 83. Consider the integral 1 ffm(z,Z')-Mz,Z') taken along the line Ijcm between two contiguous regions, the order of the suffixes in 1]^^ being the same as their order in 4m- Manifestly -'- km ^ J- mk • As the function/^ {z, Z') — /^ {z, Z') is regular everywhere along the path of integration, the integral is of the same character as the integral previously denoted by Xri^^ ^')^ ^^® ^i^® hm is a section for the function 7^^. Now take all these integrals Ikm which arise for contiguous regions, and write ^{z, Z') = tlicm, where the summation is for all pairs of suffixes that correspond to contiguous regions. The function ^ {z, z ) has each line ^^ as a section ; at every place that does not lie upon a section, $ {z, z') is regular. Next, we take a set of functions ^i {z, z), (f)^ {z, /), . . . , n{z, z), associated with the respective regions i^i, Rp{z, z) as the value of ^ {z, z') within the region Rp. A point P' in the /-plane may lie within a region ; it may lie upon the boundary of two contiguous regions ; and it may be a point of concurrence of several such boundaries. When the point P' lies within the region Rp, the function ^p{z, z) as defined is regular, because the sections of {z, z) are only the boundaries of regions. When the point P' lies on a boundary of the region Rp, say on the line Ipq so that Rq is the contiguous region, and when P' does not lie at either extremity of Ipq, the analytical continuation of <^p (z, z') through the point P' remains regular. For, writing gvq (^. ^') =fq (^. ^0 -fv (^' ^')> so that g^q (z, z') is regular for all the values of z and z' considered, the earlier investigation shews that, in crossing the section Ipq, the change in the analytical continuation of Ipq is — gpq (z, z) when Ipq, as it is crossed, is being described positively. For this position of P', every element in the sum of the functions I]cm is regular except Ipq ; and therefore the change in the analytical continuation of O (z, z) is — gpq (z, z'). But the new function (f)q (z, z') is the value of {z, z') in the region Rq ; hence <^q (Z, Z') = <^p (Z, Z') - gpq (z, /), 136 FUNCTIONS AND [CH. V and therefore where Rp and Rg are contiguous regions. When the point P' is a point of concurrence of several boundaries, the regions may be taken as in the figure. Our function (z, z') can be rearranged in its sum- mation. We group together all the integrals Ikm which have no section passing through P' ; and we call this group ^i{z, z). We group together all the remaining integrals, the section of each of which passes through P' ; and we call this gi'oup ^2(2,/). Thus • (^, /) = a>i(^, /)+<^,(^, /). The sum $1 {z, z) is regular at P', because every element / in the sum is regular. As regards the sum ^^{z, z'), our earlier investigation shews that the function ^M^')-^^\og{P'-z')\^g{z,z') is regular at P' . But the functions g {z, z'), for the various elements / in ^2{^, z') taken as in the figure, are fp {z, z) -/„ {z, z'), fy{Z, Z')-f^{Z,Z), f&{Z,Z)-fy{Z,Z'), f,{z,z')~f,{z,z'), f^{z,z')-f,{z,z'), that is, the quantity 1g {z, z) is -identically zero. Hence the sum 2 {z, z) is regular at P'. . Consequently, the function ^ {z, z) is regular at P', in this third case ; and therefore all the functions <^ {z, z), equivalent to one another at P/, are regular at that point. We thus have a set of functions ^ {z, z). Each of them is regular within its own region. Each of them is regular at any point of concurrence of the boundaries of several regions. The change in the analytical continuation, from the function <^p (z, z) belonging to a region Rp , to the function (f}q (z, z) belonging to a contiguous region Rq, is known ; we have q (Z, Z) - rfg (Z, Z') = C^r (z, z) + fr {z, z'). And so on, for each region in succession, until the whole domain considered is covered. Accordingly, we define a new function F{z, z), by the relation F{z,z')=^<^r{z,z')+f,{z,z') for every region Ry. But all these different expressions for F {z, z') are the same, because the relation before ; with each region Rjc we associate a regular function U]c (z, z'), and we take fk{z,z') = \ogUk{z,z'), so that the value of fk(z, z) is subject to additive integer multiples of '^iri, and otherwise is a regular function of z and / except at places which are zero-places of U]^ (z, /). As regards the functions u^ (z, z'), ..., Un {z, z), we assume that, over any area common to two contiguous regions Rjc and R^ or, if no area is common, along the part of their boundary which is common to them, the function Uic (Z, Z') Um {Z, Z) is regular and different from zero. Consequently the function h (^> Z') -fm iz> Z') is regular for the same range of the variables, subject to a possible additive integer multiple of l-ni. We now proceed as before. We again form the integrals ^ _ 1 [fJ^^Zy-f^{z^ ^''^~2'jriJ , Z'-z' ^ ' taken along the line ^;t^ which is the boundary common to two contiguous regions ; the order of the suffixes in I]^^ is the same as their order in l]cm, and clearly -^ hm ^^ J- mk • The function fm. (z, Z') — /^ {z, Z') is regular along the line 4^, and there is nothing to cause a change in the additive multiple of liri when once this multiple has been assigned; thus the integral is of the same character as the integral previously denoted by % {z, /), and the line 4m is a section for the integral I^^rn- Again, as before, we take ^{z, Z') = tlicm., where the summation is for all pairs of suffixes that correspond to contiguous regions. The function O {z, z') has each line Ij^m as a section. 85] SUBSIDIARY THEOREMS 139 At any point P' lying within a region, the function {z, z') is regular. At any point P', which lies on a boundary of the region Rp (say on the line Ipq so that Rq is the contiguous region) and does not lie at either extremity of Ipq, the analytical continuation of {z, z) from R^ to Rq through / is regular, the function in Rq being ^{z,^')-{fq{z,z')-fp{z,Z% where the additive multiple of ^iri is the same as in the integral Ipq. When the point P' is at h', a point of concurrence of several boundaries which may be taken as before, it is again necessary to rearrange the sum- mation of {z, /). We group together all the integrals having no section passing through h' , and call the sum of this group i {z, z). We then group together all the remaining integrals, the section of each of which passes through h' ; and we call the sum of this group 2 {z, z). Thus O {z, z') = ^1 {z, z') + ^3 {z, z'). Each element / in the first sum i {z, z') is regular at h' ; and therefore i {z, z') itself is regular at h'. As regards 2 {z, z'), our earlier investigation shews that the function ^^i^^^)-^i{'^^^^V-z')]l.g{z,z') is regular at h', the summation being over all the lines I which meet at h'. Now these functions g (z, z), for the various elements / in 2 (z, z) taken as in the former figure (§ 83), are A (^, Z') -/a {Z, /), fi {z, z') -f^ (z, z), fs (Z, Z) -fy {Z, Z'), /e (Z, Z') -ft {Z, Z), fo.{z, z')-f,{z,z'\ respectively, subject — for each of the functions g (z, z') — to an additive integer multiple of 27rt. Accordingly, the quantity -g (z, z') is some integer multiple of 27ri; let it be denoted by k . 27ri. It follows that the function ^,{z,z')-k\og{b'-z) is regular at the place b'. We have seen that ^i (z, z') is regular at b' ; hence ^(z,z')-k\og{b'-z') is regular at the place b'. 140 cousin's [ch. V At any point of concurrence of boundaries h" , other than h' , the function log (6' — z') is regular, subject to an added multiple of Svri. Consequently, the function where the summation is taken over all the points of concurrence of the boundaries of regions, is regular for all places / in the range considered ; its expression being always subject to an additive integer multiple of 1tt%. Let this function be denoted by ■^ (^, z) ; then ■^ (z, z') = ^ (z, z') - X {A; log {h' - /)}. Subject to the added multiple of liri, the function ■>if {z, z) is regular for the ^'-region considered : and its sections are the lines Ipq. Having constructed this function -v/r {z, z'), we now take functions -^^ {z, z'), ^^{z, z'^, ..., ■\frn{z, z'), associating them with the regions i^j, R2, ..., Rn respectively, and defining them by the condition that the relation i^rn {2, Z') = f {Z, Z) is satisfied within and on the boundary of Rm, for all the values of rti. When we pass across the boundary of R^ into a contiguous region Rp, we change to another function ^p {z, z). But, as we have seen, the analytical change in yfr (z, z') in passing over a line l^p is - [fp {Z, Z) -fm (Z, Z% and so the analytical continuation of -v|r^ {z, z) is ^m {Z, Z') - [fp (Z, Z) -fm{z, z')]. As this is the function y\rp (z, z'), we have y^p (z, z') = ^,n (z, z') - [fp {z, z') -fm {z, z')], there always being an additive multiple of 27ri on the right-hand side. Hence, subject to this additive multiple, we have ■^m {Z, Z) +/m {Z, Z) = l/r^ {z, z) + fp (z, z), for contiguous regions Rm, and Rp. Now pass from Rp to another contiguous region Rq, distinct from R^ ', then, again subject to an additive multiple of liri, we have ■fp (^, Z) +fp (Z, Z') = yJTg (Z, Z) +fg (z, z). And so on, for the full succession of contiguous regions, until the whole 2:' -range is covered. It follows then that, for any two regions R^^ and R^, we have the relation y\r,n (Z, Z') +fm (Z, Z) = -v/r^ {Z, Z') +f^ (Z, /), always subject to an additive integer multiple of 27ri; and each of the functions -yjr is regular within its own region. 86] SUBSIDIARY THEOREMS 141 Accordingly, we define a new function G {z, z) by the equation {z, z') = -«/r,„ {z, z') +/,ft {z, z), for every region Rm. But all these different expressions for G (z, z) are the same as one another (save for an additive multiple of 27^^ which may change from region to region), because the relation ■^m {Z, Z) +f,n (Z, Z) = a/t^ {z, z') + f^ {z, z') is satisfied for all values of m and fi. Finally, take a new function U {z, z) defined by the equation U {z, z) = e^^^^'^'K The added integer multiple of liri in G {z, z') does not affect the character of U (z, z') ; and so we have ll{z, z') = e^^''''> = Urn. (Z, Z') e*m»,2') within the region R^. We thus have established the result : — A function Viz, z') exists, regular throughout the whole finite region con- sidered, such that the quotient Um (Z, Z) is a regular function of z and z within the region R^ and is different from zero, Um (z, z) being itself a regular f miction within that region ; and this holds for all the n values of m. Again it must be remembered that n, the number of functions Um (z, /), is finite. The general theorem. 86. After these two propositions, which are general in cha;racter and the second of which is immediately useful for our purpose, we can proceed to the establishment of the general theorem, stated by Weierstrass, as to the expression of a function of two variables, of which the essential singularities occur only for infinite values of either or of both the variables. It has been proved that, in the immediate vicinity of a zero-place of a uniform analytic function f{z, /), we have f{z, z) = Pe^, where P is a polynomial in z having, as coefficients of powers of z, regular functions of /, or conversely as between z and z', and where i? is a regular function of z and / which vanishes when z=0 and z = 0. We have defined a pole of a uniform analytic function F (z, z) as a place, where a function f{z, /) of the preceding form exists such that F{z,z')f{z,z) 142 EXPRESSION OF A [CH. V is a regular function of z and /, which does not vanish at the supposed pole or in its immediate vicinity. We have defined an unessential singularity of a uniform analytic function F {z, z') as a place, where two functions f(z, z') and g (z, z) of the preceding type, and irreducible relatively to one another, are such that ■t {Z, Z) -j~ Tz is a regular function of z and z' which does not vanish at the supposed singularity. Suppose, then, that a function P (z, z) is defined as being uniform and analytic over the whole field of variation : that it has poles and unessential singularities of defined type within that field : that it has no essential singu- larities except within the infinite parts of the field of variation of the two complex variables : and that, except for the poles, and for the unessential singularities, the function otherwise is regular for finite values of the variables z and z . For the expression of the function, we need take account only of functions f{z, z) which give rise to poles, and of functions f{z, z') and g (z, z) which give rise to unessential singularities. We range these functions in two classes. In one class, we include all the denominator functions /(^;, /) ; in the other class, we include all the numerator functions g (z, z'). Let f(z, z') be typical of all the denoniinators, which occur in the expression of the function at a pole and its immediate vicinity; and let f(z, z') be typical of all the denominators, which occur in the expression of the function at an unessential singularity. We proceed to construct a function G (z, z) such that, in the immediate vicinity of any of these places, the quotient f{z, z') f{z, z') • is regular and different from zero ; the function G (z, z') exists, and is regular, in the whole finite part of the field of variation. Again, let g (z, z') be typical of all the numerators which occur in the expression of the function at an unessential singularity. Analysis, precisely similar to that used for the establishment of the function G (z, /), enables us to establish the existence of a function G {z, z) such that, in the immediate vicinity of any such place, the quotient G{z,z') is regular and different from zero ; . the function G (z, z) exists, and is regular, in the whole finite part of the field of variation. 87] UNIFORM ANALYTIC FUNCTION 143 Accordingly, we consider the possibility of the existence of the functions 0{z,z'),0{z,z'). 87. Imagine a succession of regions in the field of variation, each region enclosing the one before it in the succession. We shall take, as the boundaries of the regions, concentric circles in the respective planes ; and these may be denoted by (Cj, C/), (Ca, C^), ..., which may be unlimited in number, as we proceed to cover the whole field of variation. We also take the common centres of the circles at the respective origins. For the first region, there is only a limited number of functions /,„ {z, /), each of which is regular at, and in the immediate vicinity of, its place of definition. Hence, by § 85, there is a function, say U^, which is regular throughout the region and is such that the quotient is a regular function of z and / within the region and is different from zero ; and this holds for each of the functions fm {z, z) defined within the region. For the second region, there are all the functions /^ {z, z'), which are defined for places in the first region ; and there are the additional functions, which lie in the belt between the two regions (including the boundary of the first region). Then, again by § 85, there is a function Uo which is regular throughout the second region and is such that, (i) the quotient -pr i^ ^ regular function throughout the region and is different from zero, and (ii) the quotient fn(z,zr where fn(z, z') is any one of the newly included additional' functions, is a regular function of z and z' within the region and is different from zero ; and this holds for each of these functions /„ (^, /). And so on, from each region to the region next in succession ; we obtain a gradual succession of functions U^, U^, ..., C,., ..., each regular in its region, and having the properties, (i) that —~^ is a regular function through- out the region (CV, C/) and is different from zero, and (ii) that, for each of the functions fs{z, z) defined for the region (Cy_,.i, '(7V+i) but not for the region (C^, CV), the quotient fs (Z, Z') is regular for the region (O^+i, G'^+i) and is different from zero. 144 EXPRESSION OF A [CH. V 88. Take a converging series of positive quantities Oj, a.j, ...,a,-, •••. associating them in order with the successive regions, so that a,, is associated with the region (C,., (7/). Also, let u, "^'■' then the regular functions U^, U^, ... can be chosen so as to give \pr\< e"^ for each value of r. Suppose that U-^, ..., Ug have been chosen so as to satisfy this relation for r=l, ..., s-1. The function Ug+i/ Us is regular throughout the region (Gg, Cg) and is different from zero there; and therefore log Ug^, - log Ug is (save as to an additive integer multiple of 27ri) a regular function of z and z' throughout the region. This regular function, save as to the additive multiple of 2'Tri, can be expressed as a double power-series in 2 and z' converging absolutely within the region. Let this series be denoted by »f = M = let M be the (finite) greatest value of its modulus within the region ; and let B, and E' be the radii of the circles Cg, (7/. Choose values, ybg of w,-and Vg of n, sufficiently large to secure that JJ, N; as each of them is regular for values of z and z' such that \z\^€r.n, \z\^ern> we can express fn (z, z') in a form fn{z,z')= 2 2 C^,,(-)^^A where the double series converges absolutely. As in § 88, we can assign a positive integer fin, taking /^n to be the greater of the two integers /ig and Vs there assigned, such that (00 » 00 00 00 00 "j S 2 + 22-2 2 Cp,,<-'^^/3Ke„ for all the values of z and z' considered. Hence, denoting by (^„ {z, z') the polynomial (finiz,z')= 2 2 C^.^'^'^^A p=0 3=0 and constructing a function F,(Z,Z')=^ 2 {fn{z,z')-cl>n{^,z')}, n = N+X we have, on the right-hand side, a series which converges absolutely for the values of z and z considered. Now consider the sum F{z,z')=F,{z,z) + F,{z,z). The function F{z,z')-f,,,{z,z') is regular at all the singularities of fm(z, z)\ and so the function Fiz, z') is regular at all places in the field of variation which are not singularities of any of the functions /j (z, z'),f{z, z'), ...; and F (z, z'), at places which are singularities of a function /(2, z'), is non-regular inthe same way SiS f(z, z'). 92] EXAMPLES 149 92. As a special instance of this sum-theorem, Appell adduces the case when where s is a positive integer, a is a constant, and the different functions fmn iz, z') arise by assigning to m and to n, independently of one another, all integer values from — ao to -|- oo . We have I {z + inf + (/ + nj- + a^ I > I (^ + mj- + {z + nj \-\a\^. Also {z + nif + {z + w)2 = (^ + iV + 7n, + w.) (^ — 1>' + m — in). But and Hence, if we have and therefore z -k-iz -\-m-\-in\>\m-\-%n\ — \z-V iz' \ > (m- + 71^)^ —\z\ — \z'\, z — iz -h ?M — in \ > (m^ + n^)^ — \z\ — \z' \. \z\^^ {(m^ + n^)2 — I a I — c}, \z'\<^ {(m^ + n^)^ -\a\- c], I (z + mf + {z + nf\> {\a\ + cY ; I {z + tnf + {z + nf + a^\> [\a\ + g]- - \aY > 2c I a I + C-. Consequently, for all values of z and z' within a range that increases in- definitely with m and n, as given by the foregoing limits, \fmn {z, z) \ remains smaller than an assigned quantity ; and so for those values, fmn (z, z) is a regular function. Thus the set of conditions for the function f^n (z, z') is satisfied. When the integer s is greater than unity, the series -00 — 00 [{z + my + {z + nf 4- a^Y converges absolutely. We therefore take ra= -xs w=5o "J ■ Fiz,z)^ X 2 77 ^ "^ , . - .. ^ ^ -oo -00 \{z + mf + {z + nf + a^Y The function F{z, z) has poles at all the places z = — m,-\-ia cos 6, z' ^ —n + ia sin 6, for the continuous succession of values of and for all values of m and of n. Elsewhere, at all places in the field of variation, the function F(z, /) is regular. In this case, there is no need to take polynomials corresponding to the functions (t)n(z, z') in the general investigation. 150 appell's [ch. V When the integer s is equal to unity, the expression of the function is not so simple, because the series, of which the general term is 1 {z + my + (z' + ny + a^' does not converge absolutely. We then take all the values of m and n, which are finite in number and are such that selecting all the functions fmn {^> z) given by these values of m and n, we denote their sum by F^ {z, z'). Next, take the values of m and n which are such that (wi^ + ?z^)2 > I a I + c, and expand fmn {z, z'), for any such pair of values, in powers of z and z, valid in a range k I < i K^^ + ^^)* - I a I - c}, \z'\^\ {{m^ + w2)2 -\a\-c]. Thus For our purpose, it is sufficient to take the desired polynomial <^^nn {z, z) as equal merely to the constant term in the expansion ; for the series Fa (Z, Z') = 2S < -, — ■!—, 7 ; ; ^ , ^ ^ \{z^ my + {z + ??,) + a? m2 + w^ + o?) ' for all such values of z and z , and for the doubly infinite set of values of m and n, converges absolutely. Our required function is F{z,z)=F,{z,z') + F,{zJ). It has poles at all the places z = — 7111 -via cos 6, z' = — n + ia sin 6, for the continuous succession of values of 9, and for all integer values of m and n. At all other places in the finite part of the field of variation, the function F {z, z') is regular. 93. As an example of the product-theorem, let Ui{z, z), u-^iz, z), ... denote a set of regular functions of z and z , and let them have the property that for all integers n, greater than some definite integer N, we can assign a magnitude r„ so that u^ {z, z) is distinct from zero for values of z and z' such that \z\ I a| + c, and write ^ G,{z,z) = mi ^ XWa, e V'™J^,^)|, where the product extends over all these values of m and n, and where 2??13 + 2ft/+22 + z'2^ 1 /2jW2 + 2%2' + 22 + z'2\2 ''''"" ^^' ^'~ m? + n^ + a^ 2 V m' + n'-^-a' \) ' The required function is given by G{z,^) = G,{z,^)G^{z,z'). Ex. 2. Verify that, when a is zero, the function G {z, z') can be expressed by means of two Weierstrass's o--functions. CHAPTER VI Integrals; in particular, Double Integrals As regards the matter of thivS chapter and, above all, as regards integrals of algebraic functions of two variables, the student should pay special attention to various sections in the treatise (which usually is quoted here in Picard's name) Picard et Simart, Theorie des fonctions algehriques de deux variables independantes, t. i (1897), t. ii (1906). Other references will be found in the course of this chapter. It may be noted initially, as regards algebraic functions of two variables, that I have chosen, for reasons already stated, to take two fundamental equations defining two independent algebraic functions of the variables, instead of only a single equation defining only a single algebraic function. If three (or more) equations were taken defining the same number of algebraic functions, these would not be independent ; so it is sufficient to take not more than two fundamental equations. 94. In the theory of functions of a single variable, many important results are derived through the use of Cauchy's theorems concerning contour integrals. It is natural to attempt some extension of theorems so as similarly to derive results in the theory of functions of more than one variable. Here we shall restrict the discussion to the case of a couple of complex variables. The integral of a function of tvs^o independent complex variables may be single or may be double. The definition of a single integral is the same as in the customary theory of functions of one complex variable ; but there is the added complication through the occurrence of two complex variables. Either there is' variation, within the range of the integral, of only one of the two variables; or within that range, there is a definitely connected and simultaneous variation of both variables. Of double integrals, there are two classes. In one class, the integration with regard to each variable is entirely independent of the integration with regard to the other, so that the integrations can be performed in either order. In each integration, only one variable is subject to variation. Thus the double integral is effectively only a double operation of single integration. We have already had some examples, at an earlier stage, of this class of double integrals. 95] INTEGRALS 153 Ex. A function /(!//■, 6) is periodic in y\r, with period 27r, and is also periodic in 6, with period 2ir ; and it is regular for all values of the variables within the ranges of two complete respective periods. Let u (r, r', (p, cj)') denote the integral 4^ j j ■^^'^' ^^ {1 - 2r cos (x/. - (^) + r2} {1 - 2/ cos {d - «^') +/2} ^^^^• Prove that, when r < 1 and r' < 1, the function u {r, r\ (j), cf)') is regular ; and that, in the limit when r — l and r' = l, the function u{r, r', 91 ^' = 9z (p, q), y' = g^ (p, q)- These relations imply two equations, say U (x, y, X, y) = 0, Vix, x , y, y) = 0, which represent a two-dimensional continuum (the surface /S^, as in § 5) in our four-dimensional space. We take a simple closed area in the plane of the variables p and q, represented by an equation F{p,q) = 0; and for the double integral, we allow all values of p and q within this area,, representing them by the relation F{p..q)<0. Then the limit of the range of integration on the surface 8 is given by F (p, q) = 0; and this limit will lead to three equations of the form Ps(x,y.x',y') = 0, (5=1,2,3), representing a curve in the four-dimensional space. Now let f{z, /) be the function, to be "doubly integrated" in the sense that a meaning has to be assigned to the double integral I=\\f{z,z')dzdz'. As f{z, z) is a complex function, we resolve it into its real and imaginary parts ; let f{z,z') = P + iQ, where P and Q are real functions of x, y, x', y'. Then 7 = 1 l(P + iQ) (dx + idy) {dx' + idy) = [[[{P + iQ) dxdx' + {iP - Q) dxdy + {iP - Q) dydx - (P + iQ) dydy']. Manifestly /, whatever its value, can be a complex variable ; so writing l^l, + il„ 95] DOUBLE INTEGRAL 155 where I^ and /a are real, we have /, = U[P {dxdx' - dydy')] - \1{Q (dx dy' + dy dx' )} , I^ = [[{Q (dxdx' - dydy')] + jj{P (dxdy' + dydx')]. And now, /j and I^ are ordinary double integrals involving only real variables, for the real quantities x, y, x, y' are functions of only the real variables p and q\ and these double integrals are taken over the limited area F {p, 5') ^ in the plane of the variables p and q. Both integrals are of the form {A dxdx' + Bdxdy' + Gdydx + Ddydy'), where all the quantities concerned are real — there being, of course, limitations upon the forms of A, B, G, D and also of their differential relations to one another. When we give explicit expression to the functionality of x, y, x', y in terms of ^ and q, the integral becomes but for our purposes it will suffice to take the first form. Our object is the generalisation, if generalisation be possible, of the fundamental theorem of Cauchy which asserts that, under appropriate con- ditions as lo f{z), the integral j f(z)dz taken round a closed contour is zero : it is a consequence that the integral If {2) dz, between two points in the plane, has a value independent (subject to restrictions) of the 2^-path between the points. Suppose that, instead of the former values of x, y, x', y', we take x = h, (p, q), y = h.^ (p, q), x' = h^ {p, q), y' = K {p, q), so that we could have a new surface T different from 8 ; and suppose that, corresponding to the former equation F {p, q) = limiting the range of integration, the range of integration in T is still limited by F (p, q) = 0, and that the limiting curve connected with T in our four-dimensional space is given by the same equations P, (x, y, x', y') = 0, {s= 1, 2, 3), as the limiting curve connected with S. We thus should have two different surfaces passing through the same contour. Then the generalisation would be that the integral \\f{z, z) dzdz' should remain invariable if only the surface over which the integration extends is made to pass through an 156 POINCAR^'S [CH. VI assigned fixed contour ; or, if we take a completely closed surface through the fixed contour, the integral ji f {z, z) dzdz taken over the whole of this surface vanishes. 96. Accordingly, we consider an integral where the summation is taken over all pairs of values m, n = 1, 2, 3, 4, and where oci, x^, x^, x^ take the place of x, y, x' , y'. We define the integral for the four-dimensional space as above ; consequently, because A^^fiUXijiCtXji — I I A.jnn" [ i CtX^dXfn, X-iji, X ■11} •^m with the foregoing interpretation, we have and •^ mn d'Xjn ClXji — I \ -^ mn ^'^n ^"^m } that is, taking account of the whole integral and of the combinations of m and n instead of the permutations, we shall assume that A — — A so that we need only consider the combination I JA^j^dx^^dx^^. Moreover, this process of regarding the integral obviously involves the additional assumptions for all the values of m. Next, we take* x-^, x^, x^, x^ as expressed in terms of the three variables X, Y, Z, so that our double integral becomes that is, where ^™. 1^ (^) dYdZ + J (^^) dZdX + J (^) dXdY]^ (^dYdZ + V dZdX + ^dX d Y), V '^^■^mn^ \ y IT I ' S = ^'^■^mn^ \X~Y) ' Here Picard's proof {Traite d'ATialyse, t. ii, p. 270) is followed exactly. 96] PRELIMINARY LEMMA 15T The integral is to extend over the surface in the X, Y, Z ordinary space. We therefore require the condition necessary and sufficient that such an integral '\{^dYdZ + 7)dZdX + KdXdY), over any surface which passes through an assigned contour in the j9, q plane, shall depend solely upon the contour. This condition is well known : we must have* M + l^+^i^ 'dX^'dY^'dZ ' Accordingly, the condition is In this expression, the coefficient of A^m is az X VYTz)] '^ ar r vXX;} ^ az X Vxry which vanishes identically. As regards the derivatives of J.^,;,, we have ^-^mn S ^-^mn ^^l — s dX 1=1 dec I dX ' and so for the others. Hence, in the foregoing expression, the coefficient of ^ , and the coefficient of "^^ , both vanish identically : and the non- vanishing coefficients are the sum of terms of the form (dAmn , dAni , dAi.A j- fxi, Xm, a;„^ \ dxi dXm dxn } \X,Y,Z Consequently, the condition becomes = 1 m=l n=\ [\ OXi dXrn dx^ J \A,Y,Z,'^ * When the condition is satisfied, we can take and then the integral can be expressed in the form I [adx + ^dy + y dz) , taken round the contour in the p, q plane. The result was first enunciated as a problem hy Stokes, in the old examination for the Smith's Prizes at Cambridge in the year 1854; see Stokes Math, and Phys. Papers, vol. v, p. 320, with a note by Prof. Sir J. Larmor. 158 POINCARlfi'S EXTENSION OF [CH. VI a condition which must be satisfied identically, whatever be the surface over which the integration extends, subject to its passing through the contour. The quantities xi, Xm, x^, Xp are functions of X, F, Z such that, away from the contour, any three of them are independent of one another ; and therefore the quantities J fXi, Xjf,, Xfi\ \X, 1,Z)' except along the contour and individually at special places in space, are different from zero. It follows that we must have for all the combinations I, m, n=l, 2, 3, 4. Moreover, it is easy to see that this set of four conditions is sufficient, as well as necessary, to secure that the value of the integral _2l* I \ JLffi^ClX^CtXn depends only upon the contour. 97. Now let us apply all the conditions to the integrals /j and /,. We have Jj = I Updxdx' — Qdxdy' — Qdydx — Pdydy), and we take •^i y-> '^ ■> y ^^ "^ij '^11 '^3> ^i> respectively. We have A,, = ^, A,, = P, A,, = -Q, A,,^-Q, A,, = -P, A^=0 Consider the conditions dAjnn dAnl dAjm _ ^ dxi dXyn. dXn for the combinations I, m, n = I, 2, 3, 4. . They require the relation dx dy for I, m, 71 = 1, 2, S ; the relation dy' dx for /, m, n=2, 3, 4 ; the relation dx' dy' 97] cauchy's theorem 159 for I, m, n = 3, 4, 1 ; and the relation dx dy for I, m, n = 4, 1, 2. Similarly, we have -^2= I jiQdxdx' 4- Pdxdy' + Pdydx' — Qdydy'], so that we can take -a-12 = 0, Ai^ = Q, ^14 = P, A^s = /^, ^24 = ~ Q) -^34 = 0. The general conditions require the relation dx dy ' for the combination I, m, 7i = 1, 2, S; the relation dy' ^ dx' ' for the combination I, m, w = 2, 3, 4 ; the relation dx' dy' for the combination I, m, w = 3, 4, 1 ; and the relation dy dx for the combination I, m, ?i = 4, 1, 2. Thus all the conditions are satisfied if only ap^ag ^^__oQ dP_dQ, ^jP^_^_Q dx dy ' dy dx' dx' dy' ' dy' dx ' But, by definition, we have P + iQ =f{z, z) =f(x + iy, x + iy'), where P, Q, x, y, x, y are real ; and so these four relations are satisfied. It follows, then, that /j and J^ depend solely upon the contour ; and therefore /, = I^-^il^, also depends solely upon the contour. And we have, throughout, assumed that the quantities P and Q, — that is, also the function f{z, z) — are free from singularities. Hence we have Poincare's extension of Cauchy's theorem : — If, within the closed sarface S, which is taken in the space of three dimensions X, Y, Z, and points on which are given by equations of the form X=f{p,q), Y = f(p,qX Z=f(p,q), so that, along the surface, x = F,(X, Y,Z)=g,(p,q), y = F,{X, Y, Z) = g,{p, q), x' = F, {X, Y,Z)^g, {p, q), y' = F, {X, Y, Z) = g, (p, q), 160 POINCARE'S extension of [cH. VI the7'e is no place X, Y, Z, where the function f{z, z) ceases to he regular, the value of the integral \\f{z,z')dzdz' taken over the whole of the closed surface is zero. Again, for such a function and over such a space, the value of the integral lln..,.a. ..„ ... .. ,...„/„... ../.. « w. ., „ contour, the surface and the contour lying within the domain, depends only upon the contour. Further, it follows that the value of the integral \\f{z, z') dzdz', taken over any such closed surface, remains unaltered during deformations of the surface provided they occur in the domain of X, Y, Z, and cross no place giving rise to no singularity off(z, z').- 98. Now consider the singularities, or other deviations from regularity, of a function f{z, z'). We take the preceding surface S existing, as in § 95, in an ordinary space of three dimensions, the representation of the variables being x = F,(X, Y, Z), y = F,(X, Y, Z), x' = F,{X, Y, Z), y' = F,{X, Y, Z). The singularities oi f{z, z) may be given by a set of single equations, typified for each of them by e {z, z') = 0, or by sets of two independent equations, typified for each set by e (z, z) = 0, {z, z') = 0. The former will lead to two equations, say ^1 {x, y, X, y) = 0, ^2 {x, y, x' , y')^0; so, in our X, Y, Z space, they will be given by equations @,(X, F, ^) = 0, @,(X, Y,Z) = 0. These two equations represent a curve G in that space ; at every point on the curve there is a singularity of/" {z, z). The latter will lead to four equations, which may be regarded as defining an isolated place or an aggregate of isolated places determined by the values of X, y, x, y . Such places may or may not exist in our X, Y, Z space. Take a closed surface *S^ in the space, containing no place or places X, Y, Z, giving rise to an isolated singularity of / {z, z), to any curve C, or to any part of such a curve. The integral 1 1 f{z, z)dzdz' taken over 8 is zero. Take two closed surfaces 8 and S' in the space X, Y, Z, such that 8 can be continuously deformed into S', without passing over any place giving rise to an isolated singularity of / {z, z), or over any curve G, or any part of such a curve C. The value of the integral taken over the surface aS^ is equal to its value taken over the surface 8'. 100] cauchy's theorem 161 Take two closed surfaces S and ;S^' in the space X, Y, Z, such that they enclose places giving rise to exactly the same isolated singularities oif{z, z'), to exactly the same curves C and to exactly the same portions of curves G. The value of the integral taken over the surface 8 is equal to its value taken over the surface 8'. Thus the value of the double integral \\ f{z, z') dzdz, taken over the closed surface 8, is zero when the surface encloses no place X, Y, Z, where f (z, z) ceases to be regular. When the surface does enclose places X, Y, Z, where f (z, z') ceases to be regular, the value of the integral depends upon these enclosed places ; we cannot assert that its value is zero. 99. The theorem can be enunciated in similar terms when a two-plane representation of z and / is adopted. Thus, very specially, within a circular ring in the ^-plane and within a circular ring in the /-plane, let a function f{z, z') be everywhere regular ; then the value of f{z, z') dzdz' is the same, whether the integral be taken positively round the outer circles in the two planes, or be taken positively round the inner circles in the two planes. But such a case is exceedingly special ; and, as was indicated earlier in the lectures (§ 19), the frontier of- a domain of variation for z and / is of a more com- plicated character than in the result just enunciated. 100. We proceed to consider some of the simplest cases when the subject of integration in a double integral 1 1 f{z, z) dzdz' possesses either isolated singularities or any continuous aggregate of singularities within an assigned domain. In passing to these examples, it may be remarked that the whole subject of double integrals of uniform analytic functions, possessing singu- larities of the known types, offers a field of research, in which many of the results already obtained are of a tentatively exploratory character. In the examples that will be considered, we shall use the two-plane representation of z and /, and we shall deal only with a finite part of the whole field of variation of z and / ; that is, for all the variations, | z j and | z j will be kept finite. To these examples*, all of which involve only rational functions of z and /, we now proceed in order. Example I. Let F (z, z) denote a function that is regular everywhere within an assigned finite domain ; let a, a' denote any place within that domain. Then we consider the integral K-IS^l^l^dzdz' ]Uz-a)(z'-ar''^'' {z — a) {z' — a) * In this connection, reference should be made to Picard, Fonctions algebriques de deux variables, t. i, ch. iii. F. 11 162 INTEGRALS OF FUNCTIONS [CH. VI taken over the closed frontier given by the equations z — a\= R, \ z' — a' \= R, so that it encloses the place a, a'. The singularities of the subject of integration are given by (i) z = a, z' = any enclosed value of / ; (ii) z = any enclosed value of z, z = a . By our general theorem, we can deform the closed frontier without changing the value of the double integral, provided the deformation causes no transition through any of these places. Accordingly, let the closed frontier be deformed until it encloses only the small domain, composed of the interior of the circles z — a = re^, z —o! = r'e^'^, where r and r' are small real positive constant quantities. Then \\ r ^w'f^ .. dzdz = - [If {a + re'\ a' + re''') dddd', jj{z-a){z -a\ J J the integration extending over a ^-range from to 27r and over a ^'-range from to 27r. Now F (z, z) is regular throughout the domain ; hence F(a + re'\ a + rVn = S S —, , ^ ^^' '^ >> y.my./ng(me+ne')r. But for positive integer values of m and n, such that either 7n or ?i is greater than zero, we have and jjdedO' = 47rl Hence ''fF(a + re'\ a' + r'e^'O dOdd' = ^nr'' F {a, a') ; IS^ and therefore, with our hypothesis as to the regular character of F (z, z') within the domain, we have 4'7r'j]{z-a)'^+'{z'-a'f+^ m\n\\ dz'^'dz'^ j^=a,^w' for all integer values of in and n that are not negative. Example III. Let a, /3, j, 8 denote four constants such that a8 — ^y is not zero ; and consider the double integral dzdz' \'xz + /3z')(yz + 8z')' taken over a frontier that encloses the place 0, 0. For a given value of z', the quantity az + ^z vanishes if z = z-^, and the quantity yz + 8z' vanishes if ^ = ^^2, where ^1 = z , ^2 = z . a '7r^JJ (az + /3/) {yz + Sz') ' integration ; that i\ J{i;,l:') = aS-^y, is independent of the order of integration ; that it is , where 166 EXAMPLES OF [CH. VI when integration is effected round a curve enclosing a root of ^, where ^= az-V^z', hut not a root of ^', where ^' = jz + Zz' ; that it is j , = - j , r , when integration is effected round a curve enclosing a root of "C,' but not a root of ^i and that it is zero when integration is effected round a curve enclosing both a root of t, and a root of ^'. And, of course, the value is zero when the integration is effected round a region that does not enclose any zero of ^ or of ^'. Example IV. The preceding result cannot be applied when the initial assumption, viz. that ocS — ^y is different from zero, is not satisfied. In that case, we have to deal with dz dz \az + I3zj' When the integral is taken round the place 0, 0, in either of the ways indicated in the construction of the last result, the value of the double integral is zero. Example V. From III and IV, we infer the following results relating to the double integral 1 n dzdz 4<7r^JJ\z^ + 2 fizz' + pz^ There are two cases, according as ij? is not, or is, equal to \p. (i) Suppose that yct^ - \p is not zero. When integration is effected in either plane, round a small simple curve enclosing the root of X^' + {//- + {p? - Xp)*} / = but not the root of \z + [pL- {p? - \pf'\ z = 0, and then round a small simple curve enclosing the origin in the other plane, the value of the double integral is When integration is effected in either plane, round a small simple curve enclosing the root of \z -\- [ijl - {p? - Xpf'] z' = but not the root of X2+ {p, + {p.- - Xp)*} z' = 0, and then round a small simple curve enclosing the origin in the other plane, the value of the double integral is i(//,2_xp)-i. And when integration is effected in either plane, round a small simple curve enclosing both roots of Xz'^ + 2p,zz' + pz'^ = 0, and then round a small simple curve enclosing the origin in the other plane, the value of the double integral is zero. (ii) Suppose that p?~\p= 0. When the integral is taken round the place 0, in any of the ways indicated for the preceding case, the value of the double integral is zero. 100] DOUBLE INTEGRALS 167 Example VI. Let where 70 and Sq are dififerent from zero and (for the immediate purpose) m and n are positive real quantities, not necessarily integers. We require the value of _i nv_0^)^^ 477"- jj UV whej'e u = az + P, v = /3z + Q, when the integration is effected, first, with regard to z round a small simple closed sr-curve enclosing a root of u (but not a root of v) for a value of z', and, then, with regard to z' round a small simple closed curve, passing through that value of / and enclosing the /-origin. We also assume that aQ — j3P does not vanish identically. Now /= a/»i-i [nSo +(n + l)S^z'+ ...}- jSz'"''-' [mjo + (m + 1) j^z' + ...]. Thus, if ni < n, the lowest power in J is — ml3'Yoz''^~^ ; if m > n, the lowest power is nccSoz'^^"^ ; if m = n, =1 say, the value of / is Iz'^^-' (aSo - /Sto) + (^ + 1) ^''" (aSi - /SyO + • • . • For any small value of z', such that \z'\ is less than the modulus of the smallest root of P or Q other than z = 0, let otz, + P = 0, ^z. + Q= 0. Then the double integral 1 ff dzdsf J 4<7r^ j J a^(z — Zt) (z — z^) 27ri f f J , , dz . ^iT^ } } olQ - j3P When m < n, the value of the right-hand side is n. When m > n, the value of the right-hand side is m. When m = n, = I, the value of the right-hand side is l + k, where aSj^ — /3ji; is the first of the coefficients aSo — /Sjo, '^^i — ^ji, ••• which does not vanish. In each of the three alternatives, the value of the integral is the degree of the lowest power of z in the eliminant of az + P and /3z + Q, when z is eliminated. Moreover, luhen 7n and n are integers, the value of the integral is then the multiplicity of 0, 0, as the sole isolated simultaneous zero of the uniform functions az + P, ^z+ Q, enclosed by the frontier of integration. Example VII. Next, let U=Z^ + Z^-'f, (Z') +...+fm {2'), v = z^+ z^-^ g, (/) -\- ...+gn{z'), 168 EXAMPLES OF [CH. VI where the functions u and v are independent and have no common factor of their own form, and all the coefficients /i, ..., f^, g^, ..., gn are functions of z which are regular in the vicinity of z' = and vanish with /. We require the value of the double integral _ 1 /•/•/O^J!) 477- J J UV taken (as have been the preceding integrals) round a frontier, which encloses the place 0, 0, and encloses no other simultaneous zero of u and v. Let U = {Z - Z^) {Z - Z^) {Z- Zm), V={z-i;T){z-Q {z - ^„), where each of the quantities z-^, ..., z^., ^i, •••, Cn is a regular function of positive powers of /'*; where /i is a positive rational fraction; and where each of these quantities vanishes with z'. The eliminant of u and v is m n n n {zr-Q] r = \ s=\ if, when z,. — ^s is arranged in ascending (fractional or integral) powers of z', the lowest power of ^ has an index yu,,. g, and if m n r=l s = l the eliminant of u and v is /^^ (/)(/), where (^ (0) is not zero. The magnitude M is an integer, manifestly finite : it is the measure of the multiplicity of 0, 0, as an isolated zero common to u and V. For the range of integration, first take a value z of modulus smaller than the root of (^ {z) which has the smallest modulus. In the 2^-plane mark all the quantities z-^, ..., z^, ^i, • • • , ^n> which are functions of this value of z'; and draw a simple closed ^-curve, enclosing all the places z^, ..., z^ and none of the places ^i, ..., t,n. We take the integral round this 2^-curve ; when this first integration has been efi"ected, we integrate with regard to / along a small simple closed /-curve, through the place for the assigned value of / and enclosing the /-origin. We have where z.' = ^ and t/ = ^ ; hence dz dz But the lowest power of / in z^ — ^g is z''^'-^. Hence 1 [fJ(u, v) . . , :^ ^ ,, -7"2 ' dzdz' = S 2 iXr,, = M; that is, the value of the double integral, taken over the range indicated, is the 100] DOUBLE INTEGEALS 169 measure of the multiplicity of 0, 0, as an isolated simultaneous zero of the functions u and v, which are supposed to be independent and to he devoid of any common factor of their own form. Corollary. Two or more of the quantities z-^, ..., z^ may be equal, or they may be equal in groups ; and, similarlj^ two or more of the quantities ^j, ...,^n may be equal, or they may be equal in groups ; while, after the hypothesis as to the functions u and v, no one of the quantities ^ is equal to any of the quantities Zi, ..., ^,„. The value of the double integral over the indicated range still is M. Note 1. If the range of integration, enclosing 0, and no other simul- taneous zero of u and v, is chosen so that the ^^-curve (for a value of /) encloses all the places ^i, ..., ^n and no one of the places z-^, ..., z^, and the /-curve is drawn as before, the value of the double integral becomes — M. Note 2. We have 477^ J J uv 47r- J J uv When integi-ation is effected first with regard to z', round a curve enclosing all the roots of u = and no root of v = for an assigned value of z, and then round a 2^-curve through this value and enclosing the ^^-origin, we still have 'J {u, v) 4>7r'jj' dzdz'=M. uv In other words, the value of the double integral is independent of the order of integration. Example YIII. Let a and /3 be non-variable quantities, of finite moduli; let c, c' be a level place for two regular functions, f and g, such that f(c,c')-a = 0, g{c,c)-/3 = 0; and let f(z, z') — cc, g {z, z) — ^, be independent, and have no common factor which vanishes at c, c. Then the place c, c' is isolated; its multiplicity is the value of the double integral {f(z,z')-a}{g{z,z')-^]''''''' taken first round a small simple closed curve in the z-plane which, for an assigned small value of z , encloses all the roots of f{z, z')= a and none of the roots of g{z, /) = /3, and then round a small simple closed curve, through that value of z' and enclosing the z -origin. The result follows from the last example by writing u=f{z,z)-a, v=g{z, z')- ^■, the multiplicity of c, c' as a level place for /and g is its multiplicity as a zero for u and v*. * In connection with double integrals of the preceding types and taken over such ranges of integration, the reader should consult Picard's treatise, t. i, ch. iii, quoted p. 161. 47r^iii 170 CANONICAL FORM OF A [CH. VI Algebraic functions in general. 101. Hitherto, all the subjects of integration in the double integrals that have been considered, have been uniform functions. Bearing in mind the extraordinary importance of Riemann's investigations connected with the simple integrals of algebraic functions, we should naturally seek the general- isation of that work for algebraic functions of two variables. Into that theory I do not propose to enter in detail. In one sense, it is enough for me to refer to the long series of valuable researches by Picard*. All that will be done here is to submit one or two simple propositions, when there is a single dependent variable, partly from the standpoint of the general theory of functions and without regard to the theory of the singularities of surfaces, partly also to state the corresponding propositions when we have to deal with the case when the fundamental algebraic equations provide two dependent variables and not one alone,- the number of independent variables always being two. Suppose then that we have, in the first place, a single irreducible algebraic equation f{io,z,s') = 0, expressing w as an algebraic function of z and /; and assume that the equation is of order m in w, so that w is m-valued. Any rational function in the field of variation is of the form R (lu, z, /), where R is the quotient of two poly- nomials in all the variables w, z, z' . To this rational function R (w, z, z') a canonical and recognisable form can be given ; the proposition, stating its form, can be established in the same kind of way as for the corresponding proposition when there is only a single independent variable. Let the 711 roots of the fundamental equation f{w, z, z') = be denoted by Wi, W2, ..., Wm- Then, for any positive integer n, the quantity Wi^R (W] , Z, Z) + Wo]^R {Wo_, Z, Z')+ ... + Wn-H'R (Wm, Z, z) is a symmetric function of the roots xo^^, ..., w,„, of the fundamental equation, having rational functions of z and z' for the various symmetric combinations of the roots ; it is therefore a rational function of z and /. Denoting this rational function by Pn {z, z), we have m 2 W,^R{Wy,Z,z') = Pn{z,z'). r=l This result holds for all integers n ; hence, taking it for » = 0, 1, ... , 771 — 1, we have m equations, each linear in the m quantities R {Wi, z, z'), ..., R (wm, z, z'). * They are expounded fully in bis treatise already quoted (pp. 161, 169) ; and in that treatise full references will be found to the work of Ncsther, Enriques, Castelnuovo, Severi, Humbert, Berry, and others, in especial connection with the analytical developments associated with surfaces in ordinary real space. 101] EATIONAL FUNCTION 171 Solving these m linear equations for the m functions R {w.y, z, z), we have 1 , 1 , ..., 1 R{wi, z, z') = Po(z,/) , 1 ,..., 1 Pi (Z, Z) , W.2 , ..., 10 n The determinant on the left-hand side is the product of the differences of all the roots of the fundamental equation /"(w, z, z') = regarded as an equation in w, and is usually denoted by so that, from this definition of ^, we have ± ^(Wi, W2, . . . , Wm) = (W'l - W2) (tU^ - W3) (lU^ - W,n) ^{w., ..., W„,,). On the right-hand side, each of the quantities Pr (z, z) has, as its coefficient, a determinant of the roots ?^2) •••; '^^in\ and in each case, this determinant can be expressed as a product of t,{w.2, ..., Wm) and a symmetric function of W2, ..., Wjn- Thus the coefficient of Po{z, z) is w^Wz-.. w^y^^^w^, — w^); f ^ 1 \ the coefficient of P^{z, z') is —Wo,W3 ...w„A S — ) ^(ifa, ..., w^\ and so on. Hence dividing out by ^{w^, ..., w,„), we have (wi — Wo) (lu-i — Ws) . . . (Wi — w,„) R (wj , z, z') ^^ -^0^0 ~r -I i^i -f- ... -f- r^ji—iSni—i, where Sq, s^, ..., s^-i are the symmetric functions of Wg, ..., w^n- Now by the algebraic equation /(w, z, z) = 0, each symmetric function of W2, •••, Wjn can be expressed as a polynomial in w^, having rational functions of z for its coefficients. Also A (lU, - W2) (W, -Ws) ... (Wi - Wm) = f ^J Avhere A is the coefficient of ■i^i"* in f(w, z, z). Hence (J-\ R (lu^, z, z) = (tu^, z, z'), where @ is a polynomial in w^, which can always be made of degree ^m — 1 by use of the equation /(i(;, z, /) = ; and the coefficients in this polynomial are rational functions of z and z'. A corresponding expression holds for each of the functions R {w.j, z, z'), ..., R (Wm, z, z), all the polynomials (w, z, z) having the same coefficients in the form of rational functions of z and /. Consequently, when we denote any root of our algebraic equation f{w,z,z') = ^ simply by w, any rational function R {w, z, z') of all the variables can be expressed in the form % (w, z, z) R (w, z, z) — dw 172 CANONICAL FORM OF A [CH. VI where S {w, z, z') is a polynomial in w of degree is a polynomial in w-^, having rational functions of z and / for its dW coefficients. Multiplying the denominator and the numerator by ^ — , , we have R. ^{w„z,z)^, dWdW dwx 9wi' S (tv^ , Wi, z, z') dWdW ' 102] RATIONAL FUNCTION 175 where >!:? is a polynomial in Wi and w/, having rational functions of 2 and z' for its coefficients. But dWdW_ and therefore P _ S(w^, tv,', z, z) 1 Now is a symmetric function of %v-^ and wl,XDn, and w.i, ..., the pairs of congruous roots ; and it is therefore expressible as a^ rational function of z and /, say Similarly Ao... A,^,H is a symmetric function of all the congruous pairs of roots other than the pair Wi and w/; hence it is expressible as a polynomial function of Wj, w/, having rational functions of z and / for its coefficients, say Ao . . . A^n = Q (Wi, w-[, z, z'\ Consequently Hence n,= j^^ Q,{w^, w^, z, z) Ai T{z,z') S(iUj, Wi, z, z) Q {w^, lUi, z, /) ~~ T{z,z')J, ®{Wi, Wi, z, z') on multiplying the polynomials 8 and Q, and absorbing the rational function T (z, z') into the coefficients of the product. The same conclusion holds for every congruous pair of roots. We there- fore infer that every function, rational in the algebraic field of w, w', z, z , where w and w' are given by algebraic equations /(w, w , z, z') = 0, g {iv, w', z, z') = 0, can be expressed in the form f) (w, tu', z, z) J f,fl \W, w where @ is polynomial in w and w', having rational functions of z and z for its coefficients. Modifications in the degree of @ in w and of its degree in w may some- times be effected by the use of the equations f=0 and g = 0. These modifications, when they are possible, do not affect the denominator J, and only give equivalent expressions for the polynomial @ ; it is for this reason that the form is called canonical, even though the expression for @ may happen to be not unique. 176 CANONICAL FORM OF A [CH. VI Note. In establishing the preceding form for the rational function, two theorems concerning symmetric functions have been quoted. In actual practice, we can proceed as follows. Take t = \w-\-Xw' ; eliminate w from /and g, so that they become Fit, w\ z, z') = 0, O {t, w', z, 2') = 0, of the same degrees in t and w' combined as are / and g respectively. Eliminate to' between F=Q and 6^ = 0, so as to give an equation r=o, of degree mn in t, having rational functions (frequently polynomial functions) of z and / for its coefficients. In the product AiA2...A„j„, we have symmetric functions of the congruous pairs of roots ; let such an one be where the summation is over all the like terms obtained by permuting the congruous pairs in all possible ways. We then form the symmetric function of the roots of the equation T=0 represented by 2 1 '"i+'^i t '"^^'^^^ In its expression we select the coefficient of and remove the multinomial numerical factor nil ' ^i! ' "^2! '>H^- the result is the symmetric function required. Again, in the product A2...Am„, we have symmetric functions of all the congruous pairs of roots except only the pair Wi and w/. Let T={t-ti)T', so that «25 ■■■■> imn are the roots of T'=0. The coefficients in T' are linear in the coefficients of T and are polynomials in t^ ; thus, if T= 00 1""' + di it"™-! + 62 i;""'"^ +.-.■, T' = 6q if'""- 1 + ^1 i;'"""2 + 02 ^™™-3 + _ _ _ ^ we have (^l-ti6Q = 6i, <^2-^l<^] = ^2, 03-^102 = ^3, •■•, and therefore 01 = ^1 + ^1^0, 02 = ^2 + «l^l^o + ^^l^^o^ 03 = ^3 + h dojo+h'^dido'+ti^ ^0^ and so on. As was the case with AiA2..-A„i„, which is a sum of coefficients in a polynomial function of the coefficients of T divided by a power of ^0, so also the symmetric product A2 . . . A^ji is a sum of coefficients of powers of X and X' in a polynomial function of the coefficients of T' divided by a power of ^0 ; that is, A2 . . . A,„,j is a polynomial fimction of the coefficients of 7", itself also polynomial in ti (that is, in Wi and Wi) divided by a power of ^0- These are the two theorems used. 102] RATIONAL FUNCTION 177 Ex. For particular equations, a given rational function is most easily discussed in an initial form, not in a canonical form ; it is for the general theory that a canonical form is required, as it includes all rational functions. We may however take an example, to shew the outline of the reduction to a canonical form ; but the process is only an exercise in algebra. Let the two fundamental equations be f^vfi - w'^- A=0, g = w'^ + io"'- B = 0, where A and £ are given functions of z and z' only. Their Jacobian J, on the omission of a factor 6, is J=ww' {w + w'). We take the simple rational function Ii = Til where Z is any rational function of z and s' ; and we proceed to express it in a canonical form P {w, w\ z, ^') J where P is a polynomial in w and w\ having rational functions of z and 2' for its coefficients. The TF-eliminant of / and g is Let \o-\-Z=t; then the six values of t are given by the equation 2 {t-Zf - ZB {t~Zf - 2A {t-Zf+3B^ {t - Zf + A^- B^ = 0. Let e = 2Z<^ -3BZ*+2AZ^+ZB^Z^- + A^ - B% being the term independent of t in the last equation ; then t w + Z w + Z iv+Z w + Z = 2t(fi - 2Zw^ + {2Z^- SB) w3 + {3BZ- 2A - 2Z^) w^ + (3^ - 35Z+2^Z+ 2Z*) w^ZBZ^ - 7,BZ- 2AZ^ - 2Z^ = $, say. All terms in the right-hand side, which are of degree six and higher, can be removed by using the equation TFj = 0. These terms are 2w^ + {2wi-2Z)wi^. The term 2w-^ is to be replaced by S^wi^ + 2Awi^ - SBhvi^ - (42 _ ^3) y;j ^ and the terms {2wi — 2Z) Wi^ by (wi' - Z) {SBw^i + 2Awi^ - 352 j^^2 _ ^2 + ^3}. When these changes are made, let the expression for *i be *1 = PO 'iff + pi Wl* + p2 W^ + P3 W^ + P4 ^t*! + P5l F. 12 Consequently 178 picard's [ch. VI where the coeflQcients p are polynomial in iv', and are rational in z and z'. Then finally, absorbing the rational function of z and z' represented by - - into the coefficients of *i , we have w+Z J \e e e e e ej which is of the required tyi^e. Equivalent forms are obtained for the numerator by using the equations /=0, ^ = 0. Integrals of algebraic functions. 103. The development of the theoiy of integrals, whether single or double, of algebraic functions when there are two independent complex variables, owes its main foundations to Picard*. Here I shall only restate one or two of the simplest results for the case when there are two initial fundamental algebraic equations f(w, w', z, z) = 0, g (w, w', z, z') = 0, defining two dependent variables w and w' as algebraic functions of z and z', the quantities / and g being poljmomial in all their arguments. Writing owdw aw dw \w, w J we have seen that any rational function of all the variables can be expressed in the form {w, w', z, z') J{w, w) where © (w, w', z, z) is a polynomial in w and w' having rational functions of z and z for its coefficients. Accordingly, following Picard, we take our most general single integral of algebraic functions in the form 'Zdz'-Z'dz J{w, w') where Z and Z' possess the same general form as the preceding function @. Integrals of this form are said to be of the first kind when, on the analogy of Abelian integrals, they have no infinities anywhere in the whole field of variation. Picard proves f that no integral of the first kind exists in connection with a single equation F(w, z, z) = 0, when this single equation is quite general; and he shews J that, when such an integral does exist in cojinection with a less general single equation F (w, z, z) = 0, the form of * A full and consecutive account of his researches is contained in his treatise already quoted, t His treatise, vol. i, p. 113. + Ih., p. 118. 103] SINGLE INTEGRALS 179 the subject of integration must satisfy special preliminary relations, even though these necessary relations are not of themselves sufficient to secure the existence of the integral. Here I shall proceed only so far as to obtain the corresponding necessary preliminary relations affecting the form of the subject of integration in the foregoing single integral, if it is to exist in connection with the two equations / = 0, g = 0. The quantities Z and Z' are pol5momial in w and w' ; we proceed to shew that, if the integral is everywhere finite, they must be polynomial also in z and /, of limited order. The coefficients of the various combinations of powers of w and w' are certainly rational functions of z and z' ; let any such coefficient be S (z, z') where R and 8 denote poljoiomials in z and z', and consider the integral Zdz ! J ' Assigning any parametric value to z, let z' = c' be a zero of R (z, z') for that value of z. (If there is no such zero, i.e., if i^ is a function of z only, the zeros of R would make the integral infinite : so that, for our purpose, R would then have to be constant). For that parametric value of z, let the subject of integration be expanded in powers of z' —c' ; then, whether z' = c' does or does not give a zero value to J, the subject of integration is — for every set of values of w and w' — of the form A A A r-, — ^„ + -f-' — ^~h^^ + • • • + / ^ , + regular function of z - c', {z —c y {z — cy '■ z — c in the immediate vicinity of z'=c', the positive integer s being ^1. The integral would be infinite at / = c', unless all the quantities A-^, ..., Ag vanish. These quantities involve the parametric value of z ; they can only vanish for all parametric values by vanishing identically, that is, by having no powers of z' — c' with negative indices. Hence the polynomial R (z, z'), for any parametric value of z, can have no zero for a value of z'. It thus cannot involve / ; we have seen that it cannot be a function of z alone ; hence R {z, z') is a constant. The coefficient in question is a polynomial in z and z'. Similarly for every coefficient in either Z or Z' in the integrals I Zdz [ Z'dz J J ' J J ' Consequently the quantities Z and Z' are polynomial in all four arguments w, w' , z, z. And we know that / is polynomial in those four arguments. Next, as regards the limitations upon the orders of these polynomials Z and Z\ we shall assume that / (w, w , z, z') is a quite general polynomial 12—2 180 SINGLE INTEGRALS OF [CH. VI of order m in the four arguments combined, and that g{w, w', z, z) is a similar polynomial of order n. Then J is a polynomial of order m + n — 2. It is easy to see, by an argument similar to the preceding argument, that integrals cannot be finite for infinite values of z and of z', if the order of the polynomials Z and Z' in all the four arguments combined is greater than in + n — 4. We therefore infer, as a first condition, that if the integral is to be finite at all places in the whole field of variation, Z and Z' must be polynomial in all the four variables of order ^m + n — 4, when / is the most general poly- nomial of order m and g is the most general polynomial of order n. 104. The independent variables for the integrals have been taken to be z and z ; but any two of the variables may thus be chosen, and the integral must still remain finite. We proceed to give the corresponding and equivalent expressions. We have j-dw + ^, dw' -Y ^dz + j-, dz'^ 0, dw ow oz oz ^dw + ^, dw' +^ dz + ^, dz= 0, ow ow oz oz so that, on the elimination of dw', dw, dz, dz in turn, J (w, w') div + J (z, w')dz + J (/, lu') dz = 0, / {w', w) dw'-\- J {z, w )dz + J {z , lu ) dz' = 0, J (w, z) diu + J (w', z) dw' + J (z, z )dz' = 0, J (w, z )dn} + J {w, z) dw' + J{z , z' )dz =0. Using the first of these relations to substitute diu for dz' in the difierential element, we have Zdz' — Z'dz Z'dz Z '\ j , t/ '\ ^ i 7" [J {tu, w ) dw + J{z,w) azj J (w, w') J (w, lu) J {w, w') J {z', w') - Zdw Z'J (/, iv') + ZJ (z, w') J {z', w) J{w, w') J {z , w') The differential element now is to be Wdz-Z dw J (z', tu') ' where TT is a polynomial in all the four variables ; we therefore take ZJ{z, w) + Z'J{z', w') + WJ{w, iv') = 0. Similarly, when we make z and w' the independent variables the differential element of the integral of the first kind is Zdw'-W'dz J(z', w) ' 104] THE FIRST KIND 181 where 1^' is a polynomial in all the four variables, and ZJ{z, w) + Z'J(/, w)+ W'J(w, w) = 0. In the same way, we can take any pair out of the four as the independent variables, and thus obtain six expressions in all for the subject of integration. The six expressions are Zdz' - Z'dz Wdz - Zdw Z'dw - Wdz' J(tv,w') ' J{z',w') ' J{z,iv') ' Zdw' -Wdz W'dw-Wdw' Wdz - Z'dw J{z',w) ' J{z',z) ' J{z,w) ' and the relations connecting the polynomials are ZJ{z, lu') + Z'Jiz', w') + WJ {w, w') = 0, ZJ (z, w)+Z'J{z', w) + WJ(w', w) = 0, Z'J{z', z) + WJ{w, z) + W'J{w', z) = 0, ZJ {z, z') + WJ{w, z') + WJ{w', z') = 0, which are always subject to the two fundamental equations /=0, g = 0, and are equivalent to only two independent equations. Writing M=Z^/ + Z'^^+wf + W^,, oz cz dw ow OZ oz ow ow we can express the first of the four equations in the form \ OW J OW \ ow J ow that is. The others similarly give ow ow OW ow jf^-iyrf = 0, oz oz dz' dz' 182 FORMAL [CH. VI The fundamental equations /= and ^r = are independent of one another ; hence we must have M=0, iV-0, that is, the polynomials Z, Z', W, W are such that aw ow 02 dz dw OW oz oz But these equations are not satisfied necessarily as identities ; they need only be satisfied in virtue of the permanent equations /=0, g = 0. These relations impose limitations upon the forms of the polynomials Z, Z', W, W, which occur in the differential element of an integral of the first kind. 105. Limitations arise firom two other causes. The first of these causes lies in the requirement that the condition of exact integrability shall be satisfied. As regards this condition, we shall take it for one of the forms of the integral, and shall reduce it to an expression symmetrical in all the variables. The condition, that Zdz' - Z'dz J {w, w') shall be a perfect differential, is dz\j)^dz'\jj • Now since we have dz dw dz dw' dz dg dg dw dg_ dw' _ ^ dz dw dz dw' dz J{w, w) i^ + Jiz, w') = 0, J{w', '«^)-o- +J(z, ^y) = 0; and similarly J (w, w) ^ + J{z', w') = 0, J{w', tv) ^ + J (z, w) = 0. 105] LIMITATIONS 183 The condition of integrability is therefore ^; ',, [dZ dZ'\ [dZ T , ,, dZ' , , , ,.) 'dZ \dw' T / \ ^Z T / I \ 1 „ (dJ(w, w') J {z, w) dJ(w, w') J (z, w) dJ (w, w') \ dz J {w,w') dw J{w,w') dw' „,\dJ(w, w') J(z', w') dJ(w, w') J {z', w) dJ(w, w')) _ „ ( dz' J(u', w') dw J{w, w') dw ) ' and it suffices that this condition should be satisfied in virtue of the governing equations /= and g = 0. Now, for appropriate polynomials A and B, we have ZJ(z, tu') + Z'J(z', iv') + WJ (w, w') = Af+ Bg, identically ; and so for our purpose, where the governing equations persist, we can take dW _ dZ J{z,w') dZ' J{z',w ') Z dJ{z,w') Z' dJ{z, w') diu divJ(w,w') dwJ{w,w') J{w,w') dw J{w,w') dw ZJ(z, w') + Z'J{z',w') dJ (w, w') A df_ B dg J^ (w, lu') dw J{w, w') dw J{w, w) dw ' the omitted terms vanishing in virtue of/= and g = 0. Similarly, for appropriate polynomials C and D, we have ZJ (z, w) + Z'J {z, lu) -W'J {w, w') = Cf+ Dg ; and we similarly infer the corresponding relation dW _ dZ J{z,w) dZ' J{z\w) Z dJ(z, w) Z' dJjz, w) dw' dtu' J{tu,w') div' dw' J{iv,w') dw' J{w,w') dw' _ ZJ{z,w) + Z'J{z',w) dJ(w,w') _ G df_ __ D dg_ J^(w,w') dtv' J(w,w')diu' J(w,w')dw'' the omitted terms vanishing for the same reason as before. Also we have dJ {w, w') dJ{w', z) dJ{z, 'w) _ „ dz dw dw' identically, together with three similar relations by omitting z, w, w' in turn from the set of four variables. Moreover J(z, w) J {z , iv) + J{z' , w) J{w' , z) + J(w', w) J{z, z") = 0, 184 SINGLE INTEGRALS OF THE [CH. VI also identically. Using the foregoing relations, we have ^J{w,w') dJ{z, w) dJ{w', z) \ I dz dw' dw ] ^, { dJ(w,w') ^ dJ{z',w) ^ dJ{w',z') )^ \ dz' dw' dw I ' that is, the relation dz dw dii} J{w,w')\ dw dw dw' dw') is satisfied in connection with the governing equations /=0, 5r=0. Now we know that, in virtue of the governing equations, the quantities tz^l, ^Z% dz dz vanish ; hence polynomials F, E, H, G (any one or more of which may be zero) exist such that the equations are satisfied identically. These equations give ZJ (.. »') + Z'J (/. »') + WJ (w. «■) = {¥ % - H ^) f+(E^-G P, g dw' dw'J- \ dw' dw' ) satisfied identically. But the left-hand side is identically equal to hence, subject to the governing equations, we must have dw dw dw dw Similarly, subject to the governing equations, we have Consequently always subject to the governing equations /= 0, ^ = dw dw dw aw ^l-''^'-^^^^' -')- ^a^- ^ a^= «•'("'• »■>■ 106] FIRST KIND 185 Thus the equations become zl!l + Z'^/r+Wp- + W'^. = H/+Og 02 oz ow dw dZ dZ^ dW 9^'^j^ Q dz dz dw dw' The first two of these equations are satisfied identically; the third only needs to be satisfied in connection with/=0, g = 0. They are the extension of Picard's equations* which are given for the case when there is only a single equation f(w,z,/) = 0. Picard's equations are derived from the foregoing set, by taking g = w' =0 as the second of our fundamental equations, together with W' = 0, E = 0, H=0, G = 0; and then, owing to the order of F, the third of the equations is satisfied identically. It thus appears that, when there are two equations /= and g = 0, the exact differential can be presented in six forms ; that four quantities Z, Z', W, W, each polynomial in all the four variables, occur in these forms ; and that there are other four polynomials E, F, G, H, such that the foregoing three equations exist, the first two being satisfied identically, while the third only needs to be satisfied concurrently with the governing equations f— and g = 0. 106. It can easily be seen that, when /= is a quite general equation of order m and ^f = is a quite general equation of order n, the conditions required cannot be satisfied. Let JS^(p) denote the number of terms in the most general polynomial, which is of order p in w, w', z, z , so that NKv) =. i-, (i? + 1) (i9 + 2) (p + 3) (p + 4). We have seen (§ 102) that the polynomial Z, which (§ 103) can be of order 'm + n — 4i, is subject to modification by use of the equations /= and g = 0: * I. c, t. i, ch. V, § 4. 186 SINGLE INTEGRALS [CH. VI that is, it is subject to an additive quantity Af+ Bg, where A and B are quite general polynomials of orders w — 4 and w — 4 respectively. Hence the number of disposable constants in Z effectively is N{m-\-n-^)- N{m-^)-N{n-^). Similarly as regards Z', W, W. Again, E, F, G, H are polynomials of order ^ 2m — 5, m + w — 5, m + ^ — 5, 2?i — 5 respectively. The expression Ff+Eg is unaltered by changing F into F+Jg and E into E — Jf, where / is a quite general polynomial of order m — 5 ; hence the number of disposable constants in F and E together is N (m + n-5) + N (2m -5) - N(m- 5). Similarly the number of disposable constants in O and H together is JV(m + n- 5) + N{2n - 5) - N{n - 5). The modifications in F and G do not affect the third condition, which has to be satisfied only concurrently with /= and g ^^0. Thus the total number of disposable constants is 4{i\^(m + 7i-4) - i\^(m-4) - N{n-^)] + N{m + n- 5) + i\^(2m-5) - JV(m-5) + N{m+n-5) + N{2n- 5) - N{n-b). The number of conditions to be satisfied in connection with the first identity is iV^(2m + n — 5), and the number in connection with the second identity i& N {m -\- '2n — 5). The third relation, which affects the polynomials F and G, only needs to be satisfied subject to the equations /= and g = 0; that is, subject to an additive quantity Cf+Dg on the right-hand side, where C and D are quite general polynomials of order n — b and in — 5 respectively ; consequently, the third relation requires N{m + n-b)- N{n-b)-N{m-b) conditions. Thus the total number of conditions is iV(2m+7i-5) + N(m+ 2n-5) + N{m + n-b) - N{n-5) - N{m-b). The excess of the number of conditions to be satisfied, above the number of disposable constants, is iV(2m + ?i-5) + N{m + 2n-b) + N{m + n-b) - N(n-5) - N{m-5) -4{iV"(m + w-4) -i\r(m-4) - JSr{n-4>)} -{N{m + n- 5) + N'(2m-5)- N{7n-5)} - {N{m + n-5) + N{2n - 5) - Ii(n - 5)}. 108 J DOUBLE INTEGEALS 187 When the values of the different numbers iV are inserted, this excess is easily found to be ^ rnn {20 {m - 1) (m - 2) + 18 (m - 1) {n - 1) + 20 {n -V){n- 2) + 24} - 1, which manifestly is positive when m > 1 and n>l. Accordingly, in general, the relations cannot be satisfied by the disposable constants, and so we infer the result :— When /=0 and g = are quite general equations, no single integral of the first kind connected with them exists: a result which obviously corresponds to the theorem of Picard already (§ 103) mentioned. It follows that, if an integral of the first kind is to exist in connection with two equations /= and ^ = 0, these equations must have special forms. Ex. Shew that all the preceding conditions for the existence of an integral of the first kind, in connection with the equations /= az + hto + C2V + divzz' + ew'z^ +fivh' + giviv'z + hiuho' = 0, g = a'z' + h'w' + c'zz'^ + d'w'zzf + e'w/^ +f'w'H -\-g'w\o'z' + h'ivw'\= 0, where the coefficients a, ..., k, a', ..., h' are constants, are satisfied when Z=z, Z'=-z', W^io, W'=-iv'. 107. The second class of conditions, mentioned at the beginning of § 105 as required to be satisfied in order that the single integral may be everywhere finite, depends upon the places where we have \w, w ) which is not an identity, simultaneously with /=0, g = ^. As already indicated (§ 103), I do not propose here to enter upon any discussion of these conditions. The discussion will be difficult, but it is of supreme importance as regards even the existence of these integrals of the first order, as well as for all other single integrals. It can be initiated analytically on the lines of Picard's investigations in his treatise already quoted. It will involve the algebraical singularities of w and w as algebraic functions defined by the two fundamental equations. Double Integrals. 108. The discussion of double integrals follows a different trend. There is no limitation corresponding to the condition that must be fulfilled if the element of the integral is to be a complete differential element, as in § 105. 188 DOUBLE [CH. VI We have seen (§ 102) that, when two algebraic functions of z and z' are simultaneously given by two algebraic equations f=f{iu, w' , z, /) = 0, g = g {w, w', z, z) = 0, the most general rational function of the variables can be expressed in the form @ {lU, lu', z, z) ' ■ /(M \iv, w where © is a polynomial in w and tu', the coefficients in this polynomial being rational functions of z and z\ Thus the typical double integral, con- nected with the algebraical equations /=0 and ^ = 0, is of the form %{w,w,z,z')^^^^,_ J w, w the integration extends over a two-fold continuum. To express the integral more definitely, we take z and / as functions of two real variables 'p and q, as in § 95 ; and then the expression of the integral becomes % {lU, %v\ z, z) fz^\ , w, w where the integration can be regarded as extending over an area in the p, q plane, limited initially by a fixed curve (or curves) in that plane and finally by a variable curve (or curves) in that plane. The simplest case arises, when we have a single simple closed curve as the fixed initial limit and a single simple closed curve as the variable final limit. The first form of the preceding definition takes z and z as the independent variables for integration. As we have already suggested that it may be convenient to take any two of the four variables as the independent variables for integration, we proceed to give the equivalent forms. For this purpose we assume that, in order to express the quantities w, w', z, z in terms of real variables p and q, we take two algebraic equations F — F{w, w', z, z', p, q) = Q, G = G (w, w' , z, /, p, q) = 0, forms which will prove useful in attempting an extension of Abel's theorem for the sum of any number of algebraic integrals of a single variable. The simultaneous roots of the four equations /=0, g = 0, F=0, G=0, 108] INTEGRALS 189 are functions of p and q; so we have ^_dFdw dFdw' dFdz_ dF d/ dF dw dp dio' dp dz dp dz' dp dp ' Q_dGdjv d_Gd_v/ d_Gd_i d_Gd_z^ dG div dp dw' dp dz dp dz' dp dp ' §/; aw df_dw^ dfd_z_ d_f^d^ dw dp dw' dp dz dp dz' dp ' dg div dg dw' dg dz . dg dz dw dp dw' dp dz dp dz' dp ' \z, z ,w,w J dp \p, z ,w,w J \z,z ,w,w J dp \p,z,w,tuj \z, z ,iv,w J oq \q,z , w, w / \z, z , IV, IV J dq \q, z, tv, w Now, by the properties of determinants, we have and therefore Similarly 0. w / \p,q \p, z , w, w) \q,z,w, tu'J \z, z , w, w'J \w, i \z,z,w,wj \p,qJ \w,tv J \p,qj and therefore j(^_^\= -1 jf^^G j(f'9\ ^P>9/ j(Il^IiA1\ ^P'9^' \w, iv / \z, z, w, w'J The right-hand side is symmetrical, save as to signs, for the four variables z, z, w, w' ; hence it is equal to each of the six expressions .j[^ii:\^j(fi3\, _j('iii?v,/('4iL), ji^^\^jifii\^ \'p,qJ \w,w J \p,qJ \iv,zj \p, q J \z, z J ), qj \z , w / \p, q / \z,w/ \p,q/ \z,wJ p,q J \iv ,z J \p, q \p^ qj \z,wj' \p,qJ \z',wJ' \p,(l Accordingly, when the variables of integration in the double integral are taken to be p and q, there are six equivalent expressions of the integral ; one of them is the form first taken, and the other five are similarly constructed 190 DOUBLE [CH. VI from a comparison of the six foregoing quantities; and each of the six expressions so obtained is (save as to sign) equal to the double integral* j-/ F,G,f,g \ \p,q \z, Z , W, VJ J Double integrals of algebraic functions may be divided into various classes, following the analogy of the division of simple integrals of algebraic functions of a single variable ; but the analogy is little more than a sug- gestion, because (as has been seen in Chap, iv) a definite infinity of a function of two variables can be a one-fold continuum in the immediate vicinity of any one definite place of infinite value, and because unessential singularities (when the term is used in the sense defined in | 58) have no limited analogue even in the case of uniform functions of only a single variable. One class, how- ever, survives naturally in spite of the deficiencies in the analogy; it is composed of those integrals of algebraic functions which never acquire an infinite value, no matter how the two-fold continuum of integration is deformed. Such integrals are formally styled double integrals of the first kind. 109. The conditions, which must be satisfied by the double integral of an algebraic function connected with two given algebraic functions if it is to be of the first kind, are of four categories, according to the character of a place z, z' in relation to the subject of integration; and the four categories can be grouped in two pairs. It is manifest that a finite place z, z , which is ordinary for the equations /= and ^ = 0, and is also ordinary for the subject of integration, cannot give rise to an infinity of the integral. For near such a place w = a, w' = a, z = a, z = a', we have w = a + W, w' = a + W, z= a + Z, z' = a + Z' ; This integral can also be expressed in the form {w, w', z, z') j,KCi,.r,n Z, Z , W, IV which is the natural extension of the single integral - B{w, z) J dFdG, dd). The latter integral is fundamental in one of the proofs of Abel's theorem for the sum of a number of integrals R {iv, z) dw dz, when the upper limits of the integrals are given by the simultaneous roots of a permanent algebraic equation f {iv, z) = and a parametric algebraic equation Jn 9r\ \Zr,Zr',Wr,Wr') or, if we write = © (Wr, Wj , Zr, Zr') J ( - p,q (F C \ -^ -j = {Wr, Wr, Zr, Z./), ' \Zr, Zr, Wr, Wr / SO that 4> is a polynomial of aggregate order ^ k+ t-\- m + n~ 5, the integral (for this set of roots) becomes jj^/pdq. We assume the integral taken over any finite simple closed region in the p, q plane. * I.e., t. i, ch. vii. F. 13 194 EXTENSION OF [CH. VI Let W denote the result of eliminating w', z, z between /= 0, g = ^, F = 0, G = 0; the quantities w^, ...,w^ are the roots of W=0. The theory of elimination shews that we have a relation of the form W=Kf+Lg+MF+NG. Similarly, eliminating w, z, z , and denoting the eliminant by W , we have a relation of the form If' = K'f^ L'g + M'F+N'O, and the quantities w/, . . . , w^ are the roots of W = 0. Likewise eliminating w, w' , z', and w, w', z in turn, and denoting the respective eliminants by Z and Z', we have relations of the form Z = Ff+Qg + RF + SG, Z' = P'f+Q'g + RF + S'G; the quantities z-^, ..., z^ are the roots of Z = 0, and the quantities Zi, ..., z^' are the roots of Z' = 0. And the quantities K, L, M, N, K', L', M', N', P, Q, R, S, P', Q', R', S' are polynomials of the respective appropriate orders. In particular, if we write A= K, L, M, N K', U, M', N' P, Q,- R, S P', Q', R', S' A is a polynomial of aggregate order {mnpq — m) + (mnpq — n) + (mnpq —k) + (mnpq — I), = 4yL6 — m —n — k — l. The simultaneous combinations^ w^j '^r, Zy, z,! (for r = 1, ..., ix) are the simul- taneous roots of /=0, ^ = 0, F=Q, G = 0; these we call the congruous roots. All other combinations of the roots of W =Q, W = 0, Z = 0, Z' =0, are called non-congruous roots ; they are not simultaneous roots of /= 0, ^ = 0, ^ = 0, G = 0; but, for each such combina- tion, we have A = 0. For the sake of simplicity, we shall assume that each of the roots of W = 0, W = 0, Z=0, Z' =0, is simi^le. Now consider the quantity O ( w, w', z, z') A WWW' Ill] Abel's theorem 195 It can be expressed in a partial-fraction series of the form A ^'^ rr'ss' iw - Wr) {W - wV) {Z - Zs) {Z - z\) ' the summation being for r, r', s, s', = 1, ..., [jb, independently of one another; and ^{Wr, wV, z„ z's) A^/ss' -O. rr'ss' dW dW dZ dZ' dWy dw'r' dzg dz'o' When r = r =s = s', we can denote the coefficient A hy Ar] then A.r = dWdW dz dZ'' dwr dwJ dzr dzJ Unless all the equalities r = r' = s = s are satisfied, we have so that all the coefficients A other than A,., for r = l, ..., fi, vanish. Thus we have the identity 4)(w, %u,z, z')il _ ^ Ar W W'ZZ' r = l {W — Wr) {W' — W'r') (z — Z^) {z — z'r') ' Let both sides be expanded in ascending powers of 1/w, Ijw', 1/z, l/z'. On the left-hand side, the index of the terra of highest order in w, w , z, z in the numerator is ^k + l + m + n — b + (4/i. — in — n—k — l) ^4i/M- 5; the index of the term of highest order in w, w', z, z in the denominator is Afjb ; hence the index of the first term in the expansion ^ 5. On the right- hand side, the index of the first term in the expansion is — 4, and its coefficient is i Ar. r=l No such term can occur in the left-hand side under the assigned conditions ; hence /^ 1 A, = 0, r=l that is, ** cb A rZi dW d]r_ dZdz^ dWr dWr dZj. dZr 13—2 196 EXTENSION OF [CH. VI From the expression for W, we have U = Jo-r^ } + -Lr -5 >+ -'"r o > + -'^' r 5 , OWr OWr OWr OtUr A T;r df J. dg ,- dF ^^ dG = Kr ^ + Lr ^ + Mr ;r- +Nr ^ OZr VZf OZr vZj. = iT,./-, + Lr ^, +Mr^,^ Nr ^— , dz,' dZr dZr OZr and similarly from the expressions for W, Z, Z'. Thus dW dWr ' , , , dW dWr" , , , dZ dZr ' , , , dZ' dZr that is, Kr , Lr, Mr, N^ k;, Lr', m;, Nr' dW dW dZdz^_ dWr dWr dZr dz/ _df^ dg_ d^ dG dWr ' dWr ' dWr ' dWr df dg dF dG dWr" dWr' dWr" dWr df dg dF dG; dZr ' dZr ' dZr ' dZr df dg_ , dF dG dZr dZr dZr dZr ArJr. Consequently, we have and therefore when the double integration is taken over any simple closed region in the plane of the real variables p, q. Ill] Abel's theorem 197 This is a restricted extension of a part of Abel's general theorem on the sum of integrals. The result is true, even if the integral /] -J dpdq is not everywhere finite, that is, if the integral is not of the first kind*. The conditions, which have been imposed upon the integral, are that it is to be finite for all places which are ordinary for the equations /= 0, g = 0, all infinite places being supposed included among these ordinary places. * It should be added that, by a different method, Picard (I. c, t. i, p. 190) obtains this extension for double integrals of the first kind (that is, integrals which are everywhere finite) when there is a single fundamental equation f(w, z, z')=:0. CHAPTER VII Level Places of Two Uniform Functions 112. Hitherto, save for rare exceptions, only individual functions of two variables have been considered at any one time ; and we have seen that there exist continuous aggregates of places where a function has an assigned level value or a zero value. This property precludes us from establishing definite relations of inversion between a single function of more than one variable and the variables of that function. Such relations are highly important in various branches of the theory of functions of a single variable ; they are no less important when functions involve several independent variables. To establish them, it is necessary to have as many functions, independent of one another, as there are variables ; and therefore, for the present purpose, we shall consider two independent functions of z and z' . Moreover, quite apart from reasons that make inversion a possible necessity, we have seen that it is desirable to consider simultaneously two independent functions of z and /. We still shall limit ourselves throughout to uniform analytic functions ; and we shall begin with the discussion of the relations between two functions that are regular everywhere in the finite part of the field of variation. As we know, every such function can be expressed as a series of positive integral powers of z and z' , which (if an infinite series) converges absolutely for finite values of j ^ I and j ^' ! , and has all its essential singularities outside the finite part of the field of variation. We know (§ 53) that such a function must possess zeros somewhere in the field of variation ; but it may happen that the zeros do not occur in the finite part of the field*, and then they occur at the essential singularities. We proceed to establish the following theorem : — Two independent functions, regular throughout the finite part of the field of variation, vanish simultaneously at some place or places within the whole field. * For example, the function e^+^' cannot vanish for finite values of z and of z' ; all its zeros, a continuous aggregate, occur for those values of z and z' which make the real part of z + z' negative and infinite. 113] TWO SIMULTANEOUS FUNCTIONS 199 113. Let the two functions, everywhere regular, be denoted by f{z, z) and g (z, 2'); and let a, a be any place in the finite part of the whole field of variation for z and z . In view of the proposition to be established, it is reasonable to assume that neither y (2^, /) nor g {z, z) vanishes at a, a ; if both should vanish at a, a, the proposition needs no proof; if one of them should vanish at a, a, but not the other, the following proof will be found to cover the case. We consider the immediate vicinity of a, a, and take z = a+ u, z' =^a' + u'. Because f{z, z) and g {z, z) are regular everywhere in the finite part of the field of variation, we have expressions for them in the form /■(^, /) =/(a, a) +/(it, w')m ^M, «^')m+i + • • • > g (z, z) = g {a, a') + g{u, 11 )n + g{u, u%+i + •••, where f(u, u'),n represents the aggregate of terms of combined dimension in in u and u as contained in the power-series for/"; and similarly for the other homogeneous sets of terms inf, and for the homogeneous sets of terms in g. In the simplest cases, the integer m is unity and the integer n is unity ; in all cases, both the positive integers w and n are finite. When m= 1 and n = 1, the quantities f{U, U\, g{U, U'\, are usually independent linear combinations of u and 11 ; their determinant is the value, at a, a, of \z,z J which does not vanish everywhere, because the functions / and g are inde- pendent. If it should happen that / vanishes at a, a, so that there da ' da da' ' da then we have f{a + u, a' + u') —f(a, a) =f{u, u\ + ..., f{a + u, a' + u') -f(a, a') — K{g{a+u,a' + u) — g (a, a')] = g(u, 11% + ..., where the first set of terms g{i(, u'\ is of order higher than the first set f{u; w')a and usually is not the square of f{ii, u\. If, however, where X is a constant, then we should take a new combination f{a + u,a' + u) -f{a, a) - k {g (a + u. a' + u)-g (a, a')] - \ {/(a + u,a+ u') -f{a, a')}'- 200 MODULI OF [CH. VII Similarly for other cases. We proceed until, at some stage, we obtain two series in u and ?/, such that the lowest set of terms in one series cannot be expressed solely by means of the lowest set of terms in the other series; and this stage is attained after steps that are finite in number, because '&) does not vanish identically. Similarly, if m is greater than unity and n = l ; and if m = 1, while n is greater than unity ; and if both m and n are greater than unity. In each case, we obtain a couple of series, the aggregate of terms of lowest dimensions in the two series not being expressible solely in terms of one another. And then, because of this independence, the equations A =f{u, u')m, B^giu, u')n, where A and B are assigned quantities independent of u and u', determine a limited number of values of u and u. In particular, let I be the greatest common measure of m and n, and write m = fil, n = vl; and let E be the eliminant of /(<*, u)m and g{u, u')n, so that JiJ — n ^r in -L. ^ — "-mo ^on -r . . . . Then the equation giving values of w is (ttmo^Con"^ + . . .) M'"^ + . . . + {(- AConY " (" ^«mo)^}' = 0, and therefore, if A=kP^ = kP>'\ B = \P''=\P-^, each value of u is of the type ^ u = kP; or, for sufficiently small values of | m | , \A\, \u\, \B\, and so of | P | , we have u = kP, u'^k'P, where | k \ and | k' \ are finite, while some of the quantities k and ¥ can be zero. Manifestly, K = f{k, k')m, \ = g(k, k')n ; and, in general, we shall have u = kP + k,P^+ ... u' = k'P+k,'P' + ., from the relations A = f(u, u')m + f{u, u')m+i + ... B = g{u, U')n + g{u, U)n+i + . . . . 113] TWO UNIFORM FUNCTIONS 201 After these explanations and inferences, we proceed to shew that it is possible to choose quantities u and u' of small moduli, so that the place a + u, a' + u' is in a small domain of a, a', and so also that \f{a -\-u, a + u') I < |/(a-, a') \, \g {a-{- u, a' + iif) \<\g{a, a') I , simultaneously. Let /(a, aO = Q + iR, g {a, a')=S + iT, where Q, R, S, T are real quantities, and neither \Q i-iR] nor \S + iT\ vanishes. Now choose M a small positive quantity, in every case less than \Q+ iR\, unless \Q + iR\ happens to be zero and then we take M zero ; and choose an argument -v/r such that Q and M cos -v^ have opposite signs and, at the same time, R and ilf sin-v/r have opposite signs. (If R be zero, we can take -yjr equal to either or tt and should choose the value giving opposite signs to Q and M cos ylr. Similarly, if Q be zero, with a choice of ^tt or ^tt for yj/). Again, choose N a small positive quantity, in every case less than I *S' + *2^ I , unless \S + iT\ happens to be zero and then we take iV zero ; and choose an argument x such that S and iV cos % have opposite signs and, at the same time, T and N sin % have opposite signs. (Arrangements as to choice of x can be made similar to those for yjr, if either S or T should vanish). Then evidently \f(a,a') + Me'l^\<\f(a, a')\, \g{a, a') + N'e^^lKlgia, a')\. Now we have seen that, for sufficiently small values of M and of N, the relations Me'i'^ = f{u, u% + f(u, w'),„,+i + . . . , jS'e^^ = g{u, u')n +^(u,u%+i + ..., give a limited number of sets of values of the form u = kP + k\P''+ ...] u' = ¥P + k,'P'+...j' where | P | is a small magnitude such that Me^'^ = kP^, Ne^' = XP"^; thus I u I and | u' \ are small, of the same magnitude as \P\, while \k^P^+ ... \, 1 k^P'^ + . . . I, are small compared with [PL For such values, we have \f{a + u, a' + u) I < i/(a, a') | , \g{a + u, a +u')\ <\g (a, a') \ , which was to be proved. 202 COMMON ZEROS OF TWO UNIFORM FUNCTIONS [CH. VII Accordingly, we infer that it is possible to pass from a place a, a' to a place z, /, which may be called a place adjacent to a, a\ and which is such as to give the relations \f{z,^)\<\f{a,a')\, \g{z,^)\<\g{a,a')\, simultaneously. Within the finite part of the field of variation, the functions f{z, z') and g {z, /) are everywhere regular, so that no singularities are encountered in transitions from a place to an adjacent place. We therefore can pass from place to place within the finite part of the field of variation, always choosing the passage so as to give successively decreasing values of \f{z, z') \ and If at any place c, c', one of the two functions (but not both of them) should vanish — say/(c, c') = — then we choose the next place c-\-u, c' -Y u\ so that M is zero, that is, so that k is zero, and such that f{c + u, d + It') = 0, \g{c-\-u, d -\-'Vi^\<\g (c, c') \ . The choice is always possible for finite values of z and /, because the functions f{z, /) and g (z, /) are regular for those finite values and consequently can be expressed as regular power-series. 114. It thus follows that, by an appropriately determinate choice of successive places at every stage, each place being adjacent to its predecessor, the moduli oi f(z, z) and giz, z') can be continually decreased so long as they differ, either or both, from zero. Thus they tend to zero in value, as the successive places are chosen ; and continued decrease can be effected, so long as they are not zero. Moreover, we know that every regular function possesses a zero value or zero values somewhere within the whole field of variiation. If the zero value does not occur at some ordinary place, then (§ 53) it occurs at the essential singularity or singularities, as e.g. for the function e^^^'^'\ where P (z, z) is a polynomial in z and /, when the places for the zero values belong to the non-finite part of the field. Hence ultimately, either for finite values of z and /, or for infinite values of either of them or of both of them, a place will be attained at which both the moduli \f{z, z') \ and ] g (z, z) \ are zero. Such a place is a common zero o{ f{z,z') and g{z,z); and therefore our theorem — that two functions f{z,z') and g {z, z), regular everywhere in the finite part of the field of variation, vanish simultaneously somewhere in the whole field — is established. Ex. Consider the functions f{z,z') = e''*^', g{z,z') = ze-(^^''), both of which are regular for all finite values of z and z'. Let s-t-2' = log(r™e""^0» 115] LEVEL PLACES OF TWO UNIFORM FUNCTIONS 203 where r, 6, m, n are real constants ; then g{z, 0')=r(i-")e(i -"»)»*. When < ?i< 1, we manifestly have /(^,2')=0, g{z,^)=0, when r is zero : that is, the two suggested functions acquire zero values for some specified values of ^ (even when 3 = 0) which do not lie in the finite part of the field of variation of the two variables. 115. Next, consider the case of two uniform analytic functions, each of them devoid of essential singularities in the finite part of the field of variation, and each of them possessing continuous aggregates of poles and isolated unessential singularities. We know, from an earlier proposition (| 90), that the functions can be expressed in the forms J^^^^^- q{z,zy ^''^'^^~ s{z,/)' where P {z, z), Q {z, z), R(z, /), S{z, z) are functions of z and /, which are regular everywhere in the finite part of the field of variation. The zero-places o^f{z, z') are those of P (z, z) ; it may happen that a zero- place of P (z, z) is also a zero-place of Q {z, /), and then the place is an unessential singularity o^f{z, zf) which, among its unlimited set of values there, can acquire the value zero : that is, the zeros oif{z, zf) are given by the zeros of P (z, /). Similarly for g (z, z) and R (z, z). Hence f{z, z') and g (z, z') will vanish simultaneously somewhere in the field of variation, if the functions P {z, z') and R (z, /), everywhere regular in the finite part of the field, vanish simultaneously somewhere in the whole field. But we have proved that these regular functions P {z, z') and R {z, z') must vanish simultaneously at some place or at some places in the whole field. Hence we infer the following theorem : — Two independent functions f(z, /) and g{z, z'), which are uniform and analytic, and all the essential singularities of which occur only in the non-finite part of the field of variation, must vanish together at some place or some places in the whole field of variation. We infer also, as an immediate corollary, the following further theorem : — Two independent functions f{z, z') and g (z, /), which are uniform and analytic, and all the essential singidarities of which occur only in the non-finite part of the field of variation, must acquire assigned level values at some place or some places in the whole field of variation. For if the assigned level values be a for f{z, /) and j8 for g{z, z'), the functions f(z, /)— a and g (z, z')— /3 satisfy all the conditions imposed upon 204 MULTIPLICITY OF A [CH. VII the functions f(z, /) and g {z, z) in the earlier theorem ; the application of that earlier theorem leads to the result just stated. A corresponding result holds, as regards simultaneous poles for f{z, /) and g {z, z'). In general, a corresponding result does not hold as regards the occurrence of simultaneous unessential singularities Q>if{z, z) and g (z, /). 116. When two functions f(z, z') and g (z, z') have a common zero-place, we need to consider their relations to one another in its immediate vicinity ; we need also, if possible, to assign an integer which shall represent its multi- plicity as a common zero-place. Let a, a' be such a place, so that /(«, a') = 0, g(a,a') = 0; for places in its immediate vicinity, represented by a + u, a' + u, we have f{z, z') = Ku'u'' P {u, u') e^'«'"'' = Lu'u'tQ(u, w')eQ <"'«'' g (z, z) = K'u'u'^' R {u, u) e^(«.^'> = L'u'' 11 *' S {u, w') e'^'"''*') Here K, L, K', L' are constants ; s, t, s , t' are positive integers which can be zero separately or together ; P {u, u'), Q (u, u'), R (u, u'), S(u, u) are regular functions of u and u', which vanish with u and u'. The functions P (it, u') and R (u, u') are polynomials in u, having as their coefficients regular functions of u' which vanish with u ; the functions Q (u, u') and 8 (u, u') are polynomials in u\ having as their .coefficients regular functions of u which vanish with u. When u~^v!~^f{z, /) does not vanish with u and u , we substitute unity for each of the functions P and Q; and similarly when u~^u'~^ g{z, z) does not vanish with u and u\ we substitute unity for eacK of the functions R and 8. The order of a zero-place for a single function in each variable has already been defined. For the function/ (2', z'), it is s + m in z, t + n m. z, where m and n are the positive integers, which are the degrees of P and Q regarded respectively as polynomials in u and in u' ; and m and n are zero, only when u-h(f-^f{z, z) does not vanish with u and u'. For the function g{z, /), it is similarly s' + m! in z, t' -\- n in /, where m' and v! are the positive integers, which are the degrees of R and 8 regarded respectively as polynomials in u and in u' ; and m and n are zero, only when ir^'u'-^' g (z, z) does not vanish with u and u'. Beyond the factors ?/««<'* and u^'ti^', the relations of f{z, z) -AXid g{z, z) m the vicinity of a, a' depend upon the relations of the functions P or Q (as 117] COMMON ZERO 205 representative of /) and the functions R or S (as representative of g) to one another. Consider, in particular, the functions P (u, u') = u'^ + u^-^p, {u') + . . . + p^ {u'), where ^1, ..., p^ are regular functions oi ic', vanishing with u', and M (u, u') = w'"' + li""'-^ n ("0 + . . . + r^' (u), where r^, ..., Vm' are regular functions of «', vanishing with u'. To determine whether there are common sets of values of u and u', in the vicinity of t^ = and u = 0, where P and R vanish together, we take P = 0, P = 0, as simultaneous equations, algebraical in u. Eliminating u between them, we have (save in one case) a resultant which is a function of u' only ; also, as each of the quantities p^, ..., pj^, r^, ..., r^' is a regular function of u' vanishing with u', this resultant is of the form where M is a positive integer, chosen so that (f) {u'), a regular function of u', does not vanish when u' = 0. To the exact determination of M we shall return later. The excepted case arises when the resultant vanishes identically. When the resultant does not vanish identically, the necessary values of u', making P and R vanish together, are given by where <^ (0) is not zero and (^{u') is a regular function. We at once have u = 0, as a possibility ; the associated value of w is ti = 0. The alternative possibilities would arise through zeros of the regular function (^u) : but as (f> (0) is not zero, it is possible to assign a finite positive quantity e, less than the smallest among the moduli of the zeros of (p (u). In that case, there is no value of u within the range 1 1/ | ^ e such that (f) {u') vanishes; and then the resultant vanishes for no value of u' other than u =0: that is to say, there is no zero-place for /and g in the immediate vicinity of a, a , other than a, ol itself 117. When the resultant of the two equations P = and P = 0, which are algebraical in u, vanishes identically, the inference is that these two equations in u have common roots, one or more. Let the number of these common roots be I, and let them be the roots of an equation fT" = M^ + w^-i ^j (i/0 + . . . + A;; (it') = 0, where ^•l, ...,ki manifestly are regular functions of u' vanishing with u'. Then t7 is a factor of P save as to possible multiplication by a factor e*<"V where a {u') is a regular function of u' that vanishes with u' ; and similarly U 206 TWO FUNCTIONS [CH. VII is a factor of R, save as to a similar possible limitation. Let the quotient of P by [T be and let the quotient of R by U be where all the quantities /,, ...,fm-i, Qi, ..., gm'-i are regular functions of u', vanishing with u'. The conditions, necessary and sufficient to secure this result, are those which render the relation {u^-i + u^-i-^f, + . . . ^fm-i) (w"'' + y-'"''-^ gi + . . . + qm') an identity : viz. we must have the I independent determinants, each of m+m'—2l — l rows and ^1+ m — 21 — 1 columns (we assume m^m' for purposes of statement), which can be formed out of the array Pi-r^, V2-r^,Pz-ri, ...,pm'-'rm', Pm'+i, •••• , Pm, , 0,..., , 1 , n ,..., r^'_2 , Vm'-i, Vm', ..., , , , 0,0,0 , , gw 1 , Pi , P-2 , ■■■, Pm'-i , Pm' , ,Pm-i, Pm , ,..., ^ , J- J Pi > ! Pm—2> Pni—i > Pm > • • • > " , 0,0, , p^ vanishing identically for all values of u'. In actual practice with two given functions, we should in general experi- ence the same arithmetical difficulty as before (§§ 70, 71). Here we are concerned with the effect of the relative reducibility of the functions ; the foregoing are the I analytical conditions for this reducibility. When all the conditions for the identical evanescence of these I deter- minants are satisfied, P and R have a common factor II: and then all the 2;eros of U within the domain are also zeros of P and R. Now these zeros of U form a continuous aggregate, since C/" is a regular function ; for I values of u can be associated with any value of u in the domain so as to make U vanish. 118. It thus appears on the one hand that, when the resultant of P and R, regarded as polynomials in u, does not vanish identically, the zero-place ■a, a' is isolated : that is to say, simultaneous zero- values of P and R cannot be found, except at a, a, in a region given by \ z — a\^ e, \z' — a' \^e', 118] COMMON ZEROS 207 where e and e' are assigned positive quantities made as small as we please. And it appears on the other hand that, when the resultant of P and R, regarded as polynomials in u, does vanish identically, the zero-place a, a is not isolated. Moreover, in the case when P and R have a common factor U, we can write P = Up{u, u'), R= Uq {u, u), where all the functions P, R, IT, p, q are regular functions of u and u ; each of them vanishes when m = and u =0; and each of them is a polynomial in u, having unity as the coefficient of the highest power of u and, as coefficients of the succeeding powers of u, regular functions of u which vanish when u' = 0. From the construction of U, we may assume that p and q have no common factor ; so that the zero-place of jj and q at u = and u' = is isolated. Now J (ii^) = sj (p^) + pj (^,) + v'jm) . \u, u J \u, u I \u, u J \u, u 1 Hence the Jacobian of P and R vanishes for all the aggregate of places making U vanish, because all these places make P and R vanish. But this Jacobian does not vanish (except at a, a) for places in the domain of a, a , which make P and R vanish but leave U different from zero. Also, as f{z, z') = Ku'u'P (w, u') e^<«'«'' g {z, z') = Lu'u^'R (u, u') e^(«.«') it follows that the Jacobian of the independent regular functions f and g vanishes for all the aggregate of places making U vanish, while it does not vanish (except at a, a) for places in the domain of a, a' that make f and g vanish but leave U different from zero. These results have followed upon the selection of P {u, u') as the sig- nificant factor of f in the immediate domain of a, a', and of j^ (u, u') as the significant factor of g in the same domain. The same results follow upon a selection of Q{ii, it) and R (u, u') as the significant factors of/ and g; like- wise upon a selection of P {u, u') and S (u, u') as these factors, and upon a selection of Q {u, u) and S (u, u) as these factors. Gathering together all the results, we can summarise them as follows : — (i) Any two independent functions, uniform, analytic, and devoid of essential singularities in the finite part of the field of variation of the two variables z and, z', possess common zero-places somewhere within the field of variation : — (ii) In general, each common zero-place of two independent functions, which are uniform, analytic, and devoid of essential singularities in the finite part of the field of variation of z and z , is an isolated place, so far as concerns the vanishing of the two functions : — 208 RELATIVELY FREE FUNCTIONS [CH. VII (iii) Less generally, when two such independent functions possess a common factor, which is necessarily of the same character throughout the finite part of the field of variation and which itself vanishes at the common zero-place of the two functions, then the comrnon zero-place of the two functions is not isolated ; in its immediate vicinity, the two functions possess a continuous aggregate of zero-places which belong to the common factor : — (iv) The Jacohian J, of two independent functions f and g, does not vanish identically. It may vanish at a zero-place common to the two functions. When the common zero-place is isolated, then f g, and J do not simultaneously vanish at any other place in the immediate vicinity of that place. When the common zero-place is not isolated, then f, g, and J vanish simultaneously at a continuous aggregate of places in the immediate vicinity of the common zero-place. 119. In the preceding consideration of two functions/ (2^, /) and g{z, z') discussed simultaneously, there has been the fundamental assumption that the two functions are analytically independent of one another in the sense that neither of them can be expressed, either implicitly or explicitly, by any functional relation which, save for the occurrence of/ and g, is otherwise free from variable quantities. Were the assumption not justified, the Jacobian of the two functions would vanish identically; we then should not possess sufficient material for the consideration of the common characteristic pro- perties of/ and g as simultaneous functions of two variables. But, after the preceding explanations, two limitations can be introduced as regards a couple of functions. One of these affects them simultaneously : the other affects them individually : yet neither of them imposes limitations upon generality, for the purposes of this investigation. Our discussions will deal with any pair of regular functions, which are not merely independent in the general sense, but which possess the further quality that they have no common factor, itself a regular function and vanishing at places within the domain considered. For any such pair of regular functions, each simultaneous zero-place is isolated. The zero-place may be simple or it may be multiple ; when it is multiple, the multiplicity is represented by a definite positive integer. It will be convenient to use some epithet to imply that two independent regular functions, existing together in the domain of a place where they vanish, do not possess a common factor, which is itself a regular function in that domain and vanishes at the centre of the domain. When a common factor of that type is not possessed by a couple of such functions, they will be called free. If on the contrary they do possess a common factor of that type, they will be called tied. Accordingly, when we deal with a couple of regular functions simultaneously, they will be assumed to be both independent and free. 120] COMMON ZEROS 209 The other limitation aims at the exclusion of unessential complications, and is suggested by the most general form of a function f{z, z) in the immediate vicinity of a zero a, a', viz. f{z, z') = K{z~ ay (/ - aj P{z-a,z'- a') e^(2-«.2'-«'). Thus {z — ay is a factor off{z, z') : at another zero c, c', it could have another factor (z — cY ; that is, it would have a factor (z — a)* (z — cy. And so on, for other zeros. We shall assume that, if /(^, /) initially possesses a factor which is a function of ^ alone, then/(^, /) is modified by the removal of that factor in z alone. Similarly, of course, if it initially possesses a factor which is a function of / alone, then we shall assume it to be modified by the removal of that factor also. Any such factor of either variable alone can only contribute properties characteristic of a function of a single variable. Thus, for instance, we should not consider ^J (z) p (z), where the periods of ^j (z) are unaifected by the periods of ^o (/), as a proper quadru ply-periodic function ; we should not consider ^ (z) sin / as a proper triply -periodic function ; we should not consider sin z sin z' as a proper doubly-periodic function. It seems unnecessary to introduce an epithet to indicate the non -composite character of a function /(^, /); in what follows, we shall assume that we are dealing with functions which are of this non-composite character. Accordingly we can enunciate the theorem : — The common zero-places of two functions of z and z', which are uniform, analytic, and devoid of essential singularities in the finite part of the field of variation, and luhich are independent and free, are isolated places in the field of variation. 120. An indication has been given of the determination of the integer which shall represent the multiplicity of an isolated simultaneous zero-place of two regular functions. In the vicinity of such a place a, a', we take z = a + u, z' = a' -{- 11 ; and then, after the preceding explanations, we can assume that the integers s and t are zero iox f{z, z), and that the integers s' and i! are zero for g (z, z). Thus f{z, z') = KP {u, u') e^(«.«'), g {z, z') = LR (u, u') e^<»=«'), in the immediate vicinity of u = 0, iif = ; and F {U, U) = U»'+ U'^-^p^ (uf) + ...+pm. (ll), R (u, ll') = u"^' + u"^'-^ n {u') + . . . + r^' (u'), where all the coefiicients jSj, ..., p^n, r-^, ..., r^- are regular functions of u and vanish when it' = 0. When the eliminant of P {u, u) and R (u, a'), regarded as polynomials in u, is formed, it is a regular function of w' which vanishes when u' = 0; and so it can be expressed in a form u''^^ cf) (iif), F. 14 210 COMMON [CH. VII where M= ^ ^ p., r=l s=l which thus gives an expression for the multiplicity M. It is easily established that the quantity M, thus obtained, is an integer. The simplest case occurs when, in the expansions f{z, z) = ftio {z- a)+ ttoi (/ -a') + .. g (z, z) = Cio {z-a)+ Coi {z' - a')+ .. no one of the quantities a■^Q, aoi, Cio, Coi, aio Cqi — Cio cfoi vanishes: the value of M, for the zero a, a', is unity in this case. Note. If, instead of the functions P and R, we take Q and ;S', as repre- sentative of / and g, and construct the eliminant of Q and 8 regarded as polynomials in u, the eliminant is U^^ -v/r {u), where i/r is a regular function of w such that ^/^(O) is not zero, and if is the same integer as before. The proof is a simple matter of pure algebra. 121. All the preceding remarks apply to the simultaneous zero-places of two regular functions f{z, z) and g{z, z). It applies equally to the level values of two regular functions /(^, /) and g{z, z), say a and /3 respectively, where |aj and \^\ are finite. The functions f{z, z) and g{z, z) are inde- pendent, as before. The functions / (2^, /) - a and g (z, z) - /? will be supposed free, that is, we shall extend the significance of the epithet ' free,' as applied to f{z, z) and g (z, /), so that it applies to this case also. The functions f{z,z') — a. and g (z, z') — /S wiW also be supposed non-composite as regards 122] LEVEL PLACES 211 factors which are functions of z alone or functions of / alone, as was the case with f{z, /) and g {z, /). And, now, we can enunciate the theorem : — The common level places of tiuo regular functions, ivhich exist together in a domain of the variables, and which are independent and free, are isolated ; and the multiplicity of any level place, giving values a and /3 to f{z,. /) and g(z, z') respectively, is the multiplicity of the place, as a simultaneous zero of the functions f(z, z') — oi, g (z, z') — ^. 122. Further, consider two functions f{z, /) and g {z, /), independent of one another, not tied, and existing in a common domain ; and suppose that f(z, z') has a pole at a place p, p' , which is an ordinary place for g (z, z'), say a level place for g (z, z'), (zero being a possible level value there). Then the place is a common level place for the functions ^ (z, /) and g (z, /) ; and we know that, if (z, z') and g (z, z) — g {p, p'). It is convenient to extend the significance of the terms tied and free as applied to a couple of independent uniform functions / and g. We shall say that they are tied if, for any constant quantities a and ^, either /— a and g — ^; or/— a and (g — ^)~^; or (f—a)-^ and g — ^', or (/— a)~^ and {g — /3)~^ (being really two alternatives) possess a common factor which is a regular function of z and z' having a zero (and so an infinitude of zeros) in the domain ; and we shall say that the two independent functions f and g are free, when no common factor of that type exists for any one of the combina- tions. Moreover, we shall also assume that neither/— a nor (/— ct)~^ nor g ~ ^ nor {g — /3)~^ contains any factor, which is a regular function of z alone or of z alone and vanishes for one (or for more than one) finite value of the variable. On the basis of earlier results, we can now enunciate the following theorems : — (i) Let f{z, z) and g (z, z') be two functions, which are uniform, analytic, and devoid of essential singidarities in the finite part of the field of variation of z and /, and which are independent and free. The places luhere one of the functions acquires a level value and where the other has a pole, are isolated; and the multiplicity of the place for the two functions conjointly is the multi- plicity of the place as a level- and-zero place for one of the functions and the reciprocal of the other. (ii) The common poles of two uniform functions, which exist together in a domain of the variables, and which are independent and free, are isolated ; and the multiplicity of the common pole for the two functions conjointly is the 14—2 212 COMMON LEVEL PLACES [CH, VII uiultiplicity of the place as a common zero for the reciprocals of the two functions jointly. The theorems follow at once from an earlier theorem by considering the behaviour of the reciprocal of a function in the immediate vicinity of any pole of the function. When we extend the term level value of a uniform function to include (i) a zero value of the function, this being a unique zero, independent of the way in which the variables reach the place giving the zero value : (ii) a level value a of the function, where | a | is finite, this being a similarly unique level value of the function : (iii) an infinite value of the function, this being a unique infinity of the function arising at a pole : then all the theorems, already enunciated concerning two functions, can be summarised in the one theorem : — The common level places of two uniform functions, which are uniform, analytic, and devoid of essential singularities in the finite part of the field of variation of z and z, and which are independent and free, are isolated ; and the multiplicity of the level place for the two functions conjointly is the index of the lowest terin in the eliminant of the two functions or of their reciprocals or of either with the reciprocal of the other, expressed in the vicinity of the place. Combining this result with the investigation, which settled the order of multiplicity of the place a, a' as a level place of the functions / and g and therefore as a zero of the functions f{z,z')-a, g{z,z')-^, we have the followang corollary : — Let a, a' he an isolated common zero of multiplicity M of the functions f{z,z)-a., g{z,z')-^: then, for values of \cl \ and \ /3' \ sufiiciently small, there are common zeros, sim,ple or multiple, of aggregate multiplicity M, of the functions f{z,z')-a-o!, g{z,z')-^-^, which coalesce into the single common zero of multiplicity M of f{z,z)-a, g{z,z')-j3, when ct and /3' vanish. CHAPTER VIII Uniform Periodic FuiJctions 123. We now proceed to consider the property, of such functions as possess the property, which customarily is called periodicity. Limitation will be made at this stage to periodicity of the type that is linear and additive, though the type is only a very particular form of the general automorphic property, mentioned in Chapter il. In conformity with general usage, we say that two constant quantities w and &)' are periods, or a period-pair, or a period, of a function f{z, z') of two complex variables, when the relation f{z + a>,z'-¥co')=f{z,z') is satisfied for all values of z and of /. In such an event, the relation f{z + SCO, / + soi') =f{z, z) is satisfied for all integer values, positive and negative, of s. Moreover, it is assumed implicitly that &> and w constitute a proper period-pair ; that is to say, a relation f{z + hw, z -h k'w) =f{z, z) is not satisfied for all values of z and z except when k = h', both k and k' being integers, and that the same relation is not satisfied, even if A; = k', when the common value of k and k' is the reciprocal of an integer. In dealing with periodic functions of a single complex variable, infinitesimal periods are excluded. Speaking generally, we could say* that, if a uniform function of a single variable possessed an infinitesimal period, then within any finite region, however small, round any point, however arbitrary, the function would acquire the same value an unlimited number of times. The possibility of the existence of such functions may not be denied ; but they cannot belong to the class of analytic functions. In the case of analytic functions which are not mere constants, the result of the possession of infinitesimal periods would be to make practically any point and every point an essential singularity. Accordingly, so far as concerns functions of a single variable, the possibility of infinitesimal periods is excluded. 124. We likewise exclude the possibility of infinitesimal periods for functions of two variables ; but the exclusion can be based on different * See my Theory of Functions, § 105. 214 INFINITESIMAL [CH. VIII grounds also. For the present purpose, we shall limit ourselves to uniform analytic functions* of two variables; and we then have a theoremf, due to Weierstrass, as follows : — A uniform analytic function of two independent complex variables z and z' possesses infiniteshnal periods only if it can he expressed as a function of az + a'z', where a and a are any constants. First, suppose that our function/ (2^, /) can be expressed in a form f{z,z')=^F{az-\-a'z'). Then if we take any two quantities P and P' such that aP + a'P' = 0, we have f{z + P, / + P') =F(az + a'z' + aP + a'P') = F{az + a'z') and therefore when P and P' are constants, we may regard P and P' as a period-pair for f(z, z), supposed expressible in the given form. The only relation between P and P' is aP + a'P'=0; hence either of them can be taken infinitesimally small, and the other then is infinitesimally small also. It follows that, when a function of z and z can be expressed in the form of a function of az + a'z' alone, where a and a! are any constants, then it possesses infinitesimal periods. Further, writing az + a'z' = v, we have and therefore "^z dv ' dz' dv ' af-af,=0. dz dz Hence when the function is of, the form f(az + a'z'), so that it possesses infinitesimal periods, the foregoing relation is satisfied. Conversely, by the theory of equations of this form, the most general integral equation equivalent to this differential equation is f{z,z')=F{az + a'z'), where F is any function whatever of its single argument ; and therefore, Avhen a function/ (5, z') satisfies the relation dz dz in general (and not merely for an arithmetical pair, or for sets of arithmetical pairs, of values for z and /), it possesses infinitesimal periods. * The result holds for multiform functions and, under conditions not yet established, possibly even for functions that have an unhmited number of values for any assigned values of the variables ; see Weierstrass, Ges. Werke, t. ii, p. 69, p. 70. t It is established for the case of n variables, Weierstrass, Ges. Werke, t. ii, pp. 62—64. 124] PERIODS 215 Next, suppose that our uniform analytic function is not expressible in a form F{az+ a'z') for any constants a and a' whatever; and consider a region in the field of variation where the function / (2^, z) is regular. No relation .9/ 9/ n for non-vanishing values of a and a, is satisfied over the whole of this region ; hence we can take places z-^ and z-^, z^ and Zo within the region, such that j J";,, I , where df{z„ zi) df{z„ zl) 1/19 dz-^ ' 'bz-^ df{z„ z,') df(z,, z/) dz2 ' dz2 is finite and not zero. Also when we take places z^ + i^ and z^' + w/, z^ + u^ and Z2 + U2, Zi + Vi and ^/ + v/, z^ + v.^ and z^' + <, where all the quantities \ui\, I V I, I Wo I, I M2' I, I ^1 1, \vi\, I '^2 1> I "y/ I are infinitesimally small, the quantity I Jj2, 1 where df{z, + u,, Zy,' + u^') df{z^ + Vi, Zj +Vi) dz, ' dz^' dfiz^ -h u^, z^ + zf/) 9/ (^ 2 + v., z.j + ^ ^O fi ^•>. — f(z + h,z' + h')~f(z,z')= {Zd^+&>d^']> J z,z' dz^ ' dz2 differs from | J^., j only infinitesimally, and therefore its modulus is finite and not zero. ' Consider the possibility of the existence of two periods h and h\ What- ever these quantities may be, we have generally fz+hjZ'+h' because the subject of integration is a perfect differential. Take a combined ^-path from z to z + ?t and a ^'-path from z' to z' + h', and let ^ = z + ht, ^' = z' + h't, so that the range of integration is represented by variations of t from to 1 ; and then generally f(z + h,z' + h')-f{z,z')=hj^-^ -^dt ■ Jo 9^ Suppose now that h and h' are infinitesimal, so that the derivatives of f{z, z') differ only infinitesimally in the Grange from to 1 from their valuea at ^ = ; then we have a relation of the form 216 NUMBER OF [CH. VIII where \u\, \u'\, \v\, \v' \ are infinitesimal of the same order as | A | and \h'\, and may depend upon z and z. Accordingly, returning in particular to our two places z^ and z^, z^ and Zo!, we have J- (z, + h,z,'+h) -/ (z„z,) = h ^^ + h ~, , f{z, + h, z^' + h) -f{z,, z,) = h ^^ + h ^, , and so on for any number of places ; two will suffice for our purpose. When h and h' are periods (whether infinitesimal or not), the left-hand sides vanish. As the equations are valid, when the periods are infinitesimal, the right-hand sides also vanish ; so that we have A/i/ = 0, h'J,^' = 0. Now J12' is not zero ; hence both h and h' are zero. In other words, our uniform analytic function of two variables cannot have infinitesimal periods, unless it is expressible as a function of a single argument az + az , where a and a! are two constants. 125. Next, let w^ and w/, w. and w.^, (o^ and w^', ... be period-pairs for a uniform analytic function f{z, z) ; then we have f {z + r-i CO j^ + 1^(0.2 + rsO)s+ ..., z' + i\(Oi' + r^co^' -|- 7-3 eog' -I- ...) =/(^, /), where Vi, r^, r.^, ... are any integers, positive or negative, and independent of one another. In the case of a uniform analytic function of one variable, it is known that there are not more than two independent periods and that the ratio of these periods for a doubly periodic function cannot be real* ; the last property can be expressed by saying that if the periods are (o, = a + i^, and &>', = a' -I- i^' , the determinant ' I a, /3 is not zero. The corresponding theorem -f* in the case of uniform analytic functions of two variable^ is as follows : — A uniform analytic function of two variables z and z' cannot possess more than four independent period-pairs Wi and w/, &>, and (0.2, cog and cos, (O4 and ft)/ ; and if (Os = as + i^s, «/ = «./ + */Ss'> * When the ratio is real and commensurable, both periods are integer multiples of one and the same period ; when the ratio is real and incommensurable, there are infinitesimal periods. t It is partly due to Jacobi, Ges. Werke, t. ii, pp. 25 — 50. 126] INDEPENDENT PERIODS ' 217 for all four values of s (the parts a, /3, a', /3' beiiig real), the determinant «!, "2, as; a4 A, /32, /Ss, A «!, a2, ofs, a4 /3/, A', /s;, ys; •mMS^ not vanish. 126. As a preliminary lemma, we require the following proposition : if relations &)4 = kcOi + loio + Wlft)3 o)/ = A;&)i' + la)2 + wift)/ are satisfied among four period-pairs, where k, I, m are real quantities, then either there are not more than three linearly independent period-pairs or there are infinitesimal periods. First, suppose that k, I, m are commensurable, and that then each of them is expressed in its lowest terms. Let d denote the highest common factor of their numerators, and let M denote the least common multiple of their denominators ; and write MM' M ' where k', I', m are integers; then we have M^ d M k'co^ -j- r&)o -f m'w^ , d I 11 1,1' I , I ' &)4 = A; coi -f- 1 &).2 -r m w^ . Now Mjd is a fraction in its lowest terms, being an integer only if d is unity ; change Mjd into a continued fraction and let pjq be the last convergent before the final value ; then that M_p_ 1^ d q ~ dq' M 1 ^^-P = ^d' Now -J- (Oi and -j wl manifestly are a period-pair, and therefore also q-r ^^ and q-T (t>i \ consequently [q^-p)„, — ft)i -| — &)2 are period-pairs, integral in Wj and coi, CO2 and co.2 ; hence i/ „ m' M ^, m' , fl fM ^ fM are a period-pair, say (O5 and &).,' ; then as M ^ m' M ^, m' , 124 G>3 = ft)g, Hi 0)3 = &)3 , where M, m', //, are integers and O4, 0)3, coj, ^1^, w^, wl are constituents of pairs. But we knowi* that, in such an event there are two integral com- binations of oja, &)g, II4, and the same two integral combinations of 6O3', ooJ,, ^l, .Mm because the coefficients — and — are the same in the two relations, such that 6)3, cog, 04 are expressible as integral combinations of the first and Wg', &)g', £11 are integral combinations of the second ; that is, we have k! V — toi H — 0)2 = linear function of two periods Hj and Xlo? /^ A* k' V — &)i' H — 6)2' = same O/ and O./, /A /i . A;' V and now, in our equations, the integral coefficients — and - have no common factor. ' " Similarly for the other cases ; we can assume, in our relations MVti = k'wi + l'a}.2-\- mw-i, MD,^ = k'coi + I'w^ + mcu-^, that no two of the integers M, k', I', m have a common factor. Accordingly, we have k' \l' a fraction in its lowest terms. Expressing it as a continued fraction, and denoting by r/s the last convergent before the final value, we have ^ _!:_ + _! V s ~ si' ' * Obviously, if d=l, the period-pair W4 and W4' is unchanged, t See my Theory of Functions, § 107. 126] ^ PAIKS OF PERIODS 219 Then ± (Oi = &>! (sk' — 7^1 ') = sMfli — I ' (rwj^ + so)2 ) — sm'cos , ± (o-[ = sMCli — I' (rcoj' + sw^) — sm'cos, + (»2 = <»2 {sk' — rV) = — rMCl^ + k' (ra>i + sco.^ ) + rinfcos , ±co^' = - rMfli + k' (rft>/ + s&)/) + rrn'm-i ; and so the four period-pairs are expressible in terms of three period-pairs n4, fi/ ; 0)3, 6)3' ; r&)i -H swo, rw/ + sou.t. Thus there are not more than three linearly independent period-pairs. Next, suppose that one of the three quantities k, I, m, say k, is incom- mensurable, while the other two are commensurable. When I, m are expressed in their lowest terms, let the integer D be the least common multiple of their denominators, so that we can write J V m! Then Da>i — I'cO'i — ni(o-i = kD(Oi , Dco^' — I'coo — tn'cos = kDcoi. Now kD, like k, is incommensurable ; hence, expressing it as an infinite continued fraction, and denoting two consecutive convergents by p/q and p'/q', we have kD=P + ^-,, where the real quantity 6 is such that 1 > ^ > — 1. Thus ^-1 Aw. and r- ^ ;) «/ q qq) \q qq I are a period-pair, and therefore also ^\q qqj ^ \q qq J ^ , that is, e . e , — coi and —, (o^ q q are a period-pair. We may take q as large as we please, for the continued fraction is infinite; and the circumstances thus give rise to infinitesimal periods. Next, suppose that two of the three quantities k, I, m are incommensurable, say k and I, and that m is commensurable, equal to \|^l, where X, and /a are integers. Then our relations can be taken in the form /MCOi — \a)3 = kfJi(Oi + l/jLa)2, /ji(Oi — Xcos = kfio)^ +lfi(02. 22Q ^ LEMMA ON [CH. VIII But, writing and denoting hix and Zyu. by ^'' and V respectively, we have ci>5 = A;'a)i + Z'fWo, 0)5' = A;' ft)/ + r&jo', where k' and Z' are incommensurable, while X > — 1. Also let ti be the integer nearest to the incommensurable quantity si, and t^ be the integer nearest to the incommensurable quantity sm ; then we have si — ti = A2, S7n — t.2= A3, where Ao and A3 are incommensurable quantities, each in numerical value being less than ^. Thus 00 scii^ — rcoi — tiCOc — toco^ = — CO1 + Aoft)o -|- A3&)3, s X scOi — rtOi — tiCOo — UwJ = - ft)/ + Aoft).,' + A3 6)3''. s Again, as Ag is an incommensurable quantity, let it be expressed as a con- tinued fraction ; taking any convergent p/o", where always p and cr are integers, we have cr a- * See my Theory of Functions, § 108. t The alternative suppositions, for the last case, and for the present case, are left as an exercise. 126] PAIRS OF PERIODS 221 where y is a real quantity such that 1 >y > — 1. Also let t^ be the integer nearest to the value of o-As, and write O-As = ^3 + V, where V is an incommensurable real quantity less than \. We then have X y — a (s&)4 — ?"&)i — ^lO), — 4 0)3 ) — pw^ — igtBg = cr - &)i H — 0)2 + v ojj , / „ ' ' 4. I i '\ I 4. ' ' , y ' , T7 > . o" {SO), — roji — tift)2 ~ ^2 6)3 ) — pft)2 ~ f33 5 O- <\ 3 = cr - (Wi 4- - f»o 4- V ft), ' S cr " <\\o)z\ Our relations now have the form 0)4 = A;'ft)i -h r&)2 -|- wi'Ils, 0)4' = k'oii 4- Z'ojo -f- ni'flg, where the quantities k\ I', m fall under one or other of the cases already considered. Either we have not more than three period-pairs ; or we have infinitesimal periods ; or all the quantities k', V, m are incommensurable, while In the last event, the same kind of transformation can be adopted ; and by appropriate choice, we can form a new period-pair Vi^, XI3', such that I '^■'3 1^21 3 1 5 I ■^^S 1^21 ^^^3 I • And so on, in succession. By taking a sufficient number n of transformations, each of the preceding type, we ultimately can construct a period-pair 3 and Os', such that that is, by taking n sufficiently large, we should have an infinitesimal period. It therefore follows that, if we have two relations Adii + B(02 + Ccos + Dcoi =0, Aco^' + Bfo^ -h Gm^ + D(Oi = 0, 222 NOT MOKE THAN FOUR [CH. VIII between four period-pairs, where the coefficients A,B,G,D are real quantities, either there are not more than three period-pairs, or there are infinitesimal periods for the variables. Accordingly, when we have to deal with uniform analytic functions of two variables, there is nothing in the preceding analysis to exclude the possession of even four period-pairs, when these pairs are linearly independent in respect of combinations between their respective members. 127. For the remainder of the proposition in § 125, it is necessary to consider the possibility of the existence of five period-pairs : if this be ex- cluded, then a fo7'tiori we need not consider the existence of more than four period-pairs. For this purpose, let there be four period-pairs of the kind postulated in the theorem such that, if Ws = «s + ^'^s, &>/ = a/ + i^s, (for s = 1, 2, 3, 4), the determinant 1 ai, ao, ofs, 04 1 A> A, A, /34 I «/, <; a/, a/ ' >/, /3/, /S/, /s; does not vanish. When this last condition is satisfied, we cannot have relations nil «! + rru a^, + m^ a^ -f W4 04 = 0, ??ZiySi -f-ma/Sa +771^^3 +mi^i =0, mi a/ + m,^ ou + m^ cf/ + iiH a^ = 0, mijSi + m202 + nis^s + '>^hl3/ = 0, for any set of real quantities m^, m^, m^, m^ other than simultaneous zeros. The exclusion of the first pair of these relations excludes a relation niiQ\ + m^coo. + m-iCOs + 071^0)4, = 0, and conversely; and the exclusion of the second pair excludes a relation. rniCOi + m20)2 + nn^u)^ -V m^o)/ = 0, and conversely. Hence, after the preceding lemma, we infer that our uniform analytic functions may possess four periods, or fewer than four periods ; and they do not possess, as they cannot be allowed to possess, infinitesimal periods. Now suppose that a uniform analytic function /(^'j /) possesses, in addition to four given linearly independent period-pairs toi, co/ ; cdo, 03.2 ; 0)3, w^ ; w^, wl ; also a fifth period-pair, say Wg, a>g'. Let &>5 = as + ^'ySs , (^l = as' + ^/Ss'. 127] PAIRS OF PERIODS 223 Then, with the preceding hypothesis of the non-evanescence of the determi- nant («!, ^oy '^z, ^i) in the customary notation, the equations ttg = ?li 0£i 4- ^2 tto + '^3 "s + '^h OC4, , /3o = n,^^ + n.2l3o + ihlB^ -f n^^^ , flg' = ?ii a/ -I- n^ tto + Us tts + n^ al, /3/ = Ml /3/ -f 71./3./ + nSl + "4/3;, determine uniquely four real finite quantities n-^, 712, 1/3, n^; and they are such as to secure and to require the equations cOg = 7iift)i + ?22&)o -|- nsOii + n^co^ It therefore is necessary to consider the conditions, under which these equations are possible. The analytical consideration of the conditions follows a general march similar to that followed in the establishment of the preceding lemma. The results therefore will only be stated, without further proof. They will relate only to the most general case when no one of the six ratios % : Wg : 713 : n^, as determined by the elements of the four period-pairs is an integer; the alternative is to provide onl}^ less general cases. We find (i) when all the real quantities Wj, n^, n^, 71^ are commensurable, the formally five period-pairs can be expressed in terms of not more than four period-pairs : — (ii) when one (and only one) of these quantities is incommensurable, then an infinitesimal period-pair exists : — (iii) when two of these quantities are incommensurable, then cer- tainly one infinitesimal period-pair exists, and possibly two such pairs exist : — (iv) when three of these quantities are incommensurable, then one infinitesimal period-pair certainly exists, and three such pairs may exist : — (v) when four of these quantities are incommensurable, then one infinitesimal period-pair certainly exists, and four such pairs may exist. It therefore follows that for any uniform analytic function, which is really a function of two (and only two) independent complex variables so that it cannot possess infinitesimal periods, there may be four period-pairs, and there cannot be more than four linearly independent period-pairs*. * It is a tacit assumption, throughout the preceding investigation, that au infinitesimal period-pair w and to' for z and g' means a period-pair for which both | ca \ and | w' | are infinitesimal. 224 ONE PAIR OF PERIODS [CH. VIII 128. Now that we have established the result that a uniform analytic function of two complex variables cannot possess more than four linearly- independent pairs of periods, so that we should have for all integer values of m^, m^, m^, m^, positive or negative, we proceed to consider the various possible cases that can arise, under the significance of the result and within the alternatives admitted by the analysis leading to the result. For the present purpose, the case when there are no periods needs only to be mentioned. We then have the customary theory of the uniform analytic functions of two variables, which has been previously discussed in some detail. The remaining cases will be considered in succession. One pair of periods. 129. Let the variables z and / have the periods a and ol, and no other periods. Take new variables u and u', where z = au, ctz' — az = a.a.'u, which is an effective transformation of variables unless (i) both a a,nd o! vanish — a possibility that can be excluded — or (ii) either a or a! vanishes. If a! vanishes, we take u and / as new variables. If a vanishes, we take z and V as the variables, where z' = a'v. In all the cases, denoting the variables by u and ii, we can now take 1, as the pair of periods. Hence the field of variation of the variables is composed of a strip in the u-plane of breadth unity, measured parallel to the axis of real variables, and the whole of the w -plane ; and the uniform function in question can be expressed as a uniform function of e''^^ and u. » Two pairs of periods. 130. Let the periods be for z , = a I ~ /^ 1 respectively, in bracketted pairs ; manifestly it may be assumed that a and a' do not simultaneously vanish, and likewise that /3 and i3' do not simultaneously vanish. Choose quantities k, I, m, n, such that ka + la' = 1, k^+ 113' = 0, ma. + noi = 0, m^ + n^' = 1. 130] TWO PAIRS OF PERIODS 225 When one of the two quantities a and o.' vanishes, say a, and neither of the two quantities ^ and /3' vanishes, we take m = ; and when one of the two quantities /3 and y8' vanishes, say ^', and neither of the two quantities a and o! vanishes, we take k= 0. As will be seen, all the other possible special cases are included in the one special case that is to be considered. The values of k, I, m, n are given by k{cL^'-a.'j3)= ^', m{oilS'-a!^) = -(x', l{ai3' - a /3) = - yS , n (a/S' - a'/S) = a ; and these values are determinate and finite unless afi' - a/3 = 0. First, suppose that a/S' — a'/3 is not zero — which, of course, is the more general case. Introduce new variables u and u, such that u= kz + Iz', u' = mz + nz'; and then the period-pairs of these new variables are for u, =11 =01 u\ = Oj ' =1 respectively, in bracketted pairs. The field of variation of the variables is composed of a strip of unit breadth in the w-plane and of a strip of unit breadth in the w'-plane, the breadth of each of the strips being measured parallel to the axes of real quantities in the planes. The uniform function in question can be expressed as a uniform function of e""*^ and e"'-"'. Next, suppose that a^' — a'/3 is zero — which, of course, is a special case. As a and a may not be zero simultaneously, let a be different from zero ; and as /3 and ^' may not be zero simultaneously, let /3 be different from zero. Then there are two alternatives (i) when both of and /3' vanish : (ii) when neither a' nor /3' vanishes, and then we have a' _§^ CC-/3' "''' say, where c is not zero nor infinite. As regards (i), the variable z has periods a and 0, while the variable / is devoid of periods : and in order that a and may be effective distinct periods for z, we must as usual have the real part of ia./^ distinct from zero. The field of variation of the variables is composed of the customary a-/3 parallel- ogram in the ^■-plane, and of the whole of the /-plane ; and the uniform function in question can be expressed as a uniform function of ^ {z), p' (z), and /, where f (z) is the customary Weierstrassian doubly-periodic function with periods a and ^. F. 15 226 THREE [CH. VIII As regards (ii), we keep the original variable z\ and we introduce a variable v such that v = z' — cz. When z and / have the periods a. and a', then v has zero for its period ; and when z and z' have the periods /3 and j3', then again v has zero for its period. Accordingly, ivhen we take z and v for variables, the periods of z are a and ^, while the variable v is devoid of periods. The uniform function in question can be expressed as a uniform function of ^J {z), f' {z), and v, with the same significance as before for gj {z) and the same requirement as to the real part of ia//S. Should the requirement as to the real part of icaj^ not be satisfied, either there is an infinitesimal period, or the two pairs are equivalent to one pair only. In the former case, there is no proper uniform function with the periods ; in the latter, the periods are not effectively two pairs of periods. Three pairs of periods. 131. Taking the variables to be z and z as before, let the periods be for z , =a| ^^1 ^"^ z', ^a'i' =/3'J ' =y where manifestly no pair of quantities in a column can vanish simultaneously. Thus a can vanish, and a' can vanish ; as they may not vanish together, there are three possibilities for the a, a pair. Similarly for each of the other two pairs ; so that there are twenty-seven possibilities in all. They can be set out as follows. A. When all the quantities a', ^', y vanish, the period-tableau is 0, 0, Oy, (A); no one of the quantities a, /3, y can vanish : there is one case. B. Let two of the three quantities a, j3', y vanish, but not the third of them ; there are three possibilities. When y is the one which does not vanish, then neither a nor /3 can vanish ; and we can have two alternatives, viz. y vanishing, or y not vanishing. The period- tableaux are a, /9, 0\ (a, 13, y\ 0, 0, ry'J, (B,); lo, 0, 77,W; each is typical of three cases. 132] PAIRS OF PERIODS 227 C. Let one of the three quantities «', /3', 7' vanish, but not the other two ; there are three possibilities. When a' vanishes, then a cannot vanish : and as /3' and 7' do not vanish in that event, we can have four alternatives, viz., /3 and 7, either vanishing or not vanishing, independently of one another. The period-tableaux are /«, /3, 7\ V0,/3', 77, (CO; fa, 0, [0, ff, 7/ , (G,); /■«, /3,0\ [0,^', y'J, (C3); /a, 0, VO, /3', (C4); each is typical of three cases, D. Let no one of the three quantities a, /3', 7' vanish ; there is only a single possibility. But as regards a, /3, 7, there are eight alter- natives, viz., they may either vanish or not vanish, independently of one another. The period-tableaux are .a', /3', 77, (A); W, /3', y'J, (A); /O, 0, 7\ /O, 0, 0\ \a', ^', y'J , {!),)■ W, /S', 77 , (A). Among these, (jDj) and (A) are one case each ; (A) and (A) are, each of them, typical of three cases. 132. As regards the kinds of functions considered, no generality can be lost by assuming that a function is substantially unaltered (i) when one period-pair is interchanged with another period-pair : or (ii) when linear transformations are effected upon the variables, coupled with corresponding linear transformations upon the period-pairs : and, in particular, when the variables are interchanged provided that the periods are interchanged at the same time, each combined period-pair being conserved. Under the first of these assumptions, the three cases typified by (^j) become one case only, of which (A) will be taken as the tableau of periods. The same applies to (A), (Cj), (Co), (C3), (C4), (A), and (A), in succession. As regards (A), when we replace the variable 2 by u, where 7 / U = Z -,2 , 7 the periods for u and z' are «, /3, .0, 0, 7', the case becomes (A), and therefore needs no separate discussion. 15—2 228 THREE [CH. VIII It is convenient to consider next the case (A)- Let four quantities k, I, m, n be chosen so that koL+ la' = l, k^+ l^' = 0] ma + na = 0, m/3 + 7i^' = 1 their values are given by k{a^'-a'0)= /3', m(a/3'-a'B) = -a'] l{al3'-a'^) = -^, 7i(a/9'-a'/3)= a When a/3' — a.'^ does not vanish, the values of k, I, m, n are determinate and finite ; when it does vanish, the selection cannot be made. Accordingly, in 'the first place, suppose that ajS' — a'/S does not vanish. No generality is then lost by assuming that 7/S' — 7'/? does not vanish and also that 07' — a'7 does not vanish ; for the alternative hypothesis as to each of these magnitudes leads, by the permissible interchange of period-pairs, to the case when a/3' — a'/3 vanishes — a case yet to be considered. Now write u = kz + Iz', xi = mz + nz , ^,=kr^+ ^7 = (7^' - 7';8) - {a^' - a'yS), yu.' = m'y + ii'y' = (a7' — a'7) -=r (a/3' — a'/S), where the new variables u and u are independent of one another because kn — Im, = (o/S' — C('J3)~^, is not zero. Thus the uniform function in question becomes a uniform function of u and u', with the tableau of periods 1, 0, /. 0, 1, /.' In the second place, suppose that a^' — a'/3 does vanish. Then ' a ~ ,8 ~ ""' say. Introduce two new variables u and u, defined by the relations ? | < 1 ; thus sq/j, -r = -. 134] PAIRS OF PERIODS 231 Accordingly, a set of periods is I, 0, €- 0, i,,l When we take s very large and q/s also very large, the quantities e - , and v - , q s are infinitesimal: that is, we should have an infinitesimal period-pair^a possibility that is excluded. Thus fi and /x' cannot be simultaneously real. The most general case arises when neither /x nor /jl' is real : and we shall assume that, henceforward, we are dealing with this case. It is to be remem- bered that, in effecting the linear transformation upon the variables so that 1,0; and 0, 1 ; are two period-pairs, we have used the constants of relation. Moreover, as the periods in the tableau can be linearly combined in simultaneous pairs, we have fx,+p.l + q.O, fi' +p.O + q .1, that is, fi + p, fi' + q, as a period-pair, p and q being any independent integers ; and this period- pair can replace /x and /x' in the tableau, for any values of p and q. Let these integers be chosen so that the real parts of /jb+p and /jf + q, say R{fi+p) and R (fx' + q), satisfy the conditions O^R(fi+p) 7^ S7, (A); /«, /8, 0, S\ Vo, /3', y', B'J, (A); foe, 0, VO, /3', 7' 7> bO, (A); /a, /3, 0, 0\ Vo, &', y', B'J, (A); fa, 0, VO, 13', 7. 7. S7, (A); [oc, 0, 0, B\ Vo, 13', y', B'J, (A); fa, 0, Vo, ^', 0. , ON , B'J, 7: (A); each is typical of four cases. 234 FOUR [CH. VIII E, Let no one of the quantities a', /3', 7', B' vanish. The period- tableaux are «, /S, 7, S\ /O, /3, 7> S\ /O, 0, 7, S\ a', /3', 7', S'A (E,); W, ^', 7', SV, {E,)- U', /S', 7', ^'A (^3); /O, 0, 0, S\ /O, 0, 0, 0\ W, /S', 7', S'A(^4); U yS', 7, 87,(^5); of these, (^1) and {E^ are each one case; {E^ and (£'4) are each typical of four cases ; and ( £"3) is typical of six cases, 136. As regards the kinds of functions considered, the same assumptions, as to the interchangeability of period-pairs and as to the linear transformations of the variables without detriment to the generality of the functions, will be made as were made (§ 132) in the discussion of the triple periodicity. Consequently all the cases, of which each tableau is typical, become merged into a single case. The cases {A) and {E^ are impossible, or else the periods degenerate; there cannot be uniform functions, periodic in a single variable and having four distinct periods for that variable. The cases {B^), {B^, (Dg), {E^ are impossible, or else the periods degene- rate ; there cannot be uniform functions, periodic in a single variable and having three distinct periods in that variable. By taking a variable u instead of z, where 7 / u = z — -, z , 7 the tableau of periods in (d) is changed to a tableau of periods for u and z' represented by (C3) or {G^. Also by interchange of period-pairs, (C3) becomes (C2); hence (Cg) and {C^) are the only cases under (C) that require con- sideration. By interchange of variables and the proper interchange of periods, (Dg), {D^), {Dy) become (Cj), and so require no separate discussion ; and similarly {E^) becomes (Ci), and can therefore be omitted. By interchange of period-pairs, (A) and (A) become (D4) and so they require no separate discussion. By interchange of variables and the proper interchange of periods, {E^ becomes (A) and can therefore be omitted. Consequently, the cases that survive for further consideration are {G^, (C4), (A), (A), {E,). As regards {D^, change the variables to u and u' by the relations z = au, z' = j3'u\ 137] PAIRS OF PERIODS 235 and write ^ = aX, 8 = a/x, 7' = /3'A,', 8' = /3'/jb' ; the period- tableau for the variables u and u is '1, 0, \, /x ,0, 1, \\ [jf which temporarily will be called {F). As regards {G^, a similar change of variables, viz., z = au, z = 8''ii, leads to a special form of the period -tableau (F) in which V is zero. Assuming this included in (F), we have no new case out of (Co). As regards (C4), we have a function, which is doubly periodic in z alone with periods a and /8, and is also doubly periodic in z alone with periods 7' and 8'. The functions thus provided are undoubtedly quadruply periodic, but the periodicity has an isolated distribution; they will therefore be omitted, as not belonging to the class of functions having proper quadruple periodicity. As regards (D^) and (^1), we effect linear transformations of the variables of the type u = kz + Iz', u' — mz + nz', where the quantities k, I, m, n are determined by relations ky + ly' = l, my + ny = 0, kB+lB'=0, mB + nS' = l. Different cases arise as under (A) in the discussion of triple periodicity : and we find either (i) a period-tableau, with new variables, represented by (F) ; or (ii) cases already decided ; or (iii) cases that are impossible or degenerate. Consequently it follows that properly quadruply periodic functions, which are uniform and involve only two variables, can be modified as to their variables so that they have /I, 0, X, fj,\ \0, 1, V, /[^7 for their period-tableau. 137. Now it is a property of quadruply periodic uniform functions, on the Riemann theory, that (for this tableau) the relation (or else X, = f/) holds. Further, Appell* has proved, by analysis and reasoning quite different from those adopted for the discussion of functions on a Riemann * Liouville, 4"'' Ser., t. vii (1891), pp. 157 sqq. 236 GEOMETRICAL REPRESENTATION OF [CH. VIII surface, that this relation holds in general for a properly quadruply periodic uniform function, that is, by change of the variables and by the association of appropriate factors, the function can be made to depend upon others which possess this property. But under both theories, the property emerges from the discussion of the functions themselves, whereas the preceding investigation deals only (or mainly) with the mere transformation of the periods ; the property apparently cannot be deduced at this stage solely from the preceding considerations. Just as was the case with the triple periodicity when the period-tableau had been rendered canonical, so here also we can infer (without any reference to a property V = ytt or X= /) that all the quantities X, X\ /i, /jl' cannot be wholly real; and in the most general case they will be complex and such that neither of the quantities X'/fi, X/fi', is real. The course of the argument for the inference and its details are so similar to those in the earlier discussion that no formal exposition will be made. Moreover, the quantity X/fi is not real, nor is the quantity X'/fi' ; both statements can be established by shewing that the contrary event would lead to a zero-period for commensurable reality and to an infinitesimal period for incommensurable reality. 138. One difficulty, however, now arises; it is connected with the geometrical representation of two independent complex variables, which has already been discussed. Putting aside for the moment the method of representation in four-dimensional space, partly because of the difficulty of framing mental pictures in such a region, and partly because the representation does not by itself seem to retain sufficiently the individuality of the variables, we have the representation by means of the combined points in the ^-plane and the ^'-plane. But we cannot construct a region in the ^-plane and a region in the ^•'-plane that shall suffice for the field of variation of z and z' within their periods. Take any origins in those planes; in the 2^-plane, let the points a, b, c represent the values 1,X, fi; and in the /-plane, let the points a\ b', c represent the values 1, X', fx \ and complete the parallelograms as in the figures, so that the points a, /3, 7, h respectively represent the values X + (m, \ + lx,l-\-X,l-{-X-\-ix, and similarly in the 2-' -plane. No one parallelogram such as Oa/3cO is sufficient for the representation of z ; for there is a portion 138] QUADRUPLE PERIODICITY 237 of the parallelogram ObacO not included, and there is a portion of the paral- lelogram OaybO not included. The double parallelogram OaybacO is not sufficient, because there is a portion of the parallelogram OajScO not included ; moreover, the whole plane could not be covered once and once only by repetitions of the double parallelogram keeping unchanged the orientations of the sides. In the figure, the parallelogram Oa^cO is partly excessive and partly deficient ; for the interior of the small parallelogram between ab, by, aj3, /3c is reducible to another part of Oa^cO. The triple parallelogram OayBacO is excessive ; for much of its area (the part outside the parallelogram Oa^cO) is " reducible " to the area within that parallelogram, and also the whole plane could not be covered, once and once only, by repetitions of the triple parallelogram keeping unchanged the orientations of its sides. The same remarks apply to the /-plane, in connection with the figure as drawn. Thus, neither by means of parallelograms, nor by means of strips in the two planes of reference, is it possible to obtain definite unique and complete limited fields of variation for z and z', that shall discharge for quadruply periodic functions of two variables the same duty as is discharged for doubly periodic functions of a single variable by the customary period- parallelogram. But by taking an associated two-plane variation of the real variables x, y, x, y', the deficiency can be supplied for one purpose. This representation is as follows*. For a quadruply periodic function, with the period-tableau '1, 0, X, IX Si, 1, V, ^0 we resolve \, fM, V, /j/ into their real and imaginary parts, say X = a + ib, jji = c+id, \'=a-\-ib', /x' = c' + id' ; then every place, differing from z, z only by multiples of the periods, can be represented by X + iy + p + r{a + ib)-\- s{c + id), x' + iy + q + r (a + ib') + s(c' + id'). Take two planes, one of them to represent the variations of y and y' with reference to O'y and O'y' as rectangular axes, the other of them to represent the variations of x and x with reference to Ox and Ox' as rectangular axes. In the y, y' plane, let B be the point b, b' and JD the point d, d' ; and com- plete the parallelogram DO'BF. In the x, x plane, let OA = 1 and OC = 1 ; and complete the square COAE. Then the integers r and s can be chosen, say equal to r' and s', so that the point y -\-r'b + s'd, y' + r'b' -h sd' , * For this suggestion I am indebted to Professor W, Burnside, who communicated it to me in a letter dated 14 January 1914. 238 TEIPLY PERIODIC [CH. VIII lies within or on the boundary of the parallelogram O'BFD; let this point be Q. Then every point, which is equivalent to y, y', in the sense that its coordinates are y + rb + sd, y + rh' + sd' , is equivalent to Q and lies outside the selected parallelogram. y O' X' C E • P A ^ Again the integers p and q can be chosen, say equal to ip! and q, so that the point x-\-'p' -\- r'a + s'c, y + q' + r'a + s'c lies within or on the boundary of the square OAEC; let this point be P. Then every point, which is equivalent to a; + r'a + s'c, y + r'a' + s'c, in the sense that its coordinates are x+p + r'a + s'c, y + q + r'a' + s'c', is equivalent to P, and lies outside the selected square. It follows that, in connection with a place z, z , and with all places equivalent to it in the form z-\-'p -VrX-\-six, z' + q + r\' + s/jf, we can select a unique point Q within the y, y' parallelogram, and then associate with it another unique point P within the x, x square. We take the point-pair QP as representative of the whole set of places that, in the foregoing sense, are equivalent to z, z; it is given by the specially selected place z -{-'p + r'\ + s'fi, z' + q' + r'X' + s'jjl. Uniform triply periodic functions in general. 139. It is known (Chap, v) that a uniform function f{z, z), which can have poles and unessential singularities but which has no essential singularity lying within the finite part of the field of variation, can be expressed in the form ir(z,z) wliere (f)(z, z) and "^{z, z') are everywhere regular within the finite part of the field of variation. 139] FUNCTIONS 239 We shall therefore proceed from this result, specially for the purpose of deducing* some initial properties of triply periodic functions that are uniform. We denote the period-pairs by the tableau Now because f{z + l,z')=f{z,z'), and because the functions ^ (z, z') and -v/r (z, z') are regular, each of the equal fractions (f> (z + l, z') _ -^{z^X, z) (ji (z, z) ~ -^ {z, z') ' derived from the equation expressing the 1, periodicity of/, is devoid of zeros and of poles and of unessential singularities for finite values of the variables : hence, as in § 79, the common value of the fractions is of the form v^here g (z, z') is a regular function of the variables. Consequently (f)(z + l,z') = (f) {z, z) e^(^'^'' i/r (^ + 1, /) = -^ {z,z)e3(^'^'^ Similarly, through the 0, 1 periodicity of/, we have the relations (^ {z, / + !) = (/) {z, z) e'^<^'^'' ■f (z, z' + l) = ylr (z, z) e'^'^'^'' where also h {z, z') is a regular function of the variables. In order that the two sets of relations may coexist, we must have cf){z + l,z' + l) = (j) {z, z) e:/(^.^'+J)+/Hr,z')^ ^{z + l,z' + l)=^4> (z, z') ei/(2>^')+A(?+i.2\ and similarly for ^fr (z, z') ; therefore g (z, z' + l)—g {z, z) = k{z + 1, z') — h (z, z), (mod/ '^ir.%). Let g {z, z' + 1) — g {z, z) — Ikiri — h{z^-\, z') — h {z, z') — ^liri, where k— I is an integer : manifestly, either k or I could be taken equal to zero without loss of generality. Now suppose a function X (^, z') determined such that X{z + 1, z')—X {z, z) = g {z, z') — 2k7riz' \{z,z +1)—X {z, z) = h {z, z') - ^l-niz which two equations are consistent because of the foregoing relation between g and h. If then j)^ {z, z') = (f) {z, z') e-^(^.^'>, yjr, {z, z')^^\r {z, z') g-^f^.^'), * This particular investigation follows the earlier sections of Appell's memoir already quoted, § 137. -_ pm(z+i,z') 240 TRIPLE [CH. VIII we have where the functions ^i and i/^^ satisfy the relations l+cr')7ri/ii' — ^ ^'^mn^ '^m,n+i> for all values of m and n. Then = BGCL g(2»l+cr)!ri(jU,+A) + (2/1+0-') 7ri(|a'+A.')+2n-i/ii' , — Tir, „(2in+a-)TTiij.+ {2n+2+(r'}-n-iijL' and y Hn ^(2?n.+2+(r)TriA.+ (271+0-') TT-iX' ^m+iiTi+i — '-^'-*'OT+i,n '-' — Tifln p0n+cr)TTi{\+i2.} + (2n+a-')Tn{\.'+it')^2TriK for all values of m and w ; hence 2TTiX = 27r{/jb' (mod. 27ri), or, having regard to the existence of the period-pairs 1, and 0, 1, we infer the relation the well-known condition in the Riemann theory. Any other double grouping of terms gives rise to quadruply periodic functions. Consequently when there is a question of dealing only with triply periodic functions, there can be' only a single grouping. When the grouping is such as to affect only one of the suffixes in amn, we have seen that the resulting function is composite and can be resolved into a finite number of sums of products of simpler functions. Accordingly the grouping must be such as to affect both the suffixes in amn- The simplest difference-equation of this kind connects a,„^i,„+i with a^,?*: and so this is the grouping which has been chosen. 144. We have taken our triply periodic function in the form 6 (Z, ^') == SS (— l)"»P+»lp' qdm+a-Y^ q'(2n+(r')'^ Jj /^y^ _ ^-^\ g(2?»+o-)7rW+(2n+cr')7rlz' . and we know that, save as to a simple factor, at the utmost, 6{z, z') has 1,0; 0, 1 ; iJ', jJi! ; for its period-pairs, whatever be the form of the coefficient ^ {m, — n). The preceding discussion has indicated the reason for the choice that ultimately leads to the construction of the coefficient : but some special 144] FUNCTIONS 245 cases have to be noted and rejected from the class of triply (and only triply) periodic functions. I. Let {2n + lf ^(2m + l)Triz+(2n+l)Triz' * The symbols adopted agree with the symbols used for the double theta-functions in a memoir by the author, Phil. Trans. (1882), pp. 783 — 862 ; the reason is that, as indicated above, the functions actually become double theta-functions when the proper value is assigned to the coefficients a~. 248 EVEN OR ODD [CH. VIII where, throughout, r denotes m — n, and the coeflficient a^ is an abbreviation for ^15- Let (^ {in - w, 0, 0) =/ {m - n) =f (r) ^ (fi(m-7i,l,0) = g(m-n) = g(r) ^{m — n, 0, l) = h{m — n) = h (r) {m — n, a, a'). Let new integers m' and n be chosen so that m + 7?i' + cr = 0, n-\rn'-\-a' = 0; then (— ;^ —2') = (— IV'^+P'^^'SS (— 'l)^'P+n'p' (I ni^m'+a-)- q' {m'+a-')^ gi2m'+(T)Triz+iin'+(r')Tnz' ^ But 6 (z z'^ = S]£ (— iy«'p+»l'p'c .(7'2m'+(7)2 „'(2n'+(rVg(2ni'+ {ni — n, secures the evenness or the oddness of the functions. As regards the expression of the condition, let m —n= — r, so that m — n = r — a + a' ; the condition is (f) {— r, a, a) = (?^ — cr + a , a, a). To modify the expression of the condition, let cf) {t, a, a) = y\r{2t+ a— a', a, a), where i/r is a new form of coefficient ; then the condition is t/t (— 2?' + o- — a' , a, a') = -^ (2?^ — a -\- a' , a, a) shewing- that -v/r is an even function of the first of its three arguments. This is the necessary and sufficient condition, that each of the functions 6 (z, z') should be either odd or even. One very important class of functions is provided by limiting the co- efficients i/r still further. Let it be assumed that the function -v/r is a function of its first argument only, so that the typical coefficient, which was (f) {m — n, a, a), is •\/r (2??i — 2?i -\- a — a'), where -^ is now an even function of its only argument 2m — 1n+ a — a : the parameters a and a' enter into the coefficient solely through their occurrence in this argument. If then by any change in the function {z, z'), such as an increment of the arguments, the parameters a and a' are increased or are decreased by the same integer, the coefficient i/r is unaltered. It may be noted that the double theta-functions arise from one particular case of this last law, viz. yh. _ ^(2OT-2n+o— cK)"^ Other simple laws can be constructed, subject always to the requirement of convergence ; for our immediate purpose, we have also the requirement of merely triple periodicity. 148. Before the final postulation of the aggregate of conditions and limitations upon the coefficients, consider any function 6 {z, z'), which is triply periodic but not otherwise limited, so that it is mixed as to a quality of oddness or evenness. Let E {z, z') = e (z, z') + ei-z,- z'\ (z, z') = e (z, z') -d{-z,- /), 250 TRIPLE [CH. VIII SO that E {z, z) is certainly an even function, and (z, z) is certainly an odd function ; and let the series- expressions for E and be E (z s') = 22 ( YY^P'^''^? h (7(2)W.++n?' I picim+ay^ q' (2n+(r')- g(2m+(7}Triz+{2n+<7')iriz' Then substituting for 6 in the definition of the function E, and denoting by 0"m,n (as at first) the customary part of the coefficient of the typical term in d, we find •^m,— (m — n, cr, a') in general or t/^ (2m + a — 2n — a') less generally, excepting from the latter the single case when the expression for yjr gives quadruply periodic functions. 149] THETA-FUNCTIONS I. Let the coefficient in ^ be <^ (m — n, a, a'). We have 251 ^C; '^') \ (T , a , z J ^C; P'> ^'+y \cr, (T , zj' h: P> V 0- , (T , Z J With these half-period increments, the members of the set ^0, 0i> ^S, ^12 are interchanged among one another, as also are the members of each of the sets ^2, ^6> ^10, ^14; the law of interchange being the same as that given in the first four columns of the table on p. 254. Further, let ^ f ^' P; ^] denote the value of ^ f^' ^/ ^) when, in the \(J, a , z ) \a, a , z J latter, we take <^{m — n, a- — 1, a — 1) as the typical coefficient in place of ^ (m — n, cr, a). Also, let N = iri (z + z') + l-iri {/j, + /)• Then we have P, P, 2 +iH' a, a, z + ^fi P, p, z +^p, + ^ cr, o-', z' + \p! p, p, z +^fM \ a, a', z' + ^/jf + i P, p, z +^iii +^ ,-i\'< o-iV< ir*° o-N ^ o- + 1, 0-' + 1, / P + l, p , z o- + 1, P'^ M\ (T + 1, a, z' \<7, a + 1, z 0( \' P';') =(-iy@-(P' p;') \cr+2, a , z J ^ ' \(T, o- , z') p> P >z^ ^^_^y^^fp,.p,z a, a' +2, z J ^ ^ lo-, 0-, z\ a + 2, a' + 2, z'J ^ ^ V, @-f>' P; M] \n+ ^ 'sT •"^ ^ <^ ?S" "3? "3i 1 <5D <3? •c-i "33 "^D 1 « e ^ 1 ^ iSS Hn HN HlO) O) „ 1 e + o ■^ 5C) ^ •* ^ <3? <5i ^D ^ ) 1 ^ 1 1 O HOI cS HtM c^ ^ + r*^ <:d *c^ 1 ^ T -qT '^ 5& <5S T "3i' ?t5 T <3? 13 "3r JH S ^ t^ •^ o ^ CO ^ ^ H <^ ^iS" 1 ^ •c-s *c* 1 5^'' 1 ^ ^ 1 ,n rn O HOJ ■^ '^ ^ 5b ^ <3? 5D ^ •^ ^ "35 "35 "3i ?^ 12 CO «c a „ Hl« O 51^ •^ ■^ 'p* ^ 5C) "5i ^li ^ 53^ "^ "^ 't^ "32 ?t> „ cq m ■* o O <5i •^ <35 13 "5i -5^ 'ai <^ ^ai "35 "^ <3i ^ "33 "^ "32 + + f< i^ 152] TABLE OF ZEROS 255 But it must be remembered that each such picked zero is, for a single function, only a place in a continuous aggregate of zero-places : for any pair of functions, any simultaneous picked zero (such as 0, for 6^ and ^7) is an isolated simultaneous zero. The table* of picked zeros is as follows : — z, z' = ^0 01 d. 03 e. 05 ^6 0- ^8 ^9 ^10 ^11 ^12 ^13 ^14 ^15 0,0 1,0 hi */^) in-' X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 1^11) 2*^12 iau5i«i2 + i ^a^i + h i«i2+i 2*^125 '2*^22 5'^12 + 2> i^22 5*^12? "5'3!.29+2 i«12 + |, 4«22 + i X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Construction of functions that are strictly periodic. 152. The results of § 142 shew that each of the sixteen 6'-functions is periodic in 1 and 0, save possibly as to sign; also in and 1, save possibly as to sign ; also in jm and ^', save as to the factor exp (— liriz — liriz' — irifjb — iri^i) and save possibly as to sign. The actual periods (except for multiples of//, and p. 23. Both the tables may be compared with the table given by Konigsberger, Crelle, t. Ixiv (1865), 256 PERIODIC FUNCTIONS [CH. VIII ix , when the variable exponential factor occurs) for the functions are as follows : — 1,0 1,0 2,0 2,0 2,0 2,0 1,0 1,0 ; 0, 1 /^, /^'; ; 0, 1 2/x, 2yt.' ; ; 0,2 ^, /; ; 0,2 2m, 2/x'; ; 0, 1 A^, /^'; ; 0,1 2yLi, 2/i' ; ; 0,2 /^, /*'; ; 0,2 2/., 2/x' ; for 6q and ^i, for di and ^g for ^3 and 6^^ for ^7 and d^i for ^1 and ^^g for 65 and ^9 for 62 and ^14 for ^6 and ^10 . Hence the fifteen quotients of any fifteen of the functions by the remaining sixteenth function are actually triply periodic (save possibly as to sign) in 1, 0; 0, 1 ; /A, /i'; the squares of these quotients are actually triply periodic in the three pairs of periods. And it may be noted that the eight quotients On are actually triply periodic in 1, ; 0, 1 ; /-t, fju. The analogy of the quadruply periodic functions which arise out of the double theta-f unctions suggests that, for the triply periodic functions, we should take the quotients 6y -^ 0^2, where r has all the values 0, 1, ...,15 except r = 12. Triply periodic functions thus are secured without doubt : but it must at once be noted that the functions are tied as to their infinities. In the simplest case, when the ^-functions are regular for all finite values of the variables, the infinities of each of the fifteen quotients are the zeros of 6^2 and are these alone. But such zeros are a continuous aggregate ; and so the simultaneous poles of the fifteen quotients, taken in pairs anyhow, are not isolated points : the fifteen quotients are tied, through the common occurrence of 6^2 in the denominator. The simultaneous zeros of any two of the fifteen quotients are isolated places, being the simultaneous zeros of the ^-functions which occur in their nume- rators : and these constitute the whole of the zeros simultaneously belonging to two quotients for finite values of the variables. But, of course, the quotients indicated are, initially at any rate, not a potential aggregate of actually periodic functions. Thus, for any one of the ^-functions, it is clear that the quantities 9'-+^ lo g 6 for integers r and s, such that r + 5 ^ 2, will provide periodic functions : and so for other possible derivatives and combinations. 153] QUADRUPLY PERIODIC FUNCTIONS 257 Later (§ 161), we shall return to the " double " theta-functions which arise as a particular set of these " triple " theta-functions. A property of uniform quadruply periodic functions in combination. 153. We proceed to consider the level places of two uniform quadruply periodic* functions f (z, z) and g {z, z), having four pairs of periods in the form (I, 0, X, ti\ \S), 1, \', fi'J • Let a and /3 be two level values for /and g, so that f(z,z') = a, g(z,z)=/3. If z = a-^, / = «/ be a place where / and g acquire the values a and ^ respectively, they will acquire these respective values at the whole set of places a^ + p + rX + Sjjb, a/ -I- g + rX' + Sfx, for all integer values oi p, q, r, s. We have seen, in § 138, that, by taking an associated two-plane repre- sentation for the real variables x, y, x , y , we can choose a unique point-pair QjPi, where Qi lies in a parallelogram in the y, y plane and Pj in a square in the x, x plane, such that the point-pair Q^P^ may represent the whole foregoing set of values equivalent to a^, a/. We shall say that the whole set of values is expressible by the point-pair QiPi. Let z = a2, z = a^ be another place, not belonging to the set expressible by the point-pair QiPi, where/ and g acquire the respective values a and /3; and let the whole set of places, equivalent to a.2, a^ by the addition of periods, be expressible by the point-pair Q^P^. And so on in succession, for places and sets of places equivalent to them, each new set containing no place belonging to any of the preceding sets. Each new set will be expressible by a point-pair, in the associated two-plane representation of the real variables x, y, x', y' . We thus obtain a succession of diiferent point-pairs QjPi, Q2P2> •••> expressing the succession of distinct sets of places where the functions / and g acquire the respective level values « and ^. Each such set can be denoted by any one of the members of the set ; and from the construction of the sets, each set contains finite places in the field of variation. Let these finite places be denoted by a^, a/; a^, a^\ ...., in succession, corresponding to the point-pairs QiPi, Q2P25 — We shall say that such a finite place ^^, zj, is the irreducible level place for its set. * An attempt to establish the property for triply periodic functions, similar to that which follows for quadruply periodic functions, did not meet with success. F. 17 258 IRREDUCIBLE LEVEL PLACES OF [CH. VIII If the number of point-pairs QiPi, Q2P2, ■■• , which thus arise, is finite, then the number of irreducible level places z, z' , giving level values a and ^ to the functions / and g, is finite. If the number of point-pairs QiPi, Q2-P2. •••, which thus arise, is infinite, then within the finite y, y parallelogram and the finite x, x square, there must be at least one (and there may be more than one) limiting point-pair QP such that its immediate vicinity contains an infinite number of such point-pairs. We then, for all such point-pairs in that immediate vicinity, have an infinite number of finite places a, a, at which the functions / and g acquire the level values a. and ^ respectively. Now suppose that, for finite places in the field of variation, our functions / and g possess no essential singularities. On this hypothesis, we know (§ 121) that the level places are isolated, so that there cannot be an infinite number of those level places in the immediate vicinity of any one of them. The second alternative must therefore be rejected; and so we infer the theorem : — The numher of irreducible level places, giving level values a and ^ to two independent free uniform quadruply periodic functions, is finite. 154. It has been established for a couple of independent uniform functions in general, and therefore for a couple of independent uniform quadruply periodic functions in particular, that the level places are isolated pair-places. Any such pair-place may be simple or multiple. Whether simple or multiple, it is isolated, provided the two functions are independent and free. Further, if a, a! is a simple level place for two independent and free functions /(^r, z') and g {z, z'), such that f{z, z'.) = cc, g (z, z) = /3, so that it is an isolated level place of those functions for those values a. and yS, then there is one (and there is only one) simple level place in the immediate vicinity of a, a — say at a + h, a' + b', where | b \ and | b' \ are small — such that f{z,z') = a + a, g{z,z') = /3 + ^\ where | a j and | ^' | are sufficiently small, and |a + a'i<|a|, | /3 + /3' | < | /3 |. For, by the theorems in Chapter iv and Chapter vii, ii z = a + b, z' = a' + b', then we can write / {z, z) -a^f(a + b,a' + b') - a = ftio^-l- aoi6'-f ..., g (z, z')-/3 = g (a + b, a' + b') - /8 = c,o& + C.M + ... ; 155] TWO PERIODIC FUNCTIONS 259 and therefore, as the level place a, a is simple, the equations «10^ + «,oi6' + ...=«'] Cio& + Coi6'+ ...=I3'\ for sufficiently small values of | a' | and |/S'|, provide a single pair- value for h, h', where | h \ and j h' \ are small. Similarly, from the theorems in §§ 113, 120 — 122, we infer that, when a, a' is a multiple level place of multiplicity M for two independent and free functions f{z, z) and g {z, z), such that f{z, z) = a, g (z, z) = 0, so that it is an isolated level place of those functions of multiplicity M for those values, there are level pair-places (some perhaps simple, some perhaps multiple), in the immediate vicinity of a, a — say at a + h, a + h' where j h \ and I V j are small, — of the same multiplicity M in additive aggregate for f{z, z') = a + a, g (z, z') = ^ + ^', where i a' j and j /3' | are sufficieutl}^ small, and |a-f a'|< |o!|, |^+^'i<|/31. 155. Now consider the total finite number of irreducible level places such that the uniform quadruply periodic functions / and g acquire the values a and /3. The propositions just quoted shew that we can proceed from these values of the two functions to other values having smaller moduli : to any aggregate of level places at or near any one place a, a' for the values a and j3, there corresponds another aggregate of level places for the values a + a' and /3 + /3', the corporate multiplicity of one aggregate being the same as the corporate multiplicity of the other. We can thus proceed from one pair of level values to another pair of level values for /and g — in the argument, we have chosen a succession with decreasing moduli — without, at any step, affecting the corporate multiplicity of the level places. Moreover, in this succession, it is necessary to have only a finite range for z, and only a finite range for /, because the ranges in the y, y plane and in the x, x plane in the two-plane representation described in \ 138, giving the finite irreducible places z, z , of § 153, are finite. Hence we infer the theorem : — The nmnher of irreducible level places, at which two independent and free uniform quadruply periodic f auctions f and g, having no essential singularity for finite values of the variables, acquire finite values a. and /S, so that f{z, z) = a, g (z, z) = /3, regard being paid to possible multiplicity of each such level place, is inde- pendent of the actual level values acquired by the functions. In particular, the number of level places is the same as the number of simultaneous zero places of two such functions, regard always being paid to possible multi- plicity of occurrence at a level place or a zero place. 17—2 260 ALGEBRAIC RELATIONS BETWEEN [CH. VIII The property also holds when the level value for either of the functions or for both of the functions is a unique infinity so that the level place is a pole (an unessential singularity of the first kind) for either of the functions or for both of the functions, as the case may be ; it follows at once by con- sidering the reciprocal of the function or of the functions having the place for a pole. But care must always be exercised to make certain that the functions are free as well as independent : thus the theorem would not apply to the poles of functions, such as do -^ ^12 and 0^^ 0^^ of § 152, because the poles, so far from being isolated, are the continuous aggregates of zeros of the function d^^. But the unessential singularities (the unessential singularities of the second kind) of a single function are isolated ; and when two functions are considered simultaneously, their unessential singularities are not necessarily (and are not usually) the same places. Hence the theorem does not apply to unessential singularities. And the theorem does not apply to essential singularities. If, then, we adopt a more comprehensive definition of level places and level values, the first including ordinary places and poles, and the second including zeros, finite values, and unique infinite values, we can say that the number of irreducible level places of two independent and free uniform quadrwply periodic functions, having no essential singidarity for finite values of the variables, is independent of the actual level values, regard being paid to possible multiplicity. This integer, being the number of irreducible level places of the two functions when regard is paid to possible multiplicity, will, after Weierstrass*, be called the grade of the pair of functions. Algebraic relations between functions. 156. Now consider two uniform quadruply periodic functions f{z, z') and g (z, z') — say / and g — which are independent and free ; and let them be of grade n, so that there are n irreducible places giving level values a and ^ to / and g. Let h (z, z') be another uniform function, homoperiodic with /and g. At each of the n irreducible level places of/ and g, the uniform function h has a single definite value ; and therefore, at the aggregate of those places, there are n values of h in all. Hence there are n values of h corresponding to assigned values of/ and g; and these n values arise solely from the values of / and g, without any intervention of the variables z and z' beyond their occurrence in / and g. Consequently, there is a relation between / g, h, * Crelle, t. Ixxxix (1880), p. 7; Ges. Werke, t. ii, p. 132. 156] PERIODIC FUNCTIONS 261 which is of degree n in h ; the coefficients in this relation are functions of /' and g alone. Next, suppose that / and h, being uniform quadruply periodic functions of z and z', are independent and free ; and let them be of grade m. Also ~ suppose that g and h are independent and free ; and let them be of grade I. Then an argument, similar to the argument just expounded, leads to the con- clusion that the relation between /, g, h, already known to be of degree n in h, is of degree I in /and of degree m in ^ : it is an algebraic relation. Of the n values of h, corresponding to assigned values of / and g, it can happen that several may coincide for some not completely general assignment of values. But if this coincidence occurs for completely general values of f and g, the values of h coincide in groups of equal numbers ; and the number of values of h, corresponding to assigned values of / and g, is a factor of n. Hence we have the theorem * : — I. Between any three uniform functions, which are homoperiodic in the same four period-pairs and which taken in pairs are independent and free, there subsists an algebraic equation: the degree of this equation in each of the functions either is equal to the grade of the other two functions or is equal to some integral factor of that grade. It is assumed explicitly that the functions, in pairs, are independent and free ; and the only level places that have been used for the functions are such as give finite level values to the functions. But it may happen that two functions, independent of one another, and free for all finite values (including zero), are tied as regards infinite values. Thus the quadruply periodic functions, which arise as the quotients by di^ of the quadruple theta functions other than 6-^^, cannot be estimated for grade by their infinities ; their infinities are given by the zeros of 6^^, and (except for the irreducible isolated unessential singularities, limited in number) they are the same for all the quadruply periodic functions so framed. These functions therefore, while they are independent, are tied as regards their infinities. The foregoing theorem is still true for these uniform functions : there is nothing to traverse the argument at any of its stages. But the effect of the tie, in connection with the infinities, is to simplify the form of the algebraic equation. We can suppose that the latter has been made rational and integral. The three functions /, g, h are infinite together and only together ; and therefore the terms of the highest aggregate order in all the functions combined will, by themselves, give relations among the parts of /, g, h that govern their infinities. * This theorem, and several of the theorems that follow, were enunciated by Weierstrass for 2n-ply periodic uniform functions of n variables. The enunciations, in most instances, are not accompanied by proofs ; they are to be found in his memoirs, Berl. Monatsh. (1869), pp. 853 — 857, ih. (1876), pp. 680—693, and Grelle, t. Ixxxix (1880), pp. 1—8 ; see also his Ges. Werke, t. ii, pp. 45 — 48, 55 — 69, 125 — 133. See also Baker, Multiply periodic functions, ch. vii. 262 ALGEBRAIC RELATIONS BETWEEN [CH. VIII 157. Among the functions related to any given uniform quadruply periodic function of two variables are its two first derivatives, which mani- festly are homoperiodic with the function. Moreover, all the infinities of the original function are infinities (as to place, but in increased order) of the derivatives; and they provide all the infinities of these derivatives. The foregoing theorem, when applied to a single function, leads to the result, practically a corollary : — II. Any uniform quadruply periodic function f{z, z) and its first derivatives -^ and ^, are connected by an algebraical equation. When the equation is made rational and integral, the aggregate of the terms of highest order gives relations among the constants of the infinities of f and its derivatives. Thus a quadruply periodic uniform function of two variables satisfies a partial differential equation of the first order, just as a doubly periodic uniform function of one variable satisfies an ordinary differential equation of the first order. 158. We return to homoperiodic functions. For purposes of reference among them, we select three uniform functions /, g, h, of the character prescribed in theorem I. Now let k {z, z') — say k — be another uniform function, homoperiodic with /, g, h; and let it be untied with any of them. Then between /, g, k there subsists an algebraical equation, the degree of which in k is either n or is a factor of n : taking the degree as n, we can denote the equation by A(fg,k) = 0. Also, between f h, k there subsists an algebraical equation, the degree of which in k is either m or is a factor of m : taking the degree as m, we can denote the equation by B (f h, k) = 0. Similarly, there is an algebraical equation C{g,h,k) = 0, which is of degree I in k; and there is the original algebraical equation l)(fg,h) = 0, which is of degree I in /, of degree m in g, and of degree n in h. These equations are necessarily consistent with one another ; thus the A;-eliminants of ^ - and B=0,o{B = and C = 0, of C = and ^ = 0, all vanish in virtue of D = 0. 158] HOMOPERIODIC FUNCTIONS 263 These A;-eliminants can be formed by Sylvester's dialytic process, because all the equations are algebraic; and an added use of the process leads to another important result. The equations ^'■^ (/ ^, A;) = 0, for r = 0, 1, ...,m-2] k'B (/; h, k) = 0, „ s = 0,l, ..., n - 2j are a set of m + n — 2 equations, linear and not homogeneous in the m + n — 2 quantities k, k~, ..., A;'"^^"^"^. When these are resolved for the m + n — 2 quan- tities, we have expressions for the various powers of k (in particular, for k itself) rational in the quantities /, g, h and reducible, by means of D =0, so as to contain either /to no degree higher than ^ — 1, or ^ to no degree higher than m — 1, or h to no degree higher than n—1. Paying no special regard to these degrees, but noting the assumption made as to the degree of the equation ^ = 0, we have the theorem : — III. When f and g are uniform functions, quadruply periodic in the same periods, and are of grade n, and when h is another uniform, function, which is homoperiodic with f and g, and which takes n distinct values at the reduced point-pairs determined by given values of f and g ; then any other uniform function, luhich is homoperiodic tuith f and g, can be expressed rationally in terms of f, g, and h, provided every two of the four functions are independent and free, and provided also no one of the functions has an essential singularity for finite values of the variables. And, as before, we have a corollary to the theorem, as follows : — IV. When two uniform quadruply periodic functions f {z, z') and g {z, z) are independent and free, and when neither of them, has an essential singularity for finite values of the variables, then g(z, z') can be expressed rationally in terms of f, :^ ,^,\ and f{z, z') can be expressed rationally • . J- da da m terms of a, ^ , ^, . ■' ^ dz dz Note. But just as there was possible degeneration of degree in the equation D (f g, h) = 0, so it might conceivably happen that, owing to the equation I) (f g, h) =0, the actual expression for k might not be deter- minate. But this indeterminateness would not occur for every power of k ; and so we should then only be able to infer that some power of k is rationally expressible in terms of /, g, h. Such cases occur when the fundamental periods of the functions considered are only commensurable with one another and are not exactly the same for all the functions. The exceptions may be wider than the exceptions of the same kind in the case of doubly periodic functions of one variable, though they will cover the generalisation of such 264 HOMOPERIODIC FUNCTIONS [CH. VIII apparent (but only apparent) exceptions to Liouville's well-known theorem which might imply that en z and dn z are expressible * in the form d where P and Q are rational functions of sn z. 159. Next, consider two uniform functions fiz, z) and g (z, z), homo- periodic in the same four pairs of periods ; and, as usual, assume that they are independent and free, their grade being n, and that they have no essential singularities for finite values of the variables. Their Jacobian /, with respect to the independent variables, is dzdz' dzdz JAM) d{z,z'y It is a uniform function, homoperiodic with / and g ; consequently it satisfies an algebraical equation, which has rational functions of / and g for its co- efficients, and the degree of which in / is either n or a factor of n. Moreover, as / and g are uniform, infinities of J can arise only through infinities of / or of g or of both ; and no infinity of / can arise from finite values of / or of g, or from any integral relation between / and g satisfied by finite values of / and g. Hence, when the algebraic relation between /, /, g is completely freed from fractions, the coefficient of the highest power of / is a constant ; and the degrees in / and g of the succeeding powers of J are limited. To indicate the limits, take the simplest forms of two extreme cases : (i) when / and g are completely free as to infinities : (ii) when they are completely tied as to infinities — in such a way as are e.g. the periodic functions indicated in § 152. In the former case, consider the vicinity of a simple simultaneous pole of / and g ; then we can take, in that vicinity, U ^R where V and S have a simple simultaneous zero at the place. Then T— T where T is a uniform function, regular, and usually not vanishing at the place. The place thus is an infinity of J, as is to be expected : manifestly it is of order 4. Hence in this case, the algebraic equation (taken to be of order n in J) must be such as to provide infinities of order 4 for / ; hence the coefficient * The explanation, of course, is that snz, en z, dn z do not possess the same fundamental periods. 160] ALGEBRAIC RELATIONS 265 of J'^~^' is a polynomial in f and g of order not greater than 4?i', while for some value or values of n, among 1, 2, . . . , n, it must be of order 4?i'. In the latter case, we can take U _R J— yy 9— Y' where the infinities of the functions (now tied) are given by F = ; then "^ = y-3 '^y where TT is a uniform function, regular, and usually not vanishing with V. The place thus is an infinity of J, as again is to be expected ; manifestly it is of thrice the order for / and g. As in the preceding case, the coefficient of jn-n' jg g^ polynomial in /and g of order not greater than Sn', while for some value or values of n', among 1, 2, ..., w, it must be of order Sn'. Other orders of infinities belonging to / and g will lead to other degrees for the polynomial coefficients in the equation. In all instances, we have the theorem : — V. The Jacobian J of two uniform quadruply periodic functions f and g, which are independent and free, and which have no essential sin- gidarities for finite values of the variables, satisfies an algebraic equation ; when this equation is of degree n, the coejficient of J'^ is unity and the coefiicient of J'^~^' is a polynomial in f and g, of degree not greater than 4in', for n=l,2,...,n. Also, n is either equal to the grade of f and g, or is a factor of that grade. 160. Combining this result with the earlier theorems I and III, we have the further theorem : — VI. When f and g are uniform functions, quadruply periodic in the same periods and of grade n, and when the algebraic equation satisfied by their Jacobian J is of degree n, any uniform function, which is homo- periodic ivith them, can be expressed rationally in terms of f g, and J, provided no two of the functions are tied as to level values, and provided neither of the functions has an essential singularity for finite values of the variables. In particular, for such functions / and g, we have the relations f^=F,{fg,J), f^=Odfg,J), f^, = F.Af9,J), ^£ = G.{fg,J), where ^i, F^, Gi, G^ are rational functions of the arguments. The algebraic relation J = F1G2— F2 Gi must be satisfied in virtue of the algebraic equation between/, g, and /. 266 QUADKUPLY [CH. VIII The quadruply periodic functions which arise out of the double theta-f unctions. 161. It is desirable to have some special illustrations of the foregoing general propositions relating to periodic functions of two variables. Accordingly, we assume that the coefficients ^(m — n, or, cr') of the triple theta- functions are so specialised as to yield the double theta- functions, periodic or pseudo-periodic in four pairs of periods, always limited so as to secure the convergence of the double series. Moreover, we shall assume that our functions have no essential singularity for finite values of the variables — an assumption which requires the theta-functions to be finite (as usual) over the whole field of variation given by these finite values. We thus have ten even functions, viz., 6,^, 6^, 6^, 6^, Oi, Oq, 6^, 6^, 6-^2, dw', and six odd functions, viz., 6s, dj, $10, 6u, 0i3, ^14: all these being functions of 2 and z'. When z = and z' = 0, the six odd functions vanish. The ten even functions then acquire finite constant values which are denoted by Cq, Cj, Cg, C3, C4, Cg, Cg, Cg, Ci2, Ci5 respectively. The effects upon any function By,' , 1 of a period-increment in the various cases are given by the relations ^fp,p Z+l\ ^^_^yJp,p,Z \a;^ \ _(_-iY'n(P> P'^ p, p' z^ a,A ^ ^_,.,,_.,«„ Q (p, p', ^\ 0-, 2 /i 2 2 Og — O3 t/g T ^6 (^12 C2 /I 2 ^__ /I 2 /t 2 I /I M ^ 2 02 — Uj 03 -r t/4 vg C2" C4 ^ Co" Cg + Cg" C25" I f, 2 f, 2 — f>2f,2if,'if> 2 ' C2" C12 ^ Cg" Cg" + Cj Ci i) 2 _i_ 2 2 ^ Co" Cg + Cg" C15" C2- Cg^ + C4' Ci Ci" Cg C3" Cg C2 /^ 2 ._ /I 2 /» 2 I /^ 2 ^ 2 1 '^12 — W ^9 ^ ^2 '^15 Among the simplest relations, expressing the squares of the odd functions in terms of the even functions, are the set f,2 a 2 — ^ 2 ^ 2 _ f,2 a 2 _ c:- 6',/ = -c:j ei-\-ci ei^cie,i\ - ci di + Cx- Oi + C42 6 Cg2 6i-ci do^-ci0jo'\^ Cii 6i - cj 6.2" - C4' 6i Cii Oi + Ci" 6i - Ci" 6'i2 c,iO,'-c:" ei-cidjj as well as others derived from the relations, among the even theta-functions above given, by using the table on p. 254 for interchanges among all the functions for half-period increments. 268 HYPERELLIPTIC FUNCTIONS OF [CH. VIII Lastly, for the present purpose, it is sufficient to give the three relations Co' 6/ = c^ di + c,i d,{- - c' 6>l4^ Co' ^lo' = Ca" 65" - Ci On + C3' Ou [ , Co' OJ = - Ci/ d,^ + Ci^ 6'n^ + C/ dj connecting the squares of odd functions alone. They can be derived from the relations connecting the squares of the even functions alone, by using the same table of interchanges for half-period increments of the variables. As regards the odd functions, we write "i/. = kij^z -^ K^Z + . . . , where the expressed terms are the terms of the first order, and /x has the values 5, 7, 10, 11, 13, 14; and we have CoCgCjg/Cg = C3 Cg Ci5 ft^io + C1C4C8 /Cis\ CzCgCi^K'^ ^ CiC^^Ci^fCio-h CsCqCs A^is C0C2C9 Kii = CiCgCg ft^io + C^CeCigKis CoC2Ci2/^i4 = C^CsCq A/'io CiCgCi^Kig with exactly the same relations when k' is substituted for k. 162. All the relations thus far given, connecting the theta-functions, and connecting the quotients of the theta-functions, are quadratic in form. In each relation, there are three such quotients. Every function involves two independent variables z and z' ; and therefore it is to be expected that each of the functions is expressible algebraically in terms of two new independent variables. This expectation is justified by the detailed results and properties of the double theta-functions which give rise to the hyperelliptic functions of order two, being quadruply periodic functions ; and the actual forms can be expressed as follows. We take five constants ai, ag, Wg, 6*4, as, unequal to one another; and we write dm - an = 'mn, for all the five values of m and of n, avoiding equal values, avoiding also some other similar limitations that obviously are to be avoided. Two variables ^ and ^' are introduced ; and we write r' = Kr- a,) (r- a.)(r - cis)(^- a,) (^'- a,)]K P={(p- aO (p-a.2)(p- as)(p- a^) {p- a,)}^. 162] OKDER TWO 269 Two other variables ii and u' are introduced, being defined by the equations = 11'^^:^ — 2 I p fC ri — The variables ^ and ^' are, in general, uniform quadruply periodic functions of u and u' ; for sufficiently small values of u and u , we have 13.14.15 ^-tti ^'-a,= 12 23.24 .25 2r iC' + I ■w'2 + where the unexpressed terms are of even orders (beginning with the order 4) in u and u' combined. The fifteen quadruply periodic functions of z and z' , arising from the quotients of the double theta-functions, are algebraically expressible as follows : — ?i2 = (12.13.14.15)-ij0i (^10- ^12 = (21. 23. 24. 25) -i^2 d, - ^j2 = (- 31 . 32 . 34 . 35) -*p3 ^i2 = (-41.42.43.45)-ip4 e, -^,0 = (51. 52. 53. 54) -^^5 6u - 0,, = (13 . 14 . 15 . 23 . 24 . 25) - ijh^ e, -6'i2 = (12. 14.15.32.34. 35) -?jji3 6'i2 = (12 . 13 . 15 . 42 . 43 . 45) ^^^^^ 6'i2 = (- 12 . 13 . 14 . 52 . 53 . 54>)-^p,, d,. = (21 . 24 . 25 . 31 . 34 . ^by^jj^, e,, = (21 . 23 . 25 . 41 . 43 . 45) - ^p,, 0,, = (- 21 . 23 . 24 . 51 . 53 . 54>yip,, 0,, = (31 . 32 . 35 . 41 . 42 . 45) " * p,, e, - 6^j, = (31 . 32 . 34 . 51 . 52 . 54)-^^3, where ?i4 - ^12 = (41 . 42 . 43 . 51 . 52 . 53)-i^4 Pr' = (ar-0((^r-n for r = 1, 2, 3, 4, 5 ; and Prs ^ J T ■/ PrPs 1(^ - ar) i^-as) cr' - ar) (r - as)] K'-V for all the ten combinations of r and s from the set 1, 2, 3, 4, 5. _] •270 INITIAL TERMS IN THE [CH. VIll The constant values of the even theta-functions for zero values of the variables are related as follows : Cio — ^2 • Ci2 — Cq ■ Cio — Ci -4- Ci Co ^ Cio = Cl9 Cfi . Cio — Cl9 Cio ^ Ci2 51.52 53754; _41.42\i 43 . 45; 31 . S2Y 34 . 35. 52.13.14 \^ i2 . 53 . 54 42.13.15 \^ 12 . 43 . 45/ 41.23.25\i 21 . 43 . 45. _ 51 . 23 . 24\i ~ 21 . 53 . 54 32. 14.15 N^ 12 . 34 . 35; 31.24. 25\i J ,21 . 34 . 35, The lowest terms in the odd theta-functions are as follows '13.15.23.25\i/ 14 , 24\ "12-^^1-2' + -^ 43.45 13.14.23 . 24\4 53.54 32 . 42 . 52\* 15 ,25\ ''12-^12] pi' + ... ^12 V 12 ^ = (13.14.15.23.24.25)* '31.41.51\i u—u ~l2" + r u+ ... On 21 15.14.25.24\* 13 ,23. , ^12~T2' + -^ V 34 . 35 The relations between the two variables u and u', and the two variables z and z', are ^0-^,^^ 32.42.52 ^*^,^ 12 u "-10 — z -\ z C12 c ks k^: , /31.41.51\* 7- ^ + — z^\ — U c?i2 Cio V 21 / 164] THETA-QUOTIENTS 271 The quadruply periodic functions of z and z are quadruply periodic functions of u and v! : and conversely. Finally, derivatives of any function, of the first order with regard to u and u', are linear combinations (with constant coefficients) of its derivatives of the first order with regard to z and /. Examples of the theorems in |§ 156 — 160. 163. Adequate illustrations of the first theorem, in § 156, are provided through the homogeneous relations among the theta-functions which have just been stated. Each of them, when divided throughout by the appropriate power of ^12, gives a relation among strictly periodic functions. Many other such relations are given in the memoir by Brioschi already quoted (p. 266, note) ; and many can be deduced from the algebraical expressions for the functions p in terms of the variables t and ^'. Among them, we select the following, as being of particular use in the succeeding investigation: — rs . rt sr^ .st tr . ts where rs = ar — as, and so on, and r, s, t are any three of the integers 1, 2, 3, 4, 5; also Pr" + -^ (Prs^ - Prt^) = rl . rill, (st)prp,i + (tr)pspsi + (rs) ptpti - 0, where r, s, t, I, m are the integers 1, 2, 3, 4, 5, in any order. These examples will suffice for the present requirement. 164. We now proceed to give an example of theorem II, in § 157, by forming the partial differential equation of the first order which is satisfied by the uniform quadruply periodic function p^. From the values of u and u', expressed in terms of ^ and ^' by means of definite integrals, we have the values of ^ , ^f- , ^, , ~ . Using: the ex- du du ou du ° pression for p^^ in terms of ^ and ^', we find 2 a£i ^ 1 aj 1 d^ Pi du ^— tti du ^' — tti du 2 aj?,_ 1 [ 2t ^^, 2t' 1 272 RELATIONS AMONG [CH. VIII and therefore T .^ 1 9pi ^ X 1 9jOi ^— tti Pi du pidu ^' — «! pi 3i* pi Bi^-' Now, for the values r = 3, 4, 5 in particular, we have SO that on substituting the foregoing values of t and r'. Thus, if we write we have a = -i?3i?i3 = (23)5i + (13)5/| /3 = -;)4Pi4 = (24)^1 + (14)^/ , 7 = -i>52?i5 = (25) qi + (15) q^ ] where a, /3, 7 are temporarily used to denote the combinations of q^ and g/. Again, from the values of the functions in terms of ^ and ^', we have K + ^4(1^13' -Pi4^) = 12. 15, i>,^ + ^^(p,,^-pH^) = 12.13, ^.-f-2=34(12.15-p,0 = C,say, Ps Pi -.--2=S4(12-13-Pi0 = ^.say. Pi' ., P^" , K' ^1^ and therefore Also so that say ; and similarly 13 . 14 31 . 34 41 . 43 34 31 p3^ = 31.34 + ^^,^ + ^K = 0+^Pi\ , -^ r. 54 ^ 51 , 164] HYPERELLIPTIC FUNCTIONS 273 say. Thus -^:=A These two quadratic equations satisfied by ^^4^ can be written Gp,' - (L - j3~ - CV) p:- + /3-c' = 0, Ap,' -{N-jS'-A f /') K + ^'<>' = ^ : where 41 , 41 r ,41 ^^ „41 "="-5V '='SV ^ = °'\S1' ^='^".51- Eliminating p^- between the two equations, we find {(L-/3-^-Gc')a'-(N-/3-'-Aa')c}{{N-^-'-Aa')C-(L-^'-Gc')A] = /8- {Ac - GaJ, which is a form of tlie partial differential equation of the first order satisfied hyp,. It is desirable that the equation should be simplified ; the various re- ■^es in algebra. We find ^-C=58(12.14-;V), ductions are mere exercises in algebra. We find so that 3^ 4 -a\„'r'=-- 18 . 15 Q4 ii^ FiS (A - G) a'c'=- ''^•.. ''•'r (12 ■ 14 - jV)(13 . 14 -j9,^) (14. 15 -p^) ; ah , , 141 . do /, o T r n\ so that 14 13.15 14 S4 Af^ Pi.Q ("' - c) AG = .'T' (12.13 -^j,^) (12 . 15 -pr) (13 . 15 -p^). And Ca' -Ac' = ^\^^;^^ (12 . 13 . 14 . 15 -jh')- As regards the parts involving derivatives, we have (X-/3:)«'-(iV-/30c' 14 = - jg7Y5 {'54 (14 . 15 - pr) a^ + 35 (13 . 15 - p,') /S'^ + 43 (13 . 14 -p,') r] 14.84.45.53,^., ., „^ ,^, 13715 ' ^^^^' ~^^" ^^^^ "^ '^^ ^"^' F. 18 274 EXAMPLES OF THE [CH. VIII on substitution for oc, (3, 7 ; and, similarly, f 12 ) = - 12 . 14 . 34 . 45 . 53 |(g, + q^y - ^Y7]A7lb ^'''^'\ ' Hence the differential equation for i\ takes the form 12M3. 14M5 (q. + ^,) (Q3+ 1^3:,) = (24.^1 + 14. fy^TX/, where the various symbols in the equation (which manifestly is of the first order, and of the fourth degree, in the derivatives of yji) have the values Zi = (12.14-pr)(13.14-^,0(14.15-iJr)| X = (12.13-_pr)(12.15-_?9,=)(13.15-^r) . 2%= 12. 13. 14. 15 -_/V I The infinity of 2\ at any place being of order k, that of q^ at the place and that of qi at the place are « + 1 ; from the terms of highest order in the infinities, as they occur in the differential equation, we have (as these orders) 8a: + 4, 10« + 2, 12/c, lO/c + 2, which are the same when ic = l: that is, any infinity of p^ is simple. The result is to be expected because p^ is a constant multiple of 6-^^6^^^ : so that an infinity of pi is a zero of ^12, that is, it is simple. The terms of highest order also provide relations among the constants connected with any such infinity : but these are not our present concern. 165. The partial differential equation of the first order for any other of the fimctions p can be constructed in the same manner ; in particular, the equation satisfied by ^2 can be derived from the equation satisfied by pi, through interchanging ^1 and 2)0, qi and q2, q^ and q.,, ai and ao, where ^' du' ^- du" Note. Another proof can be fi:'amed, by noting the relations Ce'Bio' = C12' do" - Ci-dj3- - CffdJ Ce'^n- = Cjo" ^1= - Co^djs- - C/0,„" Cq" "2" = C4" C/q" C9' I7j3" Cg" "12 166] ALGEBRAIC THEOREMS 275 among the theta-functions, by using the expressions for the constants c and the quotients of the theta-functions, and by observing that ^i^o^is"" is a con- stant multiple of the quantity denoted by 7 and that diO^di^f" is a constant multiple of the quantity denoted by ^. A third proof can be framed by noting the fact that P . . 1 dpi . X 1 9/>i p—cii Pi du ^ pi du' is satisfied by ^ = ^ and p = ^', so that the quartic equation (z - a,) (z - a,) {z - a,) (z - «,) -{z- c^ |(^ - cQ ^ ^^ + (^ - «i) ^^ gf^^ = has ^ and ^' for its roots. The analytical conditions for this property of the quartic equation ultimately lead to the partial differential equation of the first order satisfied by j9i. 166. The analysis in the preceding investigation leads to a simple illustration of theorems III and IV, in § 158. It must, however, be borne in mind that those theorems refer to functions that are homoperiodic. Now the functions p^ and pi are not homoperiodic : theii' periods are only commensurable. But the functions p^- and pi' are homoperiodic : and there- fore by the theorem IV", we must have ^4- expressible rationally in terms of pi^ and its first derivatives, that is, expressible rationally in terms of pi, qi, qi- The two quadratics that occur in the investigation give Pi' _ Ac' - A'c ~ J'~ {N - ^-^ - Aa') G - (L - /3' - Cc') A' or, with the preceding notation, p, ^ (2H + 14. w,, X sr {J {p.; 'P,)Y = [^ pfPni'Pn' — i ^r j . If . Is . mr . 1116 . ur . ns . P,^ , where p,,^ii-.4iL_4l\ii_j^ 1 - li . rs si. sty \ rm . rs sm . sr) \ rn . rs sn . sr so that the square of the Jacobian oi p,. and jj^ is an even polynomial in r and s of joint degree six. Similarly, we find [J{Pr,prs)Y=^,prt'Pn,rPni ~ T9i! [p^-s" '^pr" • st — rni . rn . 6'^} [jh-s' + Ih-'' • 6""^ ~ '"'^ • t'i • *""^1 X -J^rs' + 2h'' • sn — rt . rm . sn] ; and so lor other instances of Jacobians. So long as the Jacobians are formed from any two of the fifteen functions, the algebraical equation between two functions and their Jacobian is of even degree in the Jacobian. It is easy Lo verify that |J ypnn, 'Prn)\' is an even polynomial in prm twid pm of degree six ; and I'rom general con- siderations (but without having constructed the Vespective equations) I infer that J{pv,Pst), J{Prm,Pst) each of them satisfy an equation, quartic in its own Jacobian and of the degree twelve in the term fii-ee from the Jacobian. As a last illustration, consider a special case of theorem VI in § 160. The derivative of ^Ji with respect to u, already denoted (| 164) by q^, is quadruply periodic. It is homoperiodic with p^\ but it is not homoperiodic with p., 166] GENERAL THEOREMS 277 their periods being only commensurable. But q^^, p{^, p^^ are homoperiodic : and therefore, by the theorem, q-^- is rationally expressible in terms of p^, pi, and the Jacobian oi p-c' and pi; that is, q^^ is rationally expressible in terms of p^, p.2, and J (pi, po). The actual expression can be obtained in a variety of ways, requiring mere algebra for the purpose. Proceeding from the relation already obtained for q^, we find ultimately the following result. Let 12, Ir, ... denote aa — a.2, a^ — ar, ... as usual; wo-ite A = (pi-piy - 2 . 12^{2h' + pi) + 12^ ; /c, =p,= - p;- + 12 (Ir + 2r), for r = 1, 2, 3, 4, 5 ; and, for any quantity ^, let Then a rational expression for qi is 64 . qil2' . A + 128 . 12''p,^p.J(p„ pi) = {S, + >Sfo A + A^) (3/Ci A + «i3) - (^3 + ,Sfi A) (S/c^^A + A^. Other examples can easily be indicated : these will suffice for the present purpose. 18- INDEX {The mimbers refer to the pages.) Abel's theorem partially extended to double integrals involving a couple of algebraic functions of two independent variables, 193-197. Accidental singularity, 61 ; {see . unessential singularity). Algebraic functions in general, 61, 170 et seq. ; rational functions, involving one algebraic variable, 171, and two algebraic variables, 173 ; integrals of, 178 et seq. Algebraic relations between homoperiodic functions, 261 et seq. ; illustrations of, from hyperelliptic functions, 265 et seq. Analytic function, 59. Analytical continuation, 60., 80. Appell, 147, 285, 239. Baker, H. F., 110, 131, 261, 266. Berry, 170. Borel, 77, 78, 126. Boundaries of a region for certain fields of variation, and their frontier, 20, 24. Brioschi, 266. Bromwich, 72. Burnside, W., 26, 58, 237. Campbell, 42, Canonical form of lineo-linear transformations, 26; leads to powers of the transformation, 28; of equations for quadratic frontier, 51 ; of rational functions which involve algebraic variables, 171, 173. Castelnuovo, 170. Cauchy, 4. Cauchy's theorem as to the integral of a function of a single complex variable ex- tended by Poincare to functions of two complex variables, 13, 159. Conformal representation with one variable extended to two variables, 18. Continuation of regular functions, analytical, 80. Continuity of a function, region of, 81, 82, 86. Continuous function, 59. Continuous groups. Lie's theory of, applied to determine invariants and covariants of quadratic frontiers, 40, 42. Contour integrals, as used by Cousin, 131 et seq. Cousin, 130, 147. Dautheville, 80, 126. Dependent variables, number of, 2 ; used for a kind of inversion, 4. Divisibility (relative) of two regular functions, 112. Domain, 57. Dominant function, 71. Double-integral expressions connected with coefficients in the expansion of regular functions, 64. Double integral for real variables, application of theorem by Stokes on, 157. Double integrals, defined for two complex variables, 154 ; Poincare's extension of Cauchy's theorem for functions of a single variable, 159 ; residues of, with examples, 160 et seq. , Double integrals of rational functions in- volving two algebraic variables, 187 ; equivalent forms of, 189 ; conditions that they should be of the first kind, 190 ; Abel's theorem partially extended to, 193. Double theta-funetions, 249, 253 et seq. Enriques, 170. Equivalent functions, 134, 141. Essential singularity, 61, 83, 119, 123; be- haviour of a function at and near an, 77, 83 ; functions devoid of, 125. INDEX 279 Field of variation, in general, 57 ; for periodic functions, with one pair of periods, 224 ; with two pairs of periods, 225 ; with three pairs of periods, 231 ; with four pairs of periods, 236, together with a modified two- plane representation of the variables, 237. First kind of double integrals, conditions for, 190 ; extension of Abel's theorem to, 193. First kind of single integrals of algebraic func- tions of two variables, 178 ; initial condition as to form of subject of integration, 180 ; equivalent forms of, 180, with the necessary relations, 185 ; do not exist for general equations, 187. Four-dimensional space, used to represent two variables, 5 ; used by Poincare in connection with double integrals, 153. Free functions, 208 ; properties of two, 209- 212. Frontier of a region m certain fields of variation, 20, 24 ; its analytical expression, 21 ; invariantive, for lineo-linear transforma- tions, 32 ; quadratic, 34. Functions devoid of essential singularities, everywhere, 125 ; in the finite part of the field, 130 et seq. Geometrical representation of two variables, Chapter I ; in four-dimensional space, 5 ; by means of a line in ordinary space, 7 ; by means of two planes, one for each of the variables, 13. Gordan, 25. Grade of two uniform quadruply periodic functions, 260. Hadamard, 126. Hartogs, 62, 123, 131. Hermite, 4, 131. Hobson, 1. Homoperiodic functions, algebraic relations between, 261 et seq. Humbert, 170. Hurwitz, 126. Hyperelliptic functions of order two nsed to illustrate algebraic relations between homo- periodic functions, 265 et seq. Invariant centres of lineo-linear transforma- tions, 29. Invariantive frontiers for lineo-linear trans- formations, 32 ; simplest forms of, 34, 37. Invariants and co variants of quadratic frontiers, 39 ; invariants alone, 48. Inversion, a kind of, 4. Irreducible places of quadruply periodic func- tions, 257 ; any set expressible by a single place in an associated two-plane representa- tion, 257 ; their number for level values of two functions is finite, 258, and is indepen- dent of those level values, 259. Jacobi, 14, 26. Jacobian of two homoperiodic functions, 264 ; used, in connection with the two functions, for the rational expression of other homo- periodic functions, 265 ; equation satisfied by, when they are hyperelliptic, 275. Jordan, 26. Konigsberger, 255. Krause, 266. Kronecker, 4. Laguerre, 126. Larmor, 157. Laurent's theorem extended to functions of two variables, 87-91. Level places of two uniform functions (Chapter VII) ; must exist for assigned values of the functions, 203. Level values of a regular function, 108 ; order of, 111. Levi, E. E., 123. Lie, 25, 40, 42. Line in space used to represent two complex variables simultaneously, 7 ; limitations upon use of whole line, 11 ; by means of the points where it cuts two parallel planes, 12. Lineo-linear transformations, Chapter II ; canonical form of, 26 ; powers of, 28 ; in- variant centres for, 29 ; invariantive frontiers for, 32 ; property of, when coefficients are real, 35 ; periodic, 52. Lines, Volterra's functions of, 13. Independent functions, 208. Infinitesimal periods excluded, 213-216. Integral function, 60. Integrals, of functions of two variables (Chapter VI) ; of algebraic functions, 178 et seq. Meromorphic function, 61. Multiform function, 58. Multiplicity, of a simultaneous zero of two uniform functions, 168 ; expressed as a double integral, 169 ; of a level value of two functions, as a double integral, 169. 280 INDEX Ncether, 170. Non-essential singularity, 61 ; (see unessential singularity). Order of multiplicity, of a common zero of two uniform analytic functions, 205, 209 ; of level values of two uniform analytic functions, 212. Order, of zero of a regular function, 111; of pole of uniform function, 119. Ordinary place, 60. Osgood, 62. Pairs of periods for uniform functions of two variables (see period-pairs). Periodic functions in two variables (Chapter VIII). Periodic lineo-linear transformations, 19, 28, 52. Period-pairs, if infinitesimal, are excluded, 213 ; may not be more than four for uniform function of two variables, 216-223 ; one, 224 ; two, 224, with the different cases ; three, 226, with the different cases, and the general result, 231 ; four, 232, with the different cases, 235. Picard, Preface, 5, 14, 26, 77, 78, 92, 152, 153, 156, 161, 169, 170, 178, 193, 197. Picard's theorem, on functions that cannot acquire assigned values, extended to func- tions of two variables, 78. Picard's theorem concerning single integrals of rational functions involving one algebraic variable extended to integrals of rational functions involving two algebraic variables, 180-187. Poincare, Preface, 1, 4, 5, 13, 26, 71, 126, 131, 153. Poincare's extension of Cauchy's theorem to double integrals, 159 ; with inferences, 160 ; extension to the residues of double integrals, 160, 161, with examples, 161 et seq. Pole, 61, 85 (see unessential* singularity) ; ex- pression for uniform function in the vicinity of, 119 ; sequence and order of, 120. Polynomial, when a regular function is a, 74 ; properties of, as regards singularities, 124. Prym, 266. Quadratic frontiers, 34 ; invariants and co- variants of, 39; suggested canonical form for, 51. Quadruply periodic functions, 253 et seq. ; level places of two, 257 ; satisfy an algebraic partial differential equation of the first order, 262, with example, 273. Rational, any uniform function entirely devoid of essential singularities must be, 126. Rational function connected with algebraic equations in two independent variables, most general form of : (i) when there is one equation, 171; (ii) when there are two equations in two algebraic variables, 173 ; integrals of, 178 et seq. Rational function, singularities of, 125. Reducibility (relative) of two regular functions, 115. Region of continuity of a function, 81 ; its boundary, 82, 86. Regular functions, any uniform function having essential singularities only in the infinite part of the field is expressible as the quotient of two, 147. Regular functions, 60 ; fundamental theorem relating to, 62 ; double integral expression for the coefficients in the expansion of, 64 ; one property of, 73 ; condition that it is a polynomial, 74 ; analytical continuation of, 80 ; level values of, 108 ; relative divisibility of, 112. Relative, divisibility of two regular functions, 112 ; reducibility of functions, 115. Riemann, 4, 16. Riemann's definition of a function extended to two functions, 16. Sauvage, 58. Severi, 170. Simart, Preface, 92, 152. Simultaneous poles of two uniform analytic functions exist, 204 ; usually is an isolated place, 211. Simultaneous unessential singularities of two uniform functions do not exist in general, 204. Simultaneous zero, of two regular functions, must exist, 202 ; likewise for two uniform analytic functions, 203 ; usually is an isolated place, 207, 209, but there may be exceptions, 208. Single integral, 152. Single integrals of algebraic functions in- volving two algebraic variables, 178 ; equivalent forms of, 180, with necessary relations, 185 ; first kind do not exist for general equations, 187, Singularities, 61, 82, 119 ; of a rational function, 125. Stokes, 157. INDEX 281 Tlieta-functions, triple, 240 et seq. ; even functions and odd functions, 248; double, 249, 253 et seq. Tied functions, 208. Transcendental function, 60. Triple theta-functions, 240 ; effect on, caused by increments of periods, 242, by half-period increments, 250 ; two sets of, 251 et seq. Triply periodic functions, 238. Two functions, everywhere regular in the finite part of the field, must vanish at some common place, 202 ; Ukewise, when they are uniform and analytic, 203. Two-plane representation of the real parts of the variables used for quadruply periodic functions, 237, 257. Two-plane representation of two variables, 13 ; some properties of, 14 ; limitations of, 19. Umbral symbols introduced for coefficients in homogeneous forms, 41. Unessential singularity, 61, 83, 119; ex- pression of uniform function in the vicinity of, 121 ; is an isolated place, 122. Uniform analytic function must acquire an infinite value, 72, and a zero value, 76, and an assigned finite value. 76. Uniform function, -58. Uniform periodic functions (Chapter VIII). Valentiner, 25. Vicinity of a place, 58. Vivanti, 12. Volterra, 13. Weierstrass, Preface, 4, 77, 80, 82-86, 92, 101, 105, 112, 122, 124, 141, 214, 260, 261. Weierstrass's theorem on the behaviour of a uniform continuous analytic function in the vicinity of an ordinary place, 92 ; various cases of, 96, 97, 100; example of, 102; alternative method of proceeding in one case, 105. Weierstrass's theorem on functions entirely devoid of essential singularities, 126 ; proof of, 126-129 ; on functions having essential singularities only in the infinite part of the field, 130, with Cousin's proof, 130 et seq. Weierstrass's theorem on infinitesimal periods, 214. Weierstrass's theorems on algebraic relations between homoperiodic functions, 261 et seq.; illustrated by hyperelliptic functions, 265 et seq. Zeros (selected) of the theta-functions of two variables, 255. ©ambfitigc : PRINTED BY JOHN CLAY, M.A. AT THE UNIVEESITY PKESS k / BOSTON COLLEGE 3 9031 l.j 549206 9 mil BOSTON COLLEGE SCIENCE UBRAIff MATH, UEPT, BOSTON COLLEGE LIBRARY UNIVERSITY HEIGHTS CHESTNUT HILL. MASS. Books may be kept for two weeks and may be renewed for the saxne period, unless reserved. Two cents a day is charged for each book kept overtime. ^ ^i yon cannot find what yon want, ask the Librarian who will be glad to help you. The borrower is responsible for books drawn on his card and for all fines accruing on the same.