THEORY OF DIFFERENTIAL EQUATIONS. Loubon: C. J. CLAY AND SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE. SIasgob3: 50, WELLINGTON STREET. li~eiptig: F. A. BROCKIIAUS. Aebo 30-orr: THE MACMILLAN COMPANY. l3ombav anb Carcutta MACMILLAN AND CO., LTD. THEORY OF DIFFERENTIAL EQUATIONS. PART III. ORDINARY LINEAR EQUATIONS. BY ANDREW RUSSELL FORSYTH, Sc.D., LL.D., F.R.S., SADLERIAN PROFESSOR OF PURE MATHEMATICS, FELLOW OF TRINITY COLLEGE, CAMBRIDGE. VOL. IV. CAMBRIDGE: AT THE UNIVERSITY PRESS. 1902 All rights reserved. (ambribge: PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. PREFACE. THE present volume, constituting Part III of this work, deals with the theory of ordinary linear differential equations. The whole range of that theory is too vast to be covered by a single volume; and it contains several distinct regions that have no organic relation with one another. Accordingly, I have limited the discussion to the single region specially occupied by applications of the theory of functions; in imposing this limitation, my wish has been to secure a uniform presentation of the subject. As a natural consequence, much is omitted that would have been included, had my decision permitted the devotion of greater space to the subject. Thus the formal theory, in its various shapes, is not expounded, save as to a few topics that arise incidentally in the functional theory. The association with homogeneous forms is indicated only slightly. The discussion of combinations of the coefficients, which are invariantive under all transformations that leave the equation linear, of the associated equations that are covariantive under these transformations, and of the significance of these invariants vi PREFACE and covariants, is completely omitted. Nor is any application of the theory of groups, save in a single functional investigation, given here. The student, who wishes to consider these subjects, and others that have been passed by, will find them in Schlesinger's Handbuch der Theorie der linearen Differentialgleichungen, in treatises such as Picard's Cours d'Analyse, and in many of the memoirs quoted in the present volume. In preparing the volume, I have derived assistance from the two works just mentioned, as well as from the uncompleted work by the late Dr Thomas Craig. But, as will be seen from the references in the text, my main assistance has been drawn from the numerous memoirs contributed to learned journals by various pioneers in the gradual development of the subject. Within the limitations that have been imposed, it will be seen that much the greater part of the volume is assigned to the theory of equations which have uniform coefficients. When coefficients are not uniform, the difficulties in the discussion are grave: the principal characteristics of the integrals of such an equation have, as yet, received only slight elucidation. On this score, it will be sufficient to mention equations having algebraic coefficients: nearly all the characteristic results that have been obtained are of the nature of existence-theorems, and little progress in the difficult task of constructing explicit results has been made. Moreover, I have dealt mainly with the general theory and have abstained from developing detailed properties of the functions defined by important particular equations. The latter have been used as illustrations; had they been developed in fuller detail than is PREFACE vii given, the investigations would soon have merged into discussions of the properties of special functions. Instances of such transition are provided in the functions, defined by the hypergeometric equation and by the modern form of Lame's equation respectively. A brief summary of the contents will indicate the actual range of the volume. In the first Chapter, the synectic integrals of a linear equation, and the conditions of their uniqueness, are investigated. The second Chapter discusses the general character of a complete system of integrals near a singularity of the equation. Chapters III, IV, and V are concerned with equations, which have their integrals of the type called regular; in particular, Chapter V contains those equations the integrals of which are algebraic functions of the variable. In Chapter VI, equations are considered which have only some of their integrals of the regular type; the influence of such integrals upon the reducibility of their equation is indicated. Chapter VII is occupied with the determination of integrals which, while not regular, are irregular of specified types called normal and subnormal; the functional significance of such integrals is established, in connection with Poincare's development of Laplace's solution in the form of a definite integral. Chapter -VIII is devoted to equations, the integrals of which do not belong to any of the preceding types; the method of converging infinite determinants is used to obtain the complete solution for any such equation. Chapter IX relates to those equations, the coefficients of which are uniform periodic functions of the variable: there are two V111 PREFACE classes, according as the periodicity is simple or double. The final Chapter deals with equations having algebraic coefficients; it contains a brief general sketch of Poincare's association of such equations with automorphic functions. In the revision of the proof-sheets, I have received valuable assistance from three of my friends and former pupils, Mr. E. T. Whittaker,M.A., and Mr. E. W. Barnes, M.A., Fellows of Trinity College, Cambridge, and Mr. R. W. H. T. Hudson, M.A., Fellow of St John's College, Cambridge; I gratefully acknowledge the help which they have given me. And I cannot omit the expression of my thanks to the Staff of the University Press, for the unfailing courtesy and readiness with which they have lightened my task during the printing of the volume. A. R. FORSYTH. TRINITY COLLEGE, CAMBRIDGE, 1 March, 1902. CONTENTS. CHAPTER I. LINEAR EQUATIONS: EXISTENCE OF SYNECTIC INTEGRALS: FUNDAMENTAL SYSTEMS. ART. PAGE 1. Introductory remarks........ 1 2. Form of the homogeneous linear equation of order m. 2 3-5. Establishment of the existence of a synectic integral in the domain of an ordinary point, determined uniquely by the initial conditions: with corollaries, and examples.. 4 6. Hermite's treatment of the equation with constant coefficients 15 7. Continuation of the synectic integral beyond the initial domain; region of its continuity bounded by the singularities of the equation....... 20 8. Certain deformations of path of independent variable leave the final integral unchanged.. 22 9. Sets of integrals determined by sets of initial values.. 24 10. The determinant A(z) as affecting the linear independence of a set of m integrals: a fundamental system and the effective test of its fitness...... 27 11. Any integral is linearly expressible in terms of the elements of a fundamental system.... 30 12. Construction of a special fundamental system... 32 CHAPTER II. GENERAL FORM AND PROPERTIES OF INTEGRALS NEAR A SINGULARITY. 13. Construction of the fundamental equation belonging to a singularity........ 35 14. The fundamental equation is independent of the choice of the fundamental system: Poincare's theorem... 38 X CONTENTS ART. PAGE 15, 16. The elementary divisors of the fundamental equation in its determinantal form also are invariants. 41 17. Tannery's converse proposition, with illustrations.. 44 18. A fundamental system of integrals, when the roots of the fundamental equation are distinct from one another. 50 19. Effect of a multiple root....... 52 20. Transformation of the fundamental equation: elementary divisors of the reduced form..... 54 21, 22. Group of integrals connected with a multiple root: resolution of the group into sub-groups...... 57 23. Hamburger's sub-groups: equations characteristic of the integrals in a sub-group: and the general analytical form of the integrals....... 62 24. Modification of the analytical expression of the integrals in a Hamburger sub-group....... 64 25-28. Converse of the preceding result: the integrals in a sub-group satisfy a linear equation of lower order: Fuchs's theorem relating to such integrals.. 66 CHAPTER III. REGULAR INTEGRALS: EQUATION HAVING ALL ITS INTEGRALS REGULAR NEAR A SINGULARITY. 29. Definition of integral, regular in the vicinity of a singularity: index to which it belongs...... 73 30. Index of the determinant of a fundamental system of integrals all of which are regular near the singularity... 75 31. Form of homogeneous linear equation when all its integrals are regular near a........ 77 32, 33. Converse of the preceding result, established by the method of Frobenius..... 78 34. Series proved to converge uniformly and unconditionally. 81 35. Integral associated with a simple root of an algebraic (indicial) equation..... 85 36-38. Set of integrals associated with special group of roots of the algebraic (indicial) equation, with summary of results when all the integrals are regular.... 86 39. Definition of indicial equation, indicial function: significance of integrals obtained...... 94 40. The integrals obtained constitute a fundamental system: with examples. 95 CONTENTS xi ART. PAGE 41. Conditions that every regular integral belonging to a particular exponent should have its expression free from logarithms; with examples...... 106 42, 43. Conditions that there should be at least one regular integral belonging to a particular exponent and free from logarithms......... 110 44. Alternative method sometimes effective for settling the question in ~~ 42, 43....... 113 45. Discrimination between real singularity and apparent singularity: conditions for the latter..... 117 Note on the series in ~ 34....... 122 CHAPTER IV. EQUATIONS HAVING THEIR INTEGRALS REGULAR IN THE VICINITY OF EVERY SINGULARITY (INCLUDING INFINITY). 46. Equations (said to be of the Fuzchsian type) having all their integrals regular in the vicinity of every singularity (including o ): their form: with examples...123 47. Equation of second order completely determined by assignment of singularities and their exponents: Riemann's P-function........ 135 48. Significance of the relation among the exponents of the preceding equation and function......139 49. Construction of the differential equations thus determined. 141 50. The equation satisfied by the hypergeoretric series, with some special cases........144 51, 52. Equations of the Fuchsian type and the second order with more than three singularities (i) when oo is not a singularity, (ii) when oo is a singularity..... 150 53. N1ormal forms of such equations...... 156 54. Lame's equation transformed so as to be of Fuchsian type. 160 55. B6cher's theorem on the relation between the linear equations of mathematical physics and an equation of the second order and Fuchsian type with five singularities.. 161 56. Heine's equations of the second order having an integral that is a polynomial....... 165 57. Equations of the second order all whose integrals are rational 169 Xii CONTENTS CHAPTER V. LINEAR EQUATIONS OF THE SECOND AND THE THIRD ORDERS POSSESSING ALGEBRAIC INTEGRALS. ART. PAGE 58. Methods of determining whether an equation has algebraic integrals. 174 59. Klein's special method for determining all the finite groups for the equation of the second order.... 176 60. The equations satisfied by the quotient of two solutions of the equation of the second order: their integrals.. 180 61. Construction of equations of the second order algebraically integrable......... 182 62. Means of determining whether a given equation is algebraically integrable: with examples....184 63, 64. Equations of the third order: their quotient-equations. 191 65. Painleve's invariants, corresponding to the Schwarzian derivative for the equation of the second order; connection with Laguerre's invariant...... 194 66. Association with finite groups of transformations, that are lineo-linear in two variables......197 67. Indications of other possible methods..... 198 68. Remarks on equations of the fourth order.... 199 69. Association of equations of the third and higher orders with the theory of homogeneous forms. 202 70. And of equations of the second order. 206 71, 72. Discussion of equations of the third order, with a general theorem due to Fuchs; with example, and references for equations of higher order. 209 CHAPTER VI. EQUATIONS HAVING ONLY SOME OF THEIR INTEGRALS REGULAR NEAR A SINGULARITY. 73. Equations having only some of their integrals regular in the vicinity of a singularity: the characteristic index.. 219 74. The linearly independent aggregate of regular integrals satisfy a linear equation of order equal to their number. 222 75. Reducible equations....... 223 76. Frobenius's characteristic function, indicial function, indicial equation; normal form of a differential equation associable with the indicial function, uniquely determined by the characteristic function..... 226 77. The number of regular integrals of an equation of order m and characteristic index n is not greater than m-n. 229 CONTENTS xiii ART. PAGE 78. The number of regular integrals can be less than - n. 233 79. Determination of the regular integrals when they exist; with examples....... 235 80. Existence of irreducible equations..... 247 81. An equation of order m, having s independent regular integrals, has m - s non-regular integrals associated with an equation of order m - s...... 248 82. Lagrange's equation, adjoint to a given equation.. 251 83. Relations between an equation and its adjoint, in respect of the number of linearly independent regular integrals possessed by the two equations..... 256 CHAPTER VII. NORMAL INTEGRALS: SUBNORMAL INTEGRALS. 84. Integrals for which the singularity of the equation is essential: normal integrals. 260 85. Thome's method of obtaining normal integrals when they exist.......... 262 86, 87. Construction of determining factor: possible cases.. 264 88. Subnormal integrals....... 269 89, 90. Rank of an equation; Poincare's theorem on a set of normal or subnormal functions as integrals; examples 270 91. Hamburger's equations, having z=0 for an essential singularity of the integrals, which are regular at oo and elsewhere are synectic: equation of second order. 276 92. Cayley's method of obtaining normal integrals... 281 93, 94. Hamburger's equations of order higher than the second. 288 95. Conditions associated with a simple root of the characteristic equation for the determining factor... 294 96. Likewise for a multiple root..... 298 97. Subnormal integrals of Hamburger's equations.. 299 98, 99. Detailed discussion of equation of the third order.. 301 100. Normal integrals of equations with rational coefficients. 313 101. Poincare's development of Laplace's solution for grade unity......... 317 102. Liapounoff's theorem........ 319 103-105. Application to the evaluation of the definite integral in Laplace's solution, leading to a normal integral. 323 106. Double-loop integrals, after Jordan and Pochhammer. 333 107. When the normal series diverges, it is an asymptotic representation of the definite-integral solution... 338 108. Poincare's transformation of equations of rank higher than unity to equations of rank unity.... 342 xiv CONTENTS CHAPTER VIII. INFINITE DETERMINANTS, AND THEIR APPLICATION TO THE SOLUTION OF LINEAR EQUATIONS. ART. PAGE 109. Introduction of infinite determinants: tests of convergence: properties...... 348 110. Development......... 352 111. Minors.......... 353 112. Uniform convergence when constituents are functions of a parameter......... 358 113. Solution of an unlimited number of simultaneous linear equations......... 360 114. Differential equations having no regular integral, no normal integral, no subnormal integral. 363 115. Integral in the form of a Laurent series: introduction of an infinite determinant 1 (p).... 365 116. Convergence of Q (p)...... 366 117. Introduction of another infinite determinant D (p): its convergence, and its relation to Q (p), with deduced expression of Q (p). 369 118. Convergence of the Laurent series expressing the integral. 376 119. Generalisation of method of Frobenius (in Chap. III) to determine a system of integrals.... 379 120-123. Various cases according to the character of the irreducible roots of D(p)0..... 380 124. The system of integrals is fundamental.... 387 125. The equation D (p)=0 is effectively the fundamental equation for the combination of singularities within the circle I z= R...... 389 126. General remark: examples........392 127. Other methods of obtaining the fundamental equation, to which D (p)= 0 is effectively equivalent: with an example... 398 CHAPTER IX. EQUATIONS WITH UNIFORM PERIODIC COEFFICIENTS. 128. Equations with simply-periodic coefficients: the fundamental equation associated with the period.. 403 129. Simple roots of the fundamental equation... 407 130. A multiple root of the fundamental equation... 408 CONTENTS XV ART. PAGE 131. Analytical form of the integrals associated with a root. 411 132. Modification of the form of the group of integrals associated with a multiple root...... 414 133. Use of elementary divisors: resolution of group into subgroups: number of integrals, that are periodic of the second kind..... 416 134. More precise establishment of results in ~ 132... 417 135. Converse proposition, analogous to Fuchs's theorem in ~ 25 420 136. Further determination of the integrals, with examples. 421 137. Liapounoff's method..... 425 138-140. Discussion of the equation of the elliptic cylinder, w" + (a + c cos 2z) w = 0.... 431 141. Equations with doubly-periodic coefficients; the fundamental equations associated with the periods.. 441 142, 143. Picard's theorem that such an equation possesses an integral which is doubly-periodic of the second kind: the number of such integrals... 447 144, 145. The integrals associated with multiple roots of the fundamental equations: two cases.... 451 146. First stage in the construction of analytical expressions of integrals.... 457 147. Equations that have uniform integrals: with examples. 459 148. Lame's equation, in the form w"=w{n(n+l) d(z)+B}, deduced from the equation for the potential..464 149-151. Two modes of constructing the integral of Lame's equation 468 CHAPTER X. EQUATIONS HAVING ALGEBRAIC COEFFICIENTS. 152. Equations with algebraic coefficients.... 478 153, 154. Fundamental equation for a singularity, and fundamental systems; examples... 480 155, 156. Introduction of automorphic functions.... 488 157, 158. Automorphic property and conformal representation. 491 159-161. Automorphic property and linear equations of second order 495 162. Illustration from elliptic functions..... 501 163. Equations with one singularity...... 506 164. Equations with two singularities..... 507 165. Equations with three singularities..... 508 166. General statement as to equations with any number of singularities, whether real or complex... 510 xvi CONTENTS ART. PAGE 167. Statement of Poincare's results.... 515 168, 169. Poincare's theorem that any linear equation with algebraic coefficients can be integrated by Fuchsian and Zetafuchsian functions. 517 170. Properties of these functions: and verification of Poincare's theorem....... 521 171. Concluding remarks........ 524 INDEX TO PART III..... 527 CHAPTER I. LINEAR EQUATIONS; EXISTENCE OF SYNECTIC INTEGRALS: FUNDAMENTAL SYSTEMS. 1. THE course of the preceding investigations has made it manifest that the discussion of the properties of functions, which are defined by ordinary differential equations of a general type, rapidly increases in difficulty with successive increase in the order of the equations. Indeed, a stage is soon reached where the generality of form permits the deduction of no more than the simplest properties of the functions. Special forms of equations can be subjected to special treatment; but, when such special forms conserve any element of generality, complexity and difficulty arise for equations of any but the lowest orders. There is one exception to this broad statement; it is constituted by ordinary equations which are linear in form. They can be treated, if not in complete generality, yet with sufficient fulness to justify their separate discussion; and accordingly, the various important results relating to the theory of ordinary linear differential equations constitute the subject-matter of the present Part of this Treatise. Some classes of linear equations have received substantial consideration in the construction of the customary practical methods used in finding solutions. One particular class is composed of those equations which have constants as the coefficients of the dependent variable and its derivatives. There are, further, equations associated with particular names, such as Legendre, Bessel, Lame; there are special equations, such as those of the hypergeometric series and of the quarter-period in the Jacobian theory of elliptic functions. The formal solutions of such equations F. IV, 1 2 HOMOGENEOUS [1. can be regarded as known; but so long as the investigation is restricted to the practical construction of the respective series adopted for the solutions, no indication of the range, over which the deduced solution is valid, is thereby given. It is the aim of the general theory, as applied to such equations, to reconstruct the various methods of proceeding to a solution, and to shew why the isolated rules, that seem so sourceless in practice, actually prove effective. In prosecuting this aim, it will be necessary to revise for linear equations all the customarily accepted results, so as to indicate their foundation, their range of validity, and their significance. For the most part, the equations considered will be kept as general as possible within the character assigned to them. But from time to time, equations will be discussed, the functions defined by which can be expressed in terms of functions already known; such instances, however, being used chiefly as illustrations. For all equations, it will be necessary to consider the same set of problems as present themselves for consideration in the discussion of unrestricted ordinary equations of the lowest orders: the existence of an integral, its uniqueness as determined by assigned conditions, its range of existence, its singularities (as regards position and nature), its behaviour in the vicinity of any singularity, and so on: together with the converse investigation of the limitations to be imposed upon the form of the equation in order to secure that functions of specified classes or types may be solutions. As is usual in discussions of this kind, the variables and the parameters will be assumed to be complex. It is true that, for many of the simpler applications to mechanics and physics, the variables and the parameters are purely real; but this is not the case with all such applications, and instances occur in which the characteristic equations possess imaginary or complex parameters or variables. Quite independently of this latter fact, however, it is desirable to use complex variables in order to exhibit the proper relation of functional variation. 2. Let z denote the independent variable, and w the dependent variable; z and w varying each in its own plane. The differential equation is considered linear, when it contains no term of order higher than the first in w and its derivatives; and a linear equation is called homogeneous, when it contains no term independent of w 2.] LINEAR EQUATIONS 3 and its derivatives. By a well-known formal result*, the solution of an equation that is not homogeneous can be deduced, merely by quadratures, from the solution of the equation rendered homogeneous by the omission of the term independent of w and its derivatives; and therefore it is sufficient, for the purposes of the general investigation, to discuss homogeneous linear equations. The coefficients may be uniform functions of z, either rational or transcendental; or they may be multiform functions of z, the simplest instance being that in which they are of a form ( (s, z), where q is rational in s and z, and s is an algebraic function of z. Examples of each of these classes will be considered in turn. The coefficients will have singularities and (it may be) critical points; all of these are determinable for a given equation by inspection, being fixed points which are not affected by any constants that may arise in the integration. Such points will be found to include all the singularities and the critical points of the integrals of the equation; in consequence, they are frequently called the singularities of the equation. Accordingly, the differential equation, assumed to be of order m, can be taken in the form dmw dn-lw dm-2w -dz = pl dn + P2 d- + d... +' + p, where the coefficients pi, p,, pm are functions of z. In the earlier investigations, and until explicit statement to the contrary is made, it will be assumed that these functions of z are uniform within the domain considered; that their singularities are isolated points, so that any finite part of the plane contains only a limited number of them: and that all these singularities (if any) for finite values of z are poles of the coefficients, so that their only essential singularity (if any) must be at infinity. Let ' denote any point in the plane which is ordinary for all the coefficients p; and let a domain of i be constructed by taking all the points z in the plane, such that |z- 1| la- 1, where a is the nearest to ' among all the singularities of all the coefficients. Then within this domain (but not on its boundary) we have Ps = P, (= (S ) 2,...,m), * See my Treatise on Differential Equations, ~ 75. 1-2 4 SYNECTIC [2. where P8 denotes a regular function of z -, which generally is an infinite series of powers of z- 5 converging within the domain of g. An integral of the equation existing in this domain is uniquely settled by the following theorem:In the domain of an ordinary point, the differential equation possesses an integral, which is a regular function of z - and, with its first m - 1 derivatives, acquires arbitrarily assigned values when z = r; and this integral is the only regular function of z- ' in the specified domain, which satisfies the equation and fulfils the assigned conditions*. The integral thus obtained will be called the synectic integral. SYNECTIC INTEGRALS. 3. The existence of an integral which is a holomorphic function of z- { within the domain will first be established. Let r be the radius of the domain of ~; let M1,..., Mm denote quantities not less than the maximum values of 1p1,..., ipmi respectively, for points within the domain; and let dominant functions 0b,..., (m, defined by the expressions Ms Os =, (s =,..., m), r be constructed. Then: daps. d__s dz"a z= dza z=. for every positive integer a. The dominant functions b are used to construct a dominant equation dmu dm-1u dm-2lt dm = 01 dzmz- + 02 dzm-2 + *.. + fmu, which is considered concurrently with the given equation. * The conditions, as to the arbitrarily assigned values to be acquired at v by w and its derivatives, are called the initial conditions; the values are called the initial values. t As it is a regular function of the variable, it would have been proper to call it the regular integral. This term has however been appropriated (see Chapter III, ~ 29) to describe another class of integrals of linear equations; as the use in this other connection is now widespread, confusion would result if the use were changed. + See my Theory of Functions, 2nd edn., ~ 22: quoted hereafter as T. F. 3.] INTEGRALS 5 Any function which is regular in the domain of ' can be expressed as a converging series of powers of z-; and the coefficients, save as to numerical factors, are the values of the various derivatives of the function at A. Accordingly, if there is an integral w which is a regular function of z -, it can be formed when the values of all the derivatives of w at ' are known. To dw dm-lw w, d' ' dz... the arbitrary values specified in the initial conditions are assigned. All the succeeding derivatives of w can be deduced from the differential equation in the form daw dm-lw dm-2w A,,a + Aa2 + A aW, dzw dzm- + dzn-2. + A (for a =m, m + 1,... ad inf.), by processes of differentiation, addition, and multiplication: as the coefficient of the highest derivative of w in the equation (and in every equation deduced from it by differentiation) is unity, new critical points are not introduced by these processes, so that all the coefficients A are regular within the domain of C. The successive derivatives of u are similarly expressible in the form da- dim-1u dn-2 u -Bal d + B2 d 2 + ' t Bamun, dza dzi-n dz-+. (for a=m, m+ 1,... ad inf.), obtained in the same way as the equation for the derivatives of w. The coefficients B have the same form as the coefficients A, and can be deduced from them by changing the quantities p and their derivatives into the quantities p and their derivatives respectively. The values of the derivatives of w and u at C are required. When z = g, all the terms in each quantity B are positive; on account of the relation between the derivatives of the quantities p and (, it follows that Bas > I Aas, (s = 1,..., m), dw dM~-lw when z=. Let the initial values of lw,,dz..., dm 1, when du d"-'u z=', be assigned as the values of u, dz,..., dzm- when z=; then daw dau dZa dza 6 EXISTENCE OF [3. when z =, for the values m, m + 1,... of a. If the series (U) + (z-t') K;)j + ( d2) + converges, where (d ) denotes the value of - when z =, the onverge;s0, Kldza dza series (W) + dwA (z - + )2 d2w'\ d(- ) + 2! dz<+ - fdmw\,,. cdaw where -( d) denotes the value of - when z = C, also converges; \dz x dcz it then represents a regular function of z- ' which, after the mode of formation of its coefficients, satisfies the differential equation. We therefore proceed to consider the convergence of the series for u, obtained as a purely formal solution of the dominant equation. To obtain explicit expressions for the various coefficients in this series, let z - = rx, taking x as the new independent variable. Points within the domain of ' are given by Ix < 1; and the dominant equation becomes dmu rn drn-su (1 - x). r'dx,_ When the series for u, taken in the form 00 U = S bax a=0 is substituted in the equation which then becomes an identity, a comparison of the coefficients of xk on the two sides leads to the relation (m + k))! bm+k = (m + k - 1)! (k + Mr) bm+k- + (m + k - s)! Msrs bm+k-, s=2 holding for all positive integer values of k. This relation shews that all the coefficients b are expressible linearly and homogeneously in terms of bo, b,,..., bm_,: and that, as the first m of these coefficients have been made equal to the moduli of the m arbitrary quantities in the initial conditions for w and therefore are positive, all the coefficients b are positive. Hence k + Mir bm+k > k + mk — Ic+m rn-i 3.] A SYNECTIC INTEGRAL 7 By the initial definition of M,, it was taken to be not less than the maximum value of IP11 within the domain of '; it can therefore be chosen so as to secure that Mr > m. Assuming this choice made, we then have bm+k > bm+k-1, so that the successive coefficients increase. From the difference-equation satisfied by the coefficients b, it follows that bm+k _k + M1r (m + - s)! M bm+k-s - ( m + k)I r bm+kl' bm+k-1 c + n,=2 (m +)! bm+kSo far as regards the m -1 terms in the summation, the ratio bm+k-s - bm+k-i is less than unity for each of them; Msrs is finite for each of them; and (m + k - s)!- (m + k)! is zero for each of them, in the limit when k is made infinite. Hence we have Lim bm+k =1, k=o0 bm+k-1 and therefore Lim b1+, x l+' j=X b, ixW <1, for points within the domain of 5, so that * the series ao a=O converges within the domain of. The convergence is not established for the boundary, so that it can be affirmed only for points within the domain; it holds for all arbitrary positive values assigned to bo, bi,..., bm-_. It therefore follows that, at all points within the domain of, a regular function of z - exists which satisfies the original differential equation for w, and, with its first m -l derivatives, acquires at ' arbitrarily assigned values. 4. Now that the existence of a synectic integral is established, the explicit expression of the integral in the form of a power-series in z -, this series being known to converge, can be obtained * Chrystal's Algebra, vol. ii, p. 121. 8 UNIQUENESS OF [4. directly from the equation. As ' is an ordinary point for each of the coefficients p, we have PsP= Ps-, (= 1, 2,...,m), where Ps denotes a regular function of z- '. Let a0, a,,..., a,_ dw di-lw be the arbitrary values assigned to w, dz,., dz- ' when z =; and take t = E (Z_ ' )which manifestly satisfies the initial conditions. In order that this may satisfy the equation, it must make the equation an identity when the expression is substituted therein. When the substitution is effected, and the coefficients of (z - )s on the two sides of the identity are equated, we have a relation of the form am+s _ A s! - m+s, where Am+, is a linear homogeneous function of the coefficients a,, such that c < m + s, and is also linear in the coefficients in the quantities P1 (z - ),..., Pm (z - ~); and the relation is valid for s = 0, 1, 2,..., ad inf. Using the relation for these values of s in succession, we find am, a,,+m, a,+2,... expressed (in each instance, after substitution of the values of the coefficients which belong to earlier values of s) as a linear homogeneous function of the quantities a0, ac,..., am_i: and in am+s, the expressions, of which the initial constants ax, al,..., oam_ are coefficients, are polynomials of degree s + 1 in the coefficients of the functions P (z-C),..., Pm (z - '). The earlier investigation shews that the power-series for w converges; accordingly, the determination of the coefficients a in this manner leads to the formal expression of an integral w satisfying the equation. 5. Further, the integral thus obtained is the only regular function, which is a solution of the equation and satisfies the initial conditions associated with a0, al,..., am-1. If it were possible to have any other regular function, which also is a solution and satisfies the same initial conditions, its expression would be of the form nz - 1 + Xc Z- I,,=O - Ym u THE SYNECTIC INTEGRAL 9 a regular function of z -. The coefficients would be determinable, as before, from a relation -m+As m A+ where A'm+S is the same function of a0,,..., a_, a,..., a/ +S_ as Am+s is of ao,..., am-, a,..., am+s-i. Hence aCI, = Alm = Am =am; a n+l = A/m+i = Am+i, after substitution for a',, = am+l; and so on, in succession. The coefficients agree, and the two series are the same, so that w = w'; and therefore the initial conditions uniquely determine an integral of the equation, which is a regular function of z - C in the domain of the ordinary point f. COROLLARY I. If all the initial constants ao, al,..., am_- are zero, then the synectic integral of the equation is identically zero. For in the preceding discussion it has been proved that am+S, for all the values of s, is a linear homogeneous function of a0,..., am,; hence, in the circumstances contemplated, a,+S = 0 for all the values of s. Thus every coefficient in the series vanishes; accordingly, the integral is an identical zero. COROLLARY II. The initial constants ao, al,..., am- occur linearly in the expression of the synectic integral; and each of the m variable quantities, which have those constants for coefficients, is a synectic integral of the equation. The first part is evident, because all the coefficients in w are linear and homogeneous in a0, al,..., am_. As regards the second part, the variable quantity multiplied by as is derivable from w by making as = 1, and all the other constants a equal to zero; these constitute a particular set of initial values which, according to the theorem, determine a synectic integral of the equation. Thus the synectic integral, determined by the initial values a0,..., ami-, is of the form ao U + a iU2 +... + am-ium, where each of the quantities u1, u2,..., Ur is a synectic integral of the equation. Note 1. The series of powers of z -, which represents the synectic integral, has been proved to converge within the domain 10 EXISTENCE OF [5. of ', so that its radius of convergence is a -, where a is the singularity of the coefficients which is nearest to '. All these singularities lying in the finite part of the plane are determinable by mere inspection of the forms of the coefficients: another method must be adopted in order to take account of a possible singularity when z= =o because, even though z= =o may be an ordinary point of the coefficients, infinite values of the variable affect the character of w and its derivatives. For this purpose, we may change the variable by the substitution Zx = 1, and we then consider the relation of the x-origin to the transformed equation as a possible singularity. The transformation of the equation is immediately obtained by means of the formula dzk a=i a!(a- 1)! (k- a)! dx-a inspection of the transformed equation then shews whether x = 0 is, or is not, a singularity. Or, without changing the independent variable, we may consider a series for w in descending powers of z: examples will occur hereafter. It may happen that there is no singularity of the coefficients in the finite part of the plane, infinite values then providing the only singularity. In that case, we should not take the quantity r in the preceding investigation as equal to l o —[, that is, as infinite; it would suffice that r should be finite, though as large as we please. It may happen that there is no singularity of the coefficients for either finite or infinite values of z; if the coefficients are uniform, they then can only be constants. The dominant equation is then effectively the same as the original equation; the investigation is still applicable, but it furnishes less information as to the result than a method which will be indicated later (~ 6). Note 2. The preceding proof is based upon that which is given* by Fuchs in his initial, and now classical, memoir on the theory of linear differential equations. * Crelle, t. LXVI (1866), pp. 122-125. 5.] A SYNECTIC INTEGRAL 11 The theorem can also be established by regarding it as a particular case of Cauchy's theorem, which relates to the possession of unique synectic integrals by a system of simultaneous equations. If Wa = dza (a=0, 1,..., m- 1), the homogeneous linear equation of order m can be replaced by the system dws d = w,+l, for s = 0, 1,.., n - 2, dWm1 dz -= PiWm-i + p2Wm-2 +... + pWo. These equations possess integrals, expressible as regular functions of z - I, such that wo, w,,..., wm-_ assume arbitrarily assigned values when z= ', and the integrals are unique when thus determined: which, in effect, is the theorem as to the synectic integral of the linear equation*. NYote 3. A different method for establishing the existence of the integrals, though it does not indicate fully the region of their convergence, can be based upon a suggestion made by Giinthert. It consists in the adoption of another subsidiary equation dmv dm-Iv dm-2v dzm dzm-1 Z + Z...M + *- mV, where r = ^ -,_ for = 1,..., rn. The advantage of this form of equation is that its integrals are explicitly given in the form = (1 -Z-, where o is a root of the equation (o - )... (a - m + 1)= - rM (o - 1)... ( - n + 2) + r2M2- (o- -1)... ( - n + 3) +... + (- l)m'-rm-lMm-_ + (- 1)1l"Mm. * See Part II of this Treatise, ~~ 4, 10-13. t Crelle, t. cxvIIi (1897), pp. 351-353; see also some remarks thereupon by Fuchs, ib., pp. 354, 355. 12 EXAMPLES [5. If a root a- is multiple, the corresponding group of integrals is easily obtained*. The construction of the actual proof on the foregoing lines is left as an exercise. Ex. 1. Consider the equation d2w 2z dw K dz2 1-z2 dz + l _ 2 ' where K is a constant. The singularities in the finite part of the plane are z=1, z=-1. On transforming the equation by the substitution zx= 1, so that it becomes d2w 2x dw K ____ _ __ —.- Q=O, dx2 1- x2 dx Xi (I - x) we see that x=0 (and therefore z = ) is another singularity of the coefficients: so that the preceding investigation does not apply to the immediate vicinity of x= 0. It is clear that the z-origin is an ordinary point of the coefficients of the original equation: the domain of z=0 is a circle of radius unity. The equation therefore possesses a synectic integral, which is a series of powers of z converging within the circle; it is uniquely determined by the conditions that w=a, d-=3, when z=0, where a and 3 are arbitrary constants. To dZ= obtain its expression, let a0"= E bnZn n=O be substituted in d2w dw (1 - z2) -- 2z - +KW = 0, which then must be an identity. In order that the coefficient of zn may vanish after substitution, we must have (n+2) (n+ 1) b+2 - (n2+n- K) b,=0, so that +- n-K_ ob+2=(n+2) (+il) n Now by the initial conditions, we have bo=a, bi==3; hence b (2m-1 ) (2m- 2)- b 2- 2m (2m -) 2m-2 s=m = 2 n {(2s - 1) (2s - 2)-K}; 2m! s=1 * See my Treatise on Differential Equations, ~~ 47, 48. 5.] EXAMPLES 13 and, similarly, (2+l) s= b +1= (2~+1)! IIl {2s(281)- K; the expressed products being taken for integer values of s from 1 to m. The synectic integral satisfying the initial conditions is z2m s=m z2m + l s=m a E 2 - II {((2s-1)(2s-2)-K})+ 3 (2 +) 1 {2s (2s -1) - K)}; m=o0 2m! s=1 m=o(2+- 1) s=1 both series, if infinite, converging for values of z such that I z < 1. The best known instance of this equation is that which is usually associated with Legendre's name: K then is p (p + 1), and p (in the simplest form) is a positive integer. If p be an even integer, all the coefficients b2m, for 2m > p, vanish, so that the quantity multiplying a is then a polynomial; the quantity multiplying 3 is an infinite series. If p be an odd integer, all the coefficients b2m+1, for 2m+ 1 >p, vanish, so that the quantity multiplying f is then a polynomial; the quantity multiplying a is an infinite series. In all other cases, the quantities multiplying a and /3 are, each of them, infinite series; in every instance, the series converge when I z I < 1. Ex. 2. Obtain the synectic integral of the equation d2w 1dw / b\ dz2 + z dz+ a-2 =O, (which includes Bessel's equation as a special case), with the initial conditions dw that w=a, -d =/3 when z=c, where c I > 0. Ex. 3. Determine the synectic integral of the equation of the hypergeometric series d2A dw z(1 -z) d+y-( + (a+/3 +1)) d-a/3= dw the initial conditions being that w==A, d =B, when z=. Ex. 4. Determine the synectic integrals in the domain of z=0, possessed by the equation dw with the initial conditions (i) that w=l, d =0, when z=0; dw (ii) that w=0, - =1, when z=0. Ex. 5. Prove that the synectic integral in the domain of z=0, possessed by the equation d= Weaz dz2 14 EQUATIONS WITH [5. dw with the initial conditions that w=, -= O, when z=O, is U2. 2 az3 1 1+a2 4a+a3 + 11la2 +a4 w=l+- + - + Z4 + T 5 - Z6-; wl+2. 3+! 4! + 5! 6! and if the term in w involving zn be en Zn, then n. Cn=an-2+(2n-2-n+1) atn-4+ {I 3n-2- (-4) 2n-3- n-} an-6+.... Prove also that the primitive can be expressed in terms of Bessel's functions 2i laz of order zero and argument - ea a Ex. 6. The equation with constant coefficients may be taken in the form dn'w dn-lw dm-2W dzm 1 d- 1 2i dZ n-2 it possesses a synectic integral in the form oo Zk w-= E ak l, k=0 which converges everywhere in the finite part of the plane: and a0,..., am-, are the arbitrarily assigned initial constants. Substituting in the differential equation this value of w, and equating coefficients of n, we have n. an+n= Cl am + n+-1 + 2am + n2... an. The expression of the coefficients a,,, am+l,... in terms of a0, al,.., am-_ depends (by the solution of the foregoing difference-equation) upon the algebraical equation ( (0) = 8 -- 1 - 1 _ - Cm-2 - 2... - Cm= 0. When the roots of q (0)=0 are different from one another, let them be denoted by a, a2,..., am; and in connection with the m arbitrary constants a0, al,..., a_-,, determine m new constants Al, A,..., Am, by the relations ar 2 a IrA, (r=O, 1,..., in —1). fL=1 The determination is unique: for on solving these in relations as m linear equations in Al,..., A, the determinant of the right-hand sides is 1, 1,..., 1 a1, a2.. a m a12 a22... am2 am - 1 -, - which is equal to the product of the differences of the roots and is therefore not zero. Hence, as the constants a0, al,..., am-1 are arbitrary, the m new 5.] CONSTANT COEFFICIENTS 15 constants Al,..., Am, when used to replace the former set, can be regarded as m independent arbitrary constants. With these constants thus determined, we have 7w in agm+nn A E, (~latm+n-l1+42alQnm+n-2+... +CqMnaln) Al m m m -el a a A n+n-lA.+C 2 2 am+n-2A4+... +cm 2 aYAL, /u=1,=1 / for all values of n. When n =0, we have 2 am A=-Cl am, l+2am -+... '+ ao = am; f.=l when n= 1, we have aLnm+l Ai=clamn+42am-l +... + al=anz+; tz=l and so on, the general result being that m E,, m, + n A z — am + n p=1 for all values of n. Hence oo zk qV= 2 akk=O = 2(A^a1'~af+A... 4-a2)'k+ k==0O =Alez + AzA2ea2"z+.. + eamz, the customary form of the solution, Al,..., A, being m independent arbitrary constants. Ex. 7. Apply the preceding method to obtain a similar expression in finite terms, when the roots of the equation p (0)=0 are not all different from one another. 6. A different method of discussing the linear equation with constant coefficients has been given by Hermite. Taking the equation, as before, in the form dmuw dm-lw dm-2w d = eC + c dn-l +..+ Cmw, we associate with it the expression q (0) = m - (C1am-l + C2 m-2 +... + c,). Denoting by f () any polynomial in ', let WT2 =I ez f (r) d:, e e i i integration being taken round any simple contour in the '-plane. 16 HERMITE'S METHOD FOR EQUATIONS [6. In the first place, the degree of the polynomial f(~) may be taken to be less than m. If initially it is not so, then we have f<() = fi () fp(0 A )0 on division, g (~) being a polynomial, and fi (~) a polynomial of order less than that of b, that is, less than m. Now fezg () d = o, round any simple contour in the p-plane; in the remaining integral, the polynomial is of the form indicated. Accordingly, f(g) will be assumed to be of order less than m. We have dr eZi (r= 0, 1, 2, ) taken round the same contour; so that dmlW dm-lW dM-2W dm C dz f- + C2m (m-2 + m... ' C dW dzl dez-2 f( 2irrl eZ f (~) d 2iwj cb(n) =0, because f( ) is a polynomial and the integral is taken round a simple contour in the d-plane. Thus W is a solution of the equation. The only restriction upon f(g) is that, effectively, its degree must be less than m. It may therefore be taken as the most general polynomial of degree m- 1; in this form, it will contain m disposable coefficients which can be used to satisfy the initial conditions. Let these conditions require that, when x =0, the variable w and its first m - 1 derivatives acquire values k10,,..., kl_m respectively; then we determine f(~) as follows. Since (dr Wl ) kr /() f rd:, we shall draw the simple contour in the ~-plane so as to enclose the origin; and then the preceding relation shews that, when 6.] WITH CONSTANT COEFFICIENTS 17 f(O.,(g) is expanded in descending powers of g, the coefficient of '-r-1 is k,.; so that, as it holds for r = 0, 1,..., - 1, we have f(g') k 0 k, km-l ( - ++..+ + +..., +< (~? 2+ T+ 1 and therefore /(.=.) 2 + +.***. As f(C) is a polynomial in ', all terms involving negative powers of must disappear, when multiplication is effected on the righthand side; and therefore m-l f(T) = S kr {g.m-r-l - (cJ~nL-r-2 +... + Cm-r-_)}, r=o the coefficient of km,_ being unity. If therefore w and its first s derivatives are all to acquire the value zero when z = 0, then the degree of the polynomial f(') is m - s- 2. In order to obtain the customary expression for W, let the contour be chosen so as to include all the zeros of b (g). Let ac be a zero, and let its multiplicity be nj, so that b (4) = (I- )n, 0, (O), where the roots of cp (4) are the other roots of b (4). Let f() A'l A'21 A' f() p+ () 4' ++, (0)?- = ( - (a,)2 ( 0- 1)-l (g)' A'l1, A'2,..., being constants, and fi (4) a polynomial of order m - i - 1. So far as the first n1 terms are concerned, their contribution to the value of the expression for W is given by taking a contour round a, only. We then have 1 e r d A' A d 'r-i e - r r (e- ) 27ri (-a r (r- 1)! dar(e = Alzr-1 eza, on changing the constants; and therefore the part, arising through the root a, of multiplicity n,, in the expression for the integral is (A11 + A2z +... + An,1 zn-) eza,, F. IV. 2 18 HERMITE'S METHOD FOR EQUATIONS [6. involving a number of constants equal to the multiplicity of the root. This form holds for each root in turn; and therefore the number of constants is the sum of the multiplicities, that is, it is equal to m, the degree of B ('). But nm is the number of arbitrary constants in f('), when it is initially chosen: these can therefore be replaced by the constants A in the expression (Al + A3z +... + Anzn-1) eaz the summation extending over the roots a of 0(0)= 0, and n denoting the multiplicity of a. The simplest case, of course, occurs when all the roots of q ( 0)= 0 are different from one another. The method can be applied to the equation drw - dm - 1+ dzm -cl d-1 +. + Co)w == F (z), where F(z) is any function of z. Consider F,fz; f~, f) d where q (~) has the same significance as before, f(z, S) is a polynomial in { with (unknown) functions of z as coefficients of the powers of C, and integration extends round a simple contour that includes all the roots of () = O. Then dW = f (z ' provided ) f (ZX, ) dC=O; also d2 W _e_2 f(_, ) d, provided ^ C f(Z, C)d:=0; and so on in succession, until we have dm-i1 WW zi~nif1 (ZY d, dzm-l - provided cm-2 af (z, ) dc= O. Then dmW F = -m f(z, C1C FeZ. dzm ( ) FZI~"-lS~oS 6.] WITH CONSTANT COEFFICIENTS 19 Hence, remembering that f (z, C) is a polynomial in C and that therefore fezf(z, ) d= o, we have W as a solution of the given equation if, in addition to the other conditions, which are that | f,' a- f (z, ) d~ o, for r=2, 3,..., m, we have 0 M- 1f, C) d~ = F (,). Now as the contour embraces all the roots of q (a), we have* JC r- r for r=2,..., m; so that, taking af (Z, )= (z) e-ZS where 0 (z) is a function of z at our disposal, we satisfy the m -1 formal conditions unconnected with F (z); and then 0 (z) must be such that (0 (z) d= F(z). But as ^o( -r1 ^ () Clm -1-+... + Cm), we havet dJ ( = - 2ri; and therefore 0 (z)= F (z). Hence axf(Zj )=21 e-zg F(z), so that f(z, C)=g()i+ / -e-F (u) du, where g (0) is, so far as concerns this mode of determining f(z, '), any function of (, and integration with regard to u is along any path that ends in z. When F (z) is zero,f (z, () reduces to g (); and then the solution of the differential equation shews that g (() is a polynomial in (, of degree not higher than m- 1. Accordingly, as g(C) is independent of z, we take it to be a polynomial of degree n - 1 in (, with arbitrary constants for the coefficients; and then the integral of the equation has the form w=feZrg d, + if ) f) * T. F., ~ 24, II. t T. F., ~ 24, II, Cor. 2-2 20 CONTINUATION OF THE [6. where the:-integration extends round any simple contour including all the roots of 5 (C)=-0, and the u-integration extends from any arbitrary initial point along any path (the simpler the better) to z. The single integral in the expression for W is clearly the complementary function, and the double integral is the particular integral, in the primitive of the differential equation. The expression can be developed into the customary form, in the same way as in the simpler case when F (z) vanishes. Hermite's investigation, based upon Cauchy's treatment by the calculus of residues as expounded in the Exercices de Mathe'matiques, is given in a memoir in Darboux's Bull. des Sciences Math., 2me Ser. t. in (1879), pp. 311-325: it is followed by a brief note (1. c., pp. 325-328), due to Darboux. A memoir by Collet, Ann. de l'Ec. Norm. Sup., 3me Ser. t. iv (1887), pp. 129-144, may also be consulted. THE PROCESS OF CONTINUATION APPLIED TO THE SYNECTIC INTEGRAL. 7. The synectic integral P (z - C) is known at all points in the domain of I, being uniquely determined by the assigned initial conditions at '. So long as the variable remains within this domain, the integral at z does not depend upon the path of passage from I to z, so that the path from ~ to z can be deformed at will, provided it remains always within the domain. Let ~' be any point in the domain; then the values of the integral and its first m - 1 derivatives at ~' are uniquely determined by the initial conditions at ~, and they can themselves be taken as a new set of initial conditions for a new origin g'. Accordingly, construct the domain of ~'; and, with the values at g' taken as a new set of initial values, form the synectic integral which they determine. As the new initial values are themselves dependent upon the initial values at ', the synectic integral in the domain of ^' may be denoted by P1 (z- -', ). If the domain of ~' lies entirely within that of (it then will touch the boundary of the domain of 5 internally), the series P,(z - ', >) must give the same value as P (z - '): for every point z in the domain of ~' is then within the domain of I, and it is known that the synectic integral is unique within the original domain. If part of the domain of r' lies without that of ', then in the remainder (which is common to the two domains) the series Pi must give the same value as P. But in that part which is outside, the series P, defines a synectic integral in a region where 7.] SYNECTIC INTEGRAL 21 P does not exist; it therefore extends our knowledge of the integral, and it is a continuation of the synectic integral out of the original domain. Let Z be any point in the plane; and join Z to I by any curve, drawn so as not to approach infinitesimally near any of the singularities of the coefficients in the differential equation. Beginning with ~, construct the domains of a succession of points along this curve, choosing the points so that each lies in the domain of a preceding point and each new domain includes some portion of the plane not included by any previous domain. Owing to the way in which the curve is drawn, this choice is always possible and, after the construction of a limited number of domains, it will bring Z within a selected region. With each domain we associate its own series: so that there is a succession of series, each contributing a continuation of its predecessor. We can thus obtain at Z a synectic integral of the equation, which is uniquely determined by the initial values at f and by the path from 5 to Z. Further, taking the values of the integral and its first m- 1 derivatives at Z as a set of new initial values, and taking the preceding curve reversed as a path from Z to ~, we obtain at f the original set of assigned initial values. To establish this statement, it is sufficient to choose the succession of points along the curve in the preceding construction, so that the centre of any domain lies within the succeeding domain, and to pass back from centre to centre. Stating the proposition briefly, we may say that the reversal of any path restores the initial values. By imagining all possible paths drawn from any initial point g to all possible points z that are not singular, we can construct the whole region of continuity of the integral, as defined by the differential equation and by the initial values arbitrarily assigned at g: moreover, we shall thus have deduced all possible values of the integral at z, as determined by the initial values at f. It is clear, from the construction of the domain of any point and after the establishment of a synectic integral in that domain, which can be continued outside the domain (unless the boundary of the domain is a line of singularity, and this has been assumed not to be the case), that the region of continuity of the integral is bounded by the singularities of the coefficients. As has already 22 DEFORMABLE [7. been remarked, these singularities are called the singularities of the equation. Thus all the critical points of the integral are fixed points; and if the equation be taken in the form dmw dm-lw qo dz = ql dz +* + where the functions q0,..., qm are holomorphic over the finite part of the plane and have no common factor, these critical points are included among the roots of qo, with possibly z= oc also as a critical point. The value of the integral at an ordinary point near a singularity has been obtained as a synectic function valid over the domain of the point, which excludes the singularity. In later investigations, other expressions for the integral at the point will be determined, when the point belongs to a different domain that includes the singularity. 8. Any path from r to z can be deformed in an unlimited number of ways: and it is not inconceivable that these deformations should lead to an unlimited number of values of the integral at z, as determined by a given set of initial values: but the number is not completely unlimited, because all paths fronm to z lead to the same final value at z with a given set of initial values at g, provided they are deformable into one another without crossing any of the singularities. To prove this, consider a path from T to z, drawn so that no point of it is within an infinitesimal distance of a singularity, and draw a second path between the same two points obtained by an infinitesimal deformation of the first; no point of the second path can therefore be within an infinitesimal distance of a singularity. On the first path, take a succession of points zl, z2,..., so that z1 lies within the domains of C and of z2, Z2 within the domains of z1 and z3, and so on. On the second path, take a similar succession of points z,', z2,..., near zi, z2,... respectively, in such a way that z,' lies in the part common to the domains of ~ and z, while z, is in the domain of z,'; z2' in the part common to the domains of zi and z,, while z2 is in the domain of Z2'; and so on. Join z/z1', 2Z2',... by short arcs in the form of straight lines. Now we have seen that, in any domain, the path from the centre to a point can be deformed without affecting the value of the integral at the point, provided every deformed path lies within 8.] PATHS 23 the domain. Hence in the domain of ', the path 'z, gives at zl the same integral as the path z,/'z,. This integral furnishes a set of initial values for the domain of zl; and then the path zz,2 gives at z2 the same integral as the path zz,'z2z. Consequently the path zz,2 gives at z2 the same integral as the path rz/'z1, followed by zlzz2'z2z. But the effect of z,'zx followed at once by zlzl' is nul, because a reversed path restores the values at the beginning of the path; and therefore the path 'zz,2 gives at z2 the same integral as the path 'zl/z2z. And so on, from portion to portion: the last point on the first path is z, which also is the last point on the second path; and therefore the path tz1z2...z gives at z the same integral as the path Cz,'z2,...z. Now take any two paths between ' and z, such that the closed contour formed by them encloses no singularity of the equation. Either of them can be changed into the other by a succession of infinitesimal deformations: each intermediate path gives at z the same integral as its immediate predecessor: and therefore the initial path and the final path from C to z give the same integral at z; which is the required result. If however two paths between [ and z are such that the closed contour formed by them encloses a singularity of the equation, then at some stage in the intermediate deformation the curve will pass through the singularity, and we cannot infer the continuation along the curve or the deformation into a consecutive curve as above. It may or may not be the case that the two paths from ' to z give at z one and the same integral determined by a given set of initial values; but we cannot assert that it is the case. Accordingly, we may deform a given path without affecting the integral at the final point, provided no singularity is crossed in the process. Moreover, in order to take account of different paths not so deformable into one another, it will be necessary to consider the relation of the singularities to the function representing the integral: this will be effected in a later investigation. When two paths can be deformed into one another, without crossing any singularity, they are called reconcileable; when they cannot so be deformed, they are called irreconcileable. If two irreconcileable paths lead at z to different integrals from the same initial values at I, the closed circuit made up of the two paths leads at [ to a set of values different from the initial values. 24 FORM OF THE [8. These new values can be taken as a new set of initial values: when the same circuit is described, they are not restored, so that either the old initial values or a further set of values will be obtained: and so on, for repeated descriptions of the circuit. By this process, we may obtain any number, perhaps even an unlimited number, of sets of values at ' deduced from a given initial set; and thus there may be any number, perhaps even an unlimited number, of values of the integral at any point z. Consider any path from ~ to z; and without crossing any of the singularities, let it be deformed into loops, drawn from g to the singularities and back, (these loops coming in appropriate succession), followed by a simple path (say a straight line) from ' to z. The final value of the integral at z is determined by the values at ' at the beginning of the straight line, and these values are deducible from the initial values originally assigned. Hence the generality of the integral at z is not affected by taking any particular path from r to z, provided complete generality be reserved for the initial values: and therefore, from this aspect, it will be sufficient to discuss the complete system of integrals as arising from completely arbitrary systems of initial values at an ordinary point. This investigation relates to properties of the integrals, which will be found useful in discussing the effect of a singularity upon a given integral; it will accordingly be undertaken at once. 9. It has already been remarked that the synectic integral, determined by the arbitrary constants which are assigned as the initial values of the function and its derivatives, is linear and homogeneous in those constants: so that, if [/n, /12, *.., Xlm denote the arbitrary constants, and w, denotes the synectic integral which they determine in the domain of an ordinary point ', we have Wl =- 11'Il' + L12U2 +... + -JimUm, where Ua, u2,..., um are holomorphic functions of z - A, not involving any of the arbitrary coefficients is. Take other m -1 sets of arbitrary constants /L, such that the determinant All, i2, -., /Lm, = A() say, /rnl, /-L22, **, /nr2m P7m, 3 * * * X PmmM 9.] SYNECTIC INTEGRAL 25 is different from zero. Each set of m constants, regarded as a set of initial values, determines a synectic integral in the domain of ~; as the quantities u1, U2,..., um in the expression for wI do not involve the arbitrary constants determining wI, it is clear that the expressions for these other m- 1 integrals are Ws = t/siUI + /S2U2 +... + /snUm, (s = 2,..., m). Let Mst denote the minor of ust in the non-vanishing determinant A (); then from the expressions for the m integrals w,,..., Wm in terms of ua,..., Uz, we have a (U) u = Mlt1 + M2tW2 +... + M =, (..., m). Now any other synectic integral, determined in the domain of g by assigned initial values 01, 02,..., 0,, is given by = OU + 02U2 +... + Om'U = -'wl + + %2W2 +,* + Sm, where the constants % are given by 1 ' M.r /\ (t-, a(r=l,.., ). A ()t=1 These constants?r cannot all vanish, when the constants 01, 02,... 0, are not simultaneous zeros: for the determinant of the minors Mrt is {A ()}m-1, and therefore is not zero. Accordingly, any integral can be expressed as a linear combination of any mn integrals, provided the determinant of the initial values of those m integrals and their first mn-1 derivatives does not vanish. But it is not yet clear that the integrals w1,..., wn are linearly independent of one another; until this property is established, we cannot affirm that the expression obtained is the simplest obtainable. Consider therefore, more generally, the determinant of the m integrals and their first m - 1 derivatives, not solely at ~ but for any value of z in the domain of ', say dmi-lwl dm-2wl' (Z) dzmr-l ' dzJm-2 W I ^~'dz^' dz_2,. wl dm-lWm dm-2wm dzM-l1' dZM-_2., wm 26 A SPECIAL DETERMINANT [9. When z =, it becomes the determinant of initial values denoted by A ('). We have dA (z) _ dw d rT-2 W dz dzm ' dzm-2 ' W, dmw2 dm-2w2 dzm ) dzm2, *, w2 o.o.........o............ o.... diwmi dM-2wm. Wm dzm ' dzmn-2,... = A (z), on substituting for dwml,, dm"wm their values in terms of the on substituting for dmw d derivatives of lower orders as given by the equation. Hence \ pldz f, A (z)= ()e- Now within the domain of ', the function pi is regular, being of the form P1 (z - ); hence the integral in the exponent of e is of the form R (z- ), where R is a regular function that vanishes when z=. Consequently the exponential term on the right-hand side does not vanish at any point in the domain of C; also A () is not zero; so that A (z) has no zero within the domain of '. Moreover, each of the quantities w1,..., wn is a holomorphic function of z - in that domain, so that A (z) is holomorphic also; hence A(z) has no zero and no infinity within the domain of the ordinary point C. As a matter of fact, the only points where A (z) may vanish or may become infinite are the singularities of pl. For in any region of common existence of the functions zw,..., WM, we have A (z) fpdz the path from f to z lying within that region, while z is not now necessarily in the domain of. If a be one of the singularities of P1, the expression of pl in any part of an annular region round a as centre is of the form p1 =g' (Z)+ ~a + +P ( ) X-a (V-a)2 where the number of terms in negative powers of z -a is finite or infinite, according as the singularity is accidental or essential; and g' (z) is holo 9.] FUNDAMENTAL SYSTEMS 27 morphic in the vicinity of a. Taking the simplest case as an instance, let a2=a3=... =0; then fzpnldz Z -- a eJ =d_ (2/- )aeg(z)-g() shewing that a is a zero of A (z) if the real part of a, be positive, and that it is an infinity of A (z) if the real part of a1 be negative. More generally, the nature of A (z) in the vicinity of any singularity a depends upon the character of Pl in that vicinity: in the case of the above more general form, a is an essential singularity of A (z). FUNDAMENTAL SYSTEMS OF INTEGRALS. 10. The linear independence of wI,..., w,, and the property that A (2) has a finite non-zero value at any point in the plane which is not a singularity of the equation, are involved each in the other. It is easily seen that, if a homogeneous linear relation between wI,..., Wm of the form ClW, +... + CnWm = 0 were to exist, the quantities cl,..., Cm being constants, then A (z) would vanish for all values of z. The inference is at once established by forming the m -l derived equations drwl drwm 1 +. + C dz-=, (r= 1,.., rn- ), and eliminating the m constants cl,..., c between the in equations which involve them linearly; the result of the elimination is A (z) = 0. Hence if, for any set of integrals w1,..., w,, the determinant A (z) does not vanish (except possibly at the singularities of the equation), no homogeneous linear relation between the integrals exists. To establish the inference that, if A (z) does vanish for all values of z, a homogeneous linear relation between 1w,..., Wm exists, we proceed as follows. In the first place, suppose that some minor of a constituent in dm-eiwm the first column of A (z), e.g. the minor of dlzm-_ in A (z), say 28 A FUNDAMENTAL SYSTEM [10. A1 (z), does not vanish for all ordinary values of z; and take m quantities yi,..., y,, the ratios of which are defined by the relations ylw +... + ymW = 0, dwl dwm l dz +...*+Y dz =0,....................o...o............ d7m-2w dm-2 wm Y- dzm + + + y dzm = 0. From the hypotheses that A (z)= 0 and that A1 does not vanish, it follows that dm-lwl dm-lwm... Ym zm- = 0. yl dz-l + + y dz_-l - Because of the assumption that A1 does not vanish, the ratios Yl Y2 Ym-1 yin yin Ym are determinate finite functions of z. Differentiate the first of the relations: then, using the second, we have Y' Wi +.. + Yn Wa = 0, where y.' denotes dyr/dz, for the n values of r. Differentiating the second of the relations, and using the third, we have,dwi dw,W y - +...- +y =0; and so on, up to,dm-2wl dam-2,m, y] dz,_ +... +Y y dzm- = O n obtained by differentiating the last of the postulated relations and by using the deduced relation. We thus have m - 1 relations, homogeneous and linear in the quantities y',..., y,'; in form, they are precisely the same as the m - 1 relations, which are homogeneous and linear in the quantities yi,..., y,. Hence, as AI does not vanish, we have yr Y- (r=1, 2,...,m-1), Ym yn that is, d ( = 0 dz ymO/ 10.] AND ITS DETERMINANT 29 so that Yr =constant= X (r=1, 2,... m-1), Ym Xm' where X,...,, Xm_, m are simultaneous values of y,..., m.., y, for any particular value of z: that is, the quantities X are constants. This particular value of z is at our disposal; we may assume that X, is different from zero, because the ratios of y,..., yn- to ym are determinate and finite. Now ylwU' +... + ymWm = 0; hence X1W1 +... + XmWm = 0, that is, a linear relation exists among the quantities w, if A (z) is zero, and some minor of a constituent in the first column does not vanish. Next, suppose that the minor of every constituent in the first column vanishes: in particular, let A (z) =0, for all ordinary values of z. Then A (z) is a determinant of m-1 rows and columns, constructed from m- 1 quantities w,..., wm-1 in the same way as A(z), a determinant of m rows and columns, is constructed from the m quantities wt,..., wM. The preceding analysis shews that, if some minor of a constituent in the first column of A, (z) does not vanish for all ordinary values of z, then a relation /CKW +... + Km —iWm-i = 0, where K1,..., Km- are constants, is satisfied: so that a linear relation exists among the quantities w, and it happens not to involve wi. Let the process of passing from A (z) to A, (z), from Al (z) to a corresponding minor, and so on, be continued: the successive steps are effected by removing the successive columns in A(z) beginning from the left and by removing a corresponding number of rows. At some stage, we must reach some minor which is not zero for all ordinary values of z: so that dnm-s-lw1 dza"-s- ' )... I W dm-s-wm-s Wd^zms_1,..., Wm-s 30 THE NUMBER OF [10. vanishes when s= 0,,..., r, but is different from zero when s =r + 1. Then the earlier analysis shews that a linear relation of the form piWi +... + pm-sWm-s = 0 exists, where pl,..., pm-s are constants: in effect, a linear homogeneous relation among the quantities w1,..., wm which happens not to involve Wms+,..., w,. Hence, if the determinant A (z), constructed from the m integrals w1,..., Wi, vanishes for all ordinary values of z, there is a homogeneous linear relation between these integrals. Integrals are sometimes called independent when they are linearly independent, that is, connected by no homogeneous linear relation; but the independence is not functional, because all the integrals are functions of the one variable z. A set of m linearly independent integrals w is called a fundamental system; and each integral of the set is called an element or a member of the system. The determinant A (z), constructed out of a set of m integrals, is called the determinant of the system; so that the preceding results may be stated in the form:If the determinant of a set of m integrals vanishes for ordinary (that is, non-singular) values of the variable, the set cannot constitute a fundamental system; and the determinant of a fundamental system does not vanish for any non-singular value of the variable. 11. We now have the important proposition:Every integral, which is determined by assigned initial values, can be expressed as a homogeneous linear combination of the elements of a fundamental system. Let W denote the integral determined by the assigned values at ', taken to be an ordinary point of all the coefficients in the differential equation; and let w1,..., wm be a fundamental system. Let constants c1,..., cm be deduced such that, when z=, we have m W = E CWA X=1 dW _ m dwx dz m=. dz dnm — W in dMw-1w m-1 h=l zm-1 11.] LINEARLY INDEPENDENT INTEGRALS 31 This deduction is uniquely possible; because the determinant of the quantities c on the right-hand sides is the determinant of a fundamental system, and therefore does not vanish when z=. m Thus W- e cxwx is an integral of the equation; this integral A=l and its first m - 1 derivatives vanish when z=; so that it vanishes everywhere (Cor. I, ~ 5), and therefore m W = E CAWA, A=l the constants c being properly determined as above. COR. I. Between any m + 1 branches of the general solution, there must be a homogeneous linear relation. For if m of them be linearly independent, the remaining branch can be regarded as another integral: by the proposition, it is expressible linearly in terms of the other rm. COR. II. Any system of integrals u1,..., un is fundamental if no relation exists of the form Al, +... + Aum = 0, where A&,..., A, are constants. For taking a fundamental system Wi,..., win, we can express each of the solutions u in the form U = airwl +... + amrWm, (r = 1, 2,..., ), where the coefficients a are constants. If C denote the determinant of these m2 coefficients, C must be different from zero: for otherwise, on solving the m equations to express w1 in terms of Ui,..., uim, we should have a relation of the form A ul +... +- Amum = Cwl = 0; and no such relation can exist. If, then, A, (z) denote the determinant of the set of integrals u, and if Aw (z) denote that of the fundamental system w1,..., wm, we have aL (z) = CA, (z), by the properties of determinants. Now C does not vanish, nor does Ao (z) at any ordinary point in the plane; hence A, (z) does not vanish at any ordinary point in the plane, and therefore ut,..., u, are a fundamental system of integrals. 32 A SPECIAL [11. The result may be stated also as follows: If m integrals u be given by equations Ur = airi +... + antrwr, (r = 1,..., m), where the determinant of the coefficients a is not zero, and the integrals w are a fundamental system, then the system of integrals u is also fundamental. 12. One particular fundamental system for the differential equation can be obtained as follows. Let w, be a special integral of the equation, that is, an integral determined by any special set of initial conditions, and substitute w = w1fvdz in the equation; then v is determined by the equation dm-iv dm-2v dsz_ -l q dz_-2 +... + qm-1V, where m dw1 ql = lP w, dz Similarly, let vl be a special integral of this new equation, with the appropriate conditions; then substituting v = vlfudz, we find that the equation, which determines u, is of the form dnz-2u dm-3u dz,-~-rl dza- +.. + rm-2u, where m- 1 dv1 rl= -- - - vI dz And so on. It is manifest that the quantities W1, w1fv1dz, w1(vfuldz)dz,... are integrals of the original equation. Moreover, they constitute a fundamental system; for, otherwise, they would be linearly connected by a relation of the form c1wi + c2wlfv1dz + c3w1f (vIfu1dz) dz +...= 0, that is, ci + c2fvdz + cf(vSfudz) dz +... = 0. 12.] FUNDAMENTAL SYSTEM 33 When this is differentiated, it gives c21 + c3vlf Udz +... = 0, that is, C2 + c3Sfudz +... = 0. Effecting m - 1 repetitions of this operation of differentiating and removing a non-zero factor, we find cm =0 as the result at the last stage. Using this in connection with the equation at the last stage but one, we have Cm-1 = 0. And so on, from the equations at the various stages, we find that all the coefficients c vanish. The homogeneous linear relation therefore does not exist: the system of integrals, obtained in the preceding manner, is a fundamental system. As an immediate corollary from the analysis, we infer that v1, vlfuldz,... constitute a fundamental system for the equation in v; and so for each of the equations in succession. The determinant of this particular fundamental system is simple in expression. Denoting it by A, and denoting by Al the determinant of the fundamental system of the equation in v, we have, as in ~ 9, 1 dA Adz1 dAi m dw, A, dz =1 Wq=Pl dz ' so that 1 dA 1 dA _ m dw A dz A- dz wI dz hence - = Xllm, where f a is a constant. Similarly, if At denote the determinant of the fundamental system of the equation in u, we have = 2Vlm-1; A2 F. IV. 3 34 FORM. OF THE DETERMINANT [12. and so on. The last determinant of all is the actual integral of the last of the equations; hence A= Cwlmvl rn- ulm-2..., where C is a constant. Moreover, A is the determinant of a particular system, so that C is a determinate constant. It is not difficult to prove that X= (- 1)m-, X =(- 1)m-,..., and therefore C =(-1)m (M-1); consequently, A = (- 1)-(m-) Wlm Vl ---l uM-2.. Ex. Verify the last result, as to the form of A, in the case of (i) Legendre's equation: (ii) the equation of the hypergeometric series: (iii) Bessel's equation. CHAPTER II. GENERAL FORM AND PROPERTIES OF INTEGRALS NEAR A SINGULARITY. 13. WE have seen that, within the domain of an ordinary point, a synectic integral of a linear differential equation is uniquely determined by a set of assigned initial values; and that the said integral can be continued beyond that domain, remaining unique for all paths between the initial and the final values of the variable which are reconcileable with one another. When the variable is permitted to pass out of its initial domain though returning to it for a final value, or when two paths between the initial and the final values are not reconcileable, the various propositions that have been established are not necessarily valid under the modified hypothesis: it is therefore desirable to consider the influence of irreconcileable paths upon an integral, still more upon a set of fundamental integrals. Remembering that any path is deformable without affecting the integral if, in the deformation, it does not pass over a singularity, we shall manifestly obtain the effect of a singularity, that renders two paths irreconcileable, by making the variable describe a simple circuit, which passes from the point z round the singularity and returns to that point z, and which encloses no other singularity. Let a be the singularity round which the simple closed circuit is completely described by the variable. Let wi,..., w, denote a fundamental system at z; and suppose that the effect of the 3-2 36 EFFECT OF A SINGULARITY [13. circuit is to change the m integrals into wl',..., Wi,' respectively. That the set of m new integrals thus obtained is a fundamental system can be seen as follows. If it were not a fundamental system, some relation of the form m krw,.' = 0 r=l would exist, with constant coefficients k, for all values of z in the immlediate vicinity. In that case, the quantity E kw,.' (which is an integral) is zero everywhere, together with all its derivatives, as it is continued with the variable moving in the ordinary part of the plane. Accordingly, let the integral be continued from z along the closed circuit reversed until it returns to z where, by what has been stated, it is zero. The effect of the reversal is (~ 7) to change wr' into w,: and so the integral after the reversed circuit has been described is E kwr, so that we should have m k.wT = 0, r=l contrary to the fact that w,,..., wm constitute a fundamental system. The initial hypothesis from which this result is deduced is therefore untenable: there is no homogeneous linear relation among the quantities w',..., w,', which therefore form a fundamental system. Since the system w1,..., wn is fundamental, each of the integrals w/,,..., w' is expressible linearly in terms of the elements of that system; so that we have equations of the form ws = asLwI +... + Casm, (s = 1,..., m), where the coefficients a are constants. As the system w,' is fundamental, the determinant of these coefficients is different from zero: this being necessary in order to ensure the property that w,,..., Wm are expressible linearly in terms of w',..., Wm', a fundamental system. Take any arbitrary linear combination of the system, say plW1 +... + pmWm, where the coefficients p are disposable constants; and denote this integral by u. When the variable describes the complete closed 13.] UPON A FUNDAMENTAL SYSTEM 37 circuit round the singularity, let u' denote the modified value of u, so that U' =- 1W +... + pmWm' m in = i rWr... + + pm E amrWr. r=l r=l It is conceivable that the coefficients p could be chosen so that the integral reproduces itself except as to a possible constant factor; a relation u' = Ou would then be satisfied, 0 being a constant quantity. This relation, in terms of w1,..., wm, is m m pi I alrWr +... + pm E amrWr r=l r=l = (plWl +. + + pI ), which, as it involves only the members of a fundamental system linearly, must be an identity: the coefficients of wi,..., wi must therefore be equal on the two sides. Hence we have pl (all - ) + p2tz2 +... + prmeml =l 0 pl12 + p2 (a22 - ) +.. + pmam = 0.........................................,.,.,.,,.I plaim + p2aMm +... + p (amm - ) = 0 If, therefore, 0 be determined as a root of the equation A = all - 0,,.., a = 0, a1 2, 2-, 9, am2................................. aim, a2M,..., a - 0 the preceding relations then lead to values for the ratios of the constants p for each such root. It is to be noted that, in this equation, the term, which is independent of 0, does not vanish, for it is the determinant of the coefficients a; hence the equation has no zero root. As the equation definitely possesses roots 0, it follows that integrals exist which, after a description of the simple contour round a, reproduce themselves save as to a constant factor. If it should happen that the constant factor is unity, then the effect of 38 THE FUNDAMENTAL EQUATION [13. description of the contour upon the integral is merely to leave it unchanged: in other words, such an integral is uniform in the vicinity of the singularity. PROPERTIES OF THE FUNDAMENTAL EQUATION. 14. The special significance of the equation, in relation to the singularity a, lies in the proposition that the coefficients of the various powers of 0 in A = 0 are independent of the fundamental system initially chosen for discussion. To prove* the statement, it will be sufficient to shew that the same equation is obtained when another fundamental system is initially chosen. For this purpose, let y1,..., ym, denote some other fundamental system; and suppose that, by the simple closed contour round a described by the variable, the members of the system become y,',..., yi' respectively. Then, as both these systems are fundamental, there are relations of the form yS=/= /,ly +... + /3smym, ( = 1,..., m), where the determinant of the coefficients /3 is not zero. The equation B= O, corresponding to A = 0 for the determination of the factor 0, is formed from the coefficients / in the same way as A from the coefficients a, so that the expression for B is B = I311- 0, a 21.,. m, P12, 322 - *0..,* / m2...,.,.............................. /Pin, Pm, ** X, Pm - 0 Because each of the sets w W,..., wn; y,..., ym; is a fundamental system, the members are connected by relations of the form Y. =Y 7nWl + Wr2 2... + ~Yrmnm, (r =1,..., q ), where the determinant of the coefficients, which may be denoted by r, is different from zero. The quantity yr - (^rlwl +... +,rmwm) is zero everywhere in the vicinity of z; and it is an integral, which accordingly is zero everywhere in its continuations over the * The proof adopted is due to Hamburger, Crelle, t.'LxxvI (1873), pp. 113-125. 14.] IS INVARIANTIVE 39 ordinary part of the plane. When it is continued along the simple contour round a, the variable returning to z, the integral is zero there; that is, Yr = 7-rl w +... + 7%m Wm. Hence in in 3riy Y+... + 3rm m = 2 yrsastWt, t=l s=1 and therefore m m m,m E E 3rs~stWt = E E YrsastWt. t=l s=1 t=l -=1 This relation involves only the members of a fundamental system linearly; hence it must be an identity. We therefore have m mr E 3rsYst = YE rsClst s=l s=l = art, say, the relation among the constants holding for all values of r and t. Now forming the product of the determinants F and A, we have PA = 711 7 712, 713, a 2l -0, a21, a31 721, 722, 723,. a12, a22- 0, a32,.. 731, 732, 733, a.* l3, a23, a33 — 0,................ *............*. *. *.*.........** * * - = 11 - 7110, 12 - 7120, = D, 821 - 7210, 822 - 722, * say; and similarly, forming the product of B and r, we have Br= /3-0, 132 1, 13,... 711 721, 731, *121, 322 - 0, 123, *. 712, 722, 732, 1831, /32, 833- *0,... 13, 23, 733, * ************** *@ ****.******.@.............,....... = 11 - 7n10, 12 - 7120,. = D, 821 - 7210, 822 - 7022, *. so that rA =Br, 40 INVARIANTIVE PROPERTIES [14. identically. Also r does not vanish; hence A B, for all values of 0. Accordingly, the equation A = 0 is invariantive for all fundamental systems in regard to the effect of the singularity a upon the members of the system: it is called* the fundamental equation belonging to the singularity a. We note that its degree is equal to the order of the differential equation. While the equation is thus invariantive for all fundamental systems, the actual invariance of one of its coefficients is put in evidence, either when the differential equation of ~ 2 is initially dn-w devoid of the term involving dzm_-, or after the equation has been transformed by the relation W1 fpdz W = Wemf d J dm-l W so as to be devoid of the term involving dz_1 In A =0, the term which is independent of 0 is equal to unity, a property first noted by Poincaret. For when p, is zero, the determinant A of the fundamental system is a constant, for (~ 9) its derivative vanishes; it therefore is unchanged when the variable describes a simple closed circuit round the singularity. The effect of such a circuit upon A is to multiply it by the term in A which is independent of 0: accordingly, that term is unity. The linear equation can always be modified so that the term involving the derivative of the dependent variable next to the highest is absent; and the necessary linear modification of the dependent variable leaves the independent variable unaltered. This change does not influence the law giving the effect, upon the integrals, of a description of a loop round the singularity; and the fundamental equation is independent of the choice of the fundamental system. Accordingly, the coefficients of the various powers of 0 (except the highest, which has a coefficient (- l)", and the lowest, which has a coefficient unity) are frequently called the invariants of the singularity: they are m-1 in number. * Sometimes also the characteristic equation. t Acta Math., t. iv (1884), p. 202. 15.] OF THE FUNDAMENTAL EQUATION 41 15. There is a further important invariantive property of the determinants A (0), B (0), viz.: If all minors of order n (and therefore all minors of lower order) in A (0) vanish for a particular value of 0, but not all those of order n + 1, then all minors of order n in B (0) also vanish for that value of 0, but not all those of order n +1. A minor of order n is obtained by suppressing n rows and n columns; accordingly, the number of them is J m! Y __ 2 (m - n)! n! ' ~ say. Let them be denoted by aij, bij, ci, dis when formed from A (0), B (), F, D respectively, where i and j have the values 1,..., /u, these numbers corresponding to the various suppressions of the rows and the columns. Then, regarding D as the product of A and F, we have* di = ci ajl cCi, + ci. + ci aj; and regarding D as the product of B and F, we have dij = bil Cjl + bi2c2 +... + bijcj. All the quantities aij are supposed to vanish for a particular value of 0; hence for that value all the quantities di vanish. Assigning to j all the values 1,..., / in turn, we therefore have 0= ellbil + cl2bi2 +... + CIlbi,A 0 = c21bil + c22bi2 +... + c2,bi,..................................... 0 = c, bil + c,2bi2 +... - clbi+, The determinant of the coefficients of bil, bi2,..., bi, is equal tot rx, where x= (m-l)! (m-n- 1) n! that is, the determinant does not vanish. Accordingly, we must have bil=O, bi2=0,..., bi = 0; as this holds for all values of i, it follows that all the minors of B (0) of order n vanish for the particular value of 0. * Scott's Determinants, p. 53. t ib., p. 61. 42 ELEMENTARY [15. The minors of B (0), which are of order n + 1, cannot all vanish for the value of 0; for then, by applying the result just obtained, all those of A (0), which are of order n + 1, would vanish, contrary to hypothesis. 16. A more general inference can be made. Leaving 0 arbitrary and not restricting it to be a root of the fundamental equation, the two expressions for dij give Cirajr = X cisbis, r=l s=l holding for all values of i and j. Taking this equation for any one value of j and for all the g values of i, we have /u equations in all, expressing aj,, aj2,..., aj, linearly in terms of bpq. The determinant of coefficients on the left-hand side is rF, as before, and does not vanish; so that each of the quantities ajr is expressible linearly in terms of the quantities bpq, the coefficients involving only the constituents of r. Similarly, taking the equation for any one value of i and for all the p values of j, we find that each of the quantities bpq is expressible linearly in terms of the quantities ajr, the coefficients involving only the constituents of r. If therefore all the quantities ajr have a common factor 0 - 0, and if that factor be of multiplicity a, then all the quantities bpq also have that factor common and of the same multiplicity o-; and conversely. These results associate themselves at once with Weierstrass's theory of elementary divisors*. If (0 - 0,) is the highest power of 0 - 8 in A (0), if (0 - 0,)l is the highest power of that quantity common to all its minors of the first order, if (0- 0l) is the highest power common to all its minors of the second order, and so on, then (as will be proved immediately) a- > o- > 0-2 >...; and (o 01) -, (0- 0 -2... are called elementary divisors of the determinant A (0). It follows from the preceding investigation that the elementary divisors of the fundamental equation are invariantive, as well as the equation * Berl. Monatsber., (1868), pp. 310-338; Ges. WVerke, t. ii, pp. 19-44. See also a memoir by Sauvage, Ann. de l'Ec. Norm., 2e S6r., t: vii (1891), pp. 285-340; and a treatise by Muth, Elementartheiler, (Leipzig, 1899). 16.] DIVISORS 43 itself; for they are independent of the particular choice of a fundamental system. If the earliest set of minors of the same order that do not all vanish when 0 = 0~ is of order p, so that they are of degree m - p in the coefficients in A, then the elementary divisors are (0-0 1 (- - ),. ( - )01 - 2... ( -01-2- -1, ( _ )-p-1, being p in number: and then p is one of the invariantive numbers associated with the particular singularity of the equation. As two of the properties of the invariantive equation, associated with the elementary divisors, are required, they will be proved here: for full discussion of other properties, reference may be made to the authorities quoted. It is easy to obtain the result >0'C>0>-2>..., just stated above. For aA m where Ar is the minor of ar- In A there is a factor (, for each where Ayr is the minor of ar,- 9. In A77 there is a factor (9- 9i)", for each of the quantities Arr is a first minor: therefore that factor occurs in their sum and, owing to the combination of terms, it may have an even higher index than o-1. On the left, the factor in 8- 80 has the index o- 1; hence a-1 - oi, that is, cr >l. Similarly for the other inequalities. Again, we know* that any minor of degree p which can be formed out of the first minors of A (8) is equal to the product of A p-I() by the complementary of the corresponding minor of A (0). Hence, taking p= 2, we have relations of the form A1B2-A2Bi=AC, where Al, A2, B1, B2 are minors of the first order, and C is a minor of the second order. Choose a minor of the second order which is divisible by no 'higher power of 0- 08 than (0- 01)2; the left-hand side is certainly divisible by (6- 01)2', and it may be divisible by a higher power if the terms combine: hence 2o-i <o + 0-2, that is, r - -1 > al - (T2. Similarly, we have the other inequalities of the set 1- ol > o1-02 > o2-o3 > -.. > pl so that the indices of the elementary divisors, as arranged above, form a series of decreasing numbers. * Scott's Determinants, p. 58. 44 BRANCHES OF AN ALGEBRAIC FUNCTION [17. ASSOCIATION OF DIFFERENTIAL EQUATIONS WITH ALGEBRAIC FUNCTIONS. 17. Before considering the roots of the fundamental equation, it is worth while establishing a converse* of the propositions in ~ 13, as follows: Let y,..., ym be m linearly independent functions of z, which are uniform over any simply-connected area not including any critical point of the functions: let the critical points be isolated and let each of them be such that, when a simple contour enclosing it is described, the values of the functions at the completion of the contour are given by relations of the form yr = arlY1 +... + -armym, (r = 1,..., m), where the determinant of the coefficients a is not zero, and the constants may change from one critical point to another: then the m functions are a fundamental system of integrals for a linear differential equation of order m with uniform coefficients. It is clear that, if the functions are integrals of such an equation, they form a fundamental system because they are linearly independent. On account of this linear independence, the determinant dmn —1 d-2 dm-ly, dM-2 y dm-ly, dm2y~m dzm-1 dzm-2 * "') Y does not vanish for all values of z. Let As denote the determinant which is derived from A by changing the sth column into dz m dzm As PS= A. For any contour that encloses no critical point, A and As are uniform, so that ps is uniform for such a contour. For a simple contour, which encloses the critical point a and no other, the * It is given by Tannery, Ann. de l'Ec. Norm., Ser. 2me, t. iv (1875), p. 130. 17.] AS A FUNDAMENTAL SYSTEM 45 determinant A after a single description acquires a constant factor PR, where R is the (non-zero) determinant of the coefficients in the set of relations yr = riyl ++ + rmym, (r = 1,..., m). The determinant As acquires the same factor R, in the same circumstances; and therefore ps is unchanged in value by a description of the contour, that is, it is uniform for such a contour. As this holds for each contour, it follows that ps is uniform over the plane. The mn quantities y,,..., Ym evidently are special integrals of the equation dry dn-ly dM-2y dz i = AP dzM-_ + P2 dz2- + +.. + y, which is linear and the coefficients in which have been proved uniform functions of z. COROLLARY. If all the critical points of the functions are of an algebraic character, that is, of the same nature as the critical points of a function defined by an algebraic equation, and are limited in number, then the uniform coefficients p in the differential equation are rational functions of z. For as ps is uniform, the critical point a is either an infinity, or an ordinary value (including zero). If it is an infinity, it can be only of finite multiplicity; for the critical point is one, where A and A, can vanish only to finite order because of the hypothesis as to the nature of the critical point: that is, the point is then a pole of finite order. Likewise, if it is a zero, the multiplicity of the zero is finite. This holds at each of the critical points of the functions y,..., ym; and the number of such points is finite. Moreover, every point that is ordinary for each of the functions is ordinary for A and As and, in particular, A cannot vanish there: so that no such point can be a pole of any of the coefficients p. It therefore follows* that each of these coefficients is a rational meromorphic function of z. The converse of the corollary is not necessarily (nor even generally) true: it raises the question as to the tests sufficient and necessary to secure that the integrals of a linear equation with rational coefficients should be algebraic functions of the variable. This discussion must be deferred. * T. F., ~ 48. 46 ALGEBRAIC FUNCTIONS [17. Ex. 1. The most conspicuous instance arises when the dependent variable w is an algebraic function of z, defined by an algebraic equation f(w, z)= 0, of degree m in w. Each branch of the function so defined is uniform in the vicinity of an ordinary point; in the vicinity of a branch-point, the branches divide themselves into groups; and any linear combination of them is subject to the foregoing laws of change (which take a particularly simple form in this case) when z describes a circuit round a branch-point. To obtain the homogeneous linear equation of order m which is satisfied by every root of f=0, we can proceed as follows. Let p (z) =0 be the eliminant of f=0 and - =0; so that* all the branch-points of the algeaw braic function are included among the roots of = 0, though not every root is a branch-point. By a result t in the theory of elimination, we know that the resultant of two quantics u and v of degree m and n respectively in a variable to be eliminated is of the form uv1 + Vu1, where u1 and v1 are of degrees vn- 1, n - 1 respectively in that variable; and therefore (Z)= U f+ Vaw, where U is of degree nz-2 in w and V is of degree mn-1 in w. But f is permanently equal to zero for all the values of w considered; hence (P (Z)= v,zf. aw' * T. F., ch. viin. t It is most easily derivable from Sylvester's dialytic form of the eliminant, as follows. Let u= aoxm+ alxm-1 + a2xm-2+.., V =CO Xn + C Xn-1 + C2Xn-2 +...; the eliminant is E= a, a, a,... 0., a0, a1,. 0, 0, a0,........................ C, C, C2,.. 0 Co, C1,.. 0 0,, o,........................ To the last column, add the first column multiplied by xm+n-1, the second multiplied by Xrm+n-2, and so on: a change which does not affect the value of E. The constituents in the new last column are xn-lu, Xn-2u,... U, Xu -1, m-2V1, aX2..., XV, V; expanding E by taking every term in this last column with its minor, collecting all the terms involving u into one set and those involving v into another, we have E = uv1 + vu, where v. is of degree n -1 in x and ul is of degree m -1 in x. 17.] AND DIFFERENTIAL EQUATIONS 47 Now af Vaf dw az az dz 7f + (z) aw By means of f=O which is of degree m in w, we can reduce Va so that it contains no power of w higher than the (mn - 1)th, say dw P1 dz q ()' where P1 is a polynomial in w of degree not higher than nm-1. (If the highest term in f has unity for its coefficient, then P1 is a polynomial in z also.) Again, d2w Pi 8P 1 aP1 P_ a_ dZ2 = 2(g) aw + (Z) a 02(Z) az P2 ~2 (z) on reducing to a common denominator; by means of f=0, the polynomial P2 can be made of degree not higher than mn-1 in w, and its coefficients are uniform functions of z. And so on, up to dtnw Pm dzm -~ (xz) where IPm is a polynomial in w of degree not higher than n - 1, the coefficients being uniform functions of z. We thus have dw dmw PM= d-??t Among these m equations we can, by a linear combination, eliminate the m- 1 quantities w~, w2,..., wm~- from the left-hand sides; and the result has the form dw d 02w +...+ Qmw m Qow = Q1d, dzO+Q2dZ24 +Q dzm4, where Q0, Q1,..., Qm are uniform functions of z. This is satisfied for every root w of the algebraic equation: and it is of order n. Corollary. There is one special case, when the differential equation is of order m - 1, viz., when the algebraic equation is f=, + a2wm-2 +... +a,=0, so that the term in wm-l is absent. We then have +W2 +... +m= O, 48 EXAMPLES OF [17. so that one of the in branches w can be expressed linearly in terms of the others; Tannery's result shews that the differential equation is then of order not higher than n -. In that case, it would be sufficient to take only the m -1 equations drw Pr =d- (r=l,..., m- 1). For instance, consider the algebraic equation w3 + 3w = u, where u is any function of z; it is to be expected that the linear differential equation satisfied by each of the three branches of the function defined by this cubic equation will be of the second order, say d2wV dw d- + A +B=0, where A and B are functions of z. We have (W2+ 1) =dW= U' 2++1) dZ+2W2+1) d2 (W2 + 1) d2+ 2w adW = U, -/ o w \dz / =; so, substituting in (w2 + 1) d2 + +A(w2+ 1) d-+B (W3+w)=0, and using w3+3wv=u, we have [dw 2 B (u - 2w) + Au'+ tu" = 2w (\. Multiplying the right-hand side by (w2+1)2, and the left-hand side by its equivalent 1+wu-w2, we have 2wU —2= (l +wu-w2) (dAu'+ Iu" +B (u - 2w)} = ( +wu - 2) (IAu' + i't) + B {3u +w (u2 -8) - 3w2u}, on reduction by the original algebraic equation. This will hold for each of the three roots of that equation, if 12= u (3Au' +u") +B (2 - 8) O =Au' + " + 3Bu These conditions give the values of A and B; and the equation for w is easily found to be d2w +/ ' u "\ dw 1, u'2 dz2- u2+4 u z = 9 u2+4w, where i' and i" are the first and the second derivatives of u. The equation is of the second order as indicated. Note 1. When the algebraic equation of degree m in w is of quite general form, the linear differential equation satisfied by its roots is of order m. But when the algebraic equation has very special forms, though still irresoluble, the differential equation may be of order less than m; for the 17.] DIFFERENTIAL RESOLVENTS 49 elimination of various powers of w may not require derivatives up to that of order m. The most conspicuously simple case is that in which the algebraic equation is wae = R (z), where R is a rational function of z; the differential equation is dw 1 R' (z) dz m R(z)= only of the first order. Other cases occur hereafter, in Chapter v, where quantities connected with the roots of algebraic equations of degree higher than two satisfy linear differential equations of the second order. Note 2. The differential equations considered have, in each case, been homogeneous. If we admit non-homogeneous linear differential equations, viz. those which have a term independent of w and its derivatives, then in the general case, where f (w, z) has a term in wm -l, the differential equation is of order m - 1 only. This can be seen at once from the elimination of w2, w3,..., qw-1 between P dw \ dz dd n - 1 l '_ =3dz-1 leading to a (non-homogeneous) linear equation of order m - 1. This result appears to have been first stated by Cockle*: it is the initial result in the formal theory of differential resolvents+. Ex. 2. Shew that, when the algebraic equation is 2 - 2zw- z= 0, the two linear differential equations, homogeneous and non-homogeneous respectively, are d2w 3+ 2z2 d 3 + 2z2 dz2 z +3 dz + z + Z dw 1 + 2z2 z2 dz z + zW 1 + ' Ex. 3. Obtain the differential equations satisfied by each root of (i) w3-3w2+z6=0; (ii) tw- 3z-w+z3=O. Ex. 4. Shew that any root of the equation y - ny=(n - 1)x * Phil. Mag., t. xxI (1861), pp. 379-383. t For references, see a paper by Harley, Manch. Lit. and Phil. Memoirs, t. v (1892), pp. 79-89. F. IV. 4 50 SIMPLE ROOTS OF [17. (n being greater than 2) satisfies the equation dx n-1 d_(_x-_a)_ - 1 ( 'X a where a=]- -. What is the form for =2? (Heymann.) Ex. 5. Shew that any root of the equation yt - ny =n - 1 (n being greater than 2) satisfies the equation dnt - ly n- d(Jr d —rX —(-l)n-l^-X x ag dx"- ) 1~ a,=0 d (log x)''" where the constants a,. arise as the coefficients in the algebraic equation n-1 2-=0 when the roots are X=((- -z) z~ +1, for ]k= 1,..., n-1, and a,_i= 1. (Heymann.) Ex. 6. Prove that, if y _ 5y3 + 5 - 4x + 2 = 0, then d2y 2x-1 dy Y _0 dx2 2x (x- 1) dx 25x(x- 1) and explain the decrease in the order of the differential equation. (Math. Trip., Part II, 1900.) FUNDAMENTAL SYSTEM OF INTEGRALS ASSOCIATED WITH A FUNDAMENTAL EQUATION. 18. We now proceed to the consideration of the fundamental equation A = 0 appertaining to the singularity a. The simplest case is that in which the m roots of that equation are distinct from one another, say 01, 02,..., 0,. Not all the minors of the first order vanish for any one of the roots: if they did vanish, the root would be multiple for the original equation. Hence each root 0r determines ratios of coefficients cri, c2.,..., c, uniquely, such that an integral of the equation exists, having the value 'a = CrlW1 +..+ CrmWm, and possessing the property that 'Ur= OrUr, 18.] THE FUNDAMENTAL EQUATION 51 where u.' is the value of u,. after z has described a complete simple contour round a. We thus obtain a set of m integrals. These m integrals constitute a fundamental system: otherwise a permanent relation of the form KlUl + K2U2 +... + Kjmtmt-n = 0 would exist. This quantity ZK,.u, is an integral: as it is zero and all its derivatives are zero at and near z, it is zero everywhere when continued over the regular part of the plane. Accordingly, let z describe a simple closed contour round a: when it has returned to its initial position, the zero-integral is K.cu,', that is, K111'U + tCU2 + 2.2 + c. m + KmUm = 0. Similarly, after a second description of the simple closed contour, we have K12TU1 + + C.2022t2 +... + ICKm2On2Um = 0. Let m - 1 descriptions of the contour be made in this way: we have K01 Ul + KC202ru2 + -...- + KImOmrm = 0, for r = 0, 1,..., m - 1. Unless all the coefficients K1,..., cm are zero, we have 61,,...,, I..1 A, 1,...... 1 = 0, 01 d2 ** m) 01m-l, 02m-,... 0 mm-1 that is, the product of the differences of the roots is zero. This is impossible when the roots are distinct from one another; hence the coefficients 1K,..., Kcm vanish, and there is no homogeneous linear relation among the integrals Uz,..., u,,, which accordingly constitute a fundamental system. The general functional character of these integrals is easily found. Let 0 - e2rir, so that r,A is a new constant, which is determinate save as to any additive integer; as the roots 01,..., 0m are unequal, no two of the m constants rl,..., rm can differ by an integer. Now the quantity (z - a)"~ 4-2 52 EFFECT OF A [18. acquires a factor e2 t', that is, 0f, when z describes the simple complete circuit round a. Hence the quantity ut (z - a) returns to its initial value after the variable has described the simple complete circuit round a;.and therefore it is a uniform function of z in the immediate vicinity of a, say Ok, so that =, = (z - a)1, (. As this holds for each of the integers /a, it follows that we have a system of fundamental integrals in the form (z-a)aT1(i, (z -a)T2 2,..., (z-a)?'m., where Oi, b2,..., imd are uniform functions of z in the vicinity of a, the quantities r, are given by the relations ra = 2 log 0l,, and the roots 0,, *..., O of the fundamental equation are supposed distinct from one another, no one of them being zero. As regards this result, it must be noted that the functions qb are merely uniform in the vicinity of a: they are not necessarily holomorphic there. Each such function can be expressed in the form of a series of positive and negative powers of z - a, converging in an annular space bounded by two circles having a for a common centre and enclosing no other singularity of the equation. There may be no negative powers of z-a, in which case the function ( is holomorphic at a; or there may be a limited number of negative powers, in which case a is a pole of (; or there may be an unlimited number of negative powers, in which case a is an essential singularity. Moreover, r, is only determinate save as to additive integers: it will, where possible (that is, when a is not an essential singularity), be rendered determinate hereafter; so that, in the meanwhile, the result obtained is chiefly important as indicating the precise kind of multiform character possessed by the integrals near a singularity. 19. Now consider the case in which the fundamental equation A = 0 appertaining to the singularity a has repeated roots, say X, roots equal to 0,, X, roots equal to 02, and so on, where 0,, 02,... are unequal quantities, and X, + X, +... = m. It will appear that 19.1 MULTIPLE ROOT 53 a group of linearly independent integrals is associated with each such root, the number in the group being equal to the multiplicity of the root; that each such group can be arranged in a number of sub-groups, the extent and the number of which are determined by the elementary divisors connected with the root; and that the aggregate of the various groups of integrals, associated with the respective roots of the fundamental equation, constitutes a fundamental system. GROUP OF INTEGRALS ASSOCIATED WITH A MULTIPLE ROOT OF THE FUNDAMENTAL EQUATION. Let K denote any such root of multiplicity a, and let the elementary divisors of A (0) in its determinantal form be (0- c)-Y~, (0-c) -',..., (0 K)a'-2-d'-I (-^^ -C)-1; then the minors of order r (and consequently of degree - T in the coefficients of A) are the earliest in increasing order which do not all vanish when 0 = K. Consequently, in the set of equations Plair + P2a2r +... + p. mr = PrK, (r = 1,..., m), r of them are linearly dependent upon the rest; hence taking m - which are independent, we can express m - of the constants p linearly in terms of the other r, which thus remain arbitrary. Let the latter be pi,..., p,; then the integral, given by 't = pll + p2W2 +... + PmnWm, becomes U = pW W1.. + pI Wr, where TfW = Wi + kT+lw,T+1 +... + kMwz, TW 2 = W2 + kr+i,2W T+ +... + n,2m, '*..............' ' *'.. X.........*X.. e''* i WT WT + kT+I, T WT+1 +... + klmrmW and the determinate constants k are given by p7+1 = c+~1,1p1 + k.+l,2p2 +... ~ kr+L1. p P,+2 kT+l,lpl PIp+ kT+2,1p2 +* + l~,+,TpT, P+2 = k7+2, lPI +l m,2 2 +... + 1C7+2, P pn =ckmlpl +kn,,,2p2 +.. ~ kM,7PT 54( ELEMENTARY [19. being the expressions for the m - r quantities p in terms of the 7 quantities p which remain arbitrary. Evidently each of the quantities W is an integral of the equation: and they have the property Wr =,ClW,V, for r = 1,..., I. Moreover, they are linearly independent; any non-evanescent relation of the form ETYL+... + ETWTW=0 would lead to a relation between w,,..., w,, which would be homogeneous, linear, and non-evanescent, a possibility excluded by the fact that wl,..., Wm, constitute a fundamental system. The only case, in which -r = a-, occurs when the indices a- - a,, aO, - oG-,..., r-1of the elementary divisors are each unity. In that case, we have obtained a set of integrals, in number equal to the multiplicity of the root. 20. We shall therefore assume that 7- < a; and we then use the integrals,,..., 1Y4 to modify the original fundamental system w,,..., win, substituting them for w,,..., w,. When the variable z describes a simple closed contour round a, the effect upon the elements of the modified system is to change them into VI, WV', -.., TI%', W'T14,..., W'., where WI=.KTV, 13s/ = lS WI, +.. ~ /8STW7 + /38, ~,'WT+, +. ~3s,rninnm, for r = 1,..., 7-, and s = r + 1,..., m. The fundamental equation derived from this system for the singularity a is A (f)= 0, where A (2) = Kc - fl, 0 0 0, 0 0 0, Ic-fl.., 0, 0 0,..., 0 0, 0,..., I K-fl, 0, 0,..., 0 = (x - f2)TA I (7), 20.] DIVISORS 55 where............................................ Pm,T+, / m, 7+2. *, m* m - - As c is a root of A (a) of multiplicity a-, it is root of Al (f1) of multiplicity - - r; and a question arises as to the elementary divisors of Al (f2) associated with K. The elementary divisors of A, (f), which are powers of K- 2, are (Qi-' )o-E-o- (n - _/c)c~-~2-1, (n- c)2-3- 1 -i... being, in each instance, of index less by unity than those of A (f2). This result, which is due to Casorati*, follows from the property that AI () is divisible by (2 - K)"-T'; its first minors are divisible by (2- K)l-1-(T- and not simultaneously by any higher power; its second minors are divisible by (f2- c)0-(,-2) and not simultaneously by any higher power; and so on. This property, that all the minors of Al (2) of order, are divisible by (K Q)- (r - ') and not simultaneously by any higher power, can be proved as follows+. Any minor of order / of A (a) must contain at least n- - r - p of the last m -r columns: let it contain -r —p + a of these columns, where a can range from 0 to /i. It then must contain r-a of the first r columns. Similarly, it must contain at least m-r — of the last m - r rows: let it contain m-r- +a' of these rows, where a' can range from 0 to p. It then must contain r - a' of the first r rows. The minor may be identically zero: if not, then, owing to the early columns and early rows that are retained, it is divisible by (K- )T '-, and possibly by a higher power of K- 2. Consequently, some among these minors are expressible as the product of (K - ) -" by a linear combination of minors of A1 (Q) which are of order p,; the coefficients in the combination are composed of the constants, which occur in the first T - o columns and the last m-r rows, and thus are independent of Q. But a minor of order / of A (a) is not necessarily divisible by a power of K- 9 with an index higher than a-; thus (K - ~Q)~. polynomial in = (K - ) U7 - ~. sum of minors of Al (2). It therefore follows that the power of K - 0 common to all those minors of A1 (Q) is of index not higher than - -(r-/ ). * Comptes Rendus, t. xcII (1881), p. 177. + Heffter, Einleitung in die Theorie der linearen Differentialgleichungen, pp. 250-256. 56 ELEMENTARY DIVISORS OF THE [20. Next, we know that there are some minors of the original A (Q) of order r, which do not vanish when Q=K and which therefore are not divisible by K- Sl. Clearly they cannot contain any of the first r rows in A (Q); and thus they must be composed of sets of m -r columns selected among the last vn - r rows. Take the minors of order p of aly one of these non-vanishing determinants, their number being iV2, where (mr -r)! (n-z7 - -)!,! and denote these minors by 11 hk (l/, k=1..., J), the integers h and k corresponding to the obliteration of a set of p columns and a set of /L rows out of the non-vanishing determinant of order rmn-r. Let mhk be the complementary of lMhk in its own determinant. Now take the minors of A1 (1) which are of order /: their number is jV2, and they may be denoted by aij, for i, j= 1,..., N, with the same significance in the integers as for Jfhk. Construct an expression hl ail + Wrh2 ai2 +... + )r m] aiN, = Jah, say, where Jh is a determinant of order mn -T. Then either (i), Jh vanishes identically, owing to identities of rows or columns: or (ii), Jh is equal to ~+A (1Q) and therefore is divisible by (K - )0-7, that is, certainly divisible by (K - l) - (r- ), for (~ 16) we have a - rl >- - 0... * *-C-1 T 1; or (iii) Jh, when bordered by rT-/ of the first rows, and the first columns in A (Q), is a minor of order / of A (a) and is therefore divisible by (K —2)i', so that the equivalence of the two expressions for the minor of A (Q) gives (K - I)%. polynomial in = (K - Q)7 -. J,, and therefore J, is divisible by (K - (Q)a - (T -). It thus follows that Jh is divisible by (K — ) - ('- A), in every case when it is not zero: and this holds for all values of A. Taking then 9hl ail + "nh2 Ca2 +... + mnhN aiy -- Jh, for h= 1,..., N and for one particular value of i, we have a series of N linear equations in the quantities ai,..., aiy'. The determinant of their coefficients is a power of the non-vanishing determinant of order m -, for it is a determinant of all its minors of one order: and therefore it does not vanish. Hence, so far as powers of K - l are concerned, each of the minors ail,..., aiy~ is a linear combination of J1,..., JN: all of these are divisible by (K - 12) ~-(7- -), and therefore each of the minors ail,..., aiN is certainly divisible by that power. The result holds for each of the values of i. It has been seen that the power of K -1, common to all these minors of A1 (12), has an index not greater than o- (r - f); combining the results, we infer that the highest power of K - 1, common to all the minors of Al (Q) of order a, has its index equal to o- - (r - A). 21.] FUNDAMENTAL EQUATION 57 21. The indices of the elementary divisors of A1 (f2) are '<r-(-1, <TI-(72-1, ~2-Cr3-1,..; let there be T' of them, where T' T, so that the last - T' of the indices of those of A (Q) are equal to unity, on account of the property af - >~ 0i - 02 > 0 2 - 03 >... *> ST-1> 1. Then the minors of A1 of order r' (and consequently of degree n - T- T' in the coefficients of A1) are the earliest in successively increasing order, which do not all vanish when Q2 = c; consequently, in the set of equations pl /r,T+1 + p2 ',r,7+2 + + p'rm —7 3-,n = Kr, (r = + 1,..., m), T' of them are linearly dependent upon the rest. Hence taking m- T- r' of the equations which are independent, we can express m - - T' of the constants p' in terms of the other r', which thus remain arbitrary and which may be taken to be p',..., p',. Now take an integral V = piWT+l +... + p+ m — Wm, and substitute for the various coefficients p' in terms of p',..., p'. The integral becomes v- = pW, + p;2 X 12 + p... + r', where, writing X= r + T', we have Wir = W7+rt + IX+1, WA+ +... + 1l, 2'Wn, for r=1,..., T'; and the determinate constants 1 are given by PT7+S l= IA+,Spl +... + lx+s, rP T', for s = 1,... - X, being the expressions of the constants p' in terms of p',..., p. Clearly each of the quantities WIn, W12,..., WT is an integral of the equation. Moreover, they are linearly independent of one another and of WI,..., WT; for any non-evanescent linear relation of the form F141 +... + FTWI, + F'W, +... + F/T'W1r= would lead, after substitution for W1,..., WTF, WIn,..., W1T' in terms of the original fundamental system wl,..., wi, to a nonevanescent homogeneous linear relation among the members of that system-a possibility that is excluded. 58 SUB-GROUPS OF [21. As regards the effect, which is caused upon each of these newly obtained integrals by the description of a simple contour round the singularity, we have W1i/. = W'7+ r +,r+ +.+l.. + IrnrWm' = WI,, + VT., where V,. denotes a homogeneous linear combination of W1,..., WV. Now no one of the quantities V. can be evanescent, nor can any linear combination of the form 71 l +... + 7T/ V^T be evanescent: for in the former case, we should have W12. = KWi., and in the latter (7,1 W1, +... + 7rY WT1) = K (71, W1 +... + T WV1T). As Wr and y1 W... +. + y WIT in the respective cases are linearly independent of WT,..., WT, we should thus have a new integral of the same type as Wi,..., WT; and then, instead of having some of the minors of order T in A (12) different from zero when 12 = K, all of them of that order would be zero, and we should only be able to declare that some of order r + 1 are different from zero: in other words, the number of elementary divisors of A (12) would be T + 1 instead of r. The quantities V,,..., V,1 are thus linearly equivalent to T' of the quantities W,..., W,, say to WI,..., WT/,; hence constructing the linear combinations of V,..., VT, which are equal to 1,..., WI, respectively, and denoting by w,,... wT' the linear combinations of Wn,..., WT with the same coefficients as occur in these combinations of V,,..., V,., we have a set of r' integrals w,,,..., w1,,, such that wr = Kwir, + W,., (r = 1,..., ). These integrals are linearly independent of one another, and also of W,..., WT, before obtained. They constitute the aggregate of linearly independent integrals of this type; for if there were another linearly independent of them, it would imply that Al (f2) had r' + 1 elementary divisors instead of only '. As regards the two sets of integrals already obtained, it may be noted, (i), that the set t14,..., W, can be linearly combined among themselves, without affecting the characteristic equation W,.'= c Wr.; 21.] INTEGRALS 59 (ii), that to each integral of the set wl,..., wIT, there may be added any linear combination of the integrals of the set W1,..., W,, without affecting the characteristic equation U1.' = KW19. + WV.. If the index of each of the elementary divisors of Al(n) is unity, then -'=a -, so that the number r +r' of integrals obtained is then equal to -, the multiplicity of the root of A (2) = 0 in question. In every other case, r' + T< a. 22. When 7'+ T< -, so that r' is less than the degree of A1 (12), we use the integrals w,,..., wi,, to modify the fundamental system WI,..., WT, W7,+,..*., w, substituting them for WT7+1..., W7+7, in that system. When the variable z describes a simple closed contour round a, the effect upon the elements of the modified fundamental system is to change them into W,',..., WT', w11,..., W'17/, W'A+..., Win', where T + r = X, and WI'. K W., Ws = Kc1s + Ws, Wt' 7tl W1 + ** + tT WT + yt,7+1Wll +... + YtX W17' + Yt, +liWX+] +...+ 7 t, mlm, for r=1,...,; =1,..., T' t=X 1,..., m. The fundamental equation derived through this system is A (1) =(K - I,)7+'A (f) = 0, where A2(n)= 7nA+I,)+i-, 7+- l,Q+2,., +,* A+l, m..........................................o. 7m, X+l 7M, A,+2, ***, 7Ym, m - Also K is of a root of A2(12) of multiplicity a- -r'. By a further application of the proposition (~ 20) connecting the elementary divisors of A (f) and A1 (1), the indices of the elementary divisors of A2(2), which are powers of Kc-, are seen to be a —a 0-2, o1 — 2-2, 2,-a-03 - 2,. say T" in number. The procedure from the equation A2 () = 0 to the corresponding sub-group of integrals is similar to that adopted in the case of the equation A1 (2) = 0; and the conclusion is that there 60 COMPOSITION OF A GROUP [22. exists a sub-group of T" integrals w2, w22,..., w2,,, characterised by the equations W2t = KCW2t -+ Wt, for t = 1, 2,..., r". And so on, for the sub-groups in succession. Combining these results, we have the theorem*: When a root K of the fundamental equation A (f2) 0 is of multiplicity a, and when the elementary divisors of A (12) associated with that root are (K - Q) 1 (K C - )f1 2, *..., ( -_ a group of o- linearly independent integrals is associated with that root: this group consists of a number a- o-, of sub-groups, which satisfy the equations w.' = /CW., for r = 1,..., T, Wl = KWs + Ws, for s=1,...,, w2t = KW2t + wit, for t = 1,.,, and so on. The integer T is the number of elementary divisors of A (Q); T' is the number of those divisors with an index greater than unity; r" is the number of those divisors with an index greater than two; and so on. The group of o- integrals, and m - a other integrals, all linearly independent of one another, make up a fundamental system: the m- a- other integrals being associated with the m - o- roots of A (fI)= 0 other than 12 = K. When these roots are taken in turn, we have a single integral associated with each simple root, and a group of integrals of the preceding type associated with each multiple root, the number in the group being equal to the order of multiplicity of the root. We thus have a system of integrals of the original differential equation distributed among the roots of the fundamental equation associated with the ' That part of the theorem, which establishes the existence of the group of integrals associated with a multiple root, is due to Fuchs, Crelle, t. LXVI (1866), p. 136: but the initial expression given to the members of the group was much more complicated. The part which arranges the group in sub-groups, each with its own characteristic equation, is due to Hamburger, Crelle, t. LXXVI (1873), p. 121; he takes it in an arrangement, which will be found in the next section. The association of the sub-groups with the elementary divisors of A ()) is due to Casorati, Comptes Rendus, t. xcii (1881), p. 177. 22.] OF INTEGRALS 61 singularity: that the system is fundamental is manifest from the facts, that the initial system was fundamental, and that all modifications introduced have been such as to leave it fundamental. Ex. 1. Two independent integrals of the equation d2w dw+ Z2 (+ 1) d- V +(3z+ ) =O are given by w1 = z2, w2 =z +z log z. Hence when the variable describes a simple closed contour round the origin in the positive direction, we have W1= -W-1 202' =- 27i-i1 - W2; and therefore the fundamental equation belonging to the origin (which is a singularity of the equation) is -1- 0, 0 =0, -27ri, -1-6 that is, it is (+ 1)2=0. Similarly, two independent integrals of the equation 2d2w - dw Z - - -g+.z = o0 dz2 Z dz 3 are given by = Z2, W2 = Z3. Hence after a simple closed contour round the origin, we have Wl1- = - w0, w1 = a22, where a is eSi; the fundamental equation belonging to the origin is -1-0, 0 1=0, 0, a- 0 that is, (o+ 1) (o - e5')=0. Ex. 2. Construct the linear differential equation of the third order, having zT, z- log z, zX~ for three linearly independent integrals; obtain the fundamental equation appertaining to the origin as a singularity; and from the form of the differential equation, verify Poincare's theorem (~ 14) that the product of the three roots of this fundamental equation is unity. __ 62 HAMBURGER'S [23. HAMBURGER'S RESOLUTION OF A GROUP OF INTEGRALS INTO SUB-GROUPS. 23. In the case when the roots of the fundamental equation are all distinct from one another, the general analytical character of each of the integrals of the fundamental system in the vicinity of the singularity has been obtained (~ 18). We proceed to the corresponding investigation of the general analytical character of the group of integrals in the vicinity of the singularity, when the group is associated with a multiple root of the fundamental equation. We have seen that the group of linearly independent integrals can be arranged in sub-groups of the form W1, W,..., W; WU11, U12,..., W T'; W21, lt 22,..., 127";.. ooooo o.oooo... o... X the members of each sub-group being arranged in a line and satisfying an equation characteristic of the line. Let these be rearranged in the form* ]T/i, Wll, W21, W31,.W2, W~12, J22, W32,.............................., each of the integrals in the new line satisfies an equation, and the set of characteristic equations for any line is, in sequence, the same as for any other line, so far as the members extend. When any such line is taken in the form U1, U2, U. U, where the integer p changes from line to line, the set of the characteristic equations is U1- = KU1 U12 = U.2 + U1 U3' = KU3 + U''2 Ut = KUl + U -1i * These are Hamburger's sub-groups; see note, p. 60. Their number is equal to the number of elementary divisors of A (0) connected with the multiple root. 23.] SUB-GROUPS 63 Let 27ria = log K; we have [(Z- a)]I = K- ( a), and therefore [ui (z - a)-a]' = il (z -a)-~. Thus ul (z-a)-a is unaltered by the description of a simple closed contour round a; it therefore is uniform in the vicinity of a, but it cannot be declared holomorphic in that vicinity, for a might be a pole or an essential singularity of u (z - a)-. Denoting this uniform function of z - a by #1, we have U = (Z -a)al. To obtain expressions for the other integrals, Hamburger* proceeds as follows. Introduce the function L, defined by the relation L = log (z - a); then, after the description of a simple contour, we have L'= L + 1. We consider an expression F(L)=F= + +( -) _L+ (a 1) -2+* *v + ( *2 L,) L,-2 +*D-l, where nr e f, (. -l-r)! r!' and the functions i,..., are uniform functions of z-a. Then if, for all values of n, we take y,-,n = (Z - a)a K1nAnF, where the symbolical operator A is defined by the relation AF= F(L + 1) - F(L) = F'- F, we have y'_-n = (z - a)aKn1+l/nF = (z - a)acKn+l (AnF-+ A1n+F) = KcyJ —n + y-,n-i,* Crelle, t. LXXVI (1873), p. 122. 64 GENERIC FORM OF [23. holding for all values of n. These are the characteristic equations of the modified sub-group; and therefore we can write U.-n = (Z - a)Xn"a lF, with the above notations. This is Hamburger's functional form for the integrals. 24. The integrals a1,..., u, are a linearly independent set out of the fundamental system; and the system will remain fundamental if u,,..., ', are replaced by u other functions, linearly independent of one another and linearly equivalent to ul,..., u/. A modification of this kind, leading to simpler expressions for the sub-group of integrals, can be obtained. In association with F, take a series of quantities, defined by the relations VI = A,, V2 = #2 +l #L, v3 = r3 + 22L + IL2, v, = 4 + 33,L + 32,L2 + 1L3,.... F' v, L- ++( )2kL- + ' ) L2+ *+ *2l 1)2 2+ lLThen we have A F= allv.-l- + al22v,-2 + al3V/,-3 +... + a1,,-lvl, A2F = a2lV/,-2 + a22V-_3 +... + a2, -2Vl, A3F= a3 v,-_3 + a32V4 +... +. a3,, -3Vl, A/-29F = a/_2, V2 + aa,2,2V1, At-1F = al, 1 Vl, where the constants a are non-vanishing numbers, the exact expressions for which are not needed for the present purpose. Then (z- a)av is a constant multiple of (z -a)az/-lF, that is, of ul; and it therefore is an integral of the differential equation. By the last two of the above equations, (z-a)av2 is a linear combination of (z- a)oA/-2F and (z-a)a~A-lF, that is, it is a linear combination of u2 and ul; it therefore is an integral of the differential equation. By the last three of the above equations, (z - a)av3 is a linear combination of (z - a)A-h3F, (z - a)ah-2F, (z - a)aA-1F, that is, 24.] A GROUP OF INTEGRALS 65 it is a linear combination of u1, i2, u3,; it therefore is an integral of the differential equation. Proceeding in this way, we obtain / integrals of the form (z — aav1, (z - a)av,..., (z -a)av. Moreover, these are linearly independent, and so are linearly equivalent to u1,..., u,; for, having regard to the expressions of AF,..., Al-IF, we see at once that any homogeneous linear relation among the quantities vI,..., v, would imply a homogeneous linear relation among the quantities F, AF,..., A&-OF, that is, among u,,..., u,; and no such linear relation exists. Hence Hamburger's sub-group of integrals is equivalent to (and can be replaced by) the sub-group (Z - a)a V), (Z- a)a V2..., (- a)a v, Accordingly, we now can enunciate the following result as giving the general analytical expression of the group of integrals, associated with a multiple root Kc of the fundamental equation:When a root K of the fundamental equation A (0)= 0 is of multiplicity ao, the group of a- integrals associated with that root can be arranged in sub-groups; the number of these sub-groups is equal to the number of elementary divisors of A (0) which are powers of Kc-0; the number of integrals in any sub-group is determined by means of the exponents of the elementary divisors; and a sub-group, which contains pu integrals, is linearly equivalent to the pL quantities (z - a)avi, (z- C)av2,..., (z - a))v,, where 2ria = log Kc, and the tL quantities v are of the form VI = 1- V3 = 3~+ 2#2L+ 1L2, v4 = r4+ 3 + 3(3/ + 312) + LL3,............o............o.......o... v~=~ + ~(- 1) pl-2 + -l * This form of expression for the group of integrals appears to have been given first by Jiirgens, Crelle, t. Lxxx (1875), p. 154. See also a memoir by Fuchs, Berl. Sitzungsber., 1901, pp. 34-48. F. IV. 5 66 SUB-GROUP OF INTEGRALS AND [24. where L = oz -a), ( - denotes and 27where n L) r r ( - - r)! r!' and the tj quantities q],.., qf are uniform (but not necessarily holomorphic) functions of z - a in the vicinity of the singularity. DIFFERENTIAL EQUATION OF LOWER ORDER SATISFIED BY A SUB-GROUP OF INTEGRALS. 25. The preceding form of the integrals in each sub-group of a group, associated with a multiple root of the fundamental equation, has been inferred on the supposition that the coefficients of the linear equation are uniform functions. It will be noticed that the coefficient of the highest power of L in each of the members of the sub-group is the same, being an integral of the equation,-a result which is a special case of a more general theorem. Moreover, it is of course possible to verify that each member of the sub-group satisfies the differential equation; and it happens that the kind of analysis subsidiary to this purpose leads to the more general theorem above indicated, as well as to a result of importance which will be useful in the subsequent discussion of the reducibility of a given equation. We proceed to establish the following theorem*, which is of the nature of a converse to the theorem just established: If an expression for a quantity u be given in the formn 't = On +,n-IL + n-2L2 +... + 2Ln-2 + Okn-L-1 where L= i log (z-a), and each of the quantities q is of the forrn= ( = (z - a)". uniforn function of x - a, a being a constant, then u satisfies a homogeneous linear differential equation of order n, the coefficients of which are functions of z uniform in the vicinity of z = a; moreover, 8 eu a2U a-1 u L' L' 2'"' Ln-1 are integrals of the same equation and, taken together with u, they constitute a fundamental system for the equation. * Fuchs, in the memoir quoted on the preceding page, ITS DIFFERENTIAL EQUATION 67' an —1u (It is clear that aL_~ is a numerical multiple of f01, and that the coefficient of the highest power of L in each of the announced integrals is, save as to a numerical constant, the same for all; it is a multiple of 0,, which is an integral of the equation.) It is convenient to make a slight modification in the form of u; we take n - \n - 2 = n + (n n-1) + + (n —1) -.... 1 2 ' ~..+ (. - 1) 2L)-2+ Ln-, where (lr -) -- =, so that the character of the functions and their form (except as to a mere numerical constant) are the same as those of the functions b. Further, no change, either in the property that aut a2U aL' L2 ' are integrals of the equation or in the property that, taken together with it, they constitute a fundamental system, will be caused if they are multiplied by constants: so that, if the theorem can be established for 1,,..., n-, where 1 an-lu i (n_ 1)! aL=n1! 3n-2...................o.o...o..................... (n - 2) i a25 —2 u^= ^-ijiL-3 4t3 + 2#2L + #iLI, (n - 2) (n - 2\ 12...+ (U- 2) + *rLn-2, the theorem holds for the quantities as given in the enunciation of the theorem. 5-2 68 SET OF LINEARLY [26. 26. Merely in order to abbreviate the analysis, we take n =4; with the above forms, it will be found that the analysis for any particular case such as n = 4 is easily amplified into the analysis for the general case. Accordingly, we deal with quantities u, u1, u2, 3U, where U = 4 + 33L + 32L 4+ 1L3, U -3 + 22L + 1L2, 2 = #2 + IL, U1 = #1. If u can be an integral of a linear equation of the fourth order with coefficients that are uniform functions of z-a in the vicinity of a, let the equation be d4 d3 d2 d d=- + d+ Q + R + S = 0. Let the variable z describe a simple contour round a; this leaves the differential equation (if it exists) unaltered, and so the new form of ut is an integral, say u', where u' = /K4 + 3k+ 3 (L + 1) + 3/+ 2 (L + 1)2 + /il (L + 1)3, where K is the factor common to all the functions s after the description of the circuit. As u and u' are integrals of a homogeneous linear equation, so also is V, =- - ' -- K = 3U3 + 3U2 + U1. Hence v' also is an integral, and it is given by v = 3 3 {K3 + 2K2 (L + 1) + K1L (L + 1)2} + 3 {K2~ + i( (L + 1)} + Kci: and therefore W, / I,+ W, = 6 (-V -V = 6 + '1, is also an integral. Hence w' is also an integral, and it is given by w = #2, + K (L + 1) + Ki, and therefore It, = - = 1 1, is also an integral. is also an integral. 26.] INDEPENDENT INTEGRALS 69 Thus integrals are given by t, =- 1, W-t, =-'a2, (v- 3w + 2t), =3, which proves one part of the theorem, viz. that it, u,, u2, zu are simultaneous integrals of the linear equation if it exists. 27. In order to establish the property that u, ua, U2, u3 constitute a fundamental system of the equation if it exists, a preliminary lemma will be useful; viz. if A, B, C, D be functions free from logarithms and if they be such that a simple closed contour round a restores their initial values, except as to a constant factor the same for all, then no identical relation of the kind aA + /3BL +i- CL + 8DL3= 0 can exist, in which a, 3, y, 8 are constants different from zero. For let the simple contour be described any number, N, of times in succession; and let f be the constant factor acquired by the functions A, B, C, D after a single description of the simple contour. Then we should have the relation fN [aA + 3B (L + NV) + 7C (L + N) + 8D (L - N)3] = 0, and consequently the relation aA +, B (L + N) + yC(L + LN)2 + D (L + N)3 = 0, valid for all integer values of N. Consequently, the coefficients of the various powers of N must vanish: hence 0= 8D, 0=38DL + 7C, 0 = 38DL2 + 2yCL + f3B, 0= DL3 + 7yCL2 + /BL + aA, the last of which is the original postulated relation. From the first of these relations, it follows that 8=0; then, from the second, that then, from the third, that 70 EQUATION SATISFIED BY [27. and so, from the original relation, that a =0. The lemma is thus established. It may also be proved that, if A, B, C, D be functions free from logarithms, and if they be such that a simple closed contour round a restores their initial values, except as to constant factors which are not the same for all, then no identical relation of the kind aA + i3BL + yCL]2 + 8DL3 = 0 can exist, in which a, 3, 7, 8 are constants different from zero. The proof is left as an exercise. It is an immediate inference from the course of the lemma that no relation of the form a'u + 73U3 + 72 + 8'u1 = 0 can exist, in which a', 3', y', 8' are constants different from zero; for proceeding as before, it would require 0= a'l1, 0= 3a'#2 + /'1i, o = 3 '%3 + 2/'42 + 'Yi, + 0= a'4 + /3'+t3 + ',k2 + 8'#1, which clearly are satisfied only if a'' = /3' = = 7 ' = 0. Hence there is no homogeneous linear relation among the quantities u, u1, u2, u3; and they therefore constitute a fundamental system for the linear equation if it exists. 28. If the equation exists, we must have Au=0, =0, AU=0, =0; and in the operator A, the functions P, Q, R, S are to be uniform functions of z in the vicinity of a. Let Z denote any function of z with the same characteristic properties as 1i, *2, *3, *4; then with such an operator A, we have (ZL) = LAZ + Z', A (ZL2) = L2Z + 2LZ' + Z", A (ZL) = L3AZ + 3LaZ' + 3LZ" + Z'", 28.] A SET OF INTEGRALS 71 where Z', Z", Z"' are functions of the same characteristic properties as Z, that is, as 1, 21, 2 3, 4, 4 and they are free from logarithms. Now as Au, = 0, we have A%1 = 0. As Au, = 0, we have A2 + LzA, +,/ = 0, that is, A#2 + 1 = 0. As A^3, = 0, we have A,3 + 2 (LA2 + + 2+ L5A,1k + 2Lt1/ + 1= 0, that is, by using the two preceding relations, A*3 + 22/' + #1/ = 0. As Au = 0, we have A#, + 3 (LAx3 + 3/) + 3 (L2A#2 + 2LF2' + 2//) + L3A,1 + 3L21' + 3L1"' + 1"' = 0, that is, by using the three preceding relations, A,4 + 33' + 3%2" + 1//'/ = 0. Thus there are four equations; each of them involves the coefficients P, Q, R, S linearly and not homogeneously. The required inferences will be obtained if the equations determine P, Q, R, S as functions of z, uniform, in the vicinity of a. Now each of the functions * is such that (z -a)-a~f is a uniform function of z - a in the vicinity of a; accordingly, let (z- a)-a ^* = 0, (U( = 1, 2, 3, 4), where each of the O's denotes a uniform function. Substituting (z - a)YO, for +, in each of the four equations, the factor (z - a)" can be removed after the differential operations have been performed; and then all the coefficients of P, Q, R, S, and the term independent of them, are uniform functions of z in the vicinity of a. Solving these four equations of the first degree for P, Q, R, S, we obtain expressions for them as uniform functions of z- a in the vicinity of a. (In general, this point is a singularity for each of the expressions.) It follows that, for these values of P, Q, R, S, the four quantities u1, u2, us, u are integrals of the linear differential equation of the fourth order. 72 PROPERTIES OF A SUB-GROUP [28. As already remarked, similar analysis leads to the establishment of the result for the general case; and thus the theorem is proved. COROLLARY I. It is an obvious inference from the preceding theorem that, when a group of integrals is associated with a multiple root of the fundamental equation, any (Hamburger) sub-group, containing (say) n of the integrals, is a fundamental system of a linear equation of order n with uniform coefficients. Further, it is at once inferred that the n' members of that subgroup, which contain the lowest powers of the logarithm, constitute a fundamental system for a linear equation of order n' with uniform coefficients. COROLLARY II. Similarly it may be established that one (Hamburger) sub-group containing n integrals, and another subgroup containing p integrals, constitute together a fundamental system for a linear equation of order n+p with uniform coefficients. And so on, for combinations of the sub-groups generally. Ex. Prove that if the linear equation in w has a sub-group of n integrals which, in the vicinity of a singularity a, have the form 11U = +1, 2 = 4+2 + +,L, qt3 = +3 2+2L + +1L2, where 2iriL = log (z - a), and each of the functions + is such that (z - a) - a is uniform, where e2'7i" is a multiple root of the fundamental equation with which the sub-group of integrals is associated, then if the linear equation for v be constructed, where w= w1 Jf' dz, that linear equation has a corresponding sub-group of t - 1 integrals of the form where the functions q are of the same character as the functions +. CHAPTER III. REGULAR INTEGRALS; EQUATION HAVING ALL ITS INTEGRALS REGULAR NEAR A SINGULARITY. 29. THE general character of a fundamental system of integrals in the vicinity of a singularity has now been ascertained. For this purpose, the main property of the linear equation which has been usedis that a is a singularity of the uniform coefficients; the precise nature of the singularity has not entered into the discussion. On the other hand, the functions ( which occur in the integrals are merely uniform in the vicinity of a: no knowledge as to the nature of the point a in relation to these functions has been derived, so that it might be an ordinary point, or a pole, or an essential singularity. Moreover, the index r in the expressions for the integrals is not definite; being equal to I log 0, it can have any one of an unlimited number of values 27ri differing from one another by integers. Hence, merely by changing r into one of the permissible alternatives, the character of a for the changed functions ~ may be altered, if originally a were either an ordinary point or a pole: that character would not be altered, if a originally were an essential singularity. It is obvious that the character of a for the integral is bound up with the nature of a as a singularity of the differential equation, each of them affecting, and possibly determining, the other. Accordingly, we proceed to the consideration of those linear equations of order m such that no singularity of the equation can be an essential singularity of any of the functions <, which occur in the expression of the integrals in its vicinity. In this case, the functions q, which are uniform in the vicinity of 74 REGULAR INTEGRALS [29. a and therefore, by Laurent's theorem, can be expanded in a series of positive and negative integral powers ofz - a converging within an annulus round a, will at the utmost contain only a limited number of negative powers. To render r definite, we absorb all these negative indices into r by selecting that one among its values which makes the function b in an expression (z -a)" finite (but not zero) when z = a. An integral of the form I = (z - a) [.0+ +,1 log (z -a) +... +, {log (z - a)}j], where 00, 01,..., ) are uniform functions having the point a either an ordinary point or a zero, is called* regular near a. When a value of r is chosen, such that (z -a)-ru is not zero and (if infinite) is only logarithmically infinite like c + c, log (z - a)+... + {log (z - a)K, the integral is said to belong to the index (or exponent) r: the coefficients c being constants and not all of them zero. Similarly, when the singularity a is at infinity, and there is an integral w r Z-P + log+tl. log ), where *0,.., Jr, are uniform functions having z= o for an ordinary point or a zero, the integral is said to belong to the index or exponent p. It will be possible later to consider one class of integrals that do not answer to this definition of regularity: but it is clear that regular integrals, as a class, are the simplest class of integrals, and that the first attempt at obtaining integrals would be directed towards the regular integrals, if any. Accordingly, we proceed to consider the characteristics of linear differential equations which possess regular integrals: and in the first place, we shall consider equations all of whose integrals are regular in the vicinity of one of its singularities, in order to determine the form of equation in that vicinity. * After Thomi, Crelle, t. LXXV (1873), p. 266. The use of this name for a function, which is not regular in the variable, may seem anomalous: but it is now wide-spread, and confusion might be caused by the introduction of another name. See footnote, p. 4. 29.] AND THEIR EXPONENTS 75 As subsidiary to the investigation, one or two simple properties, associated with the indices to which the functions belong, will first be proved. If a regular fitnction u (in the present sense of the term) belong to an index r and another v to an index s, then u v belongs to the index r-s: as is obvious from the definition. due If a regular function u belong to an index r, then ~ belongs to the index r-1. To prove this, let u=(z- a) z K ('log(z- a)); K=0 then du (z- a). i {r +(a) - log (za)K +,K log (z-a)} 1], dz du so that T can only belong to the index r - 1, if some at least of the coefficients of powers of log (z -a) are different from zero when z =a. These coefficients in succession are '+0 + 01 rqui + 202, r + 3 03, r + n, when z=a is substituted: they cannot all vanish, for then 00, q,..., n would vanish when z=a, so that co, cl,..., cK would all be zero, and then u du would not belong to the index r. Thus d- belongs to the index r -1. There is one slight exception, viz. when u is uniform and the index r is du zero; then - is also uniform, and it may even vanish when z = a; so that, az ~du if u were said to belong to the index 0, du could be said to belong to an index not less than 0. FORM OF THE DIFFERENTIAL EQUATION WHEN ALL THE INTEGRALS ARE REGULAR NEAR A SINGULARITY. 30. As a first step towards the determination of the form of a differential equation that has all its integrals regular, we shall obtain the index to which the determinant of a fundamental system belongs. Let the system be Wi,,2,..., Wi: and let the 76 FORM OF DIFFERENTIAL EQUATION [30. indices of the members be ri, r2,..., r respectively. We take the determinant in the form 1mVOwl-in~,u/nt —2.. of ~ 12, where C is a constant. The quantity vI is a solution of an equation, a fundamental system for which is given by d fW2\ d ft%> VI =(, = _ )('-. l=dz V )\ 72 = dz \wI It is clear that, if w1, w2,... are all free from logarithms, then v,, v2,... are also free from them. If however there be a group or a sub-group of integrals associated with a repeated root 0 of the fundamental equation, we may take (~ 23) W1/ = Owl, w2' = w1 + Ow2, so that d /w2\ d /I w2\ aVI'= + = v,; thus vI is uniform and therefore free from logarithms. Similarly, u, and all the quantities used in the special form of the determinant are free from logarithms. The indices to which v,, v2,... respectively belong are r2-r1-1, r3 —r-l1, r4-rr-1,... unless it should happen that, for instance, r2= rl. In that case, we replace w2 by w2 + awl, choosing a so as to make the new integral belong to an index higher than r2 or r1: this change will be supposed made in each case where it is required. Again, the quantity u, is a solution of an equation, a fundamental system for which is given by d (V2' d v3 -U,= - VI.. dz \v, dz \vJ The index to which ul belongs is r3-ri-l-(r2- ri - =r3 - r,-1, and so for u2,...; that is, their indices are r3-r2-1, r4-r-l,.... And so on, down the series. 30.] HAVING REGULAR INTEGRALS 77 Hence the index to which CW/lmlm —1 Ulm-2.. belongs is = mzr, + (m - 1) (r2 - r1 - 1) + (m - 2) (r - r - 1) +......+l (rm- rm- 1) = r, + r +... + rM - m (mn - 1); so that, denoting the determinant of the fundamental system by A (z) as in ~ 9, it follows that, in the vicinity of the singularity a, we have A (z) = (z - a),+rl+2+ +*..-+rm-",-) R ( - a), where R is a holomorphic function of its argument in that immediate vicinity, and does not vanish at a. 31. This result enables us to infer the form of the differential equation in the vicinity of the singularity a. Manifestly, the equation is dmw dm-lw dm-2w dzm = Pl dzml + P2 dz-_ +., if P AK (=...,n PK = A') where A is the determinant of the m integrals in the fundamental system, and AK is the determinant that is obtained from A on dm —Kn dmw, replacing the column d-,'_ for s = 1,...,, by the column -ddzrn-K dzyn for s = 1,..., m. Now consider a simple closed path round a. After it has been described, A and AK resume their initial values multiplied by the same constant factor, which is the non-vanishing determinant of the coefficients a (~ 13) in the expressions for the transformed integrals; thus PK is uniform for the circuit. Hence, when the expressions for the regular integrals are substituted in A and A,, all the terms involving powers of log (z - a) disappear. Moreover, A belongs to the index rl+... +r - m(m- 1); and so far as concerns the index to which AK belongs, it contains a column of derivatives of order K, = m - (m - /), higher than the corresponding column in A, so that AK belongs to the index ri +... + rn- I-M (m - 1) -C. 78 METHOD OF FROBENIUS [31. Hence pK belongs to the index - K and therefore, in the immediate vicinity of a, the form of p, is given by PK (Z - a) where, at a and in the immediate vicinity of a, the function PK (z - a) is a holomorphic function which, in the most general instance, does not vanish when z= a, though it may do so in special instances. As this result holds for Ec=,..., m, we conclude that, when a homogeneous linear differential equation of order m has all its integrals regular in the vicinity of a singularity a, the equation is of the form dmw P1 dm-lw P2 d m-w Pmn ---...- -+ w dzm z -a dzm-1 (z -a)2 dzm-2 (z - a)m in that vicinity, where Pi, P2,..., Pm are holomorphic functions of z- a in a region round a that encloses no other singularity of the equation. CONSTRUCTION OF REGITLAR INTEGRALS, BY THE METHOD OF FROBENIUS. 32. The argument establishing this result, which is due to Fuchs*, is somewhat general, being directed mainly to the deduction of the uniform meromorphic character of the coefficients of the derivatives of w in the equation. No account is taken of the constants in the integrals: and it is conceivable that they might require the existence of relations among the constants in the functions P1,..., P,,. Hence for this reason alone, even if for no other, the converse of the above proposition cannot be assumed without an independent investigation. The conditions, which have been shewn to apply to the form of the equation, are necessary for the converse: their sufficiency has to be discussed. Accordingly, we now consider the integrals of the equation in the vicinity of the singularityt. Denoting the singularity by a, we write z-a= x, P, (z-a)=pr(x)=p,, (r = 1,..., m); * Crelle, t. LxvI (1866), p. 146. + The following method is due to Frobeuius; references will be given later. FOR REGULAR INTEGRALS 79 so that the equation can be taken in the form dmw (r- din-1W dw Dw = x" - P1 d+ 4.. + XM-1 d +1 ) 0, valid in the vicinity of x = 0. If regular integrals exist in this vicinity, they are of the form indicated in % 18, 24, the simplest of them being of the form W = XP gXI'Y = ~ gVxP+V V=O v=O = g (Xp), say; should this be an integral, it must satisfy the equation identically. We have DxT= ma( ).( ~ + 1) m a-1..o-r + 2) p,= xaf(X, a), say. Here, f(x, a-) is a holomorphic function of x in the vicinity of the x-origin and is a polynomial of degree m in a-, the coefficient of a.c being unity: so that, if it be arranged as a power-series in x, we have fx fo (O-) + Xfj (t-) + X;2f (., where f, (a) is a polynomial in a of degree mn, and f (a-), f2(a-),... are polynomials in a of degree not higher than rm - 1. Then Dg (x, p)= Y g DxP+v v= 0 g gvX~vf (X, P + V) XP+Vgfo (p + v) + g,-lf(p + v-)+...+ gof, (p)}. V=0 If the postulated expression for w is to satisfy the equation, the coefficients of the various powers of x on the right-hand side must vanish: hence 0 =g90fo (p), 0 = g0f1 (p) ~ gifo(p + 1), 0 = g0f2(p) + g1fl (p + 1) ~ g2fo (p +2), and so on. These equations shew that the values of p, which are to be considered, are the roots of the algebraical equation f (P) =0 80 CONSTRUCTION OF [32. of degree m in p: and that, for each such value of p, So hV (p), fo (p + 1).fo (p + 2)h...fo (P) where - hv (p) is the value of the determinant fo(p+l), 0, 0,..,, (), f (p+l), o(p+2), o,..., o, /( f2( p+ 1), f (p+ 2), o(p + 3),..., 0, /3(p) fV-2(P + 1), f-3(p + 2), f^-4(p + 3),..., fo(p + v -1), f_-(p) fvl(p + l), f,_o(p+ 2), f,_V(p 3),., fl(p + -l), f,(p) so that h, (p) is a polynomial in p. If no two of the roots of the equation f (p) = 0 differ by whole numbers, then no denominator in the expressions for the successive coefficients g, vanishes; the expression g(x, p) is formally adequate for an integral, but the convergence of the series must be established to ensure the significance of the expression. If a group of roots of the equation o (p) 0 differ among one another by whole numbers, let them be p, p+e,..., p+e, where the real part of p is the smallest, and that of p + e is the largest, among the real parts of these roots; equality of roots would be indicated by corresponding equalities among the positive integers 0, c1,..., e. We then take g0 =fo (p + 1)...fo (p + c) g, and thus secure that no one of the coefficients g, becomes infinite. The condition, that the equation shall formally be satisfied, has imposed no limitation upon g0, which accordingly can be regarded as arbitrary; hence g also can be regarded as arbitrary. 33. In order to deal with both sets of cases simultaneously, the formal expression is constructed in a slightly different manner. A parametric quantity a is introduced and it is made to vary within regions round the roots offo(p)= 0, each such region round a root being chosen so as to contain no other root. The quantity go in the first set of cases, and the quantity g in the second set, REGULAR INTEGRALS 81 are arbitrary; they are made arbitrary functions of a. Quantities g,, g2,... are determined by the equations = gfo (a + 1) + gof (a), 0 = g2fo (a + 2) + glfi (a + 1) + gof2 (a),........,,,,.o...o............................ the same in form as the earlier equations other than the first: these quantities g are functions of a. Moreover, we have 9g (a) = go (a)) ho (a); fo (a + 1) o (a + 2)... fJ0 (a + v) (); in consequence of the assumption as to go (a) in the second set of cases, and of the regions round the roots of fo (p)= 0 in which a varies, it follows that the quantities g1, g2,... are each of them finite for all variations of a within the regions indicated. We thus have an expression 00 y=g (, a)= ) g^xa+=; v=O also Dy = E gDxa+v gO - g avf(x, a + v) v=0 =,+V^ t{gfo (a + v) + gv-lf1 (a + v-) +... + gof (a)} v=0 =g (a) f (a) xa, the coefficient of every power of x except xa vanishing, in consequence of the law of formation of the quantities g. 34. We proceed next to consider the convergence of the power-series for y, before bringing the equation satisfied by y into relation with the original differential equation. We denote by R the radius of a circle round the x-origin within and upon which the functions p,..., pm are holomorphic: so that the circle lies within the domain of this origin. Thenf (x, a) and its derivatives with regard to x are also holomorphic for values of x within the circle and for all values of a considered. As the first of them, say f' (x, a), is of degree in a one less than f(x, a), it is convenient to consider that first derivative: let M (a) be the greatest value of If' (x, a)l for the values of x and a, so that, as f' (, a) = E (z + 1)fV+1 (a) xV, v=0 \F. IV. 6 82 CONVERGENCE OF THE [34. we have* M+(a) I(v + l)f/+l (a): RV ) and therefore, as v + 1 is a positive integer > 1, also IfA- + (a) I: R-VM(a). By the definition of the regions of variation of a and the significance of the integer e, it follows that the quantity f0(a + v +1) is distinct from zero, for all values of P > e and for all values of a; hence, as gv+ = -f0(a+ +) {g0ofv+l (a) + g fV (a) +... + gf/ (a + v)} o (a+ V +1) from the equations that define the coefficients g, it follows that < Ifo (a + V + 1)1 gol If+ (l)} + + I If i (a + )1 } 1 +l) t-R- - + 1) + lfo(a~ + v + 1).V0 f-v M (a) + fgl(l R-V+a (a - 1) +}..... + IgVl M (C +.)} 7<+ 1 say, where ry,+ denotes the expression on the right-hand side. Evidently /v+ Ifo (a + v + 1)1 - Y. Ifo(a + P)l R-= g, M (a + v) w 7, M (a + v), and therefore ( M(a+v) 1 fo(a +) |) 7V+i l +l(lf/o(a++l)l R,fo(a 1+ l) | Let a series of quantities Fr be determined by the equation - f M^a~) +1 /o(a+v) } +l Ifo(a+^+l)l + fo (a ++l) ' for values of v'>e; and let Ir=7e-. Then all the quantities F thus determined are positive, and we have I ^+l l-^+lr ^+. Consider the series r'xe + rFe+lx+ +... + FV +...; * T. F., ~ 22. SERIES FOR THE INTEGRAL 83 its radius of convergence is determined * as the reciprocal of Lim + V= IF Now A (0) is the greatest value of the modulus of - 0 (O - 1)... (O - m + 2)pl' -... -PM, within the circle xI = B. As the functions pi'.pm' are holomorphic within the circle, there are finite upper limits to the values of lp,'I,..., lpm'l within the region, say Ml..., Mn; then M (0):; a (a + 1)... (c + m - 2) Ml +t... + M,,:;(a) say, where 0 1= a. Again fo (0) = 0 (O - 1)...(O - m + I) - 0 (O - 1)... (O - m~ + 2) PI (0) -. -PM (0), so that, if ~ (a = a"+ + a$l).( + nt 1 + a-(a + 1)... (u + m - 2) 1pI (O)1 ~... + lp, (O)1, we have Ifo(O> 1 >_ Bnn fo (8) - Om~I - m _ I A (0) - 0ml, and Ifo (0) - 01iQ4 0) the term in Om being absent from f,(0) - Om, and the term in al". being absent from (a). Moreover, as these quantities are required for a limit when v tends to infinity, the quantities a and 0 will be large where they occur; thus 0m is greater than b(a-), which is a polynomial in a only of degree m - 1. Hence I A (0) I >_ am, -(a). Returning now to the expression for IP,+ F, let / denote lal; then so that Ia ~ + 1 m (v + 1 - rn Again, la v lv + l +/3, so that 4(la + V -I ) ~( + +fP) and therefore Ifo(a+- V + 1)l (V + 1 - 18)m - (V(y+ 1 + 18). * Chrystal's Algebra, vol. ii, p. 150. 6-2 84 CONVERGENCE OF THE SERIES [34. Finally, a + v < v + 3, and therefore M (a+V)Y J|a+l (v +/3); so that M (a + ) r (v +/3) i/o0( ( + +1)1 + (v+ 1 - )"- t (v + 1 + 1+)' Now F (a) is a polynomial in a of degree m - 1, as also is b (r); hence, owing to the term (i + 1 -.3)' in the denominator on the right-hand side, we have - M (a + v) Lim M 0v ifo(a + v + 1)= for all values of /3, that is, for all values of a within its regions of variation. Again, as fo (a) is a polynomial in a of degree m, it follows that Lim o (a +) _ 1, =,fo (a + v+ 1) for all the values of a, and therefore Li fo(a +v) = 1 Lim v=io fo(a+v+l) Using these results, we have rV+ 1 Lim - and therefore the series FreX + Fe+lXE+ +... converges within the circle xix = R and for the values of a: consequently also the series Y7eX + e xflX+ +... converges for the same ranges of variation for x and a. The addition of a limited number of terms that are finite does not affect the convergence: and therefore I gvxv v=0 converges, for values of x within the circle Ixl = R, and for values of a within its regions of variation. Let any region for a be defined by the condition a - p r. Then the series converges absolutely within the x-circle of radius R and the a-circle of radius r. Let R'< R, and r'< r; and let K, IC denote any finite positive quantities which may be taken 34.] INDICIAL EQUATION 85 small: then* the series converges uniformly for values of x and a such that 1XI < R' - a-p < r'-K. Thus the series converges uniformly in the vicinity of the x-origin, for all values of a in the regions assigned to that parametric variable. By a theorem due to Weierstrasst, the uniform convergence of the series, which is a power-series in x and a function-series in a, permits it to be differentiated with regard to a; and the derivatives of the series are the derivatives of the function represented by the series within the a-regions considered. SIGNIFICANCE OF THE INDICIAL EQUATION. 35. We now associate the factor xa with the preceding series, and then we have 00 00 g (x, a) = xa S gVxV = 2 gxC+v v=0 v=0 as a series, which converges uniformly within a finite region round the x-origin and can be differentiated with regard to a term by term. (It may happen that the origin must be excluded from the region of continuity of g (x, a), as would be the case if the real part of a were negative; the origin must then be excluded from the region of continuity of the derivatives with regard to a, owing to the presence of terms such as goxalog x.) The function g (x, a) thus determined has been shewn to satisfy the equation Dg (x, a) = xo (a) go (a). As associated with the original differential equation, this result requires the consideration of the algebraical equation (hereafter called the indicial equation) Ao(p)=O of degree n. The preceding analysis indicates that two cases have to be discussed, according as a root does not, or does, belong to a group the members of which differ from one another by * The uniform convergence with regard to x is known, T. F., ~ 14, finis. The uniform convergence with regard to a is established by means of a theorem due to Osgood, Bull. Amer. Math. Soc., t. II (1897), p. 73; see the Note, p. 122, at the end of this chapter. t Ges. Werke, t. ii, p. 208; see T. F., ~~82, 83. 86 INTEGRALS ASSOCIATED WITH [3.5. whole numbers (including a difference by zero, so as to take account of equal roots). Firstly, let p be a simple root of fo (p) = 0, in the sense that it is not equal to any other root and that the difference between p and any other root is not a whole number. Then when we take a = p, all the coefficients l, g2,... in g (x, p) are finite; we have Dg (x, p) = O, that is, w=g(x, p) is an integral of the differential equation: it is associated with the simple root p of the equation fo(p)= 0, and it is a regular integral. 36. Secondly, let Po, pl,..., p, constitute a group of roots of f (p)= 0, differing from one another by whole numbers and from each of the other roots by quantities that are not whole numbers; and let them be arranged so that the real parts of the successive roots decrease: thus the real part of Po is the greatest and that of pn is the least in the group. In order to secure the finiteness of the coefficients g,, g,,..., it now is necessary to take go (a) =fo(a + )fo (a + 2)...fo(a + ) g (a), =f(a) g(a), say, where e > po - p, and g (a) is an arbitrary function of a: and now Dg (, a)- x= g (a) I fo (a + s) = xg (a) F(a), s=0 where F(a)= h tf, (a + s)}. =-0 Further, there may be equalities among the roots in the group: let po, pi, pj, pk,... be the distinct roots taken from the succession in the group as they occur, so that po is a root of multiplicity i, pi of multiplicity j - i, pj of multiplicity k -j, and so on. Then in F(a), there is a factor (a - p)i through its occurrence in fo (a); there is a factor (a - pi), through the occurrence of (a - pi)j-i in Jo (a), and the occurrence of (a - p)i in fo (a + po — pi); there is a factor (a- pj)-, through the occurrence of (a - pj)k- in fo (a), the occurrence of (a - pj)j-i in fo (a + pi- pj), and the occurrence of (a - pj)i in fo (a + po - p). Now 36.] A GROUP OF ROOTS 87 so that, for F(a), p, is a root of multiplicity i, that is, 1 at least: pi is a root of multiplicity j, that is, i + 1 at least; pj is a root of multiplicity k, that is, j + 1 at least; and so on. Hence if pK be a root in the group as arranged, it is a root of F(a) of multiplicity K + 1 at least; and therefore [ ]F (a) = for /= 0, 1,..., K. But Dg (x, a) = xag (a) F(a), and g (x, a) can be differentiated with regard to a; hence l..g - EL a, [* ak] [aD [I gw a1., = La {"ag (a)F(a)} =0, for / = 0, 1,..., K certainly, and for all other integer values of /L less than the multiplicity of p, as a root of F(a)=0. Consequently, the expression W L.ag (x, a)-P _ (x, PK) _ a' ~_-p ap ~t say, for the same values of 1J, provides a set of integrals of the equation. Moreover, each of the distinct roots in the group thus provides a set of integrals; we must therefore enquire how many of the integrals out of this aggregate are linearly independent. 37. We first consider the members of any set; they are furnished by ag (x, a) for a value p assigned to a, and for a number of values of t,, say 0, 1,..., /c. Now g(x, a)= x g^ (a) v; V=0 and therefore aDg (x, a) =xi a gYYxV +a (log x) a V v+... + (log )L gV (a)x z V, L=o a-+ v=aO Ah-O 0 L^^'^^^^^y1^^-^^^^'=O 88 SUCCESSIVE SETS ASSOCIATED WITH [37. where it will be noticed that the coefficient of the highest power of log x on the right-hand side is g (x, a). Hence the set is yo = w, =g (, a), yi = w log x + w1, y = w (log c)2 + 2w, log x + w2, y= - w (log x)K + Kw1 (log X)K-1 + I C (C - 1) W2 (log X)K-2 +. + CK-1 log x + W, where the coefficients wp are independent of logarithms. From the fact that yp contains a power of log x higher than any occurring in yo, yi,..., yp-,, it follows (by the lemma in ~ 27) that no linear relation of the form Coyo + cly +... + CKYK = 0 can subsist among the integrals. 38. Next, we consider the sets in turn, associated with the values po, pi, pj,... of a, as arranged in decreasing order of real parts. The earliest of them is given by a = po: and it contains the i members alg (x, a)1 aL ac j-apo for =, 1,..., z-1. Now fo (a) = (a - po)i (a - p,)j-i (a - pj)k-... (a - pj)l+l-1A, F (c) = (a - po)i (a- pi)j (a- pj)k... (a -p)n+l A2, and therefore f (a) = (Ca p= ) (a - pj)i... ( - p~)A3, where Al, A2, As are quantities which neither vanish nor become infinite for any of the values po, pi..., pi of a. Also go(a) = g (a) f(a), where g (a) is an arbitrary function of a; so that go (a) does not vanish for a = p,; and therefore the various quantities derived 38.] ROOTS IN A GROUP 89 from go (a) for a = Po, including go (a) itself, given by a'g,(a) for 1-k= =0, 1,..., i - 1 do not all vanish. Further [ig (X, a)] F 00 00 00 " =XPO, X"~t(log x) Z, XI'.. + (log XY'-gvxvJ L apo0 vOapbg' =O which is one of the integrals; as the quantities algo atL ---go a.g awk ) awk- apo go do not all vanish, this integral belongs to the index po; and the coefficient of the highest power of log x is g (x, po). The first set thus gives i linearly independent integrals obtained by taking /6= 0, 1,..., i - 1 in the preceding expression. That which arises from,t = 0 is W = g (X, Po) = XPog (po) j 1 + xh, (Po) + X2h, (Po) + X"h. (po) +...1) where all the coefficients are finite: thus it is a constant multiple of XPO ~ xPo~1hi (P) + XPo+2 h2 (PO) +.., an integral that is uniquely determinate. Now consider the second set: it is given by a =pi, and it contains the members aL g (w, a)] for /i=0,..., i -i1, i, i i,.,j - 1. The value ofg(x, a) is ~)00 (X a) = Xa- >' g. (a) XV V==0 Po -- Pi - C =-X( gv(a) xv +xa~PO Pi Y g+ Pi (a) Xv. v=O = As regards the first part of this expression, we note that all the coefficients g, (a) for v = 0, 1,..., Po - pi - 1 contain the factor (a - pi)i; and therefore all the derivatives allr Po Pt xOL )' g, ()icoX p 90 FORM OF THE INTEGRALS [38. for b = 0-,,..., i-1, vanish when a is made equal to pi, while they do not necessarily vanish for higher values of /. As regards the second part of the expression for g (x, a), we write it in the full form xa+Po-i po- P (a) + xa+po- Pi+l gpo+ (a) +..; when a= pi, this becomes xPo gpo- Pi (Pi) + xPo + gpo- pi+ 1 (i)., which accordingly is an integral, and it belongs to the index po, being free from logarithms. But it has been seen that the integral, which belongs to the index po and is free from logarithms, is uniquely determinate, being g (x, po); hence the foregoing integral, being the non-vanishing part of g (x, a) when a is pi, is a constant multiple of g (x, po), say Kg (x, po). It might happen that K= 0. A similar result holds for the derivatives of g(x, a), for the values pu = 1,..., i- 1. Consequently, it follows that the integrals ag (x a)] for u =O, 1,..., - 1, can be compounded from the integrals of the first set; they are i in number, but they provide no integrals additional to those in the first set; and therefore, without limiting the range of their own set, they can be replaced by the i integrals of that set. As for the remainder arising from other values of,L, they are xSp - P Xv+i/(logX) V- lXl+...+(logx) E g (pi) xV L V==0 api pi v=O j= for =, i+ 1,...,j-1. Now go (a) = g (a) (a - pi)i ( - p)j... (a - pi)l A3, so that the quantities rLa So(a)l aal-s J-a ') for the values s =0, 1,..., L in any one integral, and for the values /u = i, i + 1,..., j- 1 in the different integrals, do not all vanish. 38.] IN THE SUCCESSIVE SETS 91 All these integrals therefore belong to the index pi, and they are j - i in number. Moreover, the original set of j integrals, composed of these j - i and of the replaced i integrals, was a set of linearly independent members; and therefore we now have j - i integrals, linearly independent of one another and of the former set of i integrals. Thus our second set provides j - i new integrals, distinct from those of the first set; and each of them belongs to the index pi. The first of them is given by L=i: it is p 0[ gOaip (pi) +i/log x Z i-@ (pi)... ] which p~ X" + i log x I ig(p y=O aii y=O apii-1 which certainly contains terms not involving logx; if j- 1 > i, the second of them is p a (pi) Xv + (i + 1) log XE a (X.. P=O aiib r=O aii which certainly contains terms multiplying the first power of log x; if j - 1 > i + 1, the third of them certainly contains terms involving the second power of log x; and so on. The third set among our integrals is connected with the value a = pj, and it is given by Lag (x, a)l L j Ja=pj for / = O, 1,..., k-1. Now 00 v=O pii-2- 1x O =0 Y=O The coefficients gv (a) contain (a - p)j as factor for all integers v which are less than pi - pj; hence the quantities L ( pi - pj - 1 L xa Y gv(a)x Vj [= P ( a = pj vanish for a = 0, 1,..., j- 1, and are different from zero only for / =j, j + 1,..., k- 1. As in the case of the preceding set, the quantities X[ca { pi-pj - g () Xv =pi Lr aaaT =O Pi 92 FORM OF THE INTEGRALS [38. for u = 0, 1,..., i - 1, are linearly expressible in terms of the i integrals of the first set; while for /, = i, i + 1,...,j - 1, they are linearly expressible in terms of the j-i integrals of the second set, subject to additive linear combinations of the first set. Thus the integrals in the present set which are given by g a (, a)1 for / = 0, 1,..., j-1, provide no integrals linearly independent of the i integrals of the first set and the j -i integrals of the second set; the j integrals in this new aggregate are linearly expressible in terms of those in the old. Now the present set of integrals, for l = 0, 1,..., j- 1, j, j + 1,..., k- 1, are linearly independent of one another; and therefore the integrals for u=j, j+ 1..., k-1 are linearly independent of one another, of the i integrals of the first set, and thej - i integrals of the second set. Thus the third set provides k-j new independent integrals, given by the k-j highest values of vu. The first of them, determined by pu =j, is *[sp aga(pi) xV + j log %- a3-,'g (Pj).V + =o apj _= apvQ - ' which certainly contains terms not involving log x; if - 1 >j, the second of them, determined by 1/ =j + 1, is. [0ilgi)(pi) xV 00 ___)_ xP [L Z j+ +g (j + 1) log E a~j((PJ) +v +. O apjI + vo apj which certainly contains terms multiplying the first power of log x; if k- 1 >j + 1, the third of them certainly contains terms multiplying the second power of log x; and so on. Moreover, it is clear that all these k-j integrals belong to the index pi. The law of the successive sets is now clear. The last of them, determined by a = pi, contains the integrals LS(X, a)1 for s = 1, 1 + 1,..., n, which are linearly independent of one another and of all the integrals of the preceding sets already retained. All these integrals, being n + 1 - 1, in number, belong to the index pi. 38.] GENERAL THEOREM 93 The results thus obtained may be summarised as follows: When the equation fo(p)= has a group of roots po, Pi, Pn, which differ from one another by integers (including zero) and differ from all the other roots by quantities that are not integers; when also the distinct roots are arranged in decreasing succession of real parts, so that po is a root of multiplicity i, pi is a root of nmultiplicity j- i, pj is a root of multiplicity k -j, and so on, where p,, pi,... are distinct from one another and are arranged in decreasing succession of real parts; then, corresponding to that group of roots, there exists a group of n + 1 linearly independent integrals which are regular in the vicinity of the singularity. This group of integrals is composed of a set of i integrals, which are given by ag (X, a)LP for =, 01,..., i - 1, and belong to the index po; of a set of j - i integrals, which are given by g (, a)1 for iv =, i +1,..., j- 1, and belong to the index pi; of a set of k-j integrals, which are given by pang (a, a) L Ds l al-ps for =j_, j + 1,..., k- 1, and belong to the index pj; and so on, the last set being composed of n +1- 1 integrals, which are given by aS ( y, a)for p = 1, 1 + 1,...,, and belong to the index pl. The preceding investigation is in substantial agreement with that which is given by Frobenius*. A different proof is given by Fuchst: briefly stated, it amounts to the establishment of an integral w0 belonging to the index Po, to the transformation of the equation of order m by the substitution w = Wo J vdx * Crelle, t. LXXVI (1873), pp. 214-224. t Crelle, t. LXVI (1866), pp. 148-154; ib., t. LXVIII (1868), pp. 361-367. 94 INDICIAL EQ UATION [38. into a linear equation in v of order m- 1, and to the discussion of this new equation in a manner similar to that in which the equation of order m is discussed. Expositions of the method devised by Fuchs will also be found in memoirs by Tannery* and Fabryt. 39. All the integrals of the differential equation, which has the specified form in the vicinity of the singularity, are regular in that vicinity; their particular characteristics are governed by the roots of the equation fo (p) = 0, that is, p (p -1)...(p-m + l)- p (p - )...(p - m + 2)p,() -...- (O) =0, the differential equation in the vicinity of the singularity being of the form dnw 1m2 dm-rw dxm- _ Xm-pr (X) d, --- = 0. dxm -.= 1 -This algebraic equation is of degree m, equal to the order of the differential equation; it is called+ the indicial equation of the singularity, and the function f(x, p), of which fo(p) is the term independent of x, is called the indicial function. From the form of the integrals which belong to the roots p of the indicial equation of a singularity, and those which belong to the roots 0 of the (~ 13) fundamental equation of the same singularity, it is clear that the roots of the two equations can be associated in pairs such that 0 = e2rip. When the roots of the indicial equation are such that no two of them differ by an integer, the roots of the fundamental equation are different from one another; there is a system of m regular integrals, and the m members belong to the m different values of p. When the indicial equation possesses a group of n roots which differ from one another by integers (including zero), the corresponding root of the fundamental equation is of multiplicity n: there is a corresponding group of n regular integrals, the expressions of the members of which in the vicinity of the singularity may (but do not necessarily) involve integer powers of log x. When a root of the indicial equation occurs in multiplicity c, * Ann. de l'Ec. Norm., 2e Ser. t. iv (1875), pp. 113-182. t ThAse, Facult6 des Sciences, Paris (1885). + Cayley, Coll. Math. Papers, vol. xIi, p. 398. The names adopted by Fuchs are determinirende Fundamentalgleichung, and determinirende Function, respectively. LINEAR INDEPENDENCE OF THE INTEGRALS 95 so that the corresponding root of the fundamental equation occurs in at least multiplicity K, there is a set of K associated integrals, the! expressions of all but one of which certainly involve integer po-wers of log x. 40. Having now obtained the form of integral or integrals associated with a root of the indicial equation fo (p)= 0, we must shew that the aggregate of the integrals obtained in association with all the roots constitutes a fundamental system. First, suppose that the roots of the indicial equation are such that no two of them differ by an integer; denoting them by pI, P2,.**, Po, and the m integrals associated with these roots respectively by w,,..., wi, we have Ws = (z - a)Ps P0 (z - a), where P (z - a) is a holomorphic function that does not vanish when z = a. No homogeneous linear relation can exist among these integrals: for, otherwise, we should have some equation of the kind C1W1 + C2w2 +... + CmWm = 0. Writing 0 = e2iPs, (s=1, 2,..., In), so that no two of the quantities 01,..., 0m are equal to one another, we can, as in ~ 18, deduce the equation cl10wl1 + C202"w2 +... + CmmUWm = 0, for any number of integer values of r, from the above equation, by making z describe r times a simple contour round a. Taking the latter equation for = 1,..., - 1, the set of m equations can exist with values of Ci,..., Cm differing from simultaneous zeros, only if 1,1,..., 1 0, 01, 02,., 0m.............................. - M-1, 0. m-1 f m-I which cannot hold as no two of the quantities 0 are equal. Hence we must have cl= 0 = c =... =cm, and no homogeneous linear relation exists: the system of integrals is a fundamental system. 96 LINEAR INDEPENDENCE OF [40. Next, suppose that the roots of the indicial equation can be arranged in sets, such that the members contained in each set differ from one another by integers. With each such set of roots a group of integrals is associated, the number of integrals in the group being the same as the number of roots in the set. It is impossible that any homogeneous linear relation among the members of a group can exist: if it could, it would have the form blwl +... + bwn = 0. If wi,..., wn involve logarithms, then (~ 27) the aggregate coefficient of the highest power of log (z - a) must vanish; in the case of each integral in which the logarithm occurs, this coefficient (~ 25) is itself an integral of the equation, and therefore we should have a relation of the form b,.w +... + bsws = 0, where the quantities wr,..., w8 belong to different indices, say Pr,..., ps, no two of which are the same; and wr,..., w8 are free from logarithms. Dividing by (z - a)P, we should have an equation of the form b,(z- a)P -P(P, (z- a) +... + bPs (z - a)= 0, where P,,..., Ps are holomorphic functions of z - a, not vanishing when z = a. No one of the indices pr - Ps is zero: no two are the same: and so the preceding equation can be satisfied identically, only if br=... = b,. We therefore remove the corresponding terms from biwl +... + bnwm = 0, and proceed as before: we ultimately obtain zero as the only possible value of each of the coefficients b. If Wi,..., Wm do not involve logarithms, the argument, above applied to Wr,..., w, can be repeated: there is no linear relation. The initial statement is thus established. If the tale of the groups, the members of each of which are linearly independent among themselves, is not made up of linearly independent integrals, then an equation of the form ClWl +... + CmWm = 0 THE INTEGRALS 97 exists. Equating to zero (~ 27) the aggregate coefficient of the highest power of log z that occurs, we have, as above, a relation of the form CrWr -W2... + CsWs + pWp -... 4- CqWq +... = 0, where wr,..., w, belong to one group, wp,..., Wq belong to another group, and so on. Writing CrWr -+... - CsWs = W1, CpWp +-... + CqWq = W2,... we have w, + w2+... =. Now let 0i, = e2'iP, be the factor which, after description of a loop round a, should be associated with W1; let 02 be the corresponding factor for W2; and so on: the quantities 01, 02,... being unequal to one another, because W1, W2,... belong to different groups. Then, as in ~ 18, we deduce the equation 01 4 W1 + 02 +... = 0, after X descriptions of the loop; and this would hold for all integer values of X. As before, taking a sufficient number of these equations for successive values of X, we infer that W,=O, W,=o0,...; if these are not evanescent, they would imply relations among the members of a group, and so they can be satisfied only if cry == O=... = cq, p = O....Cq Remove therefore the corresponding terms from the relation ClWx -... *- CmWrn = 0, and proceed as before: we ultimately obtain zero as the only possible value of each of the coefficients c. Hence no homogeneous linear relation exists: the system is fundamental. Some examples illustrating the preceding method of obtaining the integrals of a linear equation will now be given. Ex. 1. Consider the integrals of the equation* D (w) =X (2 - 2) w" - (2 + 4 + 2) {(1 -x) w' +w} = in the vicinity of the origin. To obtain a regular integral, we take W n=o C; n=O * The equation is not in the exact form indicated in the text. We have m=2, P2 (0) = 0, and so a factor x has been removed; also we have multiplied by the factor 2-x2. F. IV. 7 98 EXAMPLES OF [40. substituting, we have xDw= 2a (a- 2) coxa, provided O = c (a2 - 1)-CO (a+ 1), = 2c2a (a+ 2)- 2cl (a+ 2) - o (a - 2)2, and 2 (2+a+ 1) {(n+-a- 1) c+ -c}- =(n +a-3) {(+ a- 3) Cl-Cn-2}, the last holding for n = 2, 3,.... The indicial equation is a (a- 2) = 0, giving (simple) roots a= 2, a =0, so that a factor a + can be neglected: the relations among the coefficients are equivalent to (a +2n- 1) 2n+ 1 - C2n=0, c0(a- 2)2 a (a+2n) c2+2-c+2+l= 2 +1 (a+2n)(a+2n+2)' Firstly, consider the root a = 2. We have e1 = C0, 2c2=cl, 32C2-C1, ~3C3=-C2X,... so that the integral belonging to the index 2 is Cox2(l++ +-3! +...); say the integral is u, where U = X2ex. Secondly, consider the root a= 0. From the original form of the relations, we have 1== - C, by the first relation: and the second relation is then identically satisfied, leaving c2 arbitrary. Using the reduced form of the relations for the higher coefficients, we have 03 ~ C2, c -3 2c4c==c3, 3c5 c4, 4 and therefore the integral belonging to the index 0 is I X+ x2 x3 Co(1-x)+c2x2 l +x+ +3!+.... On subtracting c2u from this integral, the remainder is still an integral, and it belongs to the index 0 in the form (say) V, =1-x. Thus the system of two integrals, regular in the vicinity of x=0, is % =xx; v, =2-x. 40.] THE GENERAL THEORY 99 This method of dealing with the root a= 0 is not quite in accord with the course of the general theory; it happens to be successful because c2 is left arbitrary. In order to follow the general theory, we note that the coefficient of c2 in the original difference-equation contains a factor a which vanishes for the present root. Hence, taking C0 = Ca, we find Ca 1a-1 Ca (a2-5a + 10), a +1'+a-2 C3a+'1 (a+2) 4=C3+a2A, where A is finite, and so on; thus + ')+c ~ 1+ X X2..}~a 2R (x, a), at ( +a )+ 2 { a a+l (a+l) (a +2)+a }+V(z n+) where R (x, a) is a holomorphic function of x which, by the general theory, is finite when a=O. According to the general theory, this quantity should give rise to two integrals, viz. [W]a-=' Ldaao Taking account of the value of c2, the first of them is zero, thus giving an evanescent integral. The second is (1 - x) - Cx2ex, or adding to this integral ~Cu, which is an integral, we have C(1 -x), thus giving 1- x as the integral. Ex. 2. Discuss in a similar manner the regular integrals of the equation xw" + w' - w = 0 in the vicinity of the origin: likewise those of the equation x (1-x) iw"-(1 +4x+2x2) w'+(3 +3x-x2)w=O in the same vicinity. Ex. 3. Consider the integrals of the equation Dwi = x2 (1 + x) w"- (1 +r 2x) (xw'-w) =O in the vicinity of the origin. Substituting the expression a0 Xa I en Xn, n=O we have Dw = (a- 1)2 COXa, 7-2 100 BESSEL'S [40. provided (a+ n- 1)2 = - (a+- 2) (a +n-3) C-^, for n= 1, 2, 3,...; these values give =^cx {i _(a - 1) (a - 2) +(a-1)2 Y, where Yis a holomorphic function of x which is finite when a= 1. The indicial equation has a repeated root a= 1; hence two regular integrals are rdw] The former is cox, say the integral is u, where u=x; the latter is CoX log x +Cox2, say the integral is v, where v=x log x+ 2. Both integrals belong to the index 1; and one of them must contain a logarithm, since the index is a repeated root of the indicial equation. Ex. 4. Consider Bessel's equation for functions of order zero, viz. Dw2 = xw"t + w' + xw = 0. Substituting w=c OXa+ cl a+l +... + Cpa+P+..., we have xDw = co a2X-, provided C1=0, (a+p+ 1)2 p + +cp-= 0, the latter holding for p = 1, 2, 3,.... When these relations are solved, the value of w is W=oXa{-(a + 2)2+ (a+2)2 (a+ 4)2 -The indicial equation is a2=0, so that a=0 is a repeated root; thus the integrals of the equation, both of them belonging to the index zero, are dw] MaJ,=05 Ld~_laoThe first of them is eo 1 -..X2 X4 +22 42 *2}2 in effect, J0 (x), on making co= 1. The second is + c {22 - 2242 (1+2)+ 22.42.2 6(1 +2+)- *} coxj2 x4 94~ 40.] EQUATIONS 101 Denoting this by K0 when c0=, we have KO=Jologx+ 1-({) (2 (S)' where, (p) denotes the value of - {log II (z)} when z=p. The two integrals, regular in the vicinity of x=0, are J0 and Ko. Ex. 5. Consider next Bessel's equation for functions of order n, viz. Dw = X2W"t w + (X2-n 2) W =0. Substituting an expression wo=coXx+clxa+l+... +Cpax+p+... in the equation, we have Dw = C (a2 - n2) Xa, provided C1 {(a + 1)2- n2} =, Cp {(a+p)2 -2}+ Cp-2=0, for p=1, 2, 3,...; we thus have 'v2 xi co 1-[ (a+2)2 n2 {(a+2)2- n2} {(a +4)2 _n2} * The roots of the indicial equation are a= + n, a= -n. When n is not an integer, the corresponding integrals are seen to be effectively J., J_. When n is zero, we have a repeated root; this case has been discussed in the preceding example (Ex. 4). When n is an integer different from zero, the two roots belong to a group; and for a= -n, the coefficient of x2n is formally infinite, so that we have an illustration of the general theory in ~~ 36-38. We take the roots in order. Firstly, let a= +n: then the integral is 7o /\2p 1 X=n ((p) p(n +p) on taking c0 equal to 2i. This is the function usually denoted by J,; 2 n (I7n) and it belongs to the index n. Secondly, when a= - n, one of the coefficients becomes formally infinite through the occurrence of a denominator factor (a + 2n)2- n2. Accordingly, we write n -1 o = C (a+2n)2-n2}, (- 1)n C= E II {(a+ 2r)2-n2}; r=l 102 BESSEL'S EQUATION [40. and then 2 x2n -2 W=C{(a+2A)2-n2} | 1X ( +2) 2+... +(-1l) n-1i _1 (a n {(a- 2r)2 - n2} r= —1l -r~?r2 y4 _ +Exa+2n 1 2. 24. (a +2n +2)2 -2 2 {(a + 2n + 2)2 - n2} {(a + 2n + 4)2 -_ n2} = 1 + W2, say: and now D = C(a2-2) {(a - + 2)2 - 2} xa. Two integrals arise through this root, viz. [W1+w 2 - a awia Ja= - ] For the first of them, we have E [11]a-=O -0 [w2]aL-n n I (); so that it provides no new integral. For the second of them, we have raFv~ 1 -2Cn n-l (X2P I (n-1-p)L~aa ]~=_~ —~~n(~-l)~_-0 I - W, L aa a=-n Xn I (n -1) p=O \2 n I( 1 -say; and 8w r x2 x^- i4 an -= 7'(logX)l X2 - +4-. aw2 -Ex(logLx) 1-22(n+1) 24(n+ 1) (n +2) 1. 2 + x ( )r —l 11(n) /X\2 + EXn S ) ()+ (r+ (lb + r)- ()(2) - W2, say: so that the integral is TW + T2. In W2, the part represented by ~ELn I (_l) I () ) ()) 2r -=0 (r) n (n + r) is a constant multiple of Jn and therefore can be omitted, owing to the earlier retention of Jn. Rejecting this part, and taking C= - - 2n - (z- 1), so that 1 1 2f-1 n (n)' the integral becomes _ ()n n-II (n-p-1) ()2p:o nr (IJp)> + 1I (r)11(n+r) {2 log x-+ (r) - i (n+r)} d,=o n (-) II (n + ) which differs, only by a constant multiple of J,, from the expression given by Hankel*. * Math. Ann.,.t. I (1869), pp. 469-471, quoted in my Treatise on Differential Equations, p. 167. 40.] EXAMPLES 103 Ex. 6. Discuss in a similar manner the integrals of the equation x(1 - x)w"+{1 - (a+b+ )x) x ' -abw=O in the vicinities of x=0, and x= 1: indicating the form for the latter vicinity when a+b=1. This equation is the differential equation of the hypergeometric series F(a, b, 1, x). When, in Legendre's equation _w_ dw (1 - Z2) d - z - (p + ) = 0, the independent variable is transformed to x, where z=1 - 2x, it becomes x (1 - x)w" +( - 2x) w'+p (p+1) w=0, which is the special case of the above given by b=p 41, a= -p. The integrals of Legendre's equation in the vicinity of x=0 and of x= 1, that is, in the vicinity of z=l and of z=-1, can be deduced from those of the hypergeometric equation; the actual deduction is left as an exercise. Ex. 7. Apply the general theory to obtain the integrals of X3Wl'- 3x2w" + 7x' - 8w = 0, which are regular in the vicinity of x=0. Ex. 8. Consider in the same way the equation D (w) = (1 + X) 3W"'- (2 + 4x) X2w"+ (4 + 10) w'- (4 + 12x) w=0. Substituting for w the expression?v= Cxa + cla +1 +... + CX+ + c +... as in the earlier examples, we have Dwp= =C (a- I) (a-2)2 xa provided c, (nt +a- 1) (n +a - 2)2 + n_ (n + a - 3)2 (n+ a- 4) =0, for n=1, 2, 3,.... The roots of the indicial equation are 2, repeated, and 1, so that they form a group the members of which differ by integers. Moreover, when a=l, the coefficient c1, which is (a- 2)2 (a-3) -c a.(a-1)2 is formally infinite; for that root, we shall take Co=C (a- 1)2. Firstly, for the repeated root a =2, we have wz = CoX{ 1 +(a- 2)2 R (x, a)}, where R is a holomorphic function of x which remains finite when a= 2. The two integrals are [aw2 [lW [W]~:'z'LaV ]a=2 104 EXAMPLES OF [4~0. it is easy to see that they are constant multiples of 'U,=X ) 'U2=X:- locr both of which belong to the index 2. Secondly, for the root a = 1, we take co = C (a - 1)2, and then D (w) = C (a - 1)3(a- 2)2 Xa, where w= C(a -1)2 Xa - C (a2)2(a-3)a a (a-1)2(a-2)3(a-3) (a - 1)3 (, a), a3 (a+ 1) where Q (x, a) is a holomorphic function of x which remains finite when a=1. In connection with this expression, three integrals are derivable, viz. [W],, FLw1;I, L )a La21 The first of these is C 2X, which is 2 Cu: it is not a new integral. The second is 2Cx2 log x - 7 CX2, which is 2Cu2 - Wu17t: it is not a new integral. The third is 2 Cx + 2Cx2 (log x)2 - 14 CX2 log x + 22Cx2+ 2CX3; adding to it 14CU2 -22Cul, the new expression is still an integral and is a constant multiple of u3, where,3 =X+x3 + X2 (log x)2, which manifestly belongs to the index 1. Ex. 9. Obtain the integrals of the equations (i) (1 ~X2)x3w"'-(2 +4X2) X2W"+(4~10x2)xw'-(4+12X2) w=0; (ii) (1 + 4x) X4W"" -(4 + 20x) x3w"' +(14 + 72x) X2w" - (32 + 168x) xw'+ (36 + 192x) w =0; which are regular in the vicinity of the origin. Ex. 10. Consider the integrals of Dw = xw"' + (a, + b,x +...)w" + (a2 + b2x +...) w'+ (a3 + b3 +...)w = 0 in the vicinity of x=0, the constant a, not being an integer. To obtain the regular integrals, we substitute W=xa 2 CXn, n=O so that x2Dw-= a (a - 1) (a- 2 +a,) cox, provided (n + a) (n + a - 1) (n + a - 2 + a,) c, =f0CA 1 - I+fAC, - 2 +...5 for values of n greater than zero, no one of the quantities fo, fl,... being of degree in n greater than 2. 40.] THE GENERAL THEORY 105 The roots of the indicial equation are a=0, 1, 2-a1. For a=2- a, the difference-equation determines coefficients c,, which lead to a series converging for values of jxi within the common region of convergence of the coefficients of w", w', w. For a=0 or 1, the difference-equation holds for values of n greater than 2 or 1 respectively; the only other conditions are al 2c2 + a2. cl + a3c0 = 0 for a=0, and a1. 2c1 + a2. C2=0 for a==1. Then the difference-equation again determines coefficients which in each case lead to a series that converges within the same region as the series that belongs to the exponent 2-al. Each of the latter integrals is a holormorphic function of x; and therefore the three integrals of the equation, which are regular in the vicinity of x=0, are:-one, a holomorphic function of x belonging to the index 0; a second, likewise a holororphic function of x belonging to the index 1; and a third, belonging to the index 2- al. Ex. 11. Discuss the regular integrals of the equation in the preceding example, when a1 is an integer. Ex. 12. Prove that the equation dnw dn- w x dn +(al +bl +...) d n+... + (a + bx +...) w=-o has m -1 integrals which are holomorphic functions of x in the vicinity of x=0, when a1 is not an integer, the various coefficients a,.+brx+... in the differential equation being holomorphic in that vicinity; and discuss the regular integrals when a, is an integer. (Poincare.) Ex. 13. Shew that the series F (a, p, o-, t, )=+l - X + a a+1) + par 2! p (p+l) (o-r+l)r(r+l) satisfies the equation xd ddv d2y d3 x2 a dy = (X- P ar) 4- + ay; and obtain the other integrals, regular in the vicinity of x =0. Verify that, when a=-r, the form of the function F, say G (p, a, x), satisfies the equation of the third order 2d3__ d2G dG 2 d +(P++ 1) x d-+ p-do - G=O; dx^ " ' dx" * dx and indicate the relation between the two differential equations. (Pochhammer.) 106 REGULAR INTEGRALS [41. REGULAR INTEGRALS, FREE FROM LOGARITHMS. 41. Alike in the general investigation and in the particular examples, it has appeared that the regular integrals are sometimes affected with logarithms, sometimes free from them. Thus if no two of the roots of the indicial equation differ by a whole number, each one of the integrals in the vicinity of the singularity is certainly free from logarithms; if a root of the indicial equation is a repeated root of multiplicity n, then the first n - 1 powers of log x certainly appear in the group of n integrals which belong to that root. When a root of the indicial equation, though not a repeated root, belongs to a group the members of which differ from one another by whole numbers, the integral belonging to the root may or may not involve logarithms: we proceed to find the conditions which will secure that every integral belonging to that root is free from logarithms. Let the group of roots be denoted by po, p,,..., pP,,..., arranged in descending order of real parts, so that p, - p, for K = 0, 1,... L - 1, is in each case a positive integer: and consider the root pp, in order to obtain the conditions under which every integral belonging to pP shall be free from logarithms. In the first place, p, must be a simple root of the indicial equation. Assuming this to be the case, we know that the integral belonging to pt is pLg (s(, a) 1 L Da a =P in the notation of ~ 38. If we further admit the legitimate possibility that, to this expression, we may add constant linear multiples of the integrals which belong to the earlier roots Po, Pi,..., Pp-I and still have an integral belonging to the root p, then, in order to secure that every integral belonging to p, shall be free from logarithms, the integrals belonging to the earlier roots must also be free from logarithms; hence, as further conditions, each of the roots p0, pi,..., p-,, of the indicial equation must be simple. These conditions also will be assumed to be satisfied. The full expression for the integral belonging to p, is the value, when a = p,, of the expression xba [ Ea Xv + P, (log ) o SV + *, * + (log )P YOgv X;' Lv=o aag P=II ' '\-u3 O aaV=0- ' o FREE FROM LOGARITHMS 107 in order to be free from logarithms, the quantities L ac ja=p for - = 0, 1,..., a - 1, and for all values v = 0, 1,... ad inf., must vanish: and if these conditions be satisfied, the above expression will acquire the desired form. The conditions will be satisfied for every value of a, if gv (a) contains (a-pt) as a factor. But (p. 81) g(h) h (a) go (a) f/0 (a + 1) 0 (a +2)... f (a+) = ) say; and go (a), which (~ 36) is equal to g (a)f(a), contains (a - p) as a factor on account of its occurrence in f (a); hence it is sufficient that Hv(a) should remain finite (that is, not become infinite) when a =pp, for all values of v. Moreover, H0 (a)= 1. Having regard to the equation by which g,(a) is determined, we obtain the relation H,f, (a + v)+Hf (a + - 1) +...... + Hf1_ (a + 1) + Hof(a)=0. All the quantities f, (a + v - 1),..., f (a) are finite for values of a that are considered; hence HJfo(a + v) is finite if Ho (= 1), H1,..., H,_, are finite, and therefore, on the same hypothesis, H, will be finite for all values of v, if it remains finite for those values of the positive integer v, which make p, + v a root of the indicial equation f(0)= 0. These values are known; in ascending order of magnitude, they are p-I1- Pt, P/-2- P, -'" po- P tOConsider them in ascending order. We have H,,, = ___________ h,, (a) v /o(a + 1)o (a + 2)...o (a+ )' When v = pL-, - p, a single factor fo(a + v) in the denominator vanishes when a = p,; and it vanishes to the first order, because p,_- is a simple root of the indicial equation. Hence, in order that H, may be finite for this value of v when a = p,, it is necessary that h, (p/) = O, when v = p_- - p,; and it is sufficient that h, (p,) should vanish to the first order. 108 REGULAR INTEGRALS [41. When v = p_-2 - p,, two factors fo (a + v - p,_ + pe1), fo (a + v) in the denominator vanish when a = p,; and each of them vanishes to the first order, because p,_- and p-.2 are simple roots of the indicial equation. Hence, in order that H, may be finite for this value of v when a = p,, it is necessary and sufficient that h, (a) should vanish to the second order when a =p,: the analytical conditions are that h (a)=0, =0, when v= p,-2- p and a = p. When v = p-_3 - p, then the three factors fo(a + - -3 + p-I), f(a + v - /-3 + P-2), fo (a + ) in the denominator vanish when a = p,; and each of them vanishes to the first order, because p,-i, pt-2, p-s3 are simple roots of the indicial equation. Hence, in order that H, may be finite for this value of v when a = p,, it is necessary and sufficient that h^(a) should vanish to the third order when a= p,; the analytical conditions are that ah,(a)- (a, ah,(a)= h, ()) = 0, 0, when v = p_3 - p and a = h. Proceeding in this way, we obtain the conditions for the successive values of v that need to be considered: the last set is that =0, (==, 1...,-), when v = p0- p, and a = p,. Such is the aggregate of conditions for a = p,. We have seen that, in order to secure the freedom from logarithms of every integral belonging to p,, every preceding integral in the set as arranged must similarly be free: and so we have, in addition, all the similar conditions for p,-, p,-2, **. PI, there being no condition for the simple root po. When all these conditions are satisfied, every integral belonging to p, is free from logarithms. Manifestly these conditions also secure that every integral belonging to the roots p,-i,, p/-2, *..., p of the indicial equation 41.] FREE FROM LOGARITHMS 109 is free from logarithms: (one integral, belonging to po, is always unconditionally free from logarithms): it being assumed that each of the roots po, pi,..., p, is a simple root of the indicial equation. The conditions thus secure that, when each of the /u + 1 greatest roots in the group of roots of the indicial equation is simple, the u. + I integrals belonging to those roots respectively are free from logarithms. The preceding investigation is based upon the results obtained by Frobenius, Crelle, t. LXXVI (1873), pp. 224-226. A different investigation is given by Fuchs, Crelle, t. LxvIII (1868), pp. 361-367, 373-378; see also Tannery, Ann. de l'Ec. Norm., t. iv (1875), pp. 167-170. Ex. 1. A simple illustration arises in Ex. 1, ~ 40, for the equation x (2 -2)w"- (x2+4x +2) {(1 - )w'+w}= 0. With the notation of the text, we have fo(a)= a (a-2), so that Po=2, /,=1, pi=O: we thus have to consider hv (a) for a p = O, v =po - P= 2. But A2 (a) go (a) \2 (a)-fo (a+ 1)/fo (a+ 2)' so that, as 4-a g2 (a)= a + a-2 (a), we have 4-a h2 (a) = 4 a - fo (a + 1)fo (a+ 2) =(4- a)(a +1)(a+2)a. The (one) condition in the present case is that h2 (a) = 0, when a=O: which manifestly is satisfied. Ex. 2. If the roots of the indicial equation are different from one another, then the integrals which belong to them certainly possess terms free from logarithms. (Fuchs.) Ex. 3. Let po, Pi,..., p, be the roots of the indicial equation which form a group, the members differing by integers and no two being equal; and assume them ranged in descending order of real parts. Denote Po-Pn by s-1; and form the equation satisfied by dW- (wx P) W=3(x- p~) 110 EXISTENCE OF AN INTEGRAL [41. then according as the indicial equation for the singularity x=0 of the equation in W has no negative roots or has negative roots which are integers, the integrals of the original equation in w are free from logarithms or are affected by logarithms. (Fuchs.) Ex. 4. Shew that the integrals of the equations 2 (i) iV" +qw'- W=0, (ii) "- (q2+2 W=0, (iii) " + (q - 20) w'+ (2 - q - w=O, where q and 0 are constants, are free from logarithms. Ex. 5. Discuss the integrals of the equation 2x2 (2 - x) w"- x (4 -x)o '+(3 - x) w=O in the vicinity of the origin. [They are x5, (x - ~x2)t.] 42. If, instead of requiring (as in ~ 41) that every integral belonging to an exponent p, shall be free from logarithms, when p, is one of a group of roots of the indicial equation of the type indicated in ~ 36, we consider the possibility that there shall be some one integral free from logarithms, belonging to the exponent and belonging to no earlier exponent in the group as arranged, no such large aggregate of conditions is needed as for the earlier requirement. Thus it is no longer necessary to specify that po,..., p,- shall be simple roots of the indicial equation; nor is it necessary to specify that, even if these roots are simple, the integrals associated with them are of the required form. The conditions that arise will be particularly associated with a = p; but they will be affected by modifications arising out of the possible multiplicity of po,..., p,/- as roots of the indicial equation. The detailed results are complicated: a mode of obtaining them will be sufficiently indicated by an investigation of the conditions needed to secure that some integral free from logarithms exists belonging to pi and not to po, with the notation of ~ 36-38. Suppose that po is a root of the indicial equation of multiplicity i; and let yl,..., yi denote the set of integrals associated with po, where the expression of ys+,, for s = 0, 1,..., i -- 1, is given by ys+ - ' (X, ~ ~ a cS:,Po ' FREE FROM LOGARITHMS 111 If pi is a root of the indicial equation of multiplicity j- i, only the first of the set of associated j- i integrals can be free from logarithms: even that this may be the case, conditions will be required. Denoting that first integral by W, we have W = ag (, ) Now W certainly belongs to the exponent pi. Its expression, in general, involves logarithms; but there is a possibility of obtaining a modification of its expression, so as to free it from logarithms, if we associate with W a linear combination of y,,..., yj with constant coefficients; and the modified integral will still belong to pi but not to po. Accordingly, consider the combination U= W- AtytA t=2 where the constant coefficients A are at our disposal; this gives a= -a)- i - at-g(x, a)i oo i ( I At i-n i- - a=o t=2 aa =p =0On=O n )! ( log ap - xPo E A At+ ( (log X)P x ~ E t - ( at-Pgv Zvf t=lv=o0o 0 p jWhat we require are the conditions that may, if possible, secure that no logarithms occur in this expression for U. The least aggregate of conditions that will secure this result is: first, gv (pi) = 0, for all values of v, which secures the disappearance of (logx)i; next, i zgn = Ai gm, (po), api for all values of m and n such that pi + n = po + m, as well as -ag o gP= o, api for p = 0, 1, Po - pi- 1, these conditions securing the disappearance of (log x)i-; next, i 1) 2n Ai-, gm(po) + (i - 1) Ai apg 2 apf ap0 112 CONDITIONS THAT AN INTEGRAL BE [42. for all values of m and n such that pi + n = po + m, as well as a - 0, for p = 0, 1,..., p- pi- 1, these conditions securing the disappearance of (log x)i-2; next, i(i- 1) (i- 2) 3n =- Ai2,gm(po) + (i- 2) Ai-l + 2 8p02 (+ -l)(i - 2) Ai gm for all values of mn and n such that pi + n = po + m, as well as 3g, apiz for p = 0, 1,..., o - pi - 1; and so on. This aggregate is both necessary and sufficient. Manifestly any attempt to reduce it to conditions independent of the constants A would be exceedingly laborious, even if possible. The difficulty arises in even greater measure when we deal with the conditions that some integral belonging to p, where / > i, and to no earlier index, should exist free from logarithms. 43. If we assume zero values for all the constants A2,..., Ai in the preceding investigation, the surviving conditions are certainly sufficient to secure the result that the integral exists, free from logarithms and belonging to its proper exponent: but the conditions cannot be declared necessary. The aggregate of this set of sufficient conditions is, in the case of pi, that the equation a — P0 L a ]api shall hold for = 0, 1,..., i- 1, and for all values of n. As in ~ 41, it can be proved that all these conditions will be satisfied if the equation hp0 (a) = 0 has a simple root equal to pi. Assuming this to be the case, then an integral exists in the form v=O L jai P x' V=o L^ aai a=P, 43.] FREE FROM LOGARITHMS 113 which is free from logarithms and belongs to pi (but not to po) as its proper exponent. If pi is a multiple root of the indicial equation, the remaining integrals belonging to pi as their proper exponent are certainly affected with logarithms. Corresponding conditions, that are sufficient (but are more than can be declared necessary) to secure the existence of an integral, free from logarithms and belonging to an exponent p, (but to no earlier exponent in its group), can similarly be found; they are inferred from the investigation in ~ 41. If the equation hn (a)= 0, when n = p,_,- p, has a simple root equal to p,; if the same equation, when n = p_-, - p, has a double root equal to p/; if the same equation, when n = p,-3 - p/, has a triple root equal to p,; and so on, up to the case of n = po - p, when the equation must have a root equal to p, of multiplicity p: then an integral exists, belonging to pa as its proper exponent (and not to any of the exponents po, p,..., p,-i), and free from logarithms. If p, is a multiple root of the indicial equation, the remaining integrals belonging to pa as their proper exponent are certainly affected with logarithms. On the preceding basis, the identification of the integrals, belonging to the group of exponents, with the sub-groups as arranged by Hamburger (~ 23, 24) can be effected. The aggregate of integrals in the group, which are free from logarithms and belong to their proper exponents, not merely indicate the number of sub-groups in Hamburger's arrangement but constitute the respective first members in the respective sub-groups. The general functional forms of the remaining integrals belonging to any exponent are (save as to a power of a factor 27ri) similar to those which occur in Jirgens' form of the integrals in a subgroup. 44. In the practical determination of the integrals of specified equations, it sometimest is convenient to begin with that root * In this connection, the following memoirs may be consulted: Jirgens, Crelle, t. LxxX (1875), pp. 150-168; Schlesinger, Crelle, t. cxiv (1895), pp. 159 -169, 309-311. t As to this process, see the remarks by Cayley, Coll. Math. Papers, t. vIII, pp. 458-462. F. IV. 8 114 CAYLEY'S [44. among the group of roots which has the smallest real part, instead of beginning with the root that has the largest real part, as in ~ 36. When the process about to be discussed is effective, it has the advantage of indicating at once the number of integrals associated with the group which are free from logarithms; but it is not always effective for this purpose, and it does not determine the integrals that are affected with logarithms. The equations determining the successive coefficients gi, 2,... in the expression 00 E gyVXa+v v=0 in the method of' Frobenius are (~ 33) 0 = gfo(a + n) + n-f, (a + n -1) +... + go.n(a), for n = 1, 2,.... Let a group of roots of the indicial equation f (a) = 0, differing from one another by integers, be denoted by po, pl,..., a, where ao is the root of the group with the smallest real part; and replace a by ra in the foregoing typical equation for the g's. Then, whenever o- + n is equal to another root of the group, the equation in its given form ceases to determine gn, as a unique finite quantity. It may happen that the equation is satisfied identically; in that case gn is arbitrary, as well as go. It may happen that the equation appears to determine gn as an infinite quantity: in that case, we modify go as in ~ 36, and gn is determinate after the modification. As often as the former case arises, we have a new arbitrary coefficient; if Kc be the number of these coefficients left arbitrary, then Kc is the number of different integrals, associated with the group of roots and free from logarithms. These integrals themselves are the quantities multiplying the arbitrary coefficients in the expression 00 E g X+V v=0 Ex. 1. As an example in which the process, of dealing first with the root of a group that has the smallest real part, is effective as indicating the 44.] METHOD 115 number of integrals free from logarithms, consider the equation d4W d3w d~w (4 + 5) dw -(7z3 + 124 + 4Z5) d3W + (29z2 + 57z3 + 30z4 + 6z5) d2 (z 4 + dZ' z~d dz2 - (74z+ 1542 +93z3+ 28z4+ 4z5) d (90+ 194z+ 125z2+43z3+ 94++5) w=0. The indicial equation is easily found to be (p -2) (p 3)2(p-5) =0, so that there certainly will be an integral belonging to the exponent 5, free from logarithms; there may be a similar integral belonging to the exponent 3, and there will certainly be an integral, belonging to that exponent and affected with logarithms; and there may be an integral belonging to the exponent 2, free from logarithms. Accordingly, take the value p=2, and substitute w = CO z2 +c 123 +c2z4 + 3z5... in the equation; for the immediate purpose, we need not consider powers higher than z5 in w, because p = 5 is the root of the indicial equation with the highest real part. The equations for determining the successive coefficients are 0=co. 0, O= c. 0 + co. O, 0=C2(- 2)+cl(2)+c (-1), 0=3~ 0+C2 ( —2)+c1 (2)+c0o (-1); from which we see that c0, cl, C3 remain arbitrary. All the other coefficients are expressible in terms of them. Consequently, the equation has three integrals free from logarithms belonging to 2, 3, 5, as their respective proper exponents. (The equation was constructed so as to have z5ez, z3ez, z3ezlogz+z4e3, z2eZ for a fundamental system; the system is easily derived by writing y= we -Z when the equation for y is Z4 (1 + Z) y" -z3 (7 + 8z) y"' +z2 (29 + 36z) y"- z (74 + 96z) y'+ (90 + 120z) y =0, which can easily be treated by the general method of Frobenius.) Ex. 2. As an example in which the process is ineffective, consider the equation Dw=(i - z) z d22+(5 - 4) zd +(6-9)=0. Taking, as usual, W= CnZn + P, n=O 8-2 116 EXAMPLE [44. we have D W= (p - 2) (p-3) Cz, provided (p+n-2)(p +n-3) c=(p + -4)2 C_l, for values n=l, 2,.... If instead of beginning with the root p=3 as in the general theory (~~ 35, 36), we try p=2, the equation for the coefficients c gives n(n - 1) c=(n- 2)2, 1, determining cl apparently as infinite. To modify this, we take co=C(p-2); the equation for cl then becomes (p - ) (p-2)c =(p -3)2 (p - 2) C, which is satisfied identically, when p=2. Thus c1 remains arbitrary; but co=O. The integral which would be obtained is, in fact, that which belongs to p=3; and the process is ineffective. There happens to be no integral belonging to p=2 (and not to p=3) free from logarithms. The actual solution is easily obtained by the general method of Frobenius. We have W=CzP -(p -2)+ _ 3)-z+(p -2)2 (p-3)2 R(z,p), where R (z, p) is a holomorphic function of z when p is either 2 or 3; and then D W= C(p - 2)2 (p 3) z For p = 3, we have the integral '1 = [ W] =3 - Cz3. For p =2, we have the two integrals w2=[ W]p=2= Cz3=wl, and 3= - = z2 + cz3 log z- 3cz3. 3 aP Jp=2 The integral belonging to the index 3 is Z3 free from logarithms; that which belongs to the index 2 is effectively z2 + Z3 log z, which is affected with a logarithm, so that the index 2 possesses no proper integral free from logarithms. DISCRIMINATION BETWEEN SINGULARITIES 117 DISCRIMINATION BETWEEN REAL SINGULARITY AND APPARENT SINGULARITY. 45. The singularity, in the vicinity of which the integrals have been considered, is a singularity of coefficients of the equation dmw P, (z) dm-1w P + (z) dzm z - a dzmn-l (z - a)m and the indices to which the integrals belong are the roots of the indicial equation for z = a, which is p(p- )... (p-m + )+ p(p-1)... (p-m+ 2)P,(a)+....+ Pm (a) = 0. In general, the integrals of the equation in the vicinity of a cease to be holomorphic functions of z -a; thus they may involve fractional powers or negative powers of z - a, and they may involve powers of log (z -a). When this is the case, a is called* a real singularity. If, on the contrary, every integral of the equation in the vicinity of a is a holomorphic function of z - a, then a is called an apparent singularity of the differential equation. The conditions that must be satisfied when a singularity of the equation is only apparent, so that it is an ordinary point for each of the integrals, may be obtained as follows. Let w., W2,..., Wm denote a fundamental system of integrals in the vicinity of the singularity a; and suppose that each member of the system is a holomorphic function of z- a in that vicinity, so that the singularity a is only apparent. Let A denote the determinant (~ 10) of this fundamental system, so that dm- -lw d-2 W dw, =dz dzm - ' d) z dm-lw2 dm-2w2 dw2 dz dz-1 dzm- ' " dz 2.......................................... dm-lwm dm-2Wm dw, dzm-1 ' dzm-2 "' dz' Wm * Weierstrass (see Fuchs, Crelle, t. LXVIII (1868), p. 378) calls the singularity wesentlich in this case,: in the alternative case, he calls it ausserwesentlich. 118 REAL AND APPARENT [45. and let Ar denote the determinant which results from A when the dm-r W, sdm r column d r is replaced by d, (for s=1,..., m). Then zm__-r dzm as every constituent in Ar and A is a holomorphic function of z - a in the vicinity of a, both Ar and A are holomorphic functions of z-a in that vicinity; neither of them is infinite there. But as in ~ 31, we have P. (z),. (z - a)r ( 1 ), and some one at least of the quantities P, (a) is not zero; hence, for that value of r, A, (a) A(a) is infinite, and therefore A (a) = 0, or the determinant of a fundamental system vanishes at an apparent singularity. Moreover, as in ~ 10, we have 1 dA _ P (z),_ (a) dG(z-a) Adz z- - z-a dz ' where G (z - a) is a holomorphic function of z - a; whence (z) = A (z - a)-P (a) eG (z-a), where A is a constant. Now A is not identically zero near a, for the system of integrals is fundamental; hence A is not zero. We have seen that A (a) = 0, and A (z) is a holomorphic function of z- a; hence P1 (a) must be a negative integer, numerically greater than zero. This condition is required, in order to ensure that a is a singularity of the equation. As each of the integrals is a function, that is holomorphic in the vicinity of a, it follows that the respective indices to which they belong must be positive integers; and therefore the roots of the indicial equation p(p- 1)... (p - m 1)+p(p - 1)...(p-m+ 2) P, (a)+..... + PP,-1 (a)+ Pm (a)= 0 must be positive integers. (When one of these is zero, then Pm (a) vanishes.) Moreover, no two of these roots may be equal; 45.] SINGULARITIES 119 for otherwise, the expressions for the integrals that belong to the repeated root would certainly include logarithms, contrary to the current hypothesis. Accordingly, let the roots be p,, p,...*, pm, a set of unequal positive integers which we shall assume to be ranged in decreasing order of magnitude: they thus form a single group the members of which differ from one another by integers. The integral belonging to p1 involves no logarithm. In order that every integral belonging to p2 may involve no logarithm, one condition must be satisfied: it is as set out in ~ 41. In order that every integral belonging to p3 may involve no logarithm, two further conditions must be satisfied they are as set out in ~ 41. And so on, for each of the roots in succession until the last: in order that every integral belonging to pM, may involve no logarithms, r - 1 further conditions must be satisfied, being the conditions set out in ~ 41. The aggregate of these conditions, and the property that the roots of the indicial equation are unequal positive integers, give the requisite character to the integrals. The condition that P, (a) is a negative integer makes a a singularity of the differential equation. When all the conditions are satisfied, the singularity is apparent. In all other cases, the singularity is real. Ex. 1. Consider whether it is possible that x=O should be only an apparent singularity of the equation Dw = 2w" - (4x + XX2) w' + (4 - KX) W = 0, where K and X are constants. The first condition, that P1 (a) should be equal to a negative integer, is satisfied: in the present instance, it is -4. To discuss the integrals, let W=Coxa +C1Xa + ++... +Ca + +n.. and substitute: then Dw= o (a- 4) (a- 1) L, provided c( (a+n-4) (a+n-1)= {X (a+n-1)+K}cni, for n=l, 2,.... The indicial equation, being (a-4)(a-1)=0, has all its roots equal to positive integers; so that another of the conditions is satisfied. The two roots form a group. 120 EXAMPLES [45. The integral, which belongs to the (greater) root 4 as its index, is a holomorphic function of x; it is easily proved to be a constant multiple of (say) u, where {4 4X+ 4X + K 5+ 4X + 5O + + K 5X+ K 6X + } x4 (1 + lX+ y2x2 +73x3+...), for brevity. As regards the other root given by a= 1, we have to assign the conditions that the integral which belongs to it contains no logarithms. In accordance with the results of ~ 41, we see that there will be a single condition; expressing it in the notation there used, we write Po=4, pi=l, V=Po-Pl =3, L=1, and we have to find hv(a) for v=3; a=pl=l. Now (~ 38) fo (a)= (a-4) (a- 1), g3(a)=A3, go(a)=A0, and h3 (a) go (a) 3 o (a+ l)o (ca+2)f (a +3)' so that h3 (a)={X (a+2)+ K {X (a+1)+ 4K}{Xa+K}. The sole condition is that h3 ()=0o; and therefore we must have K= —X, or -2X, or -3X. If K has any one of these values, the origin is only an apparent singularity of the equation. If K= - X, the independent integral belonging to the root 1 is V =.. If K= - 2X, the integral is v=x+iXx'". If K= - 3X, the integral is V = x + XX2 + X2x3. In all other cases, the origin is a real singularity of the differential equation. The result, as to the relations between X and ti, can be verified independently. As w and u are solutions of the differential equation, we have /4 4 uw" - Wu" = (- + X ) (uw' -Wu'), \x and therefore uw' - wu' = Kx4 ex, where K is a constant. Hence d W1-K e+Ax dx \u X4 (1 +Y1X+Y2X2+y3X3+,.,)2 - EXAMPLES 121 If every integral is to be holomorphic in the vicinity of the origin, it is easy to see that, as u belongs to the index 4, the only condition necessary is that the coefficient of - on the right-hand side should be zero. Thus x3- IX2. 2y1 +X (3y2- 2y2) - 2y3 + 6y2 - 43= 0, which, on substitution for 7y, 72, y3, and multiplication by -36, gives (X + K) (2X+ K) (3X + K)=0, thus verifying the condition obtained by the general method. In this example it appears that the integral, which belongs to the smaller root of the indicial equation, is, in each of the three possible instances, a polynomial in x. It must not be assumed that such a result always holds when a singularity is only apparent; this is not the case*. Ex. 2. Prove that the origin is an apparent singularity for the equation x2w" - x (4 + XX2) w + (6 + x2) w = 0, where X and p are constants; and shew that no integral, holomorphic in the vicinity of the origin, can be a polynomial in x unless u is a positive integer multiple of X. Ex. 3. Prove that z=0 and z=1 are real singularities for the equation z(1 -z) w" +(1-2z)w'-Iw=O; and that z= 1, z= - 1, are real singularities for (1 - z2) w" - 2zw'+n (n+ 1) w=O, when n is an integer. Ex. 4. Shew that z=oo is a real singularity for every equation of the form d2w d2 + R (z) = 0, where R (z) denotes a rational function of z. Ex. 5. Shew that, if z= o be an apparent singularity for each integral of the equation d2qV+ P 1 dW+ Q 1 W=O dz2 \z dz \P where P and Q are holomorphic functions of z-1 for large values of lzl, then, if zP ()=X + negative powers of z, 2Q ( =/. +........................ * See some remarks by Cayley, in the memoir quoted on p. 113, note. 122 EXAMPLES [45. X must be a positive integer equal to or greater than 2, and p must be a positive integer which may be zero. Shew also that, if X = 2, then p. must be zero. Are these conditions sufficient to secure that each integral of the equation is a holomorphic function of z-l for large values of Izl? Ex. 6. Verify that every integral of the equation d2w (/3 1 \ dw (1 1 \ dz2W+ + Id + + + - =0 is holomorphic for large values of z. Note on ~ 34, p. 85. To establish the uniform convergence of a series 2gvxV for values of a, Osgood shews (I.c., p. 85) that it is sufficient to have quantities l,, independent of a, such that provided the series 2Jin converges. Take a circle in the a-plane large enough to enclose all the regions round the roots off(p)=0 given by la- pl=r'-K'; and let this circle be of radius rl, so that r1 is a constant independent of a. With the notation of ~ 34, take constants C^, for values of v >E, such that CQ Cv{(r + ) + 1} C4-1C- ( \( - _... ) - (~ 'l f) 'r, 5 while C-I=re-ye. Then, as t (rl +v) > M(a+v), (- rl+ v)m - (ri-+ v) < fo (a+ + l) l we have Yv+1l rv+l < Cv+I for all values of v. Now, as in ~ 34 for the ratios of the r's, we find Cv+1 1 Lim v=X- Cv and therefore the series CE (R-K)E + C+1 (Rf —K)fl ++. converges, R' being less than R. Accordingly, by taking M= Cn (r '-K)n, the uniform convergence of the series EgVx is established. CHAPTER IV. EQUATIONS HAVING THEIR INTEGRALS REGULAR IN THE VICINITY OF EVERY SINGULARITY (INCLUDING INFINITY). 46. WE have seen that, if a linear differential equation is to have all its integrals regular in the vicinity of any singularity a, it is necessary and sufficient that the equation should be of the form dmw P, dcn-lw P, dm-2w PM f- -- _ _ dzmn z - a dzn -1 (z- a)2 dzm-2 + " (z - a)mw in the vicinity of that singularity, the quantities PI, P2,..., Pn being holomorphic functions of z-a in a region round a that encloses no other singularity of the equation. We can immediately infer the general form of a homogeneous linear differential equation which has all its integrals regular in the vicinity of every singularity of the equation, including z== o. As Fuchs was the first to give a full discussion* of this class of equations, it is sometimes described by his name; the equations are saidt to be of Fuchsian type or of Fuchsian class. Let al, a2,..., ap denote all the singularities of the differential equation in the finite part of the z-plane, and write * = (z - a,,)(z - )... (- a,); then the conditions are satisfied for each of these singularities by the equation dnw m QK dM-Kw dzM K rK dz1-K * See his memoir, Crelle, t. LXVI (1866), pp. 139-154. + Care must be exercised in order to discriminate between equations of Fuchsian type and Fuchsian equations. The latter arise in connection with automorphic functions and differential equations having algebraic coefficients: see Chap. x. 124 EQUATIONS OF [46. provided the functions QK are holomorphic functions of z everywhere in the finite part of the plane. To secure that the integrals possess the assigned characteristics for infinitely large values of z, we note that = zR ( ), where R is a polynomial in - and is unity when z=oo, and z therefore K = ZPK () = zKR 1, where R, is of the same polynomial character as R, and is unity when z= oc. Now suppose that, for very large values of z, the determinant A (z) of a fundamental system belongs to the index ar, so that A (z)= z-OT ) where T is a regular function of I which does not vanish when z z = oo. Then, with the notation of ~ 31, we have AK (Z) = - K (I ), where TK is of the same character as T, save that it may possibly vanish when z = o: taking account of the latter, we have A, (Z) =Z-K —e () where e is an integer > 0. Thus PK A - ~z u () z where U is a regular function of - which does not vanish when z z = oo; and therefore QK =PK#K = Z(P1)K-Be1 (-R () 0<==p^ ~ z 46.] FUCHSIAN TYPE 125 for very large values of z. But QK is a holomorphic function of z near z = oo; this property, imposed on the preceding expression, shews* that QK is a polynomial in z, of degree not higher than (p-) K. Moreover, it was proved in the last chapter that all the integrals of the equation dmw Pi dm-lw P, dzm - z-a dzn-1 (z - a)m are regular in the vicinity of z = a, when the quantities P,..., Pm are holomorphic functions of z in that vicinity. Applying this proposition to each of the singularities (including oo ) of the equation d_.w M QK dm-KW dzm K, = K dzn-K ) with the restriction upon Q,..., Qm as polynomials in z of the appropriate degrees, we infer that all its integrals are regular in the vicinity of each of the singularities (including o ). Combining the results, we have the theorem, due to Fuchst:When the m integrals in the fundamental system of a linear homogeneous equation of order m have al, a2,..., a, as the whole of their possible singularities in the finite part of the z-plane; and when all the integrals are regular in the vicinity of each of these singularities, as well as for infinitely large values of z; the equation is of the form dmw Gp-l dm-lw G2(p1) dm-2W Gmp-l) d-, - -+ +g... +- w, dzm - # dzm-n1 2 dz"1-2 m! p where * denotes II (z - aK), and G,(p-l), for t = 1, 2,..., m, is a K=l polynomial in z of degree not higher than u (p- 1). Conversely, all the integrals of this differential equation are everywhere regular, whatever be the polynomials G and 4r of proper degree. Accordingly, this is the most general form of linear equation of order m, which is of Fuchsian type. * This result may also be obtained by using the transformation zx=l and applying to the equation, transformed by the relations in ~ 5, the proper conditions for the immediate vicinity of x= 0. t Crelle, t. LXVI (1866), p. 146. 126 EXAMPLES OF EQUATIONS [46. Ex. 1. Legendre's equation is (1 -z2) w- 2zw'+n (n +1) w=0, say 2z n(n + l) 1 -2 1- 2 W.2 Its form satisfies all the necessary conditions; hence its integrals are regular in the vicinity of z= 1, z= -1; and are regular also for infinitely large values of z. Similarly, the hypergeometric equation, which is (1 -z) w" W+y- (a+,3+l) z w'- a3w =O0, has all its integrals regular in the vicinity of z= 0, z= 1, and regular also for infinitely large values of z. Bessel's equation of order zero is = -- W -w z 1 z2 = —w'-w; z its integrals are regular in the vicinity of z =0; but, on account of the order of the numerator of the coefficient of w in its fractional form, they are not regular for infinitely large values of z. The same result as the last holds for 1 n2 - z2 W" = -- W + — W, which is Bessel's equation of order n. A form of Lame's equation, which proves useful (see Chap. Ix, ~~ 148 — 151), is wo"= {A (Z) +B} w, where A and B are constants; its integrals are regular in the vicinity of any point in the finite part of the z-plane congruent with z=0, and these are all the singularities in the finite part of the plane; but they are not regular for infinitely large values of z. Ex. 2. The sum of all the exponents associated with all the singularities (including oo) of the equation of Fuchsian type obtained at the end of the preceding investigation is the integer ~ (p - 1) n (m - 1), a result first given by Fuchs*. The proof is simple. The polynomial Gp-_ is of order not higher than p - 1: say Gp_1 =AzP- I+.... The indicial equation for the singularity an is (0-1)...(8-me+i1)= G-e, ((a -1)...(- +2)+..., ' (, p. 145.) * Crelle, t. LxVI, p. 145. 46.] OF FUCHSIAN TYPE 127 the unexpressed terms on the right-hand side constituting a polynomial in 0 of order not higher than m - 2. Hence the sum of the indices for the singularity an is Im n- 1) + Gp-I (a.); _q *Xl' (an) and therefore the sum of the indices for all the singularities a,, a2,..., ap in the finite part of the plane =~prm(m-l)+~ Gp (an) P" ( ) nl = ' (a.) =~prm(m- 1)+A, because al, a2,..., ap are the roots of t=0. The indices for oo are obtainable by substituting w=Z-P (+... ); the indicial equation for co is (_- 1)mp (p+ 1)...(pr ^ - 1 ))( i)lm-1A (p + 1)...(p + m - 2)-..., so that the sum of the indices for oo is -~m(m- 1)-A. The total sum of all the indices is therefore I (p-l)m(m-i). Ex. 3. The general equation of Fuchsian type, which has all its integrals regular in the vicinity of every singularity (including co ), has been obtained. The limitations upon the form of the type are mainly as to degree, so that generally the construction of the equations, when definite singularities and definite exponents at the singularities are assigned, will leave arbitrary elements in the form. The instances when the equations are made completely determinate by such an assignment are easily found. Taking the equation as of order m, we have polynomials Gp-1 (), G2P-2(Z)... P - mp (Z) which, in their most general form, contain p+(2p- 1)+(3p-2)+...(mp-m+ 1) =pmn (n+1 )-Mr (m- 1) constants. The assignment of the positions of the singularities merely determines -: it gives no assistance to the determination of the constants in the polynomials G. Each of the p singularities in the finite part of the plane requires m exponents, as does also the point z= co; so that there are m (p+ 1) constants thus provided. But, by the preceding example, their sum is definite: and thus the total number of independent constants thus provided is m(p+l)-1. 128 EQUATIONS OF FUCHSIAN TYPE [46. If therefore the equation is to be made fully determinate by the assignment of these constants, we must have 2pm(m+ (m T- m (m-l)=m (p+l)- 1, and therefore pm (m - 1)= m -l1) (m+2). When m=l, p can have any value; that is, any homogeneous linear equation of the first order, which has its integral regular in the vicinity of each of its singularities and of z =, is completely determined by the assignment of singularities and of the exponents for the integral in the vicinity of the singularities. For such equations of the first order, let al,..., ap be the singularities in the finite part of the plane; let mi,..., mp be the indices to which the integral belongs in their respective vicinities, and let m be the index for p z=0c, so that m+ E mr-=O. The equation is r=l.d;/ P m.r dz r=l z-ar p which gives the index for z = o as equal to - 2 m., being its proper value. r=l When m>1, then p=I+, m so that, as p is an integer, m must be 2 and then p=2. Thus the only homogeneous linear equation of order higher than the first, which is of the Fuchsian type, and is completely determined by the assignment of the singularities and of the exponents to which the integrals at the singularities belong, is an equation of the second order: it has two singularities in the finite part of the plane, and it has z= o for a singularity; and the sum of the six indices to which the integrals belong, two at each of the singularities, is ~ (2- 1) 2 (2.- 1), that is, the sum is unity. The discussion of the determinate equation of the second order of the foregoing type will be resumed later (~~ 47-50). Note. If p=O, so that the equation has no singularities in the finite part of the plane, the coefficients are constants if the equation is to be of Fuchsian type. The only singularity of the integrals is at o. If p =1, m > 1, the number of arbitrary constants is less than the number of constants, due from the assignment of the indices at the finite point and at z= c: the latter cannot then all be assigned at will. For values of p greater than 1 and for values of m greater than 1, the number of arbitrary constants in a linear differential equation, which are left undetermined by the assignment of the singularities and their indices, is ==_pmn (m+1)- m (m- 1)-{mt (p + 1)- 1 =` (m-l){m(p- l)-2}, which for all the specified values of p and in, other than m = 2 and p= 2 taken simultaneously, is greater than zero. EXAMPLES 129 Ex. 4. Consider the equation, indicated in the Note to Ex. 3, all whose integrals are regular at the only finite singularity, which can be taken at the origin, and regular also at infinity: it is dmw fA dim- lW - 2 dm- 2w fm dzm ~ z dzm-1 z2 dzm-2 * zm 1 wherefl, f2,..., fg are constants. The assignment of indices for z=0 determines fi,..., f/, and so determines the indices for z=cO; and similarly the assignment of indices for z = determines those for z=0. In fact, the indicial equation for z=O is m p(p-1)...(p-m+l)=- p(p-1)...(p-m+K —l) K, K=l and the indicial equation for z= co is m (- 1)O (O + 1)...(8 +m-1)= 2 ( - 1)f — ( 1)...( + - + l)/ K=l it is at once evident that the roots can be arranged in pairs, one from each equation, in the form p + =0. As regards the integrals, it is easy to verify, in accordance with the general theory, that the integral which belongs to a simple root r of the indicial equation for z=0 is a constant multiple of zr: and that the n integrals, which belong to any n-tuple root s of that equation, are constant multiples of ZS (log z)d, for a=O, 1,..., n-1. Ex. 5. Consider the equation Dw=z (1 - z) w" +( - 2z) w'- w= 0, which* clearly satisfies the conditions that its integrals should be regular, both in the vicinity of its singularities and for large values of z. To obtain the integrals in the vicinity of z=O, we substitute QW = COza + C za+l+...C + z +..., and find zDw = co a2Z", provided (a + n)2Cn-=(a+n — ~)2cn_; so that, writing (a +i) (a4 + )...(a+m-) 2 l (a+1 (+)(a+2)...(a + m) the value of w is W=Co a (1 +Yl + 722 2+...). * It is the differential equation of the quarter-period in elliptic functions: for a detailed discussion of the equation, gee Tannery, Ann. de l'Ec. Norm. Sup., Ser. 2me, t. vmII (1879), pp. 169-194, and Fuchs, Crelle, t. LXXI (1870), pp. 121-136. F. IV. 9 130 EQUATION OF QUARTER-PERIOD [46. The indicial equation is a2=0: accordingly, the two integrals belong to the index 0, and they are given by aJa==o To particularise the integrals, we take Co=- w; the first of the integrals then becomes RK(z)=x {:+ 1 3) 2.} Yrr rl~Q kz \2.4] 7r{l+aIlz+a2Z2+...}, say: and the second of them becomes L (z), where L(z) =K (z)log z+ r amzm 2 {1 + + 1 m=1 2 2 m -2 =K(z)log z+ 27r I am3mzm" 1 1 1 1 1 1 m=l =K (z) log z + I(z), say, where 1 2 + 3 4 +* +2n - 1 2m And now the two integrals in the vicinity of the origin are K(z), L(z). To obtain the integrals in the vicinity of z= 1, we substitute z= -x, when the equation takes the form d2w dw (I - x) d + ( - 2x) -w=O, which is of the same form as in the vicinity of z=O. Accordingly, the integrals in the vicinity of z=l are given by K (x), L (). To obtain the integrals in the vicinity of z = o, we substitute 1 z=t' when the equation takes the form t2(1- t) d2 _ t2 dw +Rw= O. The indicial equation for t=O is a (a-1)+ =; we take W = t2 d, and we find the equation for u to be -d2u du t (1 -t) d2u +(1 - 2t) du- iu=O, 46.] IN ELLIPTIC FUNCTIONS 131 of the same form as in the first and the second cases. Accordingly, the integrals of the original equation in the vicinity of z= o are t KI(t), t~L(t). The integrals are thus regular in the vicinity of the three singularities 0, 1, oo. Of these, the integrals K(z), L(z) are significant in the domain 2zl< 1, say in Do; the integrals K (x), L (x) are significant in the domain Ixlz- l-1 < 1, say in D1; and the integrals ttK (t), t1L(t) are significant in the domain I t < 1, that is, jz[ > 1, say in D,. The series K (z) diverges when z=l, so that the integrals cease to be significant for such a value. The domains Do and D1 have a common portion, so that values of z exist which are defined by lzi<l, Iz-1\<l. Within this common portion, the integrals K (z), L (z), K(x), L (x) are significant: so that, as K (z) and L(z) make up a fundamental system, we have K (x)=AK(z) + BL (z), L (x)=A' (z) + B'L(z), where A, B, A', B' are constants. The values of the constants are determined as follows by Tannery. The integrals are compared for real values of z which are positive and slightly less than 1, so that, as z then approaches 1, K (z) tends to an infinite value. To obtain this infinite value, we note that, as r (n+i)> {7a'-l - )} > wn 1.i3... ()> 2 -...2n by Wallis's theorem, we have 1 1 (n + X) < an< and therefore, for real values of z between 0 and 1, we have 1 oo Xn \0 ( x> n {2V I+ < +) < K(z) < 7r I + - Y. The difference of the two quantities, between which the value of K(z) lies, is /1 l 1 Zn =1 which increases as the real value of z increases and, for z =1, is n=/1 \ nt2n 2+y 1' that is, 1 - log 2. Hence we may take K (z)=e (z)+ 2 n=1 n =- (Z)-2 log(1 - ), 9- 2 132 EQUATION OF QUARTER-PERIOD [46. where 7r >(z)> 7r-l +log 2; and the values of z are real, positive, and less than 1. The result shews that K (z) is logarithmically infinite for z= 1. Proceeding similarly with I(z) in the expression for L (z), we have, for real values of z between 0 and 1, 2 m-1 4< (z < 4 < m. m=1 2mn+ 1 + im=1 2rn The difference of the two quantities, between which the value of ~I(z) lies, is 00 / 1 1 \ m1 (2 2n+)+ 1Z which increases as the real value of z increases and, for z= 1, is m=l-1 2? 2m + 1I Now 1 1 1 1 1 O=M + + +... + - Pm-1 2 3 4 2m-1 2m <log 2; and therefore the foregoing difference is less than log 2 ( - m=i 2in '1 that is, less than (1- log 2) log 2. Hence we may take 4I(z) =4ml m -E e (z), m=1 m where, for real positive values of z that are less than 1, 0< e' (z)< (1 - log 2) log 2. The expression can be further modified. We have oo fo z^ -" z" < log 2 E -X m=l m m=l In for the values of z considered. The difference between these two series is X log 2-/ m m=l A a quantity which increases as the real value of z increases and, for z= 1, is 2 -(log 2 - ). But 1 1 log 2 -/ - +... 1 <2m+l 1 2m +l ' 46.] IN ELLIPTIC FUNCTIONS 133 and therefore the difference is 00 1 < n < 2 (I - log 2), m=1 (2m4-l 1) on evaluating the series. We may therefore take - - = 2 -log 2 - )" (z) m=l m m=1 mn = - log (1 - z) log 2- " (z), where 0 < " (z) <2 (1 - log 2). Therefore, finally, we have I (z)= - log (1 - z) log 2 - El (z), where E1 (Z)= ' (Z) + E"I (Z), so that 0 < 1 (z) < 1- (log 2)2; and the values of z considered are real, positive, and less than 1. In the region common to Do and D1, we have K(x)=AK(z)+BL(z); and therefore, for real values of z less than (but nearly equal to) 1, that is, for real, positive, small values of x, K (x) = Ae (z) - A log x- 2B log x log 2-4B (z)A+B { (z) - log x) log z. When z tends to the value 1, the term log x log z tends to the value 0: moreover, K (x) then tends to the value 7r; hence, taking account of the infinite terms on the right-hand side, we have A + 4B log 2=0. Again, when z is real, small, and positive, x is real, positive, and less than (but nearly equal to) 1; hence K (x) = (x) - I log (1 - x) =e (1 - z) - log z, so that (1-z) - log z= AK (z) + BK (z) log z + BI (z), all the terms in which are finite except those involving log z; moreover, when zIl is small, K(z)= r +zR (z), where R is a holomorphic function of z; thus B= —. 7r Consequently, 4 A= log2; 7T so that A and B are known. 134 EQUATION OF QUARTER-PERIOD [46. Similarly, for the other equation L () = A'K (z) + B'L (z), for values of x and z in the common region, we have, for real, positive values of z less than 1, that is, for real, positive values of x that are small, K (x) logx + 1 (x)= A' (z) - log (1- z)}-4B' f{ log (1 - z) log 2 + 1 (z)}; hence, taking account of the logarithmically infinite terms on both sides, we see that A' +4B' log 2 = - r. Next, taking the same equation for values of z that are small, real, and positive, so that x is real, positive, and less than 1, we have A'K (z) + B' {K (z) log z + I (z)} = K (x) log x + I (x). When x is nearly unity, K (x) = (x) - ~ log (I - x), so that K (x)log x, for x nearly equal to 1, is small: and it vanishes when x=1. Also, for those values of x, I (x)= -2 log (1 -x) log 2 - 461 (x) = - 2 log z log 2- 4e1 (x); whence, equating coefficients, we have 7rB' =-2 log 2. Thus 4 16 B'=- 4log 2, A'=- (log2)2-7 r. 77 77 Accordingly, when z lies within the portion common to the two domains Do and D1, defined by the relations Izl<i, Iz-11<i, we have L (X)= { (log 2)2- } K(z)- (4 log 2) L(z) where x=1 -z. These results shew that, for complex values of z such that z l=1, both K (z) and L (z) converge. The first of them is a known result in the theory of elliptic integrals; writing z =k2, =k'2, K (z)=K, K(x) = K', we have 2K 4 00 K'=K log 2 am,3k2m 7r K m=l an equation which is specially useful for small values of k. Similarly, for values of k nearly equal to unity, we have 2K' 4 K=' log 2 ammk'2m. 7r K m=l 46.] RIEMANN'S P-FUNCTION 135 Ex. 6. With the notation of the preceding example shew that, for values of z common to the domains D1 and D,, as defined by lzl>l, Iz-1<l, the integrals K (x), L (x), tRK (t), tdL (t) are connected by the relations 1 41og2-ir I ) K (=4 log 2 - iKr (x) -- L (x) 7r 7T tAL (t) 16 (log 2)2- 4ri log 2 - r2K() 4 log 2L (x) 7r 7r (Tannery.) Ex. 7. Denoting the integrals of the equation in Ex. 5 that are associated with the values z=0, 1, oo by K, L; K', L'; K", L"; respectively; denoting also the effect upon a function U of a simple cycle round a point a by [ U]a, and of simple cycles round a and b in succession by [U]ab, prove that [K]o = K, [L]o = L + 2rriK; [K']o (1= 8i log ) K'+ 2 iL' \ T / 2i7T [K']ol = +8i log 2 K'+r 7r 7r and express [L']o, [L']1 in terms of K', L'. (Tannery.) Ex. 8. Discuss, in the same manner as in Ex. 5, the integrals of the equations (i) z(1- z) "- - O; (ii) Z(1-Z)W"+(1-Z)+W' w=1=O; (iii) z(l-z) w" +,ad'-xw=0O. RIEMANN'S P-FUNCTION. 47. It has already been proved (Ex. 3, ~ 46) that the only linear differential equation of any order other than the first, which is made completely determinate by the assignment of its singularities and of the exponents to which the integrals belong in the respective vicinities of those singularities, is an equation of the second order which, if it have oc for a singularity, has two other singularities in the finite part of the plane. If the latter be at h, k, then the transformation z-h h c-b x-a z- k c-a x-b gives a, b, c in the x-plane as the representatives of h, k,; o in the z-plane. The transformation manifestly does not affect the order 136 RIEMANN'S [47. of the equation, its sole result being to make a, b, c (but not now o ) singularities; we shall therefore suppose this transformation made. Accordingly, we proceed to consider the properties of the function, which thus determines a differential equation; they depend upon the properties initially assigned, which are taken as follows. In the vicinity of all values of z, except a, b, c (and not excepting co when a, b, c are finite), the function is a holomorphic function of the variable. In the vicinity of any point (including the three points a, b, c), there are two distinct branches of the function; and all branches of the function in the vicinity of any point are such that, between any three of them, a linear relation A'P' + A"P" + A'"P"' =0 exists, having constant coefficients A', A", A"'. (So far as this condition affects the differential equation, it manifestly determines the order as equal to two.) As exponents are assigned to the three points, let them be a and a'for a: /3 and /3' for b: y and y' for c; these quantities being subject (~ 46, Ex. 2) to the condition a +a' + f, + 3' + 7 + ' = 1. It further is assumed that a - a', 3 -/', y - y' are not equal to integers. The branches distinct from one another in the respective vicinities are denoted by Pa and Pa; P3 and P,,; P. and P,,. From the definition of the exponents to which they belong, the functions (z - a)-Pa and (z - a)-'P,, are holomorphic in the domain of a and do not vanish when z = a. Similarly for b and c. After the earlier assumption, it follows that any branch existing in the vicinity of a can be expressed in a form CaPa + Capal, where Ca and Ca' are constants; and likewise for branches in the vicinity of b and c. The assumption made as to a-a', 8 - ', ry -y not being integers will, by the results obtained in ~~ 35-38, secure the absence of logarithms from the integrals of the differential equation: it manifestly excludes the possibility of either of the branches Pa and Pa,, Pp and Ps,, Py and P.,, being absorbed into the other. P-FUNCTION 137 Riemann* denotes the function, which is thus defined, by (a b c P a ryx 1; ac' /3' 7 J:and the function itself is usually called Riemann's P-function. It is clear that a and a' are interchangeable without affecting P; likewise /3 and 3'; likewise y and y'. Also, the three vertical columns in the symbol can be interchanged among one another without affecting P; six such interchanges are possible. Again, if P be multipliedt by (x - a) (x - b)-8-6 (x - c)E, the effect is to give a new function, having a singularity at a with exponents a + 8, a' + 8: a singularity at b with exponents /3 - -e, l/'- - e; and a singularity at c with exponents y + e, y' + e. Every other point (including oo) is of the same character as for P. Hence (x - a)' (x - \c (a b ca b C (- (-b)+e -P a /3 y X i-=P4a +8 a~ - C-ey + '-"e a' [a' y' ' J = P +3 /'-S-e y+ J the exponents on the right-hand side still satisfying the condition that the sum of the exponents shall be equal to unity. A homographic transformation of the independent variable can always be chosen so as to give any three assigned points a', b', c' as the representatives of a, b, c. Accordingly, let such a transformation be adopted as will make a and 0, b and oo, c and 1, respectively correspond to one another: it manifestly is x-a c-b X -- - x-b c-a The indices are transferred to the critical points 0, oo, 1; every other point is ordinary for the new function, as every other point was for the old. For brevity, the transformed function is denoted by (a /3' 7' x * Ges. Werke, p. 63. t The sum of the indices in the factor is made zero; otherwise x = o would be a singularity for the new function. 138 RIEMANN'S [47. where the two-term columns are to be associated with 0, oo, 1 in order. Also, since a-bx-c a — b x - a- - — 7 b a-cx-b' it follows that, except as to a constant factor, {x - a)8 (x - c), (x- b) (-c) and x' (1 - x')Y agree; and thus, as regards general character, we have a',7' /S3' -=-~, y- e ' X b (1-S)e P (a/' 13 ~ ) = P (a + - '- e +e As a-a', / - 3', y- ' are the same for the P-function on the right-hand side as for the P-function on the left, Riemann denotes all functions of the type represented by the expression on the left by P (a-a', -/3', 7 -, x'). In the transformation of the variable, the points a, b, c were made to be congruent with 0, oo, 1 in the assigned order. A similar result would follow if they had been made congruent with 0, ao, 1 in any order or, in other words, if 0, o, 1 be interchanged among themselves by homographic substitution. As is known, six such substitutions are possible, viz. /,,1 x 1 x" = x', 1-x, 1,, l-,, _ c cc) 2 cx'-i' 1-x'' or, taking account of the association of the exponents with the first arrangement, the table of singularities, exponents, and variables for the six cases is 0 co I 0 co I 0 oo '8 ' Y X 1 X a l/3 7 xy; a 1 ayG'Y l O l G 1- ' a / /3 1-x,; a y /3 J'-; a 7 a 1-,'; y' a' 3' a' y' I ' i' ' a so that P-functions of these arguments with properly permuted exponents can be associated with one another. 48.] P-FUNCTIONS 139 48. The significance of the relation a + a' +13 + 3' y +' = 1, in connection with the function, appears from the following considerations. When the singularities are taken at 0, oo, 1, the axis of real variables, stretching from - co to + oo, divides the plane into two parts in each of which every branch of the function is uniform; or, if the singularities be taken at a, b, c, then a circle through a, b, c divides the plane in the same way. In either case, taking (say) the positive side of the axis or the inside of the circle, the linear relations among the branches of the function give Pa = B1P B 2P Pa = PY + C2 P,) Pa/ = Bi'PT + B2'P, J Pa/ = C1P, + C2'Py,/ say Pa, Pa/ =( B,, B, $P,, Ph.) = (b PO, PA,), IB' B2,' Pa, Pa/ =( C1, C2?PY, P,/) = (C3$P,, P,'); 01/, C2| and with the usual notation of substitutions, let -1 P P, p,' = (bPa,, Pa,), -1 P,, p,, = (c.,Pa Pa/). Consider the effect upon any two branches, say Pa and Pa,, of circuits of the variable round the singularities. When it describes positively a circuit round a alone, they become e2iaP,a and e2Pia'Pa, respectively, so that, in the above notation, Pa, Pa/ become (e27ia, 0 3Pa, Pa'). o, e27ria't When it describes positively a circuit round b alone, then PA and Pp' become e2risPp and e27rip'Pp, respectively; and therefore Pa, Pa, become (b&e27i, 0 abPa, Pa/). 0, e2'ri' Similarly, when it describes positively a circuit round c alone, -1 Pa, Pa, become (ce27ri, 0 ]cPa, Pa/). I 0 e2riy' 140 RIEMANN'S P-FUNCTION [48. Accordingly, when z describes a simple circuit round a, b, c, the initial branches Pa, Pa, are transformed into branches -1 -1 (c0e27riY, 0 3cbi1e2frip, 0 3jb^e2Ria, 0 )Pa, Pa/), 0, e2Tiy' 1 0, e2ri' I e2, ia say (IdPa, Pa'). Such a circuit encloses all the singularities of the functions; and therefore* each of the functions returns to its initial value at the end of the circuit, so that (Z)=(1, 0). o0, 11 The determinant of the right-hand side is unity; hence the determinant of I is unity, and it is the product of the determinants of all the component substitutions. Now as (c) and (c)-l are inverse, the product of their determinants is unity; and likewise, the product of the determinants of (b) and (b)-l is unity. Hence we must have e2n+(a+a'+B+'+y+y') = 1, an equation which is satisfied in virtue of the relation a +a' + 8 + 3'+ + y + ' = 1: the sum of the exponents could be equal to any integer merely so far as the preceding considerations are concerned. In the present instance, the property, that a function returns to its initial value after the description of a circuit enclosing all its singularities, can be used in the form that the effect of a positive circuit round c is the same as the effect of a negative circuit round a and round b. Applying this to Pa, we have CP. e2y7ri + C2P e2vyi = e-2a7ri (B Pge-2,ri + B2 P,e-2"'7i); and from the expressions for P,, we have CP, + CP, = B1P, + BP,. As Pa and Pa, are linearly independent of one another, it follows that e2y7 - e2'"ri must not be zero, that is, y - y' must not be an integer. Similarly for a- a' and / - /'. Ex. Prove, by means of these relations, that C1 e(a-a')rri B1 sin (a+ +y')r r _ B2sin(a +3'+y')r C1' B' sin (a' + + ') rr B2' sin (a'+ '+ y') r ' C2 e(a-a)i B1 sin (a+/3+y)7r B2sin (a —3'+y)rr 2 B1' sin (a' + 3 + y) 7r B2' sin (a' +' + y) r ' (Riemann.) * T. F., ~ 90. 49.] DETERMINES A DIFFERENTIAL EQUATION 141 DIFFERENTIAL EQUATION DETERMINED BY RIEMANN'S P-FUNCTION. 49. As regards the differential equation, associated with these P-functions, and determined by the assignment of the three singularities a, b, c, and their exponents, we know that it must be of the form d2w A'z2 + B'z + C' dw A"z4+ B"z3+C"z2+ D"z+E" dz2 + (z - a)(z- b) (z- c) dz + - a)2 - - b)2 (z - c)2 w~ which (~ 46) secures that a, b, c, oo are points in whose vicinity the integrals are regular. Now the singularities are to be merely the three points a, b, c, so that oo must be an ordinary point of the integral. Taking the most general case, when the value of every integral is not necessarily zero for z = o, we have an integral w=KoK +K + +., Z 22' where K0 does not vanish. Substituting, we have KoA" 1 + 1[(2-A')K +A"K,+{B" +2A"(a+b+c)} K]+...=0, z z the unexpressed terms being lower powers of z; hence Ko A" = 0, (2- A')K, + A"K, + {B" + 2A" (a + b + c)} Ko= 0, that is, A"= 0, (2 -A')K,+B"Ko=0, and so on. Using the result that A"= 0, the equation may be written in the form d2w ( A B C dw dZ2 + + — + d2 \ - a z - b z-c dz (z-a) (z - b) ( - c) (B + - - -b -+ ' c). Forming the indicial equations for the singularities, we have 0(0-1) +AOd+ (a -b) (a-c) 142 DIFFERENTIAL EQUATION DETERMINED [49. as the indicial equation for a; and therefore, as its roots are to be a and a', it follows that A =1 - a- a', X= a' (a - b) (a - c). Similarly B= 1-/3 - /', =/,3(b - a) (b-c), C = 1 - - y, v = ' (c - a) (c -b). Moreover A'=A + B +C=2, on account of the value of the sum of the six exponents: the condition (2- A') K + B"Ko = 0 is thus satisfied by B"= O. All the quantities are thus determined, and the equation has the form d2w 1 -a- 1 - _- 1 - / -- dw d +, — + dz2 z- a z-b z-c dz aa' (a -b) ( - c) /3/' (b- a) (b - c) yy' (c- a) (c- b) z - a z-b + z-c w (z - a) (z- b) (z- c) 0 from the mode of construction we know that the integrals are regular in the vicinity of the singularities a, b, c, and are holomorphic for large values of z. This is the differential equation, associated with (and determined by) the function a b c 4 P a 38 7 X. The branches of the integral in the vicinity of a are Pa, Pa,; those in the vicinity of b are Pp, Pi,; and those in the vicinity of c are Py, P',. Passing to the form of the function represented by Pa /'8 y ), where the three singularities are 0, o, 1, we deduce the associated differential equation from the preceding case by taking a=O, b=oo, c=l; * First given by Papperitz, Math. Ann., t. xxv (1885), p. 213. 49.] BY RIEMANN'S P-FUNCTION 143 after a slight reduction, the equation is found to be d2w 1- a - a'-(1 + + ) z dw dz2 + z ( - z) dz ' - (aa' + /3'-_ ryy') z + /'Z2 0 + --- \ w=O-_z2 0. The branches of the integral in the vicinity of the origin are Pa, Pa,, so that z-aP,, z-a'Pa, are holomorphic functions of z, not vanishing when z = 0; those in the vicinity of z = 1 are Py, P/,, so that (z - 1)-~YP, (z - 1)-Y'P, are holomorphic functions of z-1, not vanishing when z= 1; and those in the vicinity of z = oo are Pg, Pa,, so that zAPp, zo'Pp, are holomorphic functions of -, not vanishing when z = o. Lastly, passing to the form of the functions included in P (a - a', -/', 7-7, z), we saw that they arise from the association of arbitrary powers of z and 1 - z with the above function in the form a /3' 7' Z (1 -z) P ( a z) and that they lead to a function p a+8 —, +-8-6, +6 ' +s, ^- -e, 7 + y Thus we can make any (the same) change on a and a' and, as they are interchangeable, we can select either for the determinate change; accordingly, we take -a = 0, a'- a, =1 - v, say, as the modified exponents. Similarly, we can make any (the same) change on 7 and y': we take 7-7O0, -7 =v-X —, say. Then the new values of the exponents for oo are /3 a +, =X, say, and /'+a+7y =a+7+l-(a +a'+/3 7+ry ) = 1 144 INTEGRALS OF THE EQUATION OF [49. on reduction: or the exponents are 0, 1-v, for z=O, X, /Y, for z =oo, 0, v-X —/, for z=1. Their sum clearly is unity: moreover, with the preceding hypotheses, the quantities 1 - v, - X, v - X-,/ are not integers. Specialising the last form of the equation by substituting this set of values for a, a', 3, /', y, y', we find the equation, after reduction, to be z ) d2W dw 0, z (1 - z) dZ + {v - (X +- + 1) z} dz which is the differential equation of Gauss's hypergeometric series with elements X, IA, v. Either from the original form of the P-function, or from the resulting form of the equation, the quantities X and Fp are interchangeable. 50. Taking the equation in the more familiar notation (1 - z) d2W + {7 - (a + / + 1) z} d - aw = 0, so that the exponents are 0, 1- y, for z=0; a, /3, for z= o; 0, y - a - 3, for z= 1, we use the preceding method to deduce the well-known set of 24 integrals. Denoting as usual by F(a,,, y, z) the integral which belongs to the exponent zero for the vicinity of z = 0, we have F(a, 3, r, z) = 1 + a 1. ( + 1)... assigning to the integral the value unity when z =. If z (1 - Z)e F (a', /3', y', z) be also an integral, then the exponents for each of the critical points must be the same as above; hence 8, + 1-y' =0, 1-y, for z=0 e, e + '-a'-/f'= 0, y-a- /, for z=, a'-8-e, I/'- -e =a, /3, for z=oo. Apparently there are eight solutions of these equations; but as a and / can be interchanged, and likewise a' and /3', there are only four independent solutions. These are: THE HYPERGEOMETRIC SERIES 145 I. 8=0, e=O; giving a'=a, /'= 3, 7'=y; and the integral is F (a,, 3, y, z); II. 8=1-7y, e=0; giving a'=1+a-, /3'=1+3-7, y'= 2 - y; and the integral is zl-vF(l+a-7, 1+I -y, 2 -, z); III. 8=O = 0= - a - 3; giving a' = - a, '= 7-, =r- ' = y; and the integral is (1 - F ( 7y-a, -y 7-, z); IV. = 1 — y, e = - a -3; giving a' = 1 -,, 3' = 1 - a, ' = 2 — y; on interchanging the first two elements, the integral is z- (1 - Z)-a-P F(l1 - a, 1 - 3, 2 - ry, z). Next, it has been seen (~ 47) that, in the most general case, P-functions can be associated with a given P-function, when the argument of the latter is submitted to any of the six homographic substitutions which interchange 0, 1, oo among one another, provided there is the corresponding interchange of exponents. Taking the substitution z'z= 1, the new arrangement of exponents is a, /3, for z' =0, 0, 7 —a-/, for z'=1, 0, 1-7, for z'=oo; hence, if z' (1 - z') EF(a', ', y', z') is an integral, we must have, 6 + 1-y' = a, 1, for z'=0, e, e+y -a' —3'=O, y-a —3, for z'=1, a' ----e, 3'- -e = 0, 1-7y, for z'=o. Again there are four independent solutions; they are:IX. 8 = a, e = 0; giving a' = a, 3' = 1 + a -, y' =1 + a -3; and the integral is z-aF (, 1 +a-7, +a, 1-); \ ~~~~~z/ F. IV, 10 146 KUMMER'S [50. X. 8=,3, e=0; giving a'=3, 13'=1 + -7y, 7y'=1-a+,3; and the integral is z-iF(f, 1+3-7, l-a+/3, +' ); XI. 8 =, e=y -a- 83; giving a'= - a, ' = 1 - a, '= 1- a+13; on interchanging a' and 13', the integral is ^z- 1 —) F(l-a,y-a, 1-a+/3 ); XII. $=a, e=y-a-83; giving a'= 7y-3, 3'=- 1-3, y'= 1 + a-3; on interchanging a' and 13', the integral is z- (1 —) ' F(1-/, -/3, 1+ -/), ). The remaining four sets, each containing four integrals, and belonging to the substitutions, 1 z z-1 1 = 1-z' - z-1' Z respectively, can be obtained in a similar manner*. They are:V. F(a, /F, a+3-y+l, -); VI. (1 - )l-y F(a -y + 1, /3-y + 1, a + 8 - y + 1, ); VII. 'Y- -_F(y-a, y7-f, y-a-_+l, g); VIII. (s1- )-t -Y y-a-g F(l-, 1 den —, - +, 1); in which set t denotes - z: XIII. F(a, 7E-,3, a-, +1, 0); XIV. F( p a, r-a,,8-a+1, l); XV. (l_ )-I+yr~~(a-_+ l, 1-/3, a-/s + 1,; XVI. (1 )-l+~ F(/-r+ 1, 1-a, /-a + 1, a ); in which set g denotes 1 —: XVII. (-1 f(, -(a,, r-, ); XVIII. (1 - )"F (/3, 7- a, 7, ); * The complete set of expressions, differently obtained and originally due to Kummer, are given in my Treatise on Differential Equations, (2nd ed.), pp. 192 -194; the Roman numbers, used above to specify the cases, are in accord with the numbers there used. 50.] INTEGRALS 147 XIX. - (1 - -)0F(a - +l, 1-,3, 2-y, ); XX. rl(1 - (F(3 _ y + 1, 1 2-7, ); in which set g denotes: and z-1 XXI. (I - )a(a, a-y+l, a+/3- +'l, C); XXII (1-) 0F(3, -7y + 1, a+3 -7y+l, C); XXIII. -a-f (I -0f ) (1 - a, y-a, 7-a-/3-i, + ); XXIV. yv-a-(l_)aF(l- _ -U -,/+ 1, g in which set 4 denotes The preceding investigations have been based upon the assumption, among others, that no one of the quantities 1-,y, y-a-/3, a-/3, is an integer or zero: the determination of the integrals of the differential equation Dw =z (1 - z) w" + y- (a+ 3 + 1) z} w'- aw = 0, when the assumption is not justified, can be effected by the methods of ~~ 36-38. Consider, in particular, the integrals in the vicinity of z=0, when 1 -y is an integer; there are three cases, according as the integer is zero, positive, or negative. We substitute w=c COZ ec +1+... + O+n... in the equation; and we find zDw=0 (0+y - 1) coZ, provided cn(i -6+n)(0+n- 1 +y) =c (0+n-l+a)(+ n- 1 +), so that (n - 1 +a+ 0)...(a + ) (n-1 - + 3 0)...( o+0) (=n + O)...(l+ ) (n - 1 +y +)...(y+) c0 (i) Let l-y=O, so that the indicial equation is 02=0: then the two integrals belong to the index 0, and one of them certainly involves a logarithm; and they are given by [w]0=o,' dj= The former, when we take Co= 1, is F(a, 3, 1, z), with the usual notation for the hypergeometric function; as the coefficients are required for the other integral, we write F(a, 3, 1, z)=-1+K1l+ K22+...+KZ+.... 10-2 148 THE HYPERGEOMETRIC [50. The second integral, when to it we add {+ (a) + + (a)} F (a, p, 1, z), co again being made equal to unity, becomes F(a,, 1, z) log z+ K,n n{ (n +a- 1) + (n +3- 1)-2+ (n)}, n=l where + (m) denotes m- {log n (m)}. (ii) Let 1-y be a positive integer, say p, where p>0. The indicial equation, being 6 (d-p) =0, has its roots equal to p, O. We have _(n- I +a+o0)...(a +0) (n - I ++ )...(+ 0) c Cn~ (n+8)...(li +0) (n-p+ 0)...(li —p-) + c' Of the two integr ls, that, which belongs to the greater of the two exponents, is equal to zP F (a +p, 3+p, 1 +p, z), when we take c0= 0. The other integral may or may not involve logarithms. If it is not to involve logarithms, then, as in ~ 41, the numerator of Cp must vanish when 0=0, so that (p - + a)...a (p - +/)... must vanish: in other words, either a or 3 must be zero or a negative integer not less than y. When this condition is satisfied, the integral belonging to the index zero is effectively a polynomial in z of degree -a or - as the case may be, and it contains a term independent of z. When the preceding condition is not satisfied, the integral certainly involves logarithms. In accordance with ~ 36, we take c = CO, so that zDw= C2 (0 -p) z; and now w C E; (n-1+a+6)...(a+O) (n-1+f3+0)...(13+8) e+n n=o (n + 8)...(1 +0) (n-p+ )...(1 -p+ 8) There are two integrals given by F]dwo [ ]"=O' [L0=0 The first is easily seen to be a constant multiple of zP F (a+p, 3+p, 1 +p, z) thus in effect providing no new integral. The second, after reduction, and making C=1, is (p - 1 + a)...a (p - 1 + 2)...2 p + a) ( + zP F(a +p, / +p, 1 +p, z) log z p-i (n-1 + a)...a (-1 + 3)...3 n=o n! (n-p) (n- 1-p)...a(l-p) +-( _1)i-1 (- +a)...(-1. + a"(- ) " znn: n=., n (p -.(n-p)! 5o.] EQUATION 149 where n,=-i (n- I +a)++- ( - 1 +3- (n) - - + (n-p). (iii) Let 1-y be a negative integer, say -q, where q>0. The indicial equation, being 0 (0+ q)=0, has its roots equal to 0, -q. We have (-1 +a+0 )...(a+0) (- 1 +/3+0)...(3+0) en= (n+8)...(+0) (n+q+ )...(1+q+) 0' The greater of the two exponents is 0; the integral which belongs to it, on making co =1, becomes F(a, 3, 1 +q, z). The integral which belongs to the exponent -q may, or may not, involve logarithms. If it is not to involve logarithms, then, as before, the numerator of Cq must vanish when 0= - q, so that (a- 1)...(a- q) (d- 1)...(- q) must vanish: hence either a or /3 must be a positive integer greater than 0 and less than y(= l+q). When the condition is satisfied, the integral is a polynomial in z-1, beginning with z-q, and ending with z-a or z -p, as the case may be. When the preceding condition is not satisfied, the integral certainly involves logarithms. As before, in accordance with ~ 36, we take co=(0+q)K, so that zD = K0 (0 + q)2 z; and now K (n-l +a+0)...(a+ ) (n-1 +3+0)...(3+. ) +n (n+q+0)...(1 +q+0) (n+0)...(l +08) Two integrals are given by rddz"] [W]= -, [dao=' The first is easily seen to be a constant multiple of F(a, 1,+ 1q, z), so that no new integral is thus provided. The second, after reduction, and making K= 1, is (a -...(a- ) (-1 ).(2- ) F(a, 3, 1 + q, z) log z +q- (n- l +a-q)...(a-q) (n- 1 +3-q)...(q -q) Zn-q n=o n!(n-q)...(l-q) _(-1-)q- 1 (-1 + - q)...(a-q)(n-l + 3-q)..(3.- ) z,, n=q n!(q-1)!(n-q)! where n - a (n - 1 + a - q) + t(n-l+ 1-q - (n) - i (n -q). The integrals are thus obtained in all the cases, when y is an integer. 150 EQUATIONS OF THE SECOND ORDER [50. Similar treatment can be applied to the integrals of the equation, when y-a- 3 is an integer, positive, zero, or negative, contrary to the original hypothesis as to the exponents for z=l; likewise, when a - is an integer, positive, zero, or negative, contrary to the original hypothesis as to the exponents for z=oo. These instances are left as exercises. Note. There is a great amount of literature dealing with the hypergeometric series, with the linear equation which it satisfies, and with the integrals of that equation. The detailed properties of the series and all the associated series are of great importance: but as they are developed, they soon pass beyond the range of illustrating the general theory of linear differential equations, and become the special properties of the particular function. Accordingly, such properties will not here be discussed: they will be found in Klein's lectures Ueber die hypergeometrische Function (Gottingen, 1894), where many references to original authorities will be found. EQUATIONS OF THE SECOND ORDER AND FUCHSIAN TYPE. 51. No equations of the Fuchsian type, other than those already discussed, are made completely determinate merely by the assignment of the singularities and their exponents. It is expedient to consider one or two instances of equations, which shall indicate how far they contain arbitrary elements after singularities and exponents are assigned. Suppose that an equation of the second order has p singularities in the finite part of the plane and has oo for a singularity; the sum of the exponents which belong to these p + 1 singularities is (by Ex. 2, ~ 46) equal to p- 1. Now let a homographic substitution be applied to the independent variable, and let it be chosen so that all the points, congruent to the p + 1 singularities, lie in the finite part of the plane. Thus oo is not a singularity of the transformed equation: there are p + 1, say n, singularities in the finite part of the plane: and the adopted transformation has not affected the exponents, which accordingly are transferred to the respective congruent points. Hence, when an equation of the second order and Fuchsian type has n singularities in the finite part of the plane and when infinity is not a singularity, the sum of the exponents belonging to the n points is equal to n - 2. For 51.] AND FUCHSIAN TYPE 151 such an equation of the second order, let the singularities and their exponents be a1, a2,..., 81, 182, n.., then (ar + iPr)= n - 2. r=1 Let * (Z) = (z -aa) (z - a2)...(( - a.);; then, as the equation is of the second order and as all its integrals are regular, it is of the form w //F1, + F- W = 0, *f. *2~ where F1 and F2 are polynomials in z of orders not higher than n - 1 and 2n - 2 respectively. Also, let F1 A1 A2 An +, - + ---~rz-al z 2 z - an and let F2 = F2 (z) = A "Z2n-2 + B"z2n-3 ~ C"z2n4 + The indicial equation for the point z = a. is O (0 - 1)+ ArO + (a,) =0; 1*'(ar)12 and therefore ar. + fr = 1 - Ar, so that Ar = 1- ar - fir* Hence Y Ar = n - d (ar. + fir) r=L 2 2 and therefore the polynomial F1 is of the form F, = 2zn-1 + lower powers of z. Again, oc is to be an ordinary point of an integral; hence, taking the most general case, we must have an integral Kz K2 152 EQUATIONS OF [51. where K0 is not zero: for otherwise we should have a special limitation that every integral is zero at infinity. Substituting, so as to have the equation identically satisfied, and writing n E ArarK = sc r=l (so that so = 2), we find, as the necessary conditions, 0 = KoA", O= (2-so)K, + K1A" + K, B" + 2A" ar} r=l 0 = (6- 2So + A)K, + K1 (- s + B" + 2A" E ar) n;7 " + K A j "3 Y. a +2 aaa + 2 B" a + Ca", ( \ r==l / r=l and so on. The first gives A"= 0; then the second gives B" = 0; both of these equations leaving Ko and K1 arbitrary. The third equation then gives 2K= s - C"Ko, and so on, in succession. The remaining coefficients K are uniquely determinate; they are linear in K1 and Ko, the various coefficients involving the singularities and their exponents, as well as the coefficients in F2. The equation therefore has z = oo for an ordinary point of its integrals, provided F2 is of order not higher than 2n - 4. The equation can, in this case, be expressed in a different form. Let F2_ (C xl2n +...) X2 n = Pn-4 + x + 2 +.. + n z- a z-a2 z-an where Pn-_ is a polynomial of order n - 4. (Of course, if 2n-4 is less than n, which is the case when n = 3, there is no such polynomial.) As the coefficients in F2 are not subject to any further conditions in connection with the nature of z = oc for the FUCHSIAN TYPE 153 integrals, any values or relations imposed upon \X, X2,..., X, and the coefficients in Pn-4 must be associated with the singularities. The equation now is V q //+ -- r — w + Pn-4 + Xr a- W=0. r=1 z - ar r=1 Z-ar The indicial equation for z = ar is Xr 0 (O -1) + (1- ar- r) 0 -+ (,) = 0, and its roots must be a,., 1r: thus Xr = ar /r ' (ar), and therefore the equation is S I - E Ur - +r W 2p + ar'r) (ar)r 0 r=l Z-ar c r=l - ar It follows that the only coefficients which remain arbitrary are the n- 3 coefficients in the polynomial Pn-, (where n > 4). When the polynomial Pn-4 is arbitrarily taken, the foregoing is the most general form of equation of the second order and of Fuchsian type, which has n assigned singularities in the finite part of the plane with assigned exponents, and has oo for an ordinary point of its integrals. This is the form adopted by Klein*. If a new dependent variable y be introduced, defined by the relation w = y (z - al) (z - a2)P... (z- an)p, then the exponents to which y belongs in the vicinity of ar are ar-p r, 13r-pr, the difference of which is the same as for w; but z =o will have become a singularity, unless P1 + p2+ '. + Pn > 0. Now {(ar-pr) + (fr-pr)}= -2-2 pr; r=l r=l and therefore n 4 {l-(ar-,)-(/3-r)} = 2 + 2, p,.r r=l r=l * Vorlesungen uber lineare Differentialgleichungen der zzeiten Ordnung (G6ttingen, 1894), p. 7. 154 EQUATIONS OF [51. Hence, if z= oo is not to be a singularity, the quantities pi,..., cannot all be chosen so that each of the magnitudes 1 - (r - p,) - (/3 - pr) vanishes. Conversely, if the quantities pr be chosen so that each of these magnitudes vanishes, then z = co has become a singularity of the equation; having regard to the form of w for large values of z, we see that 0 and 1 are the exponents to which y belongs for large values of z; and the differential equation for y is easily seen to be y'hr + Pn-+ - - 1 - (a - I3r) (ar,] =]0 r=l - a, where Pn,, is a polynomial of order n - 3. This equation, however, has n singularities in the finite part of the plane, and a specially limited singularity at z= o: we proceed, in the next paragraph, to the more general case. Note. The indicial equation for z=oo in the case of the equation for w is f ( +1)-q >2 (1-a-3,)= O, r=l that is, b( - 1)=0. The root > = 0 gives an integral of the form Ko(i- +...); 2 z2 and the root b = 1 gives an integral of the form K, (-+z 1 + *+); both of which are holomorphic for large values of [zl, so that all integrals are holomorphic functions of - for large values of lz. In this case, oo is not a singularity of the integrals: it can be regarded as an apparent singularity of the differential equation, and (if we please) we may consider 0 and - 1 as its exponents. Ex. Shew that the preceding equation can be exhibited in the form -+ {( -a, —, '+ a r)2+ 1 w= o, r=1 z — ar r=l(- -- r=l-+ 2 ar FUCHSIAN TYPE 155 where the n constants c,..., c,, satisfy the three relations n /n n nz n Z r= 121 ca, + 21ar.Or =, I Cra,2+2 1 ar Oar 0 r=1 r=l r=1 r=l r=1 and otherwise are arbitrary in the most general case. (Klein.) 52. Now consider the equation of the second order and of Fuchsian type, which has n singularities in the finite part of the plane, say a,, a2,..., an, with exponents a, and /,,..., an and 18n, respectively, and for which co also is a singularity with exponents ca and 3: the exponents being subject to the relation (ar + 8r) = n - I - a - 8 -r=1 Let f denote (z - a,) (z - a2)... (z - an): then the equation is of the form n A G\ W -+ ( r~i ) W/ + # W = 02 where G is a polynomial of order not higher than 2m - 2. When G is divided by *, we have a polynomial of order n -2 and a fractional part: and so we may write G n 2 h'n-2 n-2 +hn-3 nn-3 +..~. + ho + I r=i z - ar The indicial equation for z = ar now is 0 (O - 1) + A,, a + Y'r 0 0 ~Jr' (ar) so that A2. = 1 - ar - 13r, I-r = crI8r#'/ (ar): holding for r'= 1, 2,..., n. The indicial equation for z = oc is + (cS~1) Ar+hn-2 0, r=i so that a+=3 Ar -1, hn2=c2: r=1 the former being satisfied on account of the relation between the exponents. The equation thus is -n Or /38r r=i z -ar W ctazn-2 zn-3 + n ar 18r (ar ),~= o +I{aflz2 + hn,3 z +...+ho+ 21 C) T ~~~~~r=i z - a 156 A NORMAL FORM OF EQUATIONS [52. the coefficients ho, hA,..., hn_- being independent of the singularities and their exponents. When a new dependent variable y is defined by the transformation w = (z - a1) (z - a2)2... (z- a)ay, then the exponents of y for ar are 0 and 3r- ar, say 0 and Xr, this holding for r = 1, 2,..., n: and its exponents for oc are n n a+ $ ar, /3+ X a., r=l r=l =o, ' say: where n o-+T+ X, =n-1. r=l The function y is, in general character, similar to w: it has the same singularities as w, and it is regular in the vicinity of each of them but with altered exponents: and it thus satisfies an equation of the second order and Fuchsian type, which (after the earlier investigation) is 1 —X, Tn-2 + k3zn- +.. + k y"+ `C y-+ + Y =1 z- a, (z-a1)(z-a2)... (z-an) y = where kn_3,..., ko are independent of the singularities and their exponents *. This transformation of an equation,, w w+ w w=0 to an equation y + G yn- + Gn2 O, where FP_-, F _-2, G_,, G-Gn are polynomials of order indicated by their subscript index, appears to have been given first by Fuchs+. The simplest example of importance occurs for n= 2, when the hypergeometric equation is once more obtained. 53. It is well known that, when y is determined by the equation y" +Py' + Q=0, * The equation for y can be obtained by the direct substitution of the expression for w in the earlier differential equation for w. When reduction takes place, there are n - 2 linear homogeneous relations between the constants h and k. t Heffter, Einleitung in die Theorie der linearen Differentialgleichungen, (1894), p. 224. z53.] OF FUCHSIAN TYPE 157 and a new variable Y is introduced by the relation yefPdz y, the differential equation for Y is d2Y dz+ IY=O, where dP I= Q- _ -I p2. In the case of the preceding equation, the relation between y and Y is Y= {II (Z - a7,)(1l-r)} y, so that Y is a regular integral in the vicinity of all the singularities and of oo, the exponents being (1- Xr), (1 +Xr), for z=a,., (r=,...,n), and ( —1+< — ), ~(-1- 4r+T), for z=oo. From the form of P and Q, it is easy to see that I~2 = polynomial of order 2n - 2 = P + E- + r r=l Z - atj where Pn-2 is a polynomial of order n - 2, say Pn-, = CZf-2 + ln-3 zn-3 +... + l. In order that ~ (1 - X), - (1 + Xr) may be the exponents of ar for the equation Y" + I= 0, they must be the roots of Br 0 (O- 1)+ (= 4' (ar) hence B,. = (- ) (). In order that 2 (- +o- ), - (- 1 - c + r) may be the exponents of co for the same differential equation, they must be the roots of +(~ +l)+=o: hence C= 1 -(- -T)2}. 158 KLEIN'S NORMAL FORM [53. The remaining constants l0, 1,..., ln-3 are expressible as homogeneous linear functions of ko, k,,..., kn-3., so that they are independent of the singularities and the exponents: and thus the equation is d2Y + 1 - ( T)2 -2 + 1 -_3 Zn-3 +.. + l1 + 1 1- X2) +r (a r) r=1 Z - ar COROLLARY. For the original equation, oo was a singularity of the integrals with exponents ac and T. If it were only an apparent singularity of the original equation, so that the integrals are regular for large values of Izl, then we have the case indicated in the Note, ~ 51, so that we can take a, = 0, - 1. The modified equation now is d +Y 17 = 0. gz t-3 Z + -... + lo + ( dz"2 ^ r=1 Z -r ar For this differential equation and its integrals, the exponents to which the integrals belong in the vicinity of ar are ' (1 - Xr), (1 + Xr); but 3c is now a singularity of the integrals, and the exponents for z= c are 0, - 1, so that z = cc is a simple zero of one of the linearly independent integrals of the modified equation. dY. These forms of the equation, from which the term in dY is absent, are the normal forms used by Klein. The simplest example of the class of equations, not made entirely determinate by the assignment of the singularities and their exponents, occurs when there are three singularities in the finite part of the plane and o also is a singularity. By a homographic transformation of the variable, two of the singularities can be made to occur at 0 and 1, and oo can be left unaltered; let a denote the remaining singularity. Let the exponents be 0, 1-X0for z=O; 0, 1-Al for z=; 0, X for z=a; (r, T for z= oo; where T +T-A-X1 + X=0. Then the differential equation is 53.] EXAMPLES 159 where q is the (sole) arbitrary constant, left undetermined by the assigned properties. The integral of this equation, which is regular in the vicinity of z=0 and belongs to the index 0, is denoted* by F(a, q; A(, 7, ), xA; z). If a= 1, q= 1, the equation degenerates into that of a Gauss's hypergeometric series: likewise if a = 0, q = 0. Ex. 1. Verify that, when a=, the group of substitutions i 1 1 1 1) 1z Z, 1 - Z, 2 2 _ 2 z z I z-1' z-1' z Z-_ interchanges among themselves the four points 0, 2, 1, oo. Prove that, when a= -1 and when a 2, there is in each case a corresponding group of eight substitutions interchanging the points 0, 1, a, co among themselves: and that, when a= (1 +i/3) and when a= (1 - i/3), there is in each case a corresponding group of twelve substitutions. Construct these groups. (Heun.) Ex. 2. Prove that there are eight integrals of Heun's equation of the form z' (z — 1) (z- a)7 F(a, q; I' 0, X,; Z), which are regular in the vicinity of the origin and have the same exponents as F(a, q; A-, r, X0, X; z). Hence construct a set of 64 integrals for the equation when a=', which correspond to Kummer's set of 24 integrals for the hypergeometric series. Indicate the corresponding results when a=-1, 2, ~(1 I+i/3), ~ (1- i/3). Ex. 3. A homogeneous linear differential equation of order n is to have n singularities a,, a2,..., a, in the finite part of the plane and also to have oo for a singularity: the integrals are to be regular in the vicinity of each of the singularities, and the exponents of ar are to be 0, 1..., n-2, ar (for r=l,..., n), while the exponents of oo are to be 0, 1,..., n-2, a, so that a+ 2 a,=(n-1)2.?=1 Shew that the differential equation is dnw n dn- sw ~ () d -n- + E( 0) dzn ' = where + (z) = (z-a,) (z - a2)...(z-a,,), the coefficient Es (z) is a polynomial in z of order not greater than s, (for s=l,..., n), and El(z)= (a,.-+l) ( ). r=l ar- z (Pochhammer.) * Heun, Math. Ann., t. xxxIII (1889), pp. 161-179, who has developed some of the properties of these equations, and has applied them, in another memoir (l.c., pp. 180-196), to Lam6's functions. 160 EQUATIONS OF FUCHSIAN TYPE [54. EQUATIONS IN MATHEMATICAL PHYSICS AND EQUATIONS OF FUCHSIAN TYPE. 54. These equations of Fuchsian type include many of the differential equations of the second order that occur in mathematical physics; sometimes such an equation is explicitly of Fuchsian type, sometimes it is a limiting form of an equation of Fuchsian type. One such example has already been indicated, in Legendre's differential equation (Ex. 1, ~ 46). Another rises from a transformation of Lame's differential equation which (~ 148) is of the form 1 d2W + d A + (z) + B = O, wdz2 where A and B are constants*. Writing (z)= x, so that x is a new independent variable, we have d2w + I- ( 2 \A dw Ax+B dw + 4-2+ 2 + ~ — 7 7 W = 0. dx2 - x - x-ea dex (x- e) (x - e2) (x - en) The singularities of this equation are el, e2, e3, oo; the exponents to which the integrals belong in the vicinity of el, e2, e3 are 0 and I, in each case; the exponents, to which they belong for large values of x, are the roots of the equation p(p + 1)-p + A = 0. The new equation is of Fuchsian type: and, in this form, it is frequently called Lame"s equation. An equation, similar to Lame's equation, but having n singularities in the finite part of the plane, each of them with 0 and 2 as their exponents, as well as z = oo with exponents a and 3, such that (~ 52) a+/3= n —1, is sometimes called Lame's generalised equation. By ~ 52, it is of the form n I w~+w' Z 2-2 + w=O, r=l z-a (-at) r=iZar l(z- ar) r=l * This is the general form; the value -n(n+l) is assigned (I.c.) to A, in order to have those cases of the general form which possess a uniform integral. IN MATHEMATICAL PHYSICS16 161 where Gn2,- is a polynomial of order n - 2, the highest, term in which is aq3n-2 55. The equation of Fuchsian type which, next after the equation determined by Riemann's P-function, appears to be of most interest is that for which there are five singularities in the finite part of the plane, while z = oc is an ordinary point.' The interest is caused by a theorem*, due to B~cher, to the effect that when the five points are made to coalesce in all possible ways, each limiting form of the equation contains, or is equivalent to, one of the linear equations of mathematical physics. Let the points be a., a2, a., a4, a., with indices ar and fi, for r = 1,2, 3,4, 5; then and the equation (p. 1053) is wV"/ + w' - ar,-r + i+I W*(r)l =O, r=1 z- ar # ~ r=1 z -ar 5 where 4' F (z - ar), and PI is a linear polynomial Ax + B. The r=1 substantiall y distinct modes of coalescence are: (i), a4 and a, into one point; (ii), a,, and a, into one point, a4 and a, into another; (iii), a3, a4, a, into one point; (iv), a, and a, into one point, a3, a4, a, into another; (v), a2, a3, a4, a. into one point; (vi), all five into one point; and the various cases will be considered in turn. Case (i). Let the indices for a a2, a, be made 0, -1 for each point; then, as */' (a4) = 0, ifr (a,) =0 in' the present case, and 1 - 0a4 - fi4 + 1 -r -,8 = 1 *Ueber die Reihenentwickelungen der Potentialtheorie, Gbitt. gekriinte Preisschrift, (1891), P. 44; and a separate book under the same title, p. 193. See also Klein, Vorlesungent iber lineare Differentialgleiehungen der zweiten Ordnung, (1894), p. 40. F. IV. 11 162 BOCHER'S THEOREM ON [55. the equation is 4 1 w" +W EI _2 - + w=O0. r=l z - ar - ( - a)(z -a2) (z- 3) (z-a4)2 Write z - a4 =, (ar-a4) e= 1, for r= 1,2,3; the equation becomes d2w dw 1) Cx+D w O X2- + - - dx2 d - x - e+.- ( xee )x - ( e) in effect, the preceding ungeneralised Lame's equation. Case (ii). The equation becomes + / V + z - a, z - a2 z -a4 + (z- ) )2W (Z-a4)2 {.1 + (a - a2)2 (a - a4)2} =, -(z - a,) (z - as)2 (z- a4)~ P1 z - al after coalescence of the points, where 1-a' -/3' =2- ~2-2-a3-/33, 1 - a" - 3" = 2 -a4- - 8 - a, - -25, and therefore a, + a' + +' +a'" +" = 1. Writing 0 = (z- a) (z - a2) (z - a4), we have the coefficient of w. W in the form P1 (z - a,) + Gal,1 (a1 - a2)2 (a, - a4)2 (z - a,) (z - a2) (z - a4) a1,/O' (a,) Qi z - al ( - a2) (z - a4)' where Q1, like P1, is an arbitrary linear polynomial. Thus Q1 contains two arbitrary coefficients; these can be determined so that P a '/3'0' (a2) + a"8/3"' (a4) (z - a2) (z- a4) z - a2 z - a4 and then the equation becomes W"i+ Wt ll al-1 + a- P' I 1- " - I"w + W 1-a -— f+ +z - a, z - a2 z a4 w a1,0' (ali) a' 0' (a2) +a"-/"' (a4)} = + z-aO - 2 z4 0 z - a, z - a2 " - a4 EQUATIONS OF FUCHSIAN TYPE 163 Owing to the form of 0 and the relation I (a + 8) = 1, this is the equation of Riemann's P-function (~ 49). When we write a, =1; a,=-1; a,=oo; all a, = / 0, 0; a', ' =o0, 0; a", /3"=-n, n +1; the equation becomes 1) = 0, that is, (1 - Z2) WI'"- 2zw' n (n + 1) w = 0, which is Legendre's equation. Case (iii). Let a,, 3= 0,; a2, 18=0,; so that 1-a3 - /3 + 1-a4 - /4 +1 - a - 15 = 1. After the coalescence of the points, the equation is I/ I I I I PIW 0 +W 2 + + + w0 z —a, z-a2 z - a3 (z - a,) (z- a2) x - a,)3 where P1 is a linear polynomial, say {A (z - a3) + B} (a3 - a,)(a8 - a2). Now let z - a3,=; after some easy reduction, the equation becomes d dw I dX2 w d 1 + 1 x + --- x: + - a3i - a, a3- a A ~Bx +- w =0. + a. - a) a3 - a) Let a,=oc, a,- a=-1; the equation is d2W 2x-1 dw A~Bx - - j + 2 ---- w= 0. dX22:(X 1) d+ ( -1) Writing x = sin2 t, we have d2W dt2 4w(A +Bsin2t)=0, 11-2 164 LIMITING FORMS OF AN [55. which is known* as the equation of the elliptic cylinder. This equation will be discussed hereafter (~~ 138-140). Case (iv). Let a,, /3 = 0, I; a3, /2 = 0, I; so that, as in the last case, 1- a3 - 833+ I - a4 -. /4+1- + a5 - /32=1. After coalescence of the points, the equation is w"w' +a + - a) w = 0. Z-al~ a z-a. (z - a, )2 (Z - a,)' Let z-a3 =-, Pi= la(z- a3)~}(a, - a), c (a, - a,)=1; then the equation becomes d2W 1 dw a~/38x d' + w=O; dX2 X-C (x - C)2 or, taking X - C = we have d2w 1 dw (4ac' ~- +W(-~ =0, dy2 y dy Y2 which includes Bessel's equation, sometimes called the equation of the circular cylinder. Case (v). Let a1, 8, = 0, ~; then 5 (I (1rI3r), r=2 and the equation, after coalescence of the points, becomes,7 P,~P W +W 2+ + W = 0. w - a, i_ -a2) (z - a,) (z - a,)4 Let z-a.=, Pi={tt(z-a2)+/3}(a2-a.), b(a, —a)=1; then the equation is d2w dw cc + a+/ - + - __2+ w = 0. dX2 dxwx-b + * Heine, Kugelfunctionen, t. i, p. 404. EQUATION OF FUCHSIAN TYPE 165 Writing x-b =2, the equation becomes d2w d2 +- 4w (a + f3b + 3y2)= 0, dy2 which is the equation* of the parabolic cylinder. Case (vi). The equation is 2 PI w" - + - = 0: z - a (z - a)5 when we take 1 z-a=-, P= a(z - a) +, the equation becomes d2W d2 + w(a +, x)= 0. This corresponds to no particular equation in mathematical physics: it will be recognised as a very special instance of equations most simply integrated by definite integralst. Ex. Discuss, in a similar manner, the limiting forms which are obtained when the singularities of (i) the equation determined by Riemann's P-function, (ii) Lame's equation, expressed as an equation of Fuchsian type, are made to coalesce in the various ways that are possible. EQUATIONS WITH INTEGRALS THAT ARE POLYNOMIALS. 56. There is one simple class of integrals which obey the condition of being everywhere regular, so that their differential equations are of the Fuchsian type; it is the class constituted by functions which are algebraic. We shall, however, reserve the discussion of linear differential equations having algebraic integrals until the next chapter; and we proceed to a brief discussion of a more limited question. * Weber, Math. Ann., t. I, p. 33. + See Ch. vii of my Treatise on Differential Equations. 166 EQUATIONS WITH [56. We have seen that an equation of the second order and of Fuchsian type can be transformed to Dy = Iy" + Gn-,y' + Gn2y = 0. Its integrals are regular in the vicinity of each of n singularities and of infinity; the question arises whether the coefficients in the polynomials G,_- and G,-2 can be chosen so that one integral of the equation at least shall be, not merely free from logarithms or even algebraic, but actually a polynomial in z. This question has been answered by Heine*; the result is that Gn-1 can be taken arbitrarily, and G,-2 has then a limited number of determinations. If the above equation, in which f = (z - al) (z - a2)... (z - an), Gn-l = cozn-1 + zn-2 +... + Cn-2Z + Cn-1, Gn-2 = kozn-2 + k -3 +... + kn_3 + kin2, is satisfied by a polynomial of order m, say by y = gozm + g1zm-1 +... + gm-1Z + gi, then Dy = ( x, 1)m+n-2 = O, so that there are m + n - 1 relations among constants. The form of these relations shews that g1, g2,..., gm are multiples of go: to express these multiples, m of the relations are required, and when the values obtained are substituted in the remainder, we have n-1 relations left, involving the constants c and k. Assuming the points a1, a2,..., an arbitrarily taken, and the coefficients Co, c.,...,n- arbitrarily assigned, we shall have these n-1 relations independent of one another, and therefore sufficient for the determination of the n - 1 constants ko, k1,..., kn-2. The first of these relations is rm (m - 1)+ c + o + ko = 0, so that ko is uniquely determinate. Denoting by [kl, C2,..., kr]r the generic expression of a function of kc, k2,..., Ik, which is polynomial in those quantities, and the terms of highest weight in * Heine, Kugelfunctionen, t. I, p. 473. 56.] POLYNOMIAL INTEGRALS 167 which are of weight r, when weights 1, 2,..., n - 2 are assigned to k1, C2,..., kC_2, we have, from the m relations next after the first, gr =go[kIc, Lk,..., k]i, for r = 1, 2,..., m. When these are substituted in the remaining n - 2 relations, we have [kl k 2,...,. n-\-2]m+s = 0, for s=1, 2,..., n -2. These determine the n-2 constants ki, k2,..., kn_2; the number of determinations may be obtained as follows. Writing kl-= X, k2 =X22, k3 = 33,..., the equations become n- 2 equations to determine n - 2 quantities Xi, X2,..., Xn-2. In these quantities, the equations are of degrees n + 1, m+ 2,..., m + n- 2, respectively; and therefore the number of sets of values for 1, X 2 >,.., n-2 is (m+ 1)(m + 2)... (m+n-2). But the same value of k2 is given by two values of x2, independently of the other constants k; so that the sets of values of xa, x2,..., xn-2 must range themselves in twos on this account. Similarly, the same value of k3 is given by three values of x3, independently of the other constants k; hence the arranged sets of values must further range themselves in threes, on account of k3. And so on, up to k,-2. Hence, finally, the number of sets of values of k1,..., k_-2 is (m + 1)(m + 2)... (m + n- 2) 2.3...n-2 (m +n - 2)! m! (n - 2)! which therefore is the number of different quantities Gn-2 permitting the equation Try" + G-_y' + G_-2y = 0 to possess* a polynomial integral of degree m. * In connection with these equations, a memoir by Humbert, Journ. de l'Ecole Polytechnique, t. xxix (1880), pp. 207-220, may be consulted. 168 POLYNOMIAL INTEGRALS [56. This result is of importance, as being related to those special forms of Lame's differential equation which possess an integral expressible as a polynomial in an appropriate variable. This polynomial can be taken as one of the regular integrals belonging to each of the singularities; the other regular integral belonging to any singularity is, in general, a transcendental function and, in general, it involves a logarithm in its expression. Ex. 1. Shew that a linear equation of the third order, having all its integrals regular, can, by appropriate transformation of its dependent variable, be changed to the form +#yt"' P/' + y' + y'+Ry=0, where = (z- a,) (z - a*)...(Z- an), al, a2,..., an being all the singularities in the finite part of the z-plane, and where P, Q, R are polynomial functions in z of degrees n-1, 2n- 2, 2n- -3 respectively. Shew that, if P and Q be arbitrarily chosen, R can be determined so that one integral of the equation is a polynomial in z; and prove that the number of distinct values of R is (m+n2n -3)! m!(2n-3)! ' where m is the degree of the polynomial integral. Ex. 2. Determine the conditions to be satisfied if y"' + Gnly'-+ G_-2Y-O has two distinct polynomials as integrals, so that every integral is a polynomial. Ex. 3. Determine how far the constants in the equation +2y' ++Py' + Qy' + Ry = may be assumed arbitrarily if the equation is to possess two polynomial integrals. Ex. 4. Prove that the equation f (x)dj2y '(x) y- I n(n+l)x+h)-= where n is an integer, f (x) = x3 + ax2 + bx + c, and a, b, c are constants, admits of two integrals whose product is a polynomial in x. Ex. 5. Shew that the only cases, in which the differential equation of the hypergeometric series x (1-x) ad +{y-(a+/ l) x} -ay= EXAMPLES 169 possesses two integrals whose product is a polynomial in x of degree n, are as follows. If n is an even integer, then either a =-n; or 3= -n; or a+43= -n, and y=~, or -~, or -,..., or -n+~. If n is an odd integer, then either a= - n and y=, or -, or -, or - 1, or 3, or 3 - 1,..., or 3- (n -1); or f3=- n and y=1 or -, or -,..., or - +1, or a, or a- a- (n-; r a +/=r -n, and y=~, or -~, or -,., or -n+~. (Markoff.) Ex. 6. Shew that, if the square root of a polynomial of degree m can be an integral of the equation d2y s=n (1- dy dV2 s=+l X-e d (X + - _ 2 X y = O, + - n (X - e,)2 s=- 1 where the exponents X and /z are subject to the usual relation, one of the exponents X8, /s, say XA, must be half of a non-negative integer, this holding for each value of s; also rn m- 2X must be a non-negative integer; and one exponent of the singularity at infinity must be equal to - ~m. If these conditions are satisfied, how many such equations exist? (van Vleck.) Ex. 7. If the differential equation d2y n ar dy+ f ( y=x d2+ 2 _ d Y dx2 r=l -er dx II ( - er,) r=l where + (x) is a polynomial, the constants a are real and positive, and the constants e are real and distinct from one another, be satisfied by a polynomial q (x), then all the roots of <p (x) are real, and no root is less than the least or greater than the greatest of the quantities e. (Stieltjes; Bocher.) EQUATIONS WITH RATIONAL INTEGRALS. 57. The investigation in ~ 56 suggests another question: what are those linear equations, all the integrals of which are rational meromorphic functions of z? Let a1,..., am be the singularities in the finite part of the plane; let a1r, ar,..., cnr be the roots of the indicial equation for ar; and let /,...,, be the roots of the indicial equation for z =. If every integral is to be a rational function of z, all the roots air, a2r,,, anr must be integers; as no integral is to involve a logarithm, no two of them may be equal. Let the arrangement 170 EQUATIONS WITH [57. of these roots be in decreasing order of the integers. The integral belonging to the index ar involves no logarithms; in order that the integrals belonging to the indices a2r, asr,..., anr respectively may involve no logarithms, 1 +2+... +(n-l), that is, 2n ( - 1), conditions in all must be satisfied, these conditions being as set out in ~ 41. Corresponding conditions hold for each of the singularities, and also for z = oo; so that there are in (n - 1) (nb + 1) conditions of relation among the constants of the equation, in addition to the necessity that the indicial equation of each singularity shall have unequal integers for its roots. These conditions are certainly necessary; they are also sufficient to secure that any integral of the equation is a rational function of z. For considering the vicinity of a,, each integral in that vicinity is of the form (z - ar)"'aPm (z - a), where anc is the least of the roots of the indicial equation, and Pm (z - a.) is holomorphic in the vicinity of ar, for m = 1,..., n; when m = n, P (z - ar) does not vanish, and for all other values of mn it does vanish. If then a,. be zero or positive, the point z = ar is an ordinary point for every integral in the vicinity of ar; if a,,n be negative, then ar is a pole of some integral, and it may be a pole of several or of all. As this holds in the vicinity of each of the singularities and of z = oo, it follows that, in the vicinity of every singularity of the equation, including z = o, every integral is uniform and has that singularity either for an ordinary point or a pole; moreover, every integral is synectic in the vicinity of every other point: hence* the integral is a rational function, which is a polynomial if co be the only pole. Thus the conditions are necessary and sufficient. It has been seen that the indicial equation for each singularity of the differential equation must have unequal integers for its roots. When these are assigned arbitrarily, subject to the one relation (Ex. 2, ~ 46) which they are bound to satisfy, they amount * T. F., ~ 48. RATIONAL INTEGRALS 171 to (m + 1) n -1 conditions; so that the total number of necessary conditions is 2n (n - 1) (m + 1) + (m + 1) n -1 = = (n + 1)(m+ 1)- 1. If such equations are being constructed, they are necessarily of the form dnw G1 d'-w G dzw -1 G dn_ k. +... + nW = 0, dz1'4 dzn-l +n where q = (z - a)... (z - an), and Go. is a polynomial of order not greater than r (m - 1), for r = 1,..., n. Hence the total number of disposable constants is m, from the positions of the singularities, n + E {r (m - 1)+ 1}, from the constant coefficients in G,..., Gn, r= 1 that is, l-n (n + 1) (m - 1) + n + m constants in all; and therefore, in order that the equations may exist, we must have n (n + 1) (m - 1)+ n+m> I n (n+ 1) (m+ 1)-1, so that m n2 -. In obtaining this result, an arbitrary assignment of unequal integers as roots of the indicial equations has been made: and it has been assumed that these conditions are independent of the necessary conditions attaching to the coefficients, in order that the integrals of the equation may be free from logarithms. It may, however, happen that a particular assignment does not leave all these conditions independent of one another, so that we might have 4n (n + l)(m - 1) +n + m = n (n 1)(m+ 1) - X, and therefore m = n2 -1-, and still have the equation determinate. An instance is furnished by the equation 2y" - 2xy' + 2y = 0, 172 EQUATIONS WITH [5'1. which, although it has only one singularity in the finite part of the plane, so that m = 1, n = 2, has an integral AX2 + Bx. For the most general case, however, we have rn > n2 - 1. Ex. 1. Investigate all the cases in which the differential equation of the hypergeometric series has every integral a rational function of the independent variable. Ex. 2. When the equation is of the second order, and all the assignments of integer roots are quite general, the smallest value of m is 3. Let the singularities be a,,..., a,, with exponents a,, 81; a2, 02;...; a,,.3,,,; and let the exponents for z = oo be a, 3. Choosing in each case the smaller of the two indices a,. and /3, let it be a,, for r= 1,..., m; then writing m m X,=13r - ar, a+ 2 ar=a, 0 3+ Y ar = r=l r=1 we have (~ 52) a- + + 2 Xr=M - 1, r=1 which is the necessary relation among the exponents. Writing w= (z - a,)"' (z - a2)a2...(Z_ -a,) a Y, so that y also is a rational function of z, our equation in y becomes l-X I qr(7.Zm - 2 + km- 1zm- 3 +... + ko r=1 z-ara (z -,) (z - a2...(z - am) =O say M l-X CG(z) Dy =y + 2 'Y + ~C Y = 0) r=lz - ar (z) and here the integers X, 2,..., X, are, each of them, equal to or greater than unity. Substituting, in the vicinity of ar, the expression y=c, (z-a,.)0+c1 (z - ar)o+... +c,, (z- ar/~ n+..., we have (Z - ar)2 Dy= c00 (0 - h,) z8 provided c1(0+1)(0+1-Xr)+c0 { - + ' 0Y a,, -a,. 4r'(ar)) and C (ar) C,, (0n (0+n/ \v 7- - X.)/T "+ - (a,.) __ -_ I O+n-2 0 ) +Z n-1 (c 1(+n- 1)+c,-, (a2 -ar)"of s a. - a, a. - ar-aa, - a,)n - G (a,) 1 Cn-3 +c0o _ - 2 -2+__ u- - 01 s +'(a.) a. - ar a. - ar a r and the summation for s is for s =1,..., m except s = r. As Xr is a positive integer, and thus is the greater root of the modified indicial equation, there is 57.] RATIONAL INTEGRALS 173 one regular integral belonging to the exponent X,, which is a constant multiple of (z - a) {1 + (- a) + (z - a)2 +...}, -= Y say, where yv=c -Co, when 0=X,. When we write f(0)= 0 (O - Xr), and solve the equations for c1, c2,..., we find h, (0) =n f(+l)...f (+,) CO We know (~ 41) that there is a single condition to be satisfied in order that the integral belonging to the exponent 0 may be free from logarithms; as f(O0+n) vanishes to the first order for 0=0 when n=Xr, the condition is h, (0)=0. There is a corresponding condition for each of the singularities and for z= =o; so that we have mn+1 conditions, which involve the arbitrary constants ko,..., km-3, and the positions of the singularities, as well as the assigned integers Xi,..., Xm, a, r. Keeping the latter arbitrary, we see that there must be at least three singularities in the finite part of the plane: when there are only three, we obtain a limited number of determinations of the equation; if there are 3+p, then p elements are left arbitrary among an otherwise limited number of determinations of the equation*. As the equation is of the second order, it is possible to proceed otherwise. Assuming that the integral Y which belongs to the exponent X, of the singularity a, is known, and denoting by Z the integral which belongs to the exponent 0 of the same singularity, we have YZ"- Y"Z (YZ'- Y'Z) 1 - =0 -.=1 z-ar so that YZ'- Y'Z=A I (z- a,)' 1, r=1 and therefore d )=A 1 - When the right-hand side is expanded in powers of z-ar, the first term involves (z - a,)-1 r, that is, the index is negative. If Z is to be free from logarithms, the term in 1 in this expansion must have its coefficient equal Z - aC to zero-a condition which must be the equivalent of /h (0)=0. * The hypergeometric case indicated in the preceding example is given by X w =e n t a t n n f a..., G g = (z- i... (-am), which will be found to satisfy the conditions for as,..., am given in the text. CHAPTER V. LINEAR EQUATIONS OF THE SECOND AND THE THIRD ORDERS POSSESSING ALGEBRAIC INTEGRALS. 58. THE general form of equation, having all its integrals regular in the vicinity of each of the singularities (including o ), has been obtained; in the vicinity of a singularity a, each such integral is of the form (z - a) [0g + ~) log (z - a) + 0 {log (z - a)}2 +.. +,, {log (z - a)}], where each of the functions 00, b1,...,,, is holomorphic at and near a. In general, each of the functions b is a transcendental function in the domain of a: they are polynomials only when special relations among the coefficients are satisfied. When attention is paid to the aggregate of the integrals so obtained, it is to be noted that the branches of a function defined by means of an algebraic equation belong to this class. If algebraic functions are to be integrals of the differential equation, they constitute a special class; special relations among coefficients of the differential equation must then be satisfied, and, it may be, special restrictions must be imposed upon its form. Accordingly, we proceed to consider those linear equations whose integrals are algebraic functions, that is, functions of z defined by an algebraic equation between w and z. It has already been proved (~ 17) that each root of such an algebraic equation of any degree in w satisfies a homogeneous linear differential equation, the coefficients of which are rational functions of z. If the algebraic equation were resoluble into a number of other algebraic equations, necessarily of lower degree, each such component equation would lead to its own differential equation of correspondingly lower order; accordingly, we shall assume that the algebraic equation is irre 58.] LINEAR SUBSTITUTIONS 175 soluble and proceed to consider linear differential equations whose integrals are the roots of an algebraic equation. In the most general case, the degree of the algebraic equation is equal to the order of the differential equation: in particular cases (~ 17, Note 1) it can be greater than the order: and as we seek algebraic integrals, it may be expected that these particular cases will occur. The investigation can be connected with an equivalent problem that arises in a different range of ideas. It has been proved that, given a fundamental system w1, w2,..., wn of integrals of a linear equation of order m, the effect upon the system, caused by the description of a closed path enclosing one or more of the singularities, is to replace the system by another of the form W1 = all WI + C12 2 + "- + alm W M..,..........................o........... Wm = aW miW + Oa2W2 +... + OnmmWrn say (W1..., Wm) S(W,..., =, ), where S denotes a linear substitution. By making the independent variable describe an unlimited number of contours any number of times, we may obtain an unlimited number of linear substitutions; and so each integral could, in that case, be made to have an unlimited number of values. If, however, the fundamental system is equivalent to the m roots of an algebraic equation, then each of the integrals can acquire only a limited number of values at a point which are distinct from one another: that is, there can be only a limited number of substitutions in the aggregate. When therefore we know all the groups of linear substitutions in mn variables which are of finite order, only those linear differential equations which possess such groups need be considered. Accordingly, if we proceed by this method, it is necessary to construct the finite groups of linear substitutions. Further, it is clear that the investigation can be associated with the theory of invariantive forms; for the relations between w1l,.., W,' and wI,..., W constitute a linear transformation of the type under which these invariantive forms persist. Indeed, it was by this association with binary, ternary, and quaternary forms that the earliest results, relating to linear equations of the orders two, three, and four, were obtained. Some brief indications of this method will be given later (~ 69-72). 176 EQUATIONS OF THE SECOND ORDER [I59. KLEIN'S METHOD FOR EQUATIONS OF THE SECOND ORDER. 59. The determination of linear equations of the second order, whose integrals are everywhere algebraic, is effected by Klein*, by a special method that associates it with the finite groups of linear substitutions of two homogeneous variables. Let w, and w2 denote a fundamental system of integrals for the differential equation; and let W1 = awI + /3w2, Wt = YW1 + S, be any one of the linear substitutions, representing the change made upon the fundamental system by the description of a closed path. Then taking Wl W2 the quotient of two algebraic integrals, so that s itself is an algebraic function, we have W, as +/3 W~- rs + 8; thus s is subject to a homographic substitution. Accordingly, the determination of the finite groups of linear substitutions in the present case is effectively the determination of the finite groups of homographic substitutions. Let any such group containing N substitutions be represented by 0(S), 1 (),...., * _-1 (S), and let o, (s)= s, the identical substitution: every possible combination of these substitutions can be expressed as some one of the members of the group. Take a couple of arbitrary constants a and b, subject solely to the negative restrictions that a is not equal to r (b) and b is not equal to s (a), for any of the values 0, 1,..., N - of r and of s; and form the equation #o (s)- a # (s)-a *-l, (s)-a X_ o (O) -b ' (s)-) - b.._1 (s - b * Math. Ann., t. xi (1877), pp. 115-118, ib., t. xii (1877), pp. 167-179; Vorlesungen iiber das Ikosaeder, (Leipzig, Teubner, 1884), pp. 115-123. WITH ALGEBRAIC INTEGRALS 177 which is an algebraic equation of degree N in s. It is unaltered when s is submitted to any of the substitutions of the group; for such a substitution only effects a permutation of the various N fractions on the left-hand side among one another. Hence, if any root s be known, all the N roots can be derived from it by submitting it to the N substitutions of the group in turn. For quite general values of X, the N roots of the equation are distinct; but it can happen that, for particular values of X, a repeated root arises, of multiplicity v. From the nature of the equation in relation to the group of substitutions, it follows that each distinct root is of multiplicity v, so that there are N v distinct roots. To consider the effect of this property of the equation, let the latter be changed so that the numerator and denominator are multiplied by the denominators of ifr(s),..., "A-l(s). It thus can be expressed in the form G (s, a) - G (s, b) where G (s, a) is a polynomial in s of degree N, the coefficients being functions of a, and G (s, b) is a similar polynomial, its coefficients being the same functions of b. Let X1 be a value of X, such that s = a- is a root of multiplicity v, when X = XI; then the equation G(s, a) G(a1, a) G(s, b) G(a,l b) has N roots each of multiplicity v, when X = X1. But each such root is a root of multiplicity 1 of the equation root is a root of multiplicity P, - 1 of the equation d fG(s, a) G(a-, a)) ds G(s, b) G (a-,, b) that is, of the equation A (s) (, b)dG (s, a), dG(s, b) A(s)= G(s, b) - G(s, a) -0 N as there are - such roots, it follows that these repeated roots account for N(v- 1) VI 12 F. IV. 178 KLEIN'S METHOD FOR [59. of the roots of this derived equation. Moreover, we then have 4 X - X, G (s, b) G (-, b) where I3) is a polynomial in s of degree -. Let X2 be another value of X, such that s = o-2 is a root of the equation of multiplicity v, when X = X2. A precisely similar argument shews that each distinct root of the equation is of multiplicity v2; that there are N v2 distinct roots; that each such root is of multiplicity 2 - 1 for the equation A (s)= 0; that these roots account for N _x,_ 1) -(V2 -) V2 of the roots of the derived equation; and that we have G(s, b)G(o2, b) X 2 N where (2 is a polynomial in s of degree -. /2 Proceeding in this way with the various values of X that lead to multiple roots of the initial equation, we shall exhaust all the roots of the equation A (s) = 0. The degree of A (s) is 2N - 2; for if G (s, a) = sNf0 (a) + sN-lf (a) +... then G (s, b)= sNf (b) + sNlf (b) +...; and therefore A (s)= S2N-2 {f (a)f, (b)- fg (b)fi (a)} +.... But taking account of the roots of A (s) = 0, as associated with the multiple roots of the original equation for the respective values of X, we see that its degree is N N -(V- 1)+-(2-l)+...; l1 V2 and therefore N N (~(1-1)+- (z - 1) +.. = 2- 2, whence V,/ V2/ N' EQUATIONS OF THE SECOND ORDER 179 Each of the integers v is equal to or greater than 2, so that each of the quantities 1- - is equal to or greater than ~. Hence the smallest number of different integers v is two; if there were only one, the left-hand side would be < 1, while the right-hand side is > 1. The largest number of different integers v is three; if there were four or more, the left-hand side would be equal to or greater than 2, while the right-hand side is less than 2. In the first place, let there be only two integers, vP and v2; then 1 1 2 v, 2 N' From the nature of the case, v/ ~ N, v2 < N, so that I I I 1 v1 N' v2 N' hence the only possible solution is = N, V = N......................(I), and N is an undetermined integer. In the next place, let there be three integers, v,, V2, V3: then 1 1 1 2 - +- +- =1 + Vl V2 V/3 N At least one of the integers v must be 2: for if each of these integers were > 3, the left-hand side would be; 1, while the right-hand side is > 1, as N is a finite integer. Taking v = 2, we have I 1 2 -+ - + /2 /23 2 ' Another of the integers v may be 2. Let it be v2; then N= 2/3, and we have the solution = 2, 2 = 2, 3 = n, N= 2n,..............(II), where n is an undetermined integer. If neither of the integers v2 and vs be 2, one of them must be 3; for if each of them were > 4, then - + - 1 and so 12-2 180 ALGEBRAIC INTEGRALS AND [59. 2 could certainly not be equal to 2 +. Taking v2= 3, we have 1 2 ~3 - N' so that v3< 6: thus possible values of v. are 3, 4, 5. The solutions are v, = 2, V2- 3, V3=3, N=12.............(III), VI = 12 =3, 13==4, N=24.(.... PI=2, = 2-3, 1)3=5, N=601,.........(V). 60. The finite groups are thus known; the corresponding equations in s are required. The solutions will be taken in order. I. Instead of X, we take a quantity Z, defined by the relation X - XI X - X2 ' so that Z= 0 gives X = XI, that is, gives s = sI, a root repeated N times, and Z = oc gives X = X2, that is, gives 8=82, a root repeated N times. We have X- XI ~(S _ S')N G (s, b)G(s1, b)' (S - S2)N X2G(s, b)G(s2, b)' and therefore /S - SI N absorbing the constant G (sI, b) + G (S2, b) into the variable Z. II, III, IV, V. These cases are of the same general form. Instead of X, we take a quantity Z, defined by the relation X -X, X1 -X3 x x3 xI x2, then Z= 0 gives X = X2, Z=1 gives X=X1, Z=oo givesX=X,, and thus Z: Z-1: I = (X - X2) (XI - X3): (X - XI) (XI - X3): (X - X3) (XI - X2). 60.] POLYHEDRAL FUNCTIONS 181 But G (s, b)G (,, b)' (D2 V2 2 G (s, b) G (, b)' YD3v3 X -3 G(s b)G(O3, b)' and therefore Z: Z-1: 1 = A2v2 (s): Bi (s): V (), where A and B are constants which, if we please, may be absorbed into the functions i2D and (Il respectively. Now these groups are the groups that occur in connection with the polyhedral functions*: and the polyhedral functions can be associated with the conformal representation-, upon a half-plane, of a triangle, bounded by three circular arcs and having angles equal to -, -, -. The analytical results connected with these V1 V2 ^3 investigations can be at once applied to the present problem. Denoting derivatives of Z with regard to s by Z', Z", Z"',..., we have (T. F., ~ 275) 1 1 1 1 1 1 - 1- - I 1 1z// /') [-Z'" 2 /11 "^ 2 + 2- 32 — '2Z2 2 22 Z'L- LZ Z ~i2Z2 Z ~(Z ) '- 1) Z(Z-) or, taking account of the properties of the Schwarzian derivative, we have 1 (i - ^2 PI 2 ( ^1 2 (1 ^+PI 22 V32 s Z} = + - 1)+ (Z- 1) The forms of the functions for the various cases II, III, IV, V are:for II, Z: Z-: 1 = {I(n -1)}2: {1 (sn+1)}2: Sn; * T. F., ~~ 276-279, 300-302. t T. F., ~~ 274, 275. + See Ex. 3, ~ 62, of my Treatise on Differential Equations. 182 ALGEBRAIC INTEGRALS OF [60. for III, Z: Z-1: 1 -(s' + 2s2.3 - 1)3 12V3 s2 (s4 + 1)2 1 ( s4 - 2s'23 - 1)3; for IV, Z: Z-.: 1 = (ss + 14s4 + 1)3: (2 - 33s8 - 33 s4 + 1)2: 108s4 (4 - 1)2; and for V, Z: Z- 1: 1 = (20 - 228s5 + 494s'0 + 228s5 + 1)3: s30 + 1 + 522s5 (S20 1)- 0005 s1(s0 + 1)}2: - 1728s5 (sl + IIs5 - 1)5. These results* can be obtained by purely algebraic processes, from the properties of finite groups proved by Gordant. 61. These results can be applied at once to the determination of linear equations of the second order d2w dw dZ2+z~d dz2- + p + qw =0, all the integrals of which are algebraic. Denoting the quotient of two integrals w1 and w2 by s, we have~,- -fpdz / -- -fpdz wI = s se, w2=s e, w2s- 1, {s, z =2q- -p2 _ dP=21, say. As all integrals are to be algebraic, it follows that s and s'- are algebraic; accordingly, fpdz must be the logarithm of an algebraic function, which is a first condition. Further, in the equations under consideration, both p and q (and therefore also 21) are rational functions of z; and therefore {s, z} = rational function of z, * They are slightly changed from the forms in ~ 302, ~ 278 (I.c.); the change is made, so as to associate the indices v2, vl, v3 with the values Z=0, Z=1, Z=o respectively. t Math. Ann., t. xII (1877), pp. 23-46. See also Cayley's memoir, "On the Schwarzian derivative and the polyhedral functions," Coll. Math. Papers, t. xi, pp. 148-216. ~ See my Treatise on Differential Equations, ~~ 61, 62. 61.] EQUATIONS OF THE SECOND ORDER 183 and the quantity s is subject to the transformation of the finite group. Now we have seen that (1-12) (1- ) I + - -1 (, r Y V2 V2 2 v\ \VI y2 V P 32 {s, z}= Z + (z-1)2 + Z(Z- 1) in cases II, III, IV, V; and for case I, it is easy to verify directly that N2 {I,Z}= -. From the properties of the Schwarzian derivative, we have {s, Z}= s, Z} ( + {Z, }; hence, taking account of the particular form of is, Z} which is actually known, and of the generic form of Is, z} which is required, we see that, in order to satisfy the conditions, we must have Z = R (), where R is a rational function of z. Conversely, the conditions will be satisfied if Z is any rational function of z. Accordingly, the differential equation of the second order must have the coefficient of w' in the form 1 du u dz where u is an algebraic function of z; and its invariant I(z), which is q -a 2 - dP' which is q - d must be of the form 1 1 — +I dz) +- 4[Z2 (Z-+ I) j()7)~ — Z(Z- 1) dzz} or 1 1 - Z where Z is any rational function of z; the integers v,, V2, v3 in the first form are the integers of the finite groups in cases II, III, IV, 184 MODE OF OBTAINING [61. V; and N in the second form is an integer. When these conditions are satisfied, the integrals are given by W1 = S'- US, W2 = S' U -, where, for the first form, s is determined in terms of Z, the rational function of z, by the equations at the end of ~ 60; and for the second form, s Z. CONSTRUCTION OF AN INTEGRAL, WHEN IT IS ALGEBRAIC. 62. The preceding investigation is adequate for the general construction of linear equations of the second order which are integrable algebraically; there still remains the question of determining whether any particular given equation satisfies the test. When the equation is of the form d2w dw +dz + d qw= 0, inspection of the form of p at once determines whether it satisfies the condition which governs it specially. Assuming this condition to be satisfied, we construct the invariant I(z) of the equation, where (z) = -q _ p2-2 d and then, if the original equation is algebraically integrable, we must also have VI I IV1 1 1 1 I (z)= + I d) + Z2 + (Z - 1) Z + (Z - 1)d+ or else I(z) i N2 (dZ\2 where Z is a rational function of z, and the integers v, v2, V3 belong to one of four definite systems. It may happen that the identification is easy, because Z has some simple value; the simplest of all is, of course, given by ALGEBRAIC INTEGRALS 185 Z= z. When the identification is not thus obvious, it is desirable to have a method of constructing the rational function Z if it exists; when it has been constructed, the further identification is only a matter of comparing coefficients. Should this identification be completely effected, then the integration of the equation is given by the results of ~ 60. Such a method is given by Klein*, who uses for the purpose a comparison of those terms on the two sides, which are connected with the poles and have the highest negative index. A rational function is determinate save as to a constant factor, when its zeros, its poles in the finite part of the plane, and their respective multiplicities, all are known; and this constant factor is determinate, when the value of the rational function is known for any other value of the variable. Accordingly, let a denote a zero of Z of multiplicity a, and so for all the zeros; let c denote a pole of Z (and therefore also of Z- 1) of multiplicity y, and so for all the poles; and let b denote a zero of Z - 1 of multiplicity /, and so for all its zeros: then Z n (z- a)a n (b - c)r (b - a)a ' n (z - c) ) where the multiplicity 3 of b is not used directly in the expression. Consider now the right-hand side of the expression for I (z). In the vicinity of a, we have Z = (z - a)a U, where U is a regular function of z - a, not vanishing when z = a; so that 1 dZ a l dy - + R (z - a), Zdz z-a and 1 - a2 = (z - a)2 + the unexpressed terms in [Z, z} having exponents greater than- 2. In the vicinity of c, we have Z=(z - c)-Y V, Z- 1 -(z - c)-YV1, * Math. Ann., t. xiI (1877), pp. 173-176: the exposition given in the text does not follow his exactly, as he transforms the equation so as to secure that z= o is an ordinary point. 186 CONSTRUCTION OF AN [62. where V and V1 are regular functions of z - c, not vanishing when z = c; thus 1 dZ -- + S(Z ), ZdZ z-c 1 dZ (z ) Z I dz - cc 1 - 7C {z, = Z _ ~? + the unexpressed terms in {Z, z} having exponents greater than - 2. In the vicinity of b, we have Z- 1 = (z - b)W, where W is a regular function of z - b, not vanishing when z = b; so that 1 dZ B -+ Y T(z - b), Z- I dz-z z-b {Z, Z =2- (a_ - 2 ), +..., the unexpressed terms in {Z, z} having exponents greater than -2. We thus have taken account of all the highest terms with negative indices which arise through zeros or poles of Z and Z- 1. On account of the form of {Z, z}, which is z" tZ3 e it is necessary to take account of the poles and the zeros of Z'. As Z is rational, all its poles are poles of Z' and the latter has no others; so that, on this score, no new terms arise. A repeated zero of Z is a zero of Z', and all these have been taken into account; likewise for a repeated zero of Z-1. Hence we need only consider those roots of Z', which are not repeated roots of Z or of Z - 1; let such an one be t, of multiplicity r, so that = (z - t) Q (z -t), where Q is a regular function of z - t, not vanishing when z = t; then (z, u Z}= + the A ( - (Z _ t)h 2 + ' the unexpressed terms in {Z, z} having exponents greater than - 2. 62.] ALGEBRAIC INTEGRAL 187 Gathering together the terms with the largest negative index, we have, for Cases II, III, IV, V, (Z) ( + + X _( * 2+ +..., (Z -a)2 (z- b)2 (Z - 2 (Z - t)2 where the unexpressed terms have integer exponents greater than - 2; and in this expression the significance of a, b, c, for the construction of Z, must be borne in mind. Actual comparison with the form of I (z) then gives indications as to which set of values of vI, v2, V3 must be chosen, and determines the values of a, /, y. The construction of Z is then possible and, Z being known, the complete identification of the right-hand side with the known value of I_(z) is merely a matter of numerical calculation. For Case I, we have 4(*i ) 2 (- 2) +42 l() (z - a)2 + (Z - -)2 (Z -t)2' and the method of proceeding is the same as before. In particular instances, it may happen that no terms of the type (z - t)2 occur: Z' then contains no roots other than the repeated roots of Z and Z - 1. An example is given by Z7 (Z- 1)2 4z Further, it may happen that a = v2, or P/ = vi, or = 3,: so that the corresponding value of z, viz. a, b, or c, is then not a singularity of the differential equation. And, in particular, if z = Co is not a singularity of the differential equation and therefore also not a singularity of the integral, then, if the equation be integrable algebraically, the numerator of the rational function Z is a polynomial in z of the same degree as the denominator*. * This form of equation is discussed by Klein in the memoir already quoted (note, p. 185): reference should be made to it for further developments. 188 EXAMPLES OF [62. Ex. 1. The equation d2V' 2 -z+1 dz + 2 (z - i)2 is integrable algebraically. For -2 (6 - 1)2 (Z - 1)2 z2 Z (z-1)' so that Z=Z; 1 whence =2; 3_1(1 1 1 We thus have an instance of case II, when n =2. All the conditions are satisfied: and thus (~ 60) the integrals of the equation are given by (S2w- 1)2 V2 4s2 2 4 2 +2 2 hencev3 2 W1S di W22 V Ex. 2. Construct a linear differential equation of the second order in its normal form, such that the quotient s of two of its solutions is given by (88 + 14842+ 1)3 (2-1)2 108s4 (s 4,1)2 4= Ex. 3. Consider the equation 1 d2w 2Z4 - 8Z3 - 15Z2 - 8z+ 2 (+ - 1)2 + _Q = 0. qv dZ2 + 9z2 (Z2 - 1)2 + (2 +1)2 We have 2z4 - 8Z3 - 152 - 8z+ 2 3a - 1)2 9Z2 (Z2 - 1)2 S Z (Z2 + 1)2 2 5 3 Z2 (Z + 1)2 (2 - 1)2 (Z - j)2 (Z + j)2 the terms indicated constituting all the infinities of 1(z) of the second order. First, it is clear that there is only one root of Z' other than repeated roots of Z and Z- 1; it is characterised by t=l, T=1. As regards the remaining terms, the numbers vj, V2, V3 must be 2 or 3; so that we either have an instance of case II with n= 3, or we have an instance of case III. 62.] KLEIN'S METHOD 189 If it were possibly an instance of case II with n = 3, then we must have so that P, = 2, 3=1, b=i, 2\ Rg=~j~ v2-~~~~~~2, a=l, a=-i. 2..v3=3,y=1l C=0, V~ Y32 9,cO and therefore Z=A z(_ + 1)2' with the condition that Z= 1 when z= b =i, so that A=i. But then Z' - _(2Z2+ 3iz + i), Z2 (Z + 1)3 shewing that Z' does not possess a root z = t=; hence the example is not an instance of case II. If therefore the equation is algebraically integrable, it must be an instance of case III. We must have therefore Vi 29 V2 = 31 V3=3 so that __3 1 whence 3 =, 1 bi, _L(1P' i../3.. '==1, b'= -i; T V12 16 and then, either a2 2 " giving a=1, a =0, y=2, c=-1; or else V22 V32 giving a=2, a=-I, y=l, c=0. Taking the former, we have Z=A (z + 1):&) from the poles and zeros of Z; as Z= 1, when z=i, we have A =2, so that 2z Z2+1 Z= Z_ =- (Z + 1)2, (Z+ 1)2' so that Z- 1 has the roots z=i, z=-i; but (z + 1)3, 190 EXAMPLES [62. shewing that Z' does not possess the root 1; and thus the first assignment of values is not possible. Taking the latter, we have Z=A (z+ )2 Z from the poles and zeros of Z; as Z-1 when z=i, we have A =, and then Z=(Z+1)2 Z_1 -Z+1 2z ' 2z ' - z2-1 Z — = z2, so that Z-1 has z=i, z= -i for roots, and Z' has z= 1 for a root. The preliminary conditions are thus satisfied; it is easy to verify that this value of Z gives the complete value of I(z). Hence, after the results of ~ 60, the integral of the differential equation is given by the equations S4- 2s2V3- -I 2z wl=s tdZ), ^V2==^tU 2: W1S S Z2 so that the differential equation is algebraically integrable. Ex. 4. Shew that the equations d2w dw (i - z) dg+2 W +-A-oW = 0 Z(-) dz2 +(-2Z) — 5 =0, d2w dw are integrable algebraically: and obtain their integrals. Ex. 5. Taking the equation, which has three singularities in the finite part of the plane and for which infinity is an ordinary point, in the form given in ~ 49, so that, by ~ 53, 1 ( 1 (I-) (a — r=l (Z — ar)' where ~ = (z- a,) (z - a2) (z - a3), and Xi = (a - a'), = =~( - fi'), ks= ~ (- 7'); discuss the possibilities of algebraic integrability for the values -1 IN = 2 x-= 13 2-5) 3=2, In particular, shew that, if a2= - 1, a3=0, then al= - '%4- (Klein.) 63.] EQUATIONS OF THE THIRD ORDER 191 EQUATIONS OF THE THIRD ORDER WITH ALGEBRAIC INTEGRALS. 63. When we pass to the consideration of linear equations of order higher than the second which are algebraically integrable, the discussion can be initiated in the same way as for equations of the second order; but the detailed development proves to be exceedingly laborious, and it has not been fully completed for each case. Only a sketch will here be given. Dealing in particular with the linear equation of the third order, we take it in the form w"+ + 3pw" + 3qw' + rw = 0, where p, q, r are rational functions of z, subject to the limitations imposed by the regularity of the integrals in the vicinity of each singularity (oo included). If w1, w2, w, denote three linearly independent integrals, we have (~ 9) W1!, 'W2/11, W = Ae-3fPdz WI, W2, W3 W1, W2, W3s so that, as w1, w2, w3 are algebraic functions of z, it follows that p, a rational function of z, must be of the form 1 du =u dz' where u is an algebraic function of z. This is a first condition: it is the same as for the equation of the second order (~ 61): and it is easily obtained as a universal condition attaching to any linear equation which is algebraically integrable. Now substitute for w by the relation wePdz = y, and let yl, y2, y3 denote the three integrals corresponding to wI, W2, w3; owing to the character of p and the functional character of the integrals w, the integrals y are also algebraic functions of z. Thus the equation in y, being y"' + 3Qy'+R=0, 192 EQUATIONS OF THE THIRD ORDER [63. where Q=q _p2_p' R= r-3pq + 2p3-p " is to be algebraically integrable. Denoting by s and t the quotients of two integrals by a third, we have Wi2 Y2 Wi Y3 W1 Yl Wl yl The quantities s and t are algebraic functions of z for equations of the class under consideration. The effect upon a fundamental system, when the independent variable describes a circuit enclosing one or more of the singularities, is represented by relations of the form Y=a yl +b 2 + c y3 Y2= a' y +b' y2 + c' y3 Y3 a"'yl + b"y2 + c"y3 If S and T denote the corresponding integral-quotients, then a' + b's + c't a/ + b"s + c"t a +bs+ct ' a+bs+ct Now if the equation is integrable algebraically, there can exist only a limited number of different sets of values of the integrals; so that the number of sets Y1, Y2, Y3 is finite, and the number of simultaneous values of S and T is finite. If then we know all the homogeneous linear groups in three variables, or (what is the same thing) all the lineo-linear groups in two variables, which are finite, then each such finite group determines its set of values of Y1, Y2, Y3 and the set of values of S and T, and so it determines a linear equation the integrals of which are algebraic: and conversely, each such linear equation is characterised by a finite group. 64. In order to utilise the method for the present purpose on the lines adopted for the equation of the second order, it is necessary to deduce from the differential equation certain differential invariants involving s and t, these invariants being expressed in terms of Q and R. This can be done in two ways. It is clear that, as s implicitly contains five arbitrary constants, it satisfies a differential equation of order five; and that, as t is of the same functional form as s, it satisfies the same differential equation. 64.] WITH ALGEBRAIC INTEGRALS 193 On the other hand, as s and t combined contain eight arbitrary constants implicitly, it may be expected that the two differential equations, which they satisfy and which will involve both of them, will be each of the fourth order or will be equivalent to two of the fourth order. The single equation is, for some purposes, the more important in the formal theory of the linear equation, which will be left undiscussed; for the present purpose, the two equations prove to be the more important. Accordingly, we substitute syl for y2, and ty, for y3, in turn in the equation y"' + 3Qy' + y = 0; whence, remembering that yj is an integral of this equation, we have 3s'y" + 3s"y' + (3Qs + s"') y = 0) 3t'y/' + 3t"yl' + (3Qt' + t"') y = o Differentiating each of these once, and substituting for yi"' from the linear equation which it satisfies, we have 6s"y," + (4s"' - 6Qs') yl' + {s"" + 3Qs" + 3 (Q'- R) s'} = 0 6t"y1" + (4t"' - 6Qt') yl + {t"" + 3Qt" + 3 (Q' - R) t'} y, = 0) so that there are four equations, linear and homogeneous in the quantities yi", y,', yi. When the ratios of y,": y,': y, are eliminated from the first pair and the first of the second pair, we have s"',/ 4s"', 6s" -3Q s", 2s', 0 -3(R-Q') s', 0, 0 =0; // 3 s"', 3s", 3s' s, 3s", 3s' t', 3t", t' 3t", 3 t' t"', 3t", 3t' and when the same ratios are likewise eliminated from the first pair and the second of the second pair, we have t"", 4t"', 6t" -3Q t", 2t', 0 -3(R-Q') t', =0. S// 3, 3s" I, 3s", 3s' s8", 3s", 3s' t,' 3t", 3t' '", 3t", 3t' t.'" 3t", 3t' These, in fact, are the two equations, each of the fourth order, satisfied by s and t. F. IV. 13 194 INVARIANTS FOR AN EQUATION [64. Suppose now that two solutions (other than the trivial solutions, s = constant, t = constant) are known, say S=0-, t=T. Solving the first pair of the foregoing equations for y/': y1, we have 3 (ar" - &T r) y + (ao7 - 7") Y) = 0, and therefore Y1 = ('%T'- _'") -, neglecting an arbitrary constant arising as a factor on the righthand side. Hence a fundamental system of integrals of the original equation is (0 - / o"")- -~, a (a-' - 'r") ~-, T ( - - o-'T") - or the original equation can be integrated if two particular solutions of the equations in s and t are known. 65. Moreover, from the source of the two equations which serve to determine s and t, it is to be expected that, when the above two (being any two) particular solutions s= a, t = r, are known, the complete primitive of the two equations is a' + b'o' + c"r a" + b"o- + c"'c a -- bo- + CT a b + a -cTr where the constants a, b, c, a', b', c', a", b", c" are arbitrary so far as those two equations are concerned. This result can be stated in a different form. The two equations in question can be written As"" + 4Bs"' + Cs" - 3Q (As" + 2Bs') - 3 (R - Q') As' = 0, At""' + 4Bt"' + 6Ct" - 3Q (At" + 2Bt') - 3 (R -- Q') At' = 0, where A, B, C are the three determinants in s', 3s", 3s' t"' 3,,3t' Now let Ul= 8" t - s't" 6U2= Si t' - St', u63 = s1" t' - s't"", V2 = S" t" - Sltw, U4 = s-""t- s't""', V3 = s"tt " - Sl't", OF THE THIRD ORDER 195~ so that A= 9u1, B=-3-3, C = 3v,; then solving the preceding equations for Q and for B - Q' in turn, we find U3 +2v, 2 U2 2 3QU+2V - ( =) Its, t tz) U, Ul and 2 U2 (%L3 4V2) U2 z\ -27 (B- Q') -9 6 2 +8 42) J(s, t, z) a, U, u1 say. The latter equations may be regarded as the equivalent of the two equations, which have been solved; and therefore we may expect that (a + b's + c't a"+b"s+ c"t z t,z), a+bs+ct ' a bs+ct a' + b's + c't a" ~ bs + c"t N C a+bsfct ' a bs ct Z =J(s, t, Z); the actual verification, which is comparatively simple, is left as an exercise. Clearly these are generalisations of the property of the Schwarzian derivative, represented by ras +} bs= IS Z. The two invariant functions I and J were first indicated* by Painlev6; they subsequently were simplified to a form, which is the equivalent of the above, by Boulanger+. The invariance of the functions I and J, as indicated, exists for lineo-linear transformation of s and t. There is also an invariance for any transformation of the independent variable z; for we easily find the equations I (s, t, z) = I (s, t, Z) Z'2 + 2 {Z, z}, J(s, t, z) = J(s, t, Z) Z'3 -91 (s, t, Z) Z'Z" - 9 d where Z is any function of z. Also I' (s, t, z)= dz 11 (, t)j = I(s, t, Z) Z'3 + 21 (s, t, Z) Z'Z" + 2 dt {z, z}, * Conlptes Rendus, t. civ (1887), p. 1830. t See his Th~se, Contribution Zt t'6tude des 6quations difftrentielles lineiaires et homogbnes intigrables algdbriquement, (Paris, Gauthier-Villars, 1897). 13-2 196 INVARIANTS AND [65. and therefore J(s, t, z) + 9' (, t, z) = [J(, t, z) + = ' (s, 9, Z)] Z'3, or J(s, t, z) + 9I' (s, t, z) is an invariant for aly change of the independent variable z. Dropping a numerical constant, this is the function R 3dQ R -- dz ' which is the known Laguerre invariant in the formal theory; that is*, if the equation y"' + 3Qy + Ry = 0 be transformed, by the relation ( dZv' dz/-1 Y Ydz' to the form d3Y dY d3 + 3Q + RQ d = 0, then R dQ (R —2 dQ' ' dZ 2Rdz R dI j dz2 As the transformation Y dZ Y= Y dz) leaves the quotient of two integrals transformed only as by a lineo-linear substitution, it follows that the preceding function, say L (s, t, z)= J(s, t, z) + I' (s, t, z), is unchanged by lineo-linear transformations effected on s, t; also, except as to a factor Z'3, it is unchanged by transformation effected on the independent variable. Now ut3 = U4 + V3, V21 = V3, U2/ = 3 - V21, Ul = U2, so that we have L 94+ 45v3- 45U.2 (u 20( L (s, t, z)= 2u — 2u + 2v) + 20 - * See a paper by the author, Phil. Trans. (1888), pp. 383, 390. Laguerre's invariant was first announced in two notes, Comptes Bendus, t. LXXXVIII (1879), pp. 116-119, 224-227. FINITE GROUPS 197 which is the full expression of Laguerre's invariant in terms of the derivatives of s and t. 66. The next stage is to associate these invariants with the algebraic equations in two variables, which admit of one or other of the finite groups. These groups have been obtained by Jordan* and Valentinert; and references to other writers are given by Boulanger'. A method of using the results is outlined by Painleve~ as follows. Let b (s, t), f (s, t) denote two irreducible invariant functions of a finite group of order N; the functions are given by Kleinll for the group of order 168, and by Boulanger (I.c.) for the group of order 216. As these functions are invariable for each substitution of the group, and as s, t are algebraic functions of z, it follows that ( and f are rational functions of z, say (s, t) = > (z),, (S, t) = ~ (). Conversely, taking 1 and T to be arbitrary rational functions of z, these two equations give rise to N sets of simultaneous values of s and t as algebraic functions of z; and if any one set of values be represented by a-, r, all the others are obtained on transforming a and 7 by all the N- 1 substitutions of the group other than the identical substitution. These two equations are used to obtain the first four derivatives of s and t with regard to z; and with these derivatives, the two invariants I(s, t, z), J (s, t, z) are constructed. The functions so formed involve derivatives of ) and P; and the coefficients of these quantities are rational in the derivatives of b (s, t) and r (s, t). As I and J are invariantive for the group, the coefficients specified are rational functions of s and t, which must be invariantive for the group and are therefore rationally expressible in terms of b and fr, that is, in terms of (i * Crelle, t. LXXXIV (1878), pp. 89-215; Atti della R. Accad. di Napoli, t. viI (1879), No. 11. t Kj5b. Vidensk. Selsk. Skr., 6 R., t. v (1889), pp. 64-235. + In the These, already cited on p. 195, note. ~ Comptes Rendus, t. civ (1887), pp. 1829-1832, ib. t. cv (1887), pp. 58-61. II Math. Ann., t. xv (1879), pp. 265-267. 198 ALGEBRAIC [66. and P. Thus I(s, t, z) and J(s, t, z) would be expressed as rational functions of z. Accordingly, taking 3Q = I (s, t, z), R = 3 ' (.s, t, z) - I J(s, t, z), we have the differential equation y' + 3Qy' + Ry = 0. The earlier investigations shewed that its integrals are expressible in terms of s, t, and their derivatives; and we thus have a mnethod of constructing all the linear differential equations of the third order which are integrable algebraically. There is a double arbitrary element for each group, viz. the arbitrary forms of the rational functions ( and P; and there is a limited number of groups. 67. While this outline is simple enough in general description, the application to particular cases requires extremely elaborate calculations. These have been effected by Boulanger for the group of order 216; they do not appear to have been yet effected for any one of the other groups. As, however, the enumeration of the finite groups in two quantities s and t is complete, the subject offers an interesting, if a laborious, field of investigation. In the absence of the complete table of equations, for all the finite groups and for two arbitrarily assumed functions 'I and P, it is not possible to use a method, analogous to that of ~ 62, to determine whether a given equation of the third order is algebraically integrable or not: it is not even possible to recognise to which of the groups it would belong if it were algebraically integrable. Indications of two general methods of procedure have been given by Painleve and have been developed to some extent by Boulanger; but the methods, while general in description, suffer from the same kind of difficulty as the method indicated for the construction of the equations, for the calculations are exceedingly laborious. We have seen that, if two particular values of s and t, say cr and r, are known, then an integral of the differential equation is given by y = (,"r'T - -i") -. 67.] INTEGRALS 199 Hence, if we take I= Y we have a (T -O T U -- 77 OT/ _ (-tT.,,, so that the number of values, which u can acquire, is equal to N or to a submultiple of N, where N is the order of the associated group: let the number of values be n. Now if y is algebraic, every zero of y and every infinity of y are of a finite order, which is commensurable in every instance; and therefore all the infinities of u are simple poles with commensurable residues. Substituting for u in the equation y" +3Qy'+Ry = 0, we find u" + 3uu' + u3 + 3Qu + R = 0, a non-linear equation of the second order satisfied by u. This equation renders it possible to test the character of the poles and the residues of u. If these are of the appropriate type, then the equation is satisfied by a relation of the form Aoun + Alun-1 +... + An-lU + An = 0, where A0, Al,..., An are polynomials in z, and A0 is the product of the factors corresponding to the poles of u. Then there is the further test that this algebraic function u must be such that efudz is algebraic. Manifestly, the calculations will generally be too elaborate to make the method effective in practice. EQUATIONS OF THE FOURTH ORDER. 68. As pointed out* by Painleve, the processes just indicated can formally be applied to linear equations of any order: but of course, if any advance towards final conditions is to be made, it is necessary to know all the finite lineo-linear groups of transformations in a number of variables less by one than the order of the * Comptes Rendus, t. cv (1887), p. 59. 200 EQUATIONS OF THE FOURTH ORDER [68. equation. Towards this enumeration of groups in three variables, which are associated with the linear equation of the fourth order, Jordan* has constructed a characteristic numerical equation which, when completely resolved, would indicate the order and the composition of each such group: but the resolution is exceedingly long and, owing to the number of cases that must be considered, it has not been completed. In these circumstances, no detailed results of a final critical character can be obtained for an equation of the fourth order or of any higher order: the only results obtainable are of a general character, and arise through the association of groups in general with linear equations. The equation of the fourth order, which may be written w'" + 4pw"' + 6qw" + 4rw' + sw = 0, can be transformed by wefpdz - y into y"" + 6Qy" + 4Ry' + Sy = 0. We denote a system of four integrals by yi, y2, 3, y4, and we introduce three quotients s, t, u, such that Y2 =-Y, Y3=Ylt, Y4=ylu; then s, t, ui are simultaneous solutions of three equations of the fifth order in the derivatives. If' a, r, v are a special set of solutions, then yl= 0-, 0-, C0 - // T, /// // V, T, V and y2 = yi0-, Y3 = YiT, Y4 = iyV. The complete primitive of the three equations is of the form s t u a' + b'o + c' + d'v a" + b"a + c"r + d"v a"' + b"'o- + c"'r + d'"v 1 a + bo- + CT + dv' * Atti della R. Accad. di Napoli, t. vIIi (1879), No. 11, p. 25; instead of dealing with lineo-linear transformations in three variables, Jordan deals with homogeneous linear substitutions in four variables. 68.] 68.] ~WITHI ALGEBRAIC INTEGRALS20 -201 There are three functions of the derivatives of s, t, u, with regard to z, which are invariantive for substitutions such as the preceding relations expressing s, t, u, in terms of a-, 7r, v; and they are equal to 1dQ SA-LL32 Q, R d ' S-dz- Q If the determinants (S ~ (s"t~'), + (sivtlu'), 'Y- ~ (sit"I'&), I ~ (sVt"u') be denoted by p, pi, p2, 93 respectively, then Qi3P30P25 P>~ 2= I (S, t, u,) say; if, in addition, the determinants be denoted by p, and p2 respectively, then dQ 2,5+ 594 pi (693+25 P2) 3__ =1(,,m ) )R -+ (P3) 2 S t ) ) lz 2p 7 2 48 p say; and if the determinant Y, ~ (-svs"V's) be denoted by p,, then dR = Q (6p9392 + 25p394 + 69293 + 1OP2) -dz -Q2 4p 48p22 /) + -Th132(93 + 5P2) - fr =3(S, it, U, I) say. The three quantities I, (s, t, u, z), 12 (s, t, u, z), 1.3 (s, t, u, z) are unchanged when lineo-linear substitutions are effected on 8, t, u; and the combinations 1T3~ 213' - 4 '-5 12 are also unchanged, except as to a power o f Z', when z is replaced by Z, any function of z. The proofs of these various statements are left as exercises. 202 ALGEBRAIC INTEGRALS AND [69. EQUATIONS, HAVING ALGEBRAIC INTEGRALS, ASSOCIATED WITH HOMOGENEOUS FORMS. 69. It has already (~ 58) been stated that the discussion of the equations, which have algebraic integrals, has been associated with the theory of homogeneous forms: the association can be seen to occur as follows. Using the preceding notation of ~~ 63-66 for the quantities connected with any linear equation of the third order, we denote by s and t the quotients of any two by the third out of any three linearly independent integrals of the equation dY+3Q + dy~ d —; - + Ry = 0. dz3 dz If, then, all the integrals of this equation are algebraic, both s and t are algebraic functions of z; they may therefore be regarded as determined, in the most general case, by a couple of distinct algebraic equations, say i(, t, Z)=0, f(s, t, Z)=0, or by g, (s, z)= 0, g (t, z)= 0. Eliminating z between the pair of equations in whichever form they are taken, we obtain a relation of the type Fo(s, t)=0, where Fo is a non-homogeneous polynomial in s and t, because it is the eliminant of two polynomials. Replacing s and t by y2 y+ and y3 + y, respectively, and multiplying by the proper power of yi to free the equation from fractions, we have F(yl, 2, y3) = o, where F is a homogeneous polynomial in its arguments or, in other phrase, is a ternary form in y1, y2, Y3. Further, the above form of equation is obtained from d3w d2w dw dz-3 + P + 3q T + rw= o, by the transformation wep dz = y; 69.] HOMOGENEOUS FORMS 203 and therefore F(wief dz, wzefpdZ, w3efpdz) = 0, that is, F(w1, W, Ws) = 0, on rejecting the factor enJPdz, which occurs because F is a ternary form (say) of order m. Hence it follows that when the integrals of a linear equation of the third order are algebraic functions, a homogeneous relation of finite order exists among any three linearly independent integrals. Moreover, when any other set of fundamental integrals Y1, Y2, Y3 is taken, we know that yl = alY1 + a Y2+ aY3) y2= blY1 + b2Y2 + bY3, y3 = clY1 + C2Y2 + c3Y3 where the coefficients a, b, c are constants. The variables in the homogeneous ternary form are therefore subject to linear transformation; and thus the theory of ternariants can be associated with those homogeneous linear equations of the third order, which have their integrals algebraic. The various cases will arise according to the order of the form F; this order is always greater than unity, because the integrals considered are linearly independent. If, still further, we choose to combine the geometry of the ternary form with the form in its association with the equation, then the preceding algebraic relation F= 0 is the equation of an algebraic plane curve referred to homogeneous coordinates: the curve is usually called the integral curve. We may proceed similarly with an equation of the fourth order d4w d3W d2w dw dz4 + 4p d3 J+ 6q d2 + 4r + s = 0, when all its integrals are algebraic. If we choose, we may transform it by the relation wefpdz y; 204 EQUATIONS OF THE [69. the quantity efPdZ must be algebraic, because W1, W2, W3 W4 =Ce4fdz, W1, 2 W2 W3, W4 Wli, "W2 /, W3, W4l Wll', W2't, W3/", W411 where C is a non-vanishing constant; and the equation in y, which is of the form d4y +6pd iyi4 dy dz-+-' dz2 + f 4P3 d + P4 y = O, dz4 dz2 dz has all its integrals algebraic. Taking any four linearly independent solutions y, y2, Y3, y4, and writing pyl = Y2, Cryl = Y3, ryl = Y4, then as p, a, 7 are algebraic functions of z, they must be given by three equations of the form fi(p,,r,,)=O, f2(p,,,z)=, f3(p,-, 7,)=0, or of simpler equivalent forms, which are completely algebraic in character. Eliminating z between the first and second, and also between the first and third, and taking the eliminants in a form free from irrational quantities if these occur, we have two equations Fo(pI, o7)= 0, Go(p, 0, 7)=0, two non-homogeneous polynomials in p, a, r. Replacing these quantities by their values in terms of y,, y2, y3, y4, and multiplying each equation by the power of y,, appropriate to free it from fractions, we find F(yl, Y2, Y3, Y4) = 0 G (yl, y2, ys, y4) =01 where F and G are homogeneous polynomials in their arguments or, in other phrase, are quaternary forms in y1, y, y3, 4,. As in the case of the cubic, these equations imply the further equations F(w1, W2, W3, W4)= 0 G (w1, w2, w3, w4)= 0) so that, when the integrals of a homogeneous linear equation oj the fourth order are algebraic functions, two homogeneous relations of finite order exist among any four linearly independent integrals. 69.1 FOURTH ORDER 205 Again, when the variables yi, y2, y,, y4 are replaced by any other set of fundamental integrals YI, Y2, Y3, Y4, the two sets of variables are connected by homogeneous linear relations: and thus the theory of quaternariants can be associated with those homogeneous linear equations of the fourth order which have their integrals algebraic. The various cases will arise according to the orders of the forms F and G; these orders are always greater than unity, because the integrals yi, Y2, y3, y4 are linearly independent. We may also combine the geometry of quaternary forms with the forms themselves as associated with the equation. In that case, each of the equations F= 0, G = 0 is the equation of a non-planar surface in three dimensions referred to homogeneous coordinates: the two equations combined determine a skew curve, which accordingly is the integral curve. Similarly, in the case of equations of the fifth order, of which all the integrals are algebraic, we have three homogeneous nonlinear relations among any fundamental set of integrals; and there are corresponding associations with the theory of homogeneous forms in five variables and the allied geometry. And so also for linear equations of higher orders. Note 1. There cannot be two homogeneous relations among a set of three linearly independent integrals of an equation of the third order: for they would determine a limited number of sets of constant values for the ratios y: y: y3, contrary to the postulate of linear independence. Similarly, there cannot be three homogeneous relations among a set of four linearly independent integrals of an equation of the fourth order: for their existence would imply a corresponding contradiction of the same postulate. And so for other equations of higher orders. It might however happen that, for an equation of the fourth order, only a single homogeneous relation exists among four linearly independent integrals; that, for an equation of the fifth order, the number of homogeneous relations among a fundamental set of integrals is less than three; and so on. If the relations thus given in each of the respective cases are the maximum number of homogeneous relations that can exist, we can infer that not all 206 BINARY [69. the integrals of the respective equations are algebraic: and a question arises as to the significance of the respective relations. Note 2. The converse of the general argument must not be assumed valid: that is to say, the existence of a homogeneous relation between the members of a fundamental system of integrals of an equation of the third order is not sufficient to ensure the property that all the integrals are algebraic. Thus we know that a number of transcendental functions of a variable can be connected by algebraic relations: and such instances are not the only possible exceptions. 70. The preceding method of associating the theory of forms with linear equations does not apply directly when the equation is of the second order: for a homogeneous relation between two integrals would imply one or other of a limited number of constant values for the ratio of the integrals, which accordingly could not be linearly independent. This deficiency, however, is rendered relatively unimportant, because Klein's method explained in ~ 59-62 for the equation of the second order gives the complete solution of the question propounded as to the cases when all its integrals are algebraic. The results there given can be (and have been) obtained by processes directly connected with the theory of binary forms. After the preceding exposition, the analysis is mainly of formal interest, and adds little to the knowledge of the solutions regarded as functions of the independent variable. It will be sufficiently illustrated* by one or two examples. Ex. 1. We take the differential equation in the form d2y, dx2 y=O and consider the value of a homogeneous polynomial function of two integrals y, and y2, linearly independent of one another. Let this polynomial be of order n, and write f(y, 1y2)=(ao, a,,..., *nyi, /yy2)=y2fn (ao, a,,..., aJ3s, 1)n * For fuller discussion and details, see Fuchs, Crelle, t. LXXXI (1876), pp. 97 -142, ib., t. Lxxxv (1878), pp. 1-25; Brioschi, Math. Ann., t. xi (1877), pp. 401-411; Forsyth, Quart. Journ., t. xxiiI (1889), pp. 45-78. A memoir by Pepin, " M6thode pour obtenir les integrales algebriques des equations differentielles lineaires du second ordre," Rom. Ace. P. d. N. L., t. xxxiv (1882), pp. 243-389, may also be consulted with advantage. 70.] FORMS 207 say, where s is the quotient y/. Y2. When substitution is made for y, and Y2 in terms of x, let the value of f be q (x), so that f (Y1, Y2) = (x). Now if H (y1, 2) =H (f) be the Hessian off, and if H(u) be the Hessian of u, so that (f) = (/o2- a2,..yI, y2)21-4, H (u) =(aoa2-aa2,...S, 1)2n - 4, we have H(f) =y22n-4 H(u), _r ( \ 9/ i\ f a2f 2(^f \2'} H(f)=n-2(n- 1) —2 Y - ) } () = 2 (n u du We have also 2d- = constant = C, Y2 dJ 9 dx say, so that Y2 C ds Now Y21 = ( ); hence n dy2 C 1 du 1 do Y2 dx 22 U ds = dx ' Differentiating, and substituting for the second derivative of 2, we have I-n dy2 2C dy2 1 du C2 d2 (log u) d2 (log q) y22 \ dx J 23 dx u ds,Y2 ds2 dx2 Multiply by n, and add the squares of the sides of the preceding equation: then C 2 d2 (log u) 1 fdu\2} 1 (d \2 d2 (log ) + 24 d + us ds/ j 2 \dx/ + dx2 The coefficient of C2y-4 on the left-hand side is n di (t I fdu\2 =- - t7 — (n- 1) W2 ( u d2 u" ( ds/ n2 (n- 1) u2 (u) so that n2 (n-1) C2xr(Y, Y2) = n di2 - fd ti - n2l thus expressing the Hessian in terms of functions of x: let this be written H(Y, Y2) = P2X. 208 EXAMPLES [70. If now D (y1, Y2) denote the cubicovariant of f, so that I (af aH -of NH &(Y1 Y2)= { n (n - 2) a ay, a2 Y2 aY, = (a0 2a3 - 3aOala2 + 2a,13, 'JY11 Y2)3) 6, then, proceeding in a similar way, we find And so for other covariants. As a special case*, let it be required to find the value of p, if when the binary form is the quadratic a0y12 + 2ay.yl2 + a2,y22, / (x) is a root of some rational function of x. In this instance, H(y1, Y2)>=a0a2 -a12, a constant; hence c (x) is either a rational function, or is the square root of a rational function. The integration is immediate; for Y2 (as2 + 2+2 s + a2) = (x), 222 ds = Cdx, whence ds Cdx a0s2+2ais+a2 - (x)2 The value of s is thus known: and the consequent values -of y, and y2 are immediately givent..Ex. 2. Shew that, if the integrals of the equation d2y are such that (aa, a1, a2, al.yj, i2 )3= (x), and qb is a root of some rational function of x, then fr4 must be rational; and obtain the relation between l and 0 (x). Ex. 3. The integrals of the equation d2y - are such that (a0, Ial, a2, a3, a4jyj, y2)-P (X), and p (x) is a root of some rational function of x; shew that, unless q (x) is actually rational, the quadrinvariant of the binary quartic must vanish. In either case, find- the relation between I and 4 (x). (Brioschi.) * Fuchs, Crelle, t. LXXXI (1876), p. 116. t See my Treatise on Differential Equations, ~ 62. 70.] OF BINARY FORMS 209 Ex. 4. Find the value of I in the equation d2y when, in the relation ay1n + by2n= q (x) connecting two integrals, the function q is supposed known. Ex. 5. Shew that, if two integrals of the equation d2y = dy + Qy dx2 dxQY are connected by a relation Ayl2 + Bly2 i- +,2 +DO=0, where A, B, C, D are constants, then dQ 2PQ=O. dx Assuming the condition satisfied, integrate the equation. (Appell.) Ex. 6. Two integrals of the equation d2y P y dx2 x+ Qy are connected by a relation of the form Ayl3 + Byl2y2 + Cyy22 + Y23 + E=O, where A, B, C, D, E are constants: prove that d2-5P d - 2Q d-3Q2+ 6P2Q=. dX2 dx dx Shew that the quantity on the left-hand side of this conditional equation is invariantive for change of the independent variable; and hence, assuming the condition satisfied, shew that the equation can be transformed so as to become a particular case of Lame's equation (Chap. ix). (Appell.) EQUATIONS OF THE THIRD ORDER AND TERNARIANTS. 71. Returning now to the differential equation of the third order in the form d3y + 3Q dy dz 3& + Ry =0, and supposing that all its integrals are algebraic, we proceed to consider the equation F (yl, 2, y3) = 0, F. IV. 14 210 EQUATIONS OF THE THIRD ORDER [71. where F is a homogeneous polynomial in any three linearly independent integrals. For this purpose, it will be convenient to have an equivalent simpler form of the equation which is given by a known transformation*, viz. we have d3U d+ l = 0, where dt dt3 dQ = YdZUC~ Kciz) - 2d' {tz}=3Q. If we take dt the last of these relations may be replaced by the equation W+ Q = o. dz2 The equation among any three integrals is F (u,, U, u) = 0. Consider the simplest case; it arises when n = 2, so that F is then a quadratic polynomial involving six terms. Writing ate = alu + (aU2 + a3U3, where a2, a2, a, are umbral symbols, the equation can be symbolically represented by au2 = 0. We have auau, = 0, au a,, + au, = 0, where u' is du/dt, and so for a". Differentiating again, and replacing u"' by - lu, we have - Iau2 + 3auau, = 0, that is, au, au = 0, on using the original equation. Similarly, on differentiating this result, - Ia,,a,u + au,,2 = 0, that is, au6,2 = O, * See a paper by the author, Phil. Trans., (1888), p. 441. 71.] AND TERNARIANTS 211 on using the first derivative of the original equation. Differentiating once more, we have Iaau,,l = 0, so that either I = 0 or auau/, = 0. If I is not zero, then we must have a au,, = 0, and therefore, by the second derivative of the original equation, au2= 0. Hence, on the present hypothesis, we have au=0, aua=,=O, aC2= 0, aCOau" = au,,, aa 0,,2 = 0. Now each of these equations is linear and homogeneous in the six real coefficients that occur in au2; eliminating these coefficients, we obtain, as equal to zero, a determinant which is the fourth power of U1, U2, 3, U, U2, U3 U1, U2/, U3/ and the latter ought therefore to vanish. But because U1, u,, Us are linearly independent, this determinant (being the determinant of a fundamental system) does not vanish-it is a non-zero constant in the present case. Accordingly, the hypothesis that I is not zero is invalid. Hence I= 0; and therefore, on returning to the original equation, we have R-_3 - 2 dz Writing dP 3Q=4P, R=2 d our original equation becomes d +4P +2 dy=0. dz3 dz dz Y Any three linearly independent integrals are connected by a quadratic relation (yl, y2, Y3)= 0. 14-2 212 TERNARIANTS [71. To obtain the integrals, we note that one value of u is a constant, say unity; thus dz y = d= 02 y= dt where d2O d + PO= 0. dz2 Thus three integrals of the original equation are 012, 0102, 02, where 06 and 02 are two linearly independent integrals of the latter equation of the second order. It may be noted that three independent integrals of the u-equation are 1, t, t2; so that dt dt dt Y1 do 1, Y2 = dzt, Y3dz-t2, and therefore y22 - yY3 = 0, thus verifying the existence of the quadratic relation obtained in a canonical form. Assuming 0 known, we have dt 1 dz 02' so that t= I; t=J 02 and thus three integrals of the original equation are 02 02 dz, 02 d} The comparison of these integrals with 012, 0102, 022 is immediate; for it is a well-known theorem that, if 01 is a solution of an equation d + P0= 0O then another solution, which is linearly independent of 01, is given by 0 [dz 012 Denoting this by 02, the above three integrals are at once seen to be 0,2, 0102, 022. EXAMPLES 213 Ex. 1. Prove that, if u be a solution of the equation d3y o _ dP +P #+ I Y=dOY'O dX3+ dx -1, jx the primitive can be expressed in the form i=Au+Buexpa f) + Cuexp ( adxf) where A,' B, C are arbitrary constants, and a is a determinate constant. What is the primitive when a vanishes? (Math. Trip. Part i, 1895.) Ex. 2. Prove that, if three linearly independent integrals of the equation d3y dX3 =I be connected by a relation F(zj, Y2, fl)=0, where F is a homogeneous polynomial of the third degree, then I must satisfy the equation (561'2 - 48111") 1"" + 54ff"'2 - 1441'I"]"' + 182; 731J1"' ++ 2421"13 113 2.36.72 4=. - 7. 36212I'I" + 8421113 +2. 5 4= 0. 25 Ex. 3. Prove that, if both the fundamental invariants* of an equation of the fourth order vanish, so that it can he taken in the form y"" + IOP11" + 10P'y' + (3P" +9P2) y = 0, then four linearly independent integrals are given by 013, 01202, 01022, 02, where 01 and 02 are linearly independent integrals of d20 d2+ PO= 0. Shew also that, if the relations fl, 21, f3 =0 Y2, f3, Y4 * These arise in the same manner as for the cubic. If the equation d4Y+6P d2Y+4P dy dX4 2 dX2 2d+ be transformed by the relations dzx d30 d=z-2, ~3P62=0- y0=U3, dx dX2 into dzu + 4Q3 du + Q4u=0, then e,=( 3 d Q dQ3 = 28 3+ 6d2(, P2); 3-2 P,-2 2 -'T6 invariants dzar Q 4 - dQ dQ3, and the fundamental invariants are Q31 Q4-2 See my memoir quoted d2 p. 210, note. 214 EQUATIONS OF THE THIRD ORDER [71. subsist among four linearly independent integrals of an equation of the fourth order, (so that the integral curve is a twisted cubic), the equation must be of the above form. Ex. 4. Construct the equation of the fourth order having 0,1, 010b2, 0201, 02(02 for a set of linearly independent integrals, where 06 and 02, 5 and q2, are linearly independent integrals of the respective equations d20 d_ + 0Q = dx+PO, -dX2 Hence infer the form of a quartic equation when a single homogeneous quadratic relation subsists among a fundamental system of integrals. Ex. 5. Shew that the equation y"" +ry"' + 4sy" '+ (6s' + 4rs) y' +2 (s" +rs') y= O is satisfied by y= 02, where 0 is an integral of " +s0=0; and hence integrate the equation. (Fano.) Ex. 6. Shew that, if five linearly independent integrals of an equation of the fifth order are connected by the relations /y, 2Y, y3 4 l=0, 2 Y2 Y3, Y41 Y5 the equation can be taken in the form d +20s y + 30 ds - + 18 d2642) + (4 dS +64s d y= dx5 dx3 di dxn r dx2 dx dx s and thence integrate the equation as far as possible. (Fano.) 72. Consider now the more general case when three linearly independent integrals of the equation d3Y3 dy + 3Qd + Ry = O dz3 dz are connected by an irresoluble relation F(yl, y2, ) y= 0, where F is a homogeneous polynomial of order greater than two: the question is as to the character of the integrals of the equation. For the discussion, it is assumed that the differential equation has its integrals regular and free from logarithms: it thus is of Fuchsian type. Let K denote any non-evanescent covariant of the quantic F; such a covariant is the Hessian, which would vanish only if F 72.] AND TERNARIANTS 215 contained a linear factor. Let z describe any contour, which encloses any one of the singularities, and return to its initial value; the effect upon the fundamental system of integrals y, y,, y3 is to change them into another fundamental system Y1, Y2, Y3, the two systems being connected by relations Yr= arl +/3ry2 + 77 Y3, (r = 1, 2, 3). The determinant of the coefficients a, S3, y (say A) is different from zero in every such case; in the present case, owing to the absence of the term in d2 from the equation, we have (~ 14) A=1, by Poincare's theorem. Now the preceding relations constitute a linear transformation of the variables in the foregoing homogeneous forms; hence if pu be the index of K, and K denote the same function of Yf, Y2, Y3 as K is of y,, y2, y3, we have - K, for /J is necessarily an integer. It thus appears that the value of K is unaltered by the description of the contour. This holds for each of the singularities, as well as for z = o; hence K, when expressed as a function of z, is a uniform function. To obtain the form of K in the vicinity of any singularity a, we take account of the fact that the equation is of Fuchsian type: hence in the vicinity we have, for any integral y, (z - a)-Py = holomorphic function of z - a, where IPl is a finite quantity. Now K is of finite order in the variables y,, y2, y; accordingly substituting for them, and remembering that K is a uniform function of z, we have (z - a) —K = holomorphic function of z - a, where a is an integer, positive or negative. This holds for each of the singularities, the number of which is limited when Q and R are rational functions of z; it holds also for z= o. Hence K is not merely a uniform function, but it is a rational function, of z. 216 APPLICATION OF [72. It therefore follows that every covariant of the quantic F is a rational function of z, exceptions of course arising in the case when the covariant in an invariant, so that it is a mere constant. Take then any two covariants, say the Hessian H, and any other, say K: we have F=O, H=-, K=#, where / and * are rational functions of z. These are three algebraical equations to determine y1, y2, Y3 in terms of z; and therefore the differential equation is integrable algebraically, a theorem first announced* by Fuchs. A case of exception arises, when the Hessian is a constant: the quantic F is then of the second order so that the case has already been discussed; the integration of the original equation depends upon the integrals of a linear equation of the second order. As an illustration, consider the equation y"' + 3Qy' + Ry=o, when a fundamental set of integrals is connected by a homogeneous cubic relation. We assume that the equation is of Fuchsian type. Taking the cubic in the canonical form, we have F=y13 +qy23 +Y33 + 61g1Y2Y3 = 0, I being a constant. The Hessian is a rational function, say 4 (1 +813); so that H= -2 (y13 +Y23 +y33) - (1 + 213) Y1Y2Y3 = (1 + 813), and therefore Y1Y21Y3= - - (, Y13 +'23 + +y33-1. Taking the other symmetric covariantt of the cubic, which also is a rational function, we have ' = (1 + 813)2 {y16 +y26 +y36 10 (23Y33 +Y33Y13 + 3Y23)}, and, is equal to a rational function; so that, taking account of the above value of 13 +y23 +Y3, we can write y13y2+ 3 Y + 33Y13 = Thus Y13, y23, y33 are the roots of s3 - 61gq2 + 43 + = 0, * Acta Math., t. I (1882), p. 330. t Cayley, Coll. Math. Papers, t. xi, p. 345. COVARIANTS 217 an irreducible cubic. So far as the coefficients are concerned, they are known to be rational functions of z; the denominator of each such function is known, because its factors arise through the singularities of the equation and the multiplicity of any factor can be determined through the associated indicial equation; and the degree of the numerator has an upper limit, determined by the behaviour of the integrals for large values of z. Hence qb and, can be regarded as known, save as to a polynomial numerator in each case. We have r,3 =61lq12 _4 q 3 _3 j ' = Aq2 + Bq + C,r" =Al?2+Bi +Cl.'m. = A2D2 + B27 + C2 the last three being obtained, after differentiation, by repeated use of the cubic equation for 7, and the quantities A, B, C,... being functions of p,, and their derivatives. Now writing y=q- in the differential equation, we find 1 ~2'111 -_ f/ + 3 + Q,2q + Rq3 = 0. When the above values are substituted and the result is reduced by means of the cubic equation, so that no power of r7 higher than the second occurs, we have an equation of the form Yir2 + Y2? + Y3 = 0, where Y1, Y2, Y3 involve 0, + and their derivatives, and are linear in Q, R. As the cubic is irreducible, so that this equation holds for each root, we have Y1 =0, Y2=0, Y3 =, three equations to determine qp and +. There consequently exists a relation among the remaining quantities, viz. Q and R: and this must be equivalent to the condition (~ 71, Ex. 2), which must be satisfied in order that the equation F=0 may exist. Similar results hold for the cubic equation, when the homogeneous relation between the integrals is of order greater than three; and corresponding results hold for linear differential equations of higher orders. In fact, if a general homogeneous relation of finite order higher than the second subsists among a fundamental system of integrals of a linear differential equation of order n, thein the equation is integrable algebraically: the proof follows the lines of the preceding proof exactly. This range of investigations will not, however, be pursued further, as it becomes mainly formal in character, depending upon 218 APPLICATION OF COVARIANTS [72. the theory of covariants and upon the application of the theory of groups to linear differential equations. An excellent account of what has been achieved, together with many references, is given in a memoir* by Fano who has made many contributions to the subject; a memoirt by Brioschi contains some investigations connected with ternariants; and other detailed references are given in Schlesinger's treatise+, which contains an ample discussion of the subject. * Math. Ann., t. LIII (1900), pp. 493-590. + Ann. di Mat., 2a Ser., t. xIII (1885), pp. 1-21. 4 Theorie der linearen Differentialgleichungen, ii, 1 (1897), pp. viii-xi. The discussion is to be found in chapters 2-6 of the tenth section of the treatise. CHAPTER VI. EQUATIONS HAVING ONLY SOME OF THEIR INTEGRALS REGULAR NEAR A SINGULARITY. 73. IT has been seen that, if all the integrals of an equation are to be regular in the vicinity of each singularity, the coefficients in the equation must be rational functions of z of appropriate form and degree. It may, however, happen that the coefficients are rational functions of z but are not of the appropriate form and degree: in that case, it is not the fact that all the integrals are regular, and it may even be the fact that none of the integrals are regular. This deviation from regularity need not occur at each singularity of the equation: a fundamental system may be entirely regular in the vicinity of one (or more than one) of the singularities, and may not possess its entirely regular character in the vicinity of some other. The conditions necessary and sufficient to secure that all the integrals are regular in the vicinity of a singularity a have already (Ch. III) been obtained. If these conditions are not satisfied, then the composition of the fundamental system in the vicinity of the singularity a is no longer of an entirely regular character: we desire to know the deviations from regularity. It may also happen that not all the coefficients are rational functions of z; in that case, if uniform, they are transcendental functions and possess at least one essential singularity, say c. Further, owing either to a possibly excessive degree of the numerator in a rational meromorphic coefficient or to a possibility that z = oo is an essential singularity of some one or more of the coefficients, it can happen that the conditions for regularity of integrals near z = cc are not satisfied. The fundamental system 220 EQUATIONS HAVING [73. is then not entirely regular near c or for large values of lz, in the respective cases indicated, and it may even be devoid of any regular element; the same question as to its composition arises as in the corresponding hypothesis for the singularity a. Accordingly, for our present purpose we assume that the coefficients in the differential equation are everywhere uniform: that (unless as otherwise stated) they may have any number of poles, and that they may have one or more essential singularities. When a is a pole of one (or more than one) of the coefficients, and is not an essential singularity of any of them, we have one of the cases just indicated; when oo is a pole of coefficients, not being an essential singularity of any one of them, we have another. We write 1 z-a=x, z=-, x X' in these respective cases; and then our differential equation takes the form d"w dm-lw dm-2w dw dxcm + -l P2 dxm -2 + d x _ + pM-1 dx + pm' = 0, where the point x = 0 is a pole of some (and it may be of all) the coefficients. If all the integrals were regular in the vicinity of x = 0, then xrp, for r= 1, 2,..., m would be a uniform function of x that does not become infinite when x = 0. As some of the integrals are to be not regular in the vicinity of x= 0, the multiplicity of the origin as a pole of Pr must be greater than r, for some value or values of r. Let py = x- P (x), (r = 1,..., m), where rr is a positive integer (which may be zero for particular coefficients), and Pr (x) is a uniform function of x which does not become infinite when x = 0: also it will be assumed that, unless Pr vanishes identically,,r has been chosen so that Pr (0) does not vanish, so that wr measures the multiplicity of the pole of Pr at the origin. Then one or more than one of the quantities 33r - r (r=1,..., m) is a positive integer greater than zero. As in ~ 23, let L=-.logx; ONLY SOME INTEGRALS REGULAR 221 and suppose that ox + A-1L + A_2L2 +... + 1 L^-l1 + b0LV is an integral of the equation, regular in the vicinity of x = 0 and belonging to an exponent a; then it is known (~ 25-28) that o0 is a regular integral also belonging to the exponent a, so that < o = X== o, where >o0 is a uniform function of x which does not vanish when x = 0. As this expression, when substituted for w, should make the equation satisfied identically, the aggregate coefficient of the lowest power of x must vanish (as, of course, must all the other aggregate coefficients). The lowest power of x in the respective terms has for its index M-m, L-m-(m-l), u -S2 - (ma-2) W, -..., - n —:, H-am: and for any other integral, belonging to an exponent a, the corresponding numbers would be a- m -, - -(m - 1), - (m - 2),..., a - m — 1, - m. Let s + (m- s)== s, (s= 0, 1,..., m), and consider the set of integers H0, IX1,..., Hn. Of these, let the greatest be chosen. It may occur several times in the set; when this is the case, let the first occurrence be at IU, as we pass in the order of increasing subscripts, so that II < Hn, for r = 0 1,..., n-1, Hn >_n +, r = O, 1,..., m-n. Then n is called* the characteristic index of the equation: when n = 0, all the integrals are regular. The lowest power of x after substitution of the expression for the regular integral has L - Tin for its index; it arises through dm-nw pn -neW and later terms in the differential equation; as the coefficient of this lowest power must vanish, the exponent I must * Thom6, Crelle, t. LXXV (1873), p. 267. 222 CHARACTERISTIC INDEX AND INDICIAL EQUATION [73. satisfy an algebraic equation of degree m - n. Similarly for an exponent ca to which any other regular integral belongs; it also is a root of the same algebraic equation; and each such exponent satisfies that same algebraic equation of degree m - n, which accordingly is called the indicial equation. But it must not be assumed (and, in fact, it is not necessarily the case when n > 0) that the number of regular integrals is equal to the degree of the indicial equation. It is clear that, in all cases where n > 0, the degree of the indicial equation is less than m. 74. Suppose now that the given differential equation of order m has a number s of regular integrals, which are linearly independent of one another, where s < m: (the case s = mn has already been discussed): and that there do not exist more than s linearly independent integrals. After the earlier discussion of fundamental systems, it is clear that any regular integral of the equation is expressible as a homogeneous linear combination of the s integrals, with constant coefficients; also that, if every regular integral of the equation is expressible as such a combination of s (and not fewer than s) such integrals, the number of regular integrals linearly independent of one another is s. Further, a linear relation among the integrals of the equation, involving a number of regular integrals and only a single one that is not of the regular type, cannot exist; for the single non-regular integral would involve an unlimited number of negative powers of x, while each of the others occurring in the linear relation involves only a limited number of such negative powers. A linear relation might exist among the integrals of the equation, involving a number of regular integrals and two integrals that are not of the regular type. We then regard the relation as shewing that the deviation from regularity is the same for the two integrals: and in constituting the fundamental system for the equation, we could use the relation as enabling us to reject one of the non-regular integrals, because it is linearly expressible in terms of integrals already retained. So also for a linear relation with constant coefficients between regular integrals and more than two integrals of a non-regular type. Again, suppose that our differential equation of order m has an aggregate of n integrals, regular in the vicinity of = 0 and THEORY OF REDUCIBILITY 223 linearly independent of one another; and let it be formed of subgroups of integrals of the type ^+ll + ( A (2-,) A-,_l,lL2..+ +... +x2,L^-1 + 1,,LA, for X = 0, 1, 2,..., Ig, where (KA+ l)='n. Then, after ~~ 25-28, we know that these n linearly independent integrals constitute a fundamental system for a linear differential equation of order n, the coefficients of which are functions of x, uniform in the vicinity of x = 0; let it be dny d'-ly d-2y dy d +r - d n+ r, d+_ + - d l y + r = O. Now this equation, being of order n, cannot have more than n linearly independent integrals: and its fundamental system in the vicinity of x = 0 is composed of the n regular integrals of the original equation. Hence, by ~ 31, we must have r, = a-x-R,, (), =, 2,..., n), where R,(x) is a holomorphic function of x in the vicinity of x = 0, such that R, (0) is not infinite. Accordingly, the aggregate of the n linearly independent regular integrals of the original equation are the n integrals in a fundamental system of a linear equation of order n of the foregoing type. REDUCIBILITY OF EQUATIONS. 75. If therefore some (but not all) of the integrals of the given equation of order m are of the regular type, it has integrals in common with an equation of lower order. On the analogy of rational algebraic equations, which possess roots satisfying an algebraic equation of the same rational form and of lower degree, the differential equation is said to be reducible. Consider two equations dPy d"ly dy Pmy = M (y) = Po d-in + P dxm-1 + - d + Pm y dy) d+-ly dyy N (y)= Qo d -n Q + 1 dy_ + +Q y =) dX1 dxn — dx 224 REDUCIBILITY OF [75. where m > n; and take an expression d'y dy L (y) = R, + RI -4... + I, + RI y, where the coefficients R0, R,,..., RI are at our disposal, and 1 = mn - n. Let these disposable coefficients be chosen, so as to make the order of the equation M (y)-L I{N(y)} = O as low as possible. By taking the 1 + 1 relations P = RoQo, p, = R,Qo + Ro (1Qo' ~ Qj), P2 = R2Q0 + RI t(I - 1) Qo"' + - 1) }, PI = RQo + RI_, (Qo'+ Q1) + RI-2 (Qo" + 2Q' + Q2) +..., which determine Ro,..., Rt, we can secure that the terms involving derivatives of y of order higher than n -1 disappear. Accordingly, writing dky dk-2y K (y)= So dk + dk2 +... + Sy, where So, SI,..., Sk are determinate quantities and kn n-1, we have M- LN= K, where K is of order less than N. Moreover, if Po,..., PM, Qo,..., Qn are uniform functions of x, having x = 0 either an ordinary point or only a pole, the same holds of the coefficients R and the coefficients S; so that L and K are of the same generic character as M and N. From this result several conclusions can be drawn. I. Any integral, common to the equations M==0, N= 0, is an integral of the equation K = 0. If, therefore, every integral of N= 0 is also an integral of M= 0, it follows that K= 0 must possess n linearly independent integrals; as its order is less than n, the equation is evanescent, and we then have M (y)= L {N(y)}. 75.] AN EQUATION 225 II. Any integral, common to the equations N = 0, K = 0, is an integral of the equation M= 0; and therefore, in connection with the first part of the preceding result, the integrals common to M = 0, N = 0 constitute the integrals common to N = 0, K = 0. The process of obtaining the integrals (if any), common to two given equations M= 0 and N= 0, can thus be made a kind of generalisation of the process of obtaining the greatest common measure of two given polynomials. Proceeding as above, we have M =LN +K N = L1K + K1 K =L2K1 + K2 Ks-2 = LsKs- + Ks where K1, K.,..., K, are of successively decreasing orders. Then unless an evanescent quantity K of non-zero order is reached, sooner or later a quantity K is reached which is of order zero, that is, contains no derivative. In the former case, let Kr+ be evanescent; then the integrals of the equation Kr= 0 constitute the aggregate of integrals common to M= O, = 0O. In the latter case, let K8 be the quantity of order zero; then the integrals common to M= 0, N = 0 are integrals of K= yf(z)= 0. Now f(z) is not zero, for otherwise Ks would be evanescent; and therefore we have y=0, the trivial solution common to all homogeneous linear equations. We then say that M = 0, N = 0 have no common integral. III. An equation having regular integrals is reducible. For one such integral exists in the form y = xf(x), where 101 is finite, andf(x) is holomorphic in the vicinity of x = 0, while f(O) is not zero. We have 1 dy 0 f'(x) y dx x f(x) -= 1R (), F. Iv. 15 F. IV. 15 226 CHARACTERISTIC FUNCTION [75. where R (x) is a holomorphic function in the vicinity of x = 0, such that R (O) is not zero. Thus the given differential equation has an integral satisfying the equation dx - yR () = 0, that is, it has an integral common with an equation, which is of the first order and is of the same form as itself: in other words, the equation is reducible. But it is not to be inferred that such equations are the only reducible equations. IV. If an equation M=0 has p (and not more than p) linearly independent regular integrals, it can be expressed in the form M (y)=L {N(y)} = 0, where N is of order p, and L is of order m- p. For the p regular integrals are known (~~ 25-28, 74) to satisfy an equation of the form N= 0, of order p. Every integral of N = 0 is an integral of M = 0; whence, by I., the result follows. 76. We proceed to utilise the last result in order to obtain some conclusions as regards the regular integrals (if any) of a given equation, say, dmw d'-lw dw P (w) = d + d-.+ + pm-l d + prnw = O. The result of substituting xP for w in P (w), where p is a constant quantity, is P(P)= P(P-)- (-~+1) p p(p-1)...(p -r+2)...+pp-i+Pm; this is called* the characteristic function of the equation P = 0 or of the operator P. We have _pp (xp) p (p-1)...(p-in + 1) + Pp (p-)...(p - m + 2) + pf x-_ +.... + Pm-1 + m; * Frobenius, Crelle, t. Lxxx (1875), p. 318. 76.] INDICIAL EQUATION 227 when the right-hand side is expanded in ascending powers of x, it contains (owing to the form of the coefficients p) only a limited number of powers with negative indices. The highest powers of x-1, arising out of the m + 1 terms in x-PP (XP), have exponents m, soT + m - 1, W.2 + m - 2,..., _m-, + i1, fan, that is, IIno, HI,..., H.Let n be the characteristic index of the equation, so that IIn is the greatest integer in the set: if several of the quantities II be equal to this greatest integer, then Hn is the first that occurs as we proceed through the set from left to right. Denoting the value of fHn by g, let =-r q.r (') =, (r = 1,,,m) so that q, (0) is not zero, and no one of the quantities qr (0) is infinite. Then X-PP (XP) = X-9 G (p, x), where G is a polynomial in p and is holomorphic in x in the vicinity of x=0. Moreover, expanding G(p, x) in ascending powers of x, we have G (p, X) = go (p) + g, (p) +.., where each of the coefficients g is a polynomial in p, of degree not higher than m; the degree of g0 (p) is m - n, and the degree of gg-m (p) is m. Also, go (p) is the quantity called (~ 39) the indicial function; the equation go (p)=0 is called the indicial equation. Now take N (w) = xgP (w) dmw dm-lw dw =_ q4ox da + tx~-l dxm — +.. + qm-lx dx + qmw, where q0 = xsg-t; the equation P =0 can manifestly be replaced by the equivalent N (w) =0, which is taken to be the normal form for the present purpose. We have x-PN (XP) = G (p, x) = go (p) + xg (p) +..., 15-2 228 NORMAL FORM OF EQUATION [76. which thus contains only positive powers of x when the equation is in its normal form, and which has the indicial function for the term independent of x. We have seen that, if P (w)= 0 possess regular integrals, it is a reducible equation: and the operator P can then be represented as a product of operators. Consider, more generally in the first instance, two operators A and B, each in its normal form; and let C, also an operator, denote AB. Further, let the characteristic functions of A, B, C, respectively be A (xP)= xPf(x, p) = XP f, (p) x = i fp (p)XI+ tL=0 <x=0 B (xP)= xPg (x, p) = P S g, (p) x = 2g (p) x++P 1A=0 tk=0 C (xP) =P h (x, p) = XP Z h (p) xa =E h, (p) x +P where the summations in f(x, p) and g (x, p) include no negative powers of x, because A and B are in their normal forms. Now, as C= AB, we have C (x) = AB (xP) =A. { g,(p) x+P} /x=o = g, (p) A (x+xP) = g (p)AYk (a + p) X+E+P, /i=O A=O and therefore h,( p) = S ZE g (p)fA (I+p) x +.,u=0 A=O As X and, are incapable of negative values, there are no negative values for a; and therefore C is in a normal form. Also h (p) = go (p)fo (p), so that the indicial function of C is the product of the indicial functions of its component operators: and Or,, (p) = I gt (P)f/-p (G + P) /1=0 Further, if C be known to possess a component factor B which, when operated upon by A, produces C, then A can be obtained. For, take B and C in their normal forms: the equation 2 h, (p) x= 2 Eg(p)fA (x + p) x G=O0 P=0 76.] CHARACTERISTIC INDEX 229 then holds. The values of X are clearly 0, 1,..., so that A is then in its normal form; and the successive quantities fx are given by the equation h(p)= gax(p) A (- - + p), A=0 for a-=0, 1,..., p, the values obtained being polynomials in p, because C is known to be composite of A and B. Of course, this merely gives the characteristic function of the operator; but the characteristic function uniquely determines the operator. For let f(x, p) be a finction, which is a polynomial in p, and the coefficients of which are functions of x: and let the degree of the polynomial be m. Then we have* m -1 f(, p)= 2 Um-np(p-1)...(p-m+ n+ 1)+ o, n=O where, taking finite differences in the form Af(x, p)= f(x, p + 1)- f(x, p), we have n! Un = tAnf(w, p)}p=O. Thus xPf (x, p) = xP {U p ( -) (p - m + n +~1) I )... } [~f (xx, w) M) M ' which is the characteristic function of the operator dm d-i+ d umxmyl d-x + Um-m-lX d- +... + U1X d + Uo the operator is determined by the characteristic function. CHARACTERISTIC INDEX, AND NUMBER OF REGULAR INTEGRALS. 77. Now let the equation of order m, taken in its normal form, be dmw dm-lw dw N (w) = qoxt dm + qxxm- dxm- ' + qm.. x d- + qm = 0; and suppose that it possesses s (and not more than s) regular integrals, linearly independent of one another. These s integrals * Boole's Finite Differences, 2nd ed., p. 35. 230 CHARACTERISTIC INDEX AND [77. are a fundamental system of an equation, of order s and of Fuchsian type; when this equation is taken in its normal form, let it be d8?w v d8-w dw S (w)=, + - Xs-] +. + -l + a8 =,0 Sw dxSy i dx-l' ~x where 1, 02,..., a.S are holomorphic functions of x in the vicinity of x =0. As all the integrals of S = 0 are possessed by N =0, there exists a differential operator T of order m - s, such that r = TS; because NV and S are in their normal forms, T also is in its normal form, so that we can take dm-s dm-s-l d m m-S _______Im _______+I _____-1 _ 4 r y + qox ddxmn-s -- 1 - dxn-8- +. + m-s —l dx + 'm-s, where Tr, 2...,,m-s are holomorphic functions of x in the vicinity of x = 0. If then T (x)= P 0 (x, p), the indicial function of T is the coefficient of x~ in 0 (x, p), which is a polynomial in p and contains no negative powers of x. This coefficient may be independent of p; in that case, the characteristic index of T is m - s. Or it may be a polynomial in p, say of degree k in p, where k > 0; the characteristic index of T then is m - s -k. Because N = TS, the indicial function of N is the product of the indicial functions of T and S; so that the indicial function of S, which gives all the regular integrals of N, is a factor of the indicial function of the original equation. The degree of the indicial function of S is equal to s, because S = 0 is an equation of order s of Fuchsian type; the degree of the indicial function of N is m-n, where n is the characteristic index of NV= 0. Hence s + k = m -n, that is, s =m - n - k m - n; so that (assuming for the moment that k may be either zero or greater than zero) an upper limit for the number of regular integrals which an equation can possess is given by m- n, 77.] THE NUMBER OF REGULAR INTEGRALS 231 where m is the order of the equation, and n is its characteristic index (supposed to be greater than zero). It is known that, when n = 0, the number of regular integrals is equal to m. COROLLARY I. An equation, whose indicial function is a constant, so that its indicial equation has no roots, has no regular integrals; for its characteristic index is equal to its order. But such equations are not the only equations devoid of regular integrals. COROLLARY II. When k is equal to zero, then s is equal to m-n, so that the number of regular integrals of the equation is actually equal to the degree of the indicial function. The necessary and sufficient condition for this result is that the equation, which is reducible, must be capable of expression in the form N = TS, where the indicial function of T is a constant, and the degree of the indicial function of S is equal to the order of S. This result, which is of the nature of a descriptive condition, appears to have been first given in this form by Floquet*. Other forms, of a similar kind, had been given earlier by Thomet and by Frobeniusi (see ~ 83, post). Note. On the basis of the preceding analysis, it is easy to frame an independent verification that the characteristic index is not greater than m- s. For in the operator ', the quantity Tm-s-k does not vanish when x = 0; and all the quantities Tr, such that X < m- s-, do vanish when x = 0. Hence, when we take N as expressed in the form N= TS, the coefficient of ds+kw dxs+k is the first (in the succession from left to right) in which rT-s-k occurs; it also contains q0, T,..., rm-s-k-,, all of them occurring * Ann. de l'Ec. Norm. Sup., 2e Ser., t. viii (1879), Suppl., pp. 63, 64. t Crelle, t. LXXVI (1873), p. 285. + Crelle, t. LXXX (1875), pp. 331, 332. 232 NUMBER OF REGULAR INTEGRALS [77. linearly. When x= 0, all of these except 7T,-s-k vanish, and T,-s-k does not vanish; and therefore qm-s-k does not vanish when = 0. In the coefficient of dtrw dxtd where /A > s + k, the quantities (q,,..., *, m occur linearly: each of these vanishes when x= 0, and therefore qm_ does vanish when x = 0. As this holds for all values of /, it follows that qm-s-k is the first of the quantities q which does not vanish when x 9; hence the characteristic index of N is m - s - k, that is, it is m - s, where s is the number of regular integrals possessed by the equation N = 0. Ex. 1. If w=w, be an integral, regular and free from logarithms, of an equation P=0, which is of order in and has s regular integrals, and if a new dependent variable u be given by w = wl udz, shew that u satisfies an equation Q = 0, which is of order m -1 and has s-1 regular integrals; and obtain the relation between the characteristic index of P=O and that of Q=O. (Thome.) Ex. 2. The equation dmw m dm - rw + Pr dm- = has m-s integrals, regular in the vicinity of z=0 and linearly independent of one another, and z=0 is a pole for pl,... p,; shew that it is a pole (not an essential singularity) for each of the remaining coefficients p. (Thome.) Ex. 3. If, in the equation in the preceding example, Pl,..., pr are arbitrarily assigned, subject to the condition that z=0 is a pole or an ordinary point, prove that the remaining coefficients p can be determined so as to permit the equation to possess m-s arbitrarily assigned regular integrals, linearly independent of one another. (Thomne.) Ex. 4. Prove that the condition, necessary and sufficient to secure that an equation NV=O, of order m and having an indicial function of degree m -y, shall have mn-y-8 linearly independent regular integrals, is that N shall be a product of the form QMD, where the indicial functions of Q, M, D are of degrees 8, 0, n -y- respectively, and D is of order m-y-8. Is there any limitation upon the order of M? (Cayley.) Ex. 5. Shew that an equation QD=O has at least as many regular integrals as D=0, and not more than Q=0 and D=0 together; and that, if all the integrals of D=0 are regular, then QD=0 has as many regular integrals as Q=0 and D=0 together. 77.] AND DEGREE OF INDICIAL FUNCTION 233 Hence (or otherwise) shew that, if an equation P=0 has all its integrals regular, then P can be resolved into a product of operators, each of the first order and such that, equated to zero, it has a regular integral. Is this resolution unique? (Frobenius.) 78. In the two extreme cases, first, where the degree of the indicial function is equal to the order of the equation, and second, where its degree is zero, the number of regular integrals is equal to that degree. The preceding proposition shews that, in the intermediate cases, the degree merely gives an upper limit for the number of regular integrals. It is natural to enquire whether the number can fall below that upper limit. As a matter of fact, it is possible* to construct equations, the number of whose regular integrals is less than the degree of the indicial function. Taking only the simplest case leading to equations of the second order, consider the two equations O'= dy v= dy U= + ky + h =, V d-j+ky=0, of the first order; and form the equation dV dU U -V = 0o, dx dx which manifestly is of the second order, say dy+ dy dx +pd x+ qy = O, where 1 dh dk k dh p=kh dx- ' dx -hdx' If we can arrange so that x = 0 is a pole of p of order n, where n > 2, then x = 0 in general will be a pole of q of order n + 1; and the indicial function will then be of the first degree. Consider now the equation of the second order. Since U= V+ h, it can be written h V = 0, dx dhx which is satisfied by V=Ah, where A is any arbitrary constant. * Thome, Crelle, t. LXXIV (1872), pp. 211-213. 234 THOMI'S THEOREM [78. Let Y be an integral of the equation of the second order. It may be an integral of V= 0; if it is not, then, when we take Y Y = - iiA' we have dxy + kyl + h = O, that is, yi is an integral of U= 0. Thus any integral of the equation of the second order either is an integral of V = 0 or is a constant multiple of an integral of U= 0. If, then, U= 0 and V= 0 are such that they possess no regular integral, the differential equation of the second order can possess no regular integral; at the same time, its indicial function is of the first degree. The equation V= 0 will not have a regular integral, if x = 0 is a pole of k of order greater than unity; and the equation U= 0 will then not have a regular integral, if h is a rational function of x. Ex. 1. The aggregate of conditions can be satisfied simultaneously in many ways. For instance, take P=, h=x; then 1 1 3+2x k=x2+, q=- ' The differential equation of the second order is d2. y 1 dyc/ 3+2x dx + x2 dx -_ y = ~; its indicial equation is of the first degree, and it has no regular integrals: or the number of its regular integrals is less than the degree of its indicial equation. The conclusion can otherwise be verified; for it is easy to obtain two linearly independent integrals in the form 1 1 1 x x no linear combination of which gives rise to a regular integral. Ex. 2. Shew that the equation +1,5 + 2x has no regular integrals: and verify the result by obtaining the integrals of the equation. (Thome, Floquet.) 79.] DETERMINATION OF THE REGULAR INTEGRALS 235 DETERMINATION OF SUCH REGULAR INTEGRALS AS EXIST. 79. When the degree of the indicial function of an equation of order m is less than m, no precise information is given as to the number of regular integrals possessed by the equation. The further conditions, sufficient to determine whether a regular integral should or should not be associated with any root of the indicial equation, can be obtained in a form, which is mainly descriptive for the equation of general order and can be rendered completely explicit for any particular given equation. Let the equation be dmw 7m-lw N(w)= qoL d= m + qlxm-l dxn — +... + qmw = 0, of characteristic index n. Let E (0) be the indicial function, and let a- be one of its zeros, so that E(a)= 0. Then, if a regular integral is to be associated with a-, it must be of the form at = X( (Co + C1x + C2x2 +... + CpXP +...). This expression, when substituted in the equation, must satisfy it identically, so that, after substitution, the coefficient of x+P must vanish for every value of p: and therefore fo (p) Cp +f/ (p)Cp+, +... +f, () cp+,= 0, where the number of terms in this difference-relation depends upon the actual forms of qo, q,,..., qm. Of the coefficients fo, fi,...,f,, the first is fo (p) = E( + p), which is of degree m-n in p; of the remainder, one at least, viz. fg-m, is of degree m in p, where g has the same significance as in ~76. The successive use of this difference-relation, together with the equations for the earlier coefficients, the first of which is CoE () = 0, leads to the values of all the quantities c,^- co, for the successive values of /; and thus a formal expression for m is obtained that 236 DETERMINATION OF THE [79. satisfies the equation. If, however, the expression is an infinite series, it has no functional significance when it diverges: that this frequently, even generally, is the case, may be inferred as follows. For if c.++ c,, with indefinite increase of p, tends to a limit that is not infinite, so also would c,+2 c,+, +3 + c,+2, and so on; and therefore CfA+a C5A+1 C_+2 C/_+a C A C C~+1 C+a —1 for finite values of a, also would tend to a limit that is not infinite. Now a number of the quantities.f (i) fo (,) for various values of 0, undoubtedly tend to zero as au increases indefinitely; some of them may have a finite limit: but one at least is infinite, viz. fs(Fl) fo(g) ' because the numerator is of degree n higher than the denominator, both of them being polynomials in /u. Consequently, the expression f] (P) C1+1, 2 (i) C4+ 2 A () C9r+1r fo( ) C () f G o() ' f0(_) C acquires an infinite value as, increases without limit. The difference-relation requires the value of the expression to be always - 1, so that the hypothesis leading to the wrong inference must be untenable. Therefore c,+ - c5, with indefinite increase of,a, does not tend to a limit that is finite, and therefore the series diverges*. There is then no regular integral to be associated with the root a. * It is not inconceivable that, for special values of m and of n, and for special forms of the coefficients q, as well as for a special value of the limit c,+l- c,, the infinite parts of the expression fr (/A) c+r r=lfo (/F) Cpt might disappear, and the expression itself be equal to -1. In that case, the series would converge: and an exception to the general theorem would occur. But it is clear that such an exception is of a very special character: it will be left without further attempt to state the conditions explicitly. 79.] REGULAR INTEGRALS THAT EXIST 237 As the series thus generally diverges when it contains an unlimited number of terms, the regular integral is-thus generally illusory. The only alternative is that the series should contain a limited number of terms: and then the regular integral would certainly exist. Accordingly, let it be supposed that the series contains k + 1 terms, so that C1 C2 Ck Co co co are quantities known from the difference-relation, and that Ck+, Ck+2,... ad inf. all vanish. If we secure that ck+lC, ck+2,, ck+ all vanish, then every succeeding coefficient must vanish in virtue of the differencerelation; and these 7 relations will then secure the existence of a regular integral to be associated with the exponent a-. Taking p =, k-1,..., k-T + 1 in succession, we find the T necessary conditions to be f0 (k) Ck = 0, that is, fo (k)= 0, and generally r 2 fs (k - r) ck_,.+ = 0, s=0 for values r= 1, 2,..., -- 1. The first of these is E (a + k) = 0, so that the indicial equation, which possesses a root o-, must possess also a root a- + k, where k is a positive integer. (In the special instance, when k = 0, no condition is thus imposed: in the general instance, when k is a positive integer greater than zero, it is easy to verify that E (a + k) is the indicial function for x = o.) When the aggregate of conditions, which will not be examined in further detail, is satisfied in connection with a root of the indicial equation, a regular integral exists, belonging to that root as its exponent; and there are as many regular integrals, thus determined, as there are sets of conditions satisfied for each root of the indicial equation. Explicit expressions for the various coefficients c can be derived, when the explicit forms of the quantities q are known: but the general results involve merely laborious calculation, and would hardly be used in any particular case. The results are therefore, 238 MODE OF OBTAINING [79. as already remarked, mainly descriptive: and so, in any particular case, it remains chiefly a matter for experimental trial (to be completed) whether a regular integral is necessarily associated with a root of the indicial equation. For this purpose, and also for the purpose of discussing the regular integrals associated with a multiple root of the indicial equation, a convenient plan is to adopt the process given by Frobenius (Chap. III) when all the integrals are regular. We substitute an expression W = CoXP + C1xP+1 +... + Cx P+ -... in the equation dmw dir-lw N (w) = q ~L + qdxm-1 ddx,-_ +... + qw, of characteristic index n. After the substitution, the first term is cE (p) P, where E(p) is the indicial function, of degree m - n; and we make all the succeeding terms vanish, by choosing the relations among the constants c appropriate for the purpose. We thus have N(w) = cE(p)xP; and the relations among the constants c are of the form c,E (p + p) = aC,_ - c,_ + c +..._ c- + c. + a,0Co, where the constants a,,_-,..., a,0 are polynomials in / and, when this relation is the general difference-relation between the coefficients c, one at least of these polynomials a,,, is of degree mL in /A. When the difference-relation is used for successive values of p, we obtain expressions for the successive coefficients c, which give each of them as a multiple of Co by a quantity that is a rational function of p/. When these coefficients are used, we have the formal expression of a quantity w which satisfies the equation N (w) = co E(p) XP. Unfortunately for the establishment of the regular integrals, this formal expression does not necessarily (nor even generally) converge: for, in the difference-relation among the constants c, the right-hand side is a polynomial of degree m in p, while the lefthand side is a polynomial of degree m - n in p, so that the series c^ P+i would, as in the preceding investigation, generally diverge. 79.] REGULAR INTEGRALS 239 But while this is the fact in general, it may happen that the series would converge when p acquires a value occurring as a root of the equation E(p)= O. In that case, the series satisfies the equation N (w)= 0: in other words, it is a regular integral of the differential equation. Further, if the particular value of p be a multiple root of the indicial equation, it can happen that the series aw converges for this particular value of p; and then N( ) =- {coE(p)xP} =0, because the value of p is a multiple root of E= 0: in other words, aw is then a regular integral of the differential equation. And so possibly for higher derivatives with regard to p, according to the multiplicity of the root of E = O. The whole test in this method is therefore as to whether the series c XP+~ converges for the particular value (or values) of p given as the roots of the indicial equation. The method of dealing with a repeated root of the indicial equation has been briefly indicated. Corresponding considerations arise, when E =0 has a group of roots differing among one another by integers. In fact, all the processes adopted (in Ch. III) when all the integrals are regular, are applicable when only some of them are regular, provided the various series, whether original or derived, are converging series. The deficiency, that arises through the occurrence of diverging series, represents the deficiency in the number of regular integrals below w - n. As already stated, the tests necessary and sufficient to discriminate between the convergence and divergence of the various series are not given in any explicit form, that admits of immediate application. 240 EXAMPLES [79. Ex. 1. Consider the equation &Xy" +xy'- (3 + 2x) y= 0, constructed in ~ 78, Ex. 1. The indicial equation is p-3= 0, so that there is not more than one regular integral; if it exists, it belongs to an exponent 3. To determine the existence, we substitute y=c0x3+cx4+ c2x5 +... in the original equation; that it may be satisfied, we must have 0={(n+2) (n+l)-2}) cn+ncn for n= 1, 2,.... We at once find,= - (n+3) ce_-, and therefore Cn =(- l)(n+3)!. The series E c,x3 +? diverges, and therefore the one possible regular integral 7=O does not exist; that is, the original equation possesses no regular integral, although the indicial equation is of the first degree. If there were a regular integral, it would satisfy an equation dy where u is a holomorphic function of x; and the original equation could then be written x2 d- ) dy _ ) - where v is some holomorphic function in the vicinity of x=0. It might be imagined that, as the indicial equation is of degree unity (a property that does not forbid the existence of a regular integral), it would be possible to obtain the regular integral through a determination of u, and that the divergence of the series in the preceding analysis is due to the operator 2 d dx X - - V, which annihilates only expressions that are not regular. That this is not the case may easily be seen. We have d \ dy dY dy du (x$2 d _ V) }(x dy _ y) =) (d2 + (X2 - Vx - X2U) dX (2 - ) so that, if the resolution be possible, we have X2 _ - X2u- =X, du x2 -U — vu=3+2x. dx EXAMPLES 241 Substituting in the second of these the value of v given by the first, we find x2 - + x2-xu+u = 3 + 2x, dx as an equation to determine u, supposed a holomorphic function of x. Let = a - ax + a2x2 -... be substituted; in order that the equation for u may be satisfied, we have ao = 3, ao2 -a0 -a = 2, and, for values of n higher than zero, (n+2a - 1) a+ 2 (a1a, _ +a2an_2+*.) -- an+1=0. Hence ao=3, a1=4, a2=24, and so on. The relation giving an+,, when taken for successive values of n, shews that all the coefficients a are positive; hence an + 1 > ( + 2a -1) a > (n+5) a,, that is, 30a > (n+4)!, and so the series for u diverges: in other words, there is no function u, and the hypothetical resolution of the equation is not possible. Note. This argument is general; it does not depend upon the particular coefficients for the special equation that has been discussed. Ex. 2. Consider the equation Dy= xy" + x2 (1 - x- 2x2)y"-x (5+4x+4x2) ' +(9+ 10x+4x2)y=O, which is in the normal form. The characteristic index is 1; the indicial equation is (0- 1)-50+9=0, that is, (0-3)2=0, so that the number of regular integrals cannot be greater than two, and such as exist belong to the exponent 3. To determine these regular integrals (if any), we adopt the Frobenius method of Ch. III. Taking y= oxP +c xP+ +... +ex e +n+..., we have y = Co (p - 3)2 xP, provided p2_ 2p-5 l=-co p-2 and, for values of n greater than unity, 0=(p+n- 3) en+{p2 + p(2n- 4)+n2-4n- 2}) c,,- 2 (p+n) c_2, a factor p+n-3 having been removed, because it does not vanish for these values of n. Let (p + n- 3) cn-2 _c 1= kn, F. IV. 16 242 242 ~DETERMINATION OF REGULAR INTEGRAL [9 [79. so that Also the difference-equation for the coefficients c becomes so that II~p-) 1(p n) Hence, writing n= II (p +n~3)U~ in the relation (p~n- 3) cn-2c,, 1=kn, and substituting the value of kn, we have -u l.(_1nI1(p~n11(p)n4 p-3 o Adding the sides of this equation, taken successively for n, n - 1,.,3, 2, and noting that U=1-11 (p-2)cl we have -2(_15+2p~p2)H(p-3)e0, We thus have a value of y in the form Y=Y n p+n n~=O where (-1)mH1(p~m)IH(p+m-4) and this satisfies the relation D Y= co (p -3)2 Xp. It is clear that formal solutions of the original differential equation are [Y]=3) a Of these, the first is Yo Co 11 (n)' in effect, a constant multiple of X3e2x; and the second is yo log x +a diverging series, 79.] WHEN IT EXISTS 243 because a series, in which n(n) I ( -- ( + If + 3)1 I (m-1 ) 1_ m=2 2_____________________n 6 II(n) is the coefficient of xn +3, manifestly diverges*. It thus appears that, although the indicial equation for x=0 is of the second degree, the differential equation possesses only one integral which is regular in that vicinity; and this integral is a constant multiple of x3e2x. This regular integral satisfies the equation x - (3+ 2) y = 0, xso that the original equation must be reducible. It is easy to verify that it can be expressed in the form d2 d d0 {X3 d +2 x- d — (3 + 2x)}- ' dX-(3+ r2) -} Ex. 3. As an example which allows the convergence of the series for the regular integral to occur in a different way, consider the equation x2y"- ( - +2x 2) y +( - 2x +x2)y=0. The indicial equation is p =0, so that one regular integral may exist. To determine whether this is so or not, we substitute y= ao + alx+ 2 +..., which (if it exists) belongs to the exponent zero. Comparing coefficients, we find a = a0, 2a2=ao, and, for all values of n that are greater than unity, (n+ 1) an+l=(n2+n 1) an- 2n-1 +an-2. Let -a = ma,,, - am; then Cn+ l-(? t+ 1) C-Cn —_l. In general, the values of c (and the consequent values of a) as determined by the last equation, lead to diverging series; but in our particular case, 1 =ai- a0= 0, C2 = 2a2 - a = 0, so that c3=0, c4=0, and generally cm=O, that is, * The series in yo is saved from divergence because, in it, these coefficients are multiplied by the factor p - 3, which vanishes for the special value of p and which therefore removes the quantities that cause the divergence in the second integral. 16-2 244 EXISTENCE OF [79. and therefore a0 a, — =m! ' so that a regular integral exists. It is a constant multiple of ex. Ex. 4. Consider the equation D (y) = (X5 + x6) y""/ + (x3 + 4x4 + 4x5) y"' - (2x2 + 33 + 2x4) y" +(3x + 62+ 4x3) ' - (3 +6x 4x2) y = 0. The characteristic index is unity; hence the number of regular integrals is not greater than three. To determine them, if they exist, we take an expression y oxP+clxPf+lf +.. +CnXPP+n+..., and form D (y), choosing relations among the coefficients c such that all terms after the first in the quantity D (y) vanish. We thus find D (y) = o (p - 1)2 (p _ 3) xP, provided Clp2(p-2)+-o (p -) (p-2)(p2 +p-3)=0, and, for values of n greater than unity, c, (p+ n- 1)2 (p + -3) + c_l (p +n- 2) (p+n- 3) {(p + )2 - (p +) - 3} +cn 2(p+n- 3)2 (p+n - 4) (p+n)=0. The indicial equation is (p- 1)2 (p - 3)=0, of degree 3 as was to be expected (=4-1), because the characteristic index is 1. The roots form a single group; if a regular integral exists belonging to the root 3, it will be free from logarithms; if two regular integrals exist belonging to the root 1, one of them may or may not be free from logarithms, and the other will certainly involve logarithms. Consider the root p=3. As p+ n -3 then vanishes for no one of the values of n, we may remove it from the difference-equation, so that the latter becomes n (p + n- )2 + c_- 1 (p + n- 2) {(p + n)2 - (p+n)- 3} -+cn_2(p+n-3)(p+n-4) (p+n)=0. Taking cr(p+n- 1)2+ cn_ (p+n-2) (p+n-3)=kn, we at once find kn+ (p - n) k 1=0. We require the value of k2. We have, for p=3, 1= - 2c, 16c2 + 5 1cl + 10co = 0, so that k2 = 16c2 + 6c, = 80co. Now kn,=(- 1)n (n + 3) (n+ 2)... 62 = 2 (- l)" (n+ 3)! co; 79.] REGULAR INTEGRALS 245 so that, writing cn,-(-1)n an, we have (n + 2)2 an - n (n+ 1) an- =(n + 3)! 2 co. As al and a2 are positive, it follows that all the coefficients a are positive; and clearly an >(nl+ )! Sco, so that the series X3 (C- ax + a2x2-...) diverges; and there is no regular integral belonging to the root 3. Moreover, the coefficient of c%, being (p+n -1)2, does not vanish when p=3 for any value of n; hence, if two regular integrals exist belonging to the root unity of the indicial equation, one of them will certainly be free from logarithms. Consider now the repeated root p=l. As p + -3 vanishes for this value of p when n= 2, the difference-equation is then evanescent for n= 2 and it does not determine c2. For other values of n, the quantity p +-n -3 does not then vanish, so that it may be removed. We then have, for values of n >3, the same form of equation as before, viz. c (p+n- 1)2+ c-1 (p +n-2) {(p+n)2-(p +n)-3} +c_2(p + n- 3) (p+ n- 4)(p +n)=O. Also 1 -(p - 1) (p2p - 3),C the value p=l not yet being inserted because we have to differentiate with regard to p. The difference-equation for n =3 gives 9c3+ 18c2 = 0, so that C3= - 2C2. For values of n > 4, let p=r - 2, so that the value of Co is 3; take n-2=m, so that the values of mr are > 2; and write cn=bm; then the difference-equation becomes bm (m -+ 1)2+bm_l (-+m - 2) {((r+m)2 - (a +m) - 3} + bm-2 (+m- 3) (c +m- 4) (c+ -)=0. Here o-=3, m>2; c2=bo, c3=b1=-2bo: so that this equation is now exactly the same as in the former case for p= 3. The series thence determined is X3 (bo-ax +a2x2-...) with the earlier notation; it certainly diverges unless b =0. If b=0, every coefficient vanishes, and the series itself vanishes. As we require regular integrals, we shall therefore assume bo=0, that is, c2=O; and then all the remaining coefficients vanish, so that we have Y=cO [XP- xP+l (p-1) (p2+p-3 ) -, 246 EXAMPLES [79. an expression which is such that D Y=Co(p-1)2 (p - 3). Accordingly, are integrals of the equation Dw= 0. The former is cox: one regular integral thus is =X. The latter is co (x log x + 2); another regular integral is w =x+x logx. The original differential equation accordingly has two regular integrals. Ex. 5. Shew that the equation x2 (1 + x)2y" - (1 + 2 + 2x2 - x4) y' - (1 + 2 + 32 + 2x3)y =0 has one integral regular in the vicinity of x=0; and express the equation in a reducible form. Ex. 6. Shew that the equation x2 (1 + 2x + 2x2 + 4) y"'+ (1 +6x + 6x2- 3x - 2x6) y" -(2+12x+15X2 + 63 - X) y' +(1 +6x +8 x2 443 + x4)y =0 has two regular integrals in the vicinity of x=0, in the form e, xex; and obtain the integral that is not regular. Ex. 7. Shew that the equation x2y"+(3x- l)y'+y=o has no integral, that is regular in the vicinity of x=0; express the equation in a reducible form, and thence obtain the integral by quadratures. (Cayley.) Ex. 8. An equation P=0 can be expressed in the form QD=0, where D=0 has no regular integrals; can P=0 have any regular integrals? Illustrate by a special case. Ex. 9. In the equation dny dn-?1 d 4-P 1 dxn-. 4- PY =O, the coefficients P are polynomials in x of degree p, and p < n: shew that it possesses n-p integrals, which are integral functions of x. (Poincare.) 80.] IRREDUCIBLE EQUATIONS 247 EXISTENCE OF IRREDUCIBLE EQUATIONS. 80. We have seen that an equation is reducible when it is satisfied by one or more of the integrals of an equation of lower order, in particular, by the integral of an equation of the first order. The main use so far made of this property has been in association with the regular integrals of the equation: but it applies equally if the equation possesses non-regular integrals that satisfy an equation of lower order. It is superfluous to indicate examples. It must not be assumed, however, that every equation is reducible by another, if only that other be chosen sufficiently general. On the contrary, it is possible to construct an irreducible equation of any order m, as follows*. We construct an appropriate characteristic function which, as is known (~ 76), uniquely determines the equation. Take a polynomial in p of degree wn, say h (x, p); let the coefficients of the powers of p be holomorphic functions of x, not all vanishing when x=0; and let the function, subject to these limitations, be so chosen that, when arranged in powers of x in the form h (x, p) = Ao (p) + xh(p)+ X2h% (p)..., ho (p) is independent of p and not zero, and h, (p) is of degree m in p. Then if N = 0 is the equation determined by h(x, p) as its characteristic function, N= 0 is irreducible. Were N reducible, an equation S= 0 of lower order s would exist such that each of its integrals satisfies N =0; and then an operator Q, of order m - s, could be found such that N= QD. We take Q and D in their normal form; and so N is in its normal form. Now Q(x = Xp {7o (p) + xw (p) + x (p) +...I, D (xP) = xP {o (p) + X (p) + x2 (p) +...}, D (MP)= ~" CO(p)+ ZU1(p) +2C2rz(p)+...}, * Frobenius, Crelle, t. LXXX (1875), p. 332. 248 REDUCIBILITY OF EQUATIONS [80. the right-hand sides of which are polynomials in p of degrees m -s and s respectively. Then, as in ~ 76, we have ho (p)= o (p) o (P), hi (p) = 'o (p) l (p)+ (p) o(p + 1). Now ho (p) is a constant, being independent of p; hence, owing to the polynomial character of Q (p) and D (xP) in terms of p, the two quantities 0o(p) and qo(p) are constants. Accordingly, 7o(p + 1) is a constant; and therefore the degree of o0 (p) l(P) + ~l(p)(p+ l) in p is the degree of r7 (p) or i (p), whichever is the greater. But the degree of' (p) is not greater than s, and that of, (p) is not greater than m- s; so that, as s > 0, the degree is certainly less than m. But the expression is equal to ha (p), which is of degree m. Hence the hypothesis adopted is untenable; and the equation N= 0, as constructed, is irreducible. EQUATIONS HAVING REGULAR INTEGRALS ARE REDUCIBLE. 81. Suppose now that, by the preceding processes or by some equivalent process, the regular integrals of the equation N = 0 have been obtained, s in number, and that the equation of which they constitute a fundamental system is S= 0, of order s: a question arises as to the other m-s integrals of a fundamental system of N= 0. Let N= TS, where T and S (and therefore also N) are taken in their normal forms. The s regular integrals of N, say yl,, 2..., ys, all satisfy S = 0; and no one of the m-s non-regular integrals of 1A, say w,, w2,..., w,,, satisfies S = 0, for this equation has all its integrals regular. Let S (Wr) = r (r=1,..., -s); then, as r (wr) = 0, we have T (Ur)= 0. Now wv is not a regular expression; hence Ur is not regular, that is, it contains an unlimited number of positive and negative 81.] HAVING REGULAR INTEGRALS 249 exponents when it is expressed as a power-series. Accordingly, the m-s quantities u are integrals of the equation T(u) =0, which is of order m- s and has no regular integrals; and the m-s non-regular integrals of N= 0 are given by S (Wvr)= Ur, it being sufficient for this purpose to take the particular integral and not the complete primitive of the latter equation. The case which is next in simplicity to those already discussed arises when s = m - 1, so that the original equation then possesses only one integral which is not regular. The equation T =0 is then of the first order. With the limitations laid down, the normal form of T is d qox dx- + ql, where q0 and q, do not become infinite when x= 0. As the integral of T(u) = 0 is not regular, it follows that q, does not vanish and that go does vanish when x = 0; so that, if qo = x Q (x), where a is a positive integer > 1 and Q(x) is a holomorphic function in the vicinity of x = 0, such that Q (0) is not zero, the equation determining u is du ql =0 udx + i Q (x) say 1 du aoa a (a- 1) d- + x+- + +... +- + R (x)= 0, u dx Xa+l XaL X where R (x) is a holomorphic function of x in the vicinity of x = 0. This gives a al aa-1 u = x-re x x P1 (x) where P, is a holomorphic function of x in the vicinity of x= 0; and then to determine w, the non-regular integral of N=O, we need only take the particular integral of S (w) = u, 250 REDUCIBILITY [81. where din-' di-2 d S = q0 wM-1 dn + qix M-2 dx qM-m dx~n I dXM-2 + '+ M2Xd in which q0, qj,..., qm_- denote holomorphic functions of x, and q, does not vanish. Writing ao a, + +a a XWa-i X wU - vet the equation for v takes the form di-lv Pi dM-2v p2 din-3v q0 ax- ++ ~ q0 dxin-i 1ctd~-2 Xi~2a~ dxm —3 + Pm-1iV = X-0'i n+1p (,), Xy(mi) (d1+) where q., Pi, p2,..., p,m- are holomorphic functions of x, such that q, and pm-i do not vanish when x = 0. In some cases it happens that a particular integral of this equation exists, in the form of a converging power-series represented by where P (x) is a holomorphic function of X: in each such case, the non-regular integral of the original equation is X (M-1) a-a-r eQ P (Xc). But, in general, the particular integral of the v-equation is not of the same type as the regular integrals of the original equation: and then the non-regular integral of the preceding equation cannot be declared to be of that type. Ex. An illustration is furnished by the equation in Ex. 6, ~ 79, viz. c2 (1 + 2x+ 2X2 + X4) y"~ + (I + 6xv4 6X2 - 3X4 - 2Xv6) y" -(2 + 12cv+ 15X2 +G' -c X6) y'(1 +G(Ix+8X2 +4X3+X4)>y=O. It has two regular integrals, viz. y,==ex, y2=xex; and these constitute the fundamental system of y" - 2y'+ y = 0, or xv~y" -2xv2y'+xv2y-=O0 THE ADJOINT EQUATION 251 in the normal form. To have the given equation in the normal form, we multiply throughout by x2; and then it must be the same as {2 (1 +2x+2x2 + X) d +p} (x" -22y' + x2y) = 0, when p is properly determined. We easily find that = 1 + 4x 42 +x4- 25; and so the equation for determining u, where = x2y" - 2x2y' + x2y, y being the non-regular integral, is X2 (1 +2x +22+ 4) d- +(1 + 4x + 42 + 4- 2xa) =0. dxC Hence 1 du 1 + 4x + 4x2 +4- 2x6 X4 dx x2 (1 + 2x+22 + x4) 1 2 2+4x+4x3 =- 2 X+ 1 +2x+2x2+x4' so that _1 +2x+2v2 + xe X2 Hence the non-regular integral of the original equation arises as the particular integral of 1 +2x?+2x2+x4 y -2y' + y - + X4. i X Let y=vex; the equation for v is easily found to be v" - 2v' 1 + I + v 1+2x+2x2 + 4 1+ 2+2x2 +4 satisfied by v = 1: and therefore the non-regular integral is y ex. THE ADJOINT EQUATION, AND ITS PROPERTIES. 82. Of the properties characteristic of a linear equation, not a few are expressed by reference to the properties of an associated equation, frequently called Lagrange's adjoint equation. It is a consequence of the formal theory of our subject, as distinct from the functional theory to which the present exposition is mainly limited, that Lagrange's is only one of a number of covariantive equations associated with the original. As its properties have been studied, while those of the others remain largely undeveloped, 252 PROPERTIES OF LAGRANGE'S [82. there may be an advantage in giving some indication of a few of its relations to the original linear equation. The latter is taken in the customary form P (w) - P W(n) + Plw(n-l' + P2w(n-2) ~... ++ P, w = 0, where w(r) is the rth derivative of w with respect to z; and from among the various definitions of the adjoint equation, we choose that which defines it to be the relation satisfied by a qiantity v in order that vP (w) may be a perfect differential. Now, on integrating by parts, we find JVprW(nr)dz -= VpW(n-r-1) _-d ( Pr) W(n-r-2) _ ~~~j- dz IvP~a~i~E~ d2 dn-r dr + d2 (VPr) W (n-r —3) - + 1)n-rfw d (VPr) dz, + Wz2 ( d -dln-r for all the values of r; hence, writing Po =Po, dz p1 = Plv - ~(vP0), d __ P2 dz (VP,dz) (VP,), d dn-1 pn-1 =PnV - d (vP+-2... + (+ 1)n-1 d(-P0), 1 (w, V) = p0w(n-) + pw(n-2) +. - - + pn-iw, d d2 dn p (v) PnV - dz (Pn-i v) ~dz2 (Pn2V) -. + 1)n dzn (vP,), we have fP(w) d= R (w, v)+fWP (v) dz, and therefore d vP (w) - wp (v) = t- {R(w, v)j. It is clear that, in order to make vP (w) a perfect differential, whatever be the value of w, it is necessary and sufficient that v should satisfy p (V) =0, 82.] ADJOINT EQUATION 253 a linear equation of order n, commonly called Lagrange's adjoint equation; and further that, if v is regarded as known, then a first integral of the equation P (w)= 0 is given by R (w, v) = a, a being an arbitrary constant, and R being a function manifestly linear in w and its derivatives. Further, since fwp(v) dz= -R (w, v) + P(w) dz, it is clear that wp (v) is a perfect differential if P (w) = 0, shewing that the original equation is the adjoint of the Lagrangian derived equation: or the two equations are reciprocally adjoint to one another. Ex. Shew that, if w,..., wu, be a fundamental system of integrals of the equation P(w)=O, then a fundamental system of integrals of the, adjoint equation p (v)=O is given by 1 -fPdz l(n- 2)) W2( 2), W(-2) V1 Vn~p- e ~ w1(3,, 2))),.., VI), P,=e Wl(n- 3)) W2(n - 3), W(n-3) WI W2,..., Wn W1 W2.. W Shew also that the product of the respective determinants of the two sets of fundamental integrals depends only upon P0. One immediate corollary can be inferred from the general result, in the case when the equation P (w)= 0 is reducible. Suppose that P(w) = PP (w)= P (W), say, where W = P2 (w); then we have jvP (w) dz = fvPi (W) dz =R, (W, v) +JWP,(v)dz, where P. is the adjoint of Pi, and R1 is of order in W and in v one unit less than PI. Again, writing V= P1 (v), 254 THE ADJOINT EQUATION [82. we have IWP,(v) dz =fVP, (w) dz =R2(V,v)fwP2(V)dz, where P2 is the adjoint of P2, and B2 is of order in V and in v one unit less than P2. Combining these results, we have fvP(w) dz =Bi(W, V)~BR2 (Vr, V) +jwP2 (V) dz = R(w, v)+fwA (v) dz, where B is of order one unit less than P in w and in v. It follows that PP1~',(v)= O is the adjoint of P (w) = PIP2 (w)= O, where P1, _P1 are adjoint to one another, and likewise P2, P2. By repeated application of this result, we see that the adjoint of P (w) = PIP2... PI (w)=O is given by prprl... P2P1 (V) =0. Hence the adjoint of a composite equation is compounded of the adjoints of the factors taken in the reverse order. Manifestly an equation and its adjoint are reducible together, or irreducible together. The expression R (w, v) is linear in the derivatives of w, up to order n - 1 inclusive, and also in those of v, up to the same order: it may be called the bilinear concomitant* of the two mutually adjoint equations. For further formal developments in respect to adjoint equations and the significance of the bilinear concomitant, reference may be made to Frobeniust, Halphen+, Dini~, Celsij, and Darboux ~. I Begleitender bilinearer Differentialausdruck, with Frobenius. t Crelle, t. LXXXV (1878), pp. 185-213; references are given to other writers. + Liouville's Journal, 4e S6r., t. i(1885), pp. 11-85. A Ann. di Mat., ga Ser., t. ii (1899), pp. 297-324, ib., t. ii (1899), pp. 125-183. Ann. de IVic. Norm., 3e S~r., t. VIii (1891), pp. 341-415. 'I Thtorie gtndrale des surfaces, t. ii, pp. 99-121. 82.] EXAMPLES 255 Ex. 1. Prove that, if a linear equation of the second order is seif-adjoint, it is expressible in the form d (' dhw + Qw=0; that if a linear equation of the third order, in the form d3W +3 d2W 3 dw 1w = 0, dZ3 + "dz2 d is effectively the same as its adjoint equation, then P=O1 2d and find the conditions that a linear equation of the fourth order should he self-adjoint. Es. 2. Prove that, if the equations gown) n 1Wn 1 2! gn2?V n2+... =0, Yo ~n)+ nlv~ - ) + 1)nY2 V(n- 2) +... = 0, are adj oint to one another, then 7o =g0, go= Y01 Y1 = -g1+g',o g1 = 71 +70, 72- =g2 - 2gl' +gO', g2 = Y/2 2y,'+ 70", Y3 = -g3 +3g2' -3g1"+g90"... g3= -73+ 372 -3,",O' and ohtain the expression of the bilinear concomitant. (llalphen.) Es. 3. Let Z1 I 23,..., z,7, denote any x arbitrary functions of x, such that the determinant Q= Z1, dXz1.. d.V -1 2,dz2 dn - 1z2 Z2'dx' dx ' d'n dn1' does not vanish identically; and suppose that these functions of x are regular in a given region of the variable, as well as the coefficients a of the equation ao ~ a ~~+..+ ay = X. Further, let a set of quantities p be constructed according to the law dp 0 dp1 dpn-I p0=2za., p, =2za,- j-, = a -dX I pn. a 256 PROPERTIES OF LAGRANGE'S [82. and let the last of them be denoted by - Z, so that there are n functions Z corresponding to the n functions z. Shew that, if Q (c) is the value of Q when the last column of the latter is replaced by constants cl,..., c,, if Q (x, xi) is its value when the last column is similarly replaced by zl (xl), z2 (X), **.. z, (x1), and if Q (x, xj) is its value when the last column is similarly replaced by Z, (xi), Z2 (x),..., Z (x), then — ' aQ Q (c) + aX (xi) Q (X, 1i) dXi + ay (xl) Q (x, 2) dx}, where a is a value of x within the given region and the constants c are determined in association with a. Indicate the form of this result when z,..., zn are a fundamental system of the equation, which is the adjoint of the left-hand side of the above equation. Also shew how, in even the most general case, it can be used as a formula of recurrence to obtain an infinite converging series of integrals as an expression for y. (Dini.) 83. Consider an expression P (w) and its Lagrangian adjoint p (v), and let R (w, v) denote their bilinear concomitant; then vP (w) - wp (v) = - R (w, v)}, which holds for all values of v and w. Accordingly, let = Z-P-s-1, v = zP, where s is any integer; then d ZpP (Z-p-S-1) _ z-P-s-1p (Zp) = R (z-P-S-, Zp)} Now the left-hand side is a series of powers of z, having integers for indices; as it is equal to the right-hand side, which is the first derivative of a similar series of powers, the left-hand side must be devoid of a term in z-1. Let P (Z), = f,. (T) Z+, be the characteristic function of P (w); then the coefficient of z-1 in zPP (z-P-8-1) is fs (- p - s - 1). Further, let p (zP), = 10 (p) P+", be the characteristic function of p(v); then the coefficient of z-' in z-p-S-lp (ZP) is (s (p). Hence s (p)=fs (- - S 1), 83.] ADJOINT EQUATION 257 and therefore fs (p) = (-p- 1 -- 1); so that, if if (p) ZP+p be the characteristic function of a given equation, then Cf/ (- - - 1) ZP is the characteristic function of the adjoint equation. When P(w) is in its normal form, all the coefficients f (p) vanish for negative values of t,, but fo(p) is not zero. Hence fi, (-p- - 1) vanishes for negative values of /u, but not fo(-p - 1); and therefore the adjoint expression p (v) is in its normal form. Moreover, their indicial functions fo (p), f, (p) are such that fo(p)=0o (-p- 1), o(p)=fo(- P - 1), so that they are of the same degree, or the characteristic indices are the same. Hence if an equation has all its integrals regular in the vicinity of a singularity, the adjoint equation also has all its integrals regular in the vicinity of that singularity; for the characteristic index is then zero for the original equation, and it therefore is zero for the adjoint equation. Similarly, if an equation has all its integrals non-regular in the vicinity of a singularity, the adjoint equation also has all its integrals nonregular in the vicinity of that singularity; for the characteristic index is then equal to the order of the original equation, and it therefore is equal to the (same) order of the adjoint equation. On the basis of these two results, we can obtain a descriptive condition necessary and sufficient to secure that, if a differential equation of order m has an indicial function of degree m - n, the number of its regular integrals is actually equal to m - n. Let P= 0 be the differential equation, with an indicial function of degree m-n. Let R=0 be the differential equation of order m- n, which has the aggregate of regular integrals of P = 0 for its fundamental system; its indicial function is of degree m-n. Then (~ 75, IV) the equation P=0 can be expressed in the form P=QR=0, * Thom6, Crelle, t. LXxv (1873), p. 276; Frobenius, Crelle, t. LXXX (1875), p. 320. F. IV. 17 258 LAGRANGE'S [83. where Q is a differential operator of order n. Because the degrees of the indicial functions of P and R are equal to one another, it follows (from ~ 76) that the degree of the indicial function of Q is zero, that is, the indicial function of Q is a constant, and therefore (~ 77, Cor. I) the equation Q = 0 has no regular integral. Now construct the equations which are adjoint to P = 0, Q = 0, R= 0 respectively; and denote them by p =0, q = 0, r =0. Because R and r are adjoint, and because all the integrals of R =0 are regular, it follows that all the integrals of r=0 are regular; and conversely. Similarly, because Q and q are adjoint, and because Q 0 has no regular integral, it follows that q =0 has no regular integral; and conversely. Further, by ~ 82, we have p = rq, so that the equation adjoint to P = 0 is p =rq = 0, and this equation possesses all the integrals of q = O, an equation whose indicial function is a constant. Hence it is necessary that the equation adjoint to P = 0 should possess all the integrals of an equation of order n, having a constant for its indicial function, if P = 0 is to have m-n linearly independent regular integrals. But this descriptive condition is also sufficient to secure this result. For, as the condition is satisfied, we have p = rq, where the indicial function of q is a constant; hence, with the preceding notation, we also have P = QR, and the indicial function of Q is a constant. Accordingly, as the indicial function of P is of degree m - n, it follows (~ 76) that the indicial function of R is of degree m-n; and therefore (Ch. III), as the order of R = 0 is m-n, all its integrals are regular. But P = 0 possesses all the integrals of R= 0; and therefore it has m - n regular integrals. We therefore infer the theorem: In order that an equation of order m, having an indicial function of degree m - n, may possess m - n regular integrals, it is necessary 83.] ADJOINT EQUATION 259 and sufficient that the adjoint equation should possess all the integrals of an equation of order n, having an indicialfunction which is a constant. This result was first established by Frobenius*; and it may be compared with the corresponding result obtained by Floquet (~ 77). The special case, when n = 1, had been previously discussed by Thomet, who obtained the result that an equation of order m, having an indicial function of degree m - 1, possesses m - 1 regular integrals, if the adjoint equation has an integral of the form e _n/ CXn7o n=O where G () is a polynomial in -, and a is a constant. We shall not pursue this part of the formal theory of linear differential equations further: we refer students to the authorities already (~ 82) quoted, as well as to Thomle+, Floquet~, and Grunfeld ii. * Crelle, t. LXXX (1875), pp. 331, 332. t Crelle, t. LXXV (1873), pp. 278, 279. + A summary of many of the memoirs upon linear differential equations by Thome, published in Crelle's Journal, will be found in Crelle, t. xcvI (1884), pp. 185-281. ~ Ann. de l'Ec. Norm. Sup., 2e Ser., t. vIII (1879), Supplement, p. 132. II Crelle, t. cxv (1895), pp. 328-342, ib., t. cxvII (1897), pp. 273-290, ib., t. CXXII (1900), pp. 43-52, 88. 17-2 CHAPTER VII. NORMAL INTEGRALS; SUBNORMAL INTEGRALS. 84. IT is now necessary to consider those integrals of the differential equation in the vicinity of a singularity, which are not of the regular type. Suppose that such an integral, or a set of such integrals, is associated with a root 0 of the fundamental equation (~ 13) of the singularity which, as in the last chapter, will be transformed to the origin by the substitution 1 z-a=x, z=-, x according as it is in the finite part of the plane, or at infinity. Let p denote any one of the values of 2i log 0; then it is known that an integral exists in the form where h is a uniform function of x in the vicinity of the origin. As this integral is not of the regular type, the function c will contain an unlimited number of negative powers, so that the origin is an essential singularity of (: in the case of the integrals considered earlier, the origin was either a pole or an ordinary point. Accordingly, when q is expressed as a power-series, it will contain an unlimited number of negative powers: it may contain an unlimited number of positive powers also, and in that case it has the form of a Laurent series. Classification of such integrals might be effected in accordance with a classification of essential singularities; but the discrimina 84.] NORMAL INTEGRALS 261 tion that thus far has been effected among essential singularities is of a descriptive type *, and has not led to functions whose general expressions are characteristic of various classes of singularities. Accordingly, it is possible to choose one function after another with differing forms of essential singularity, and to construct (where practicable) the corresponding linear equations possessing integrals with the respective types of singularity: but there is no guarantee that such a process will lead to a complete enumeration. There is one such function, however, which is simpler than any other, and yet is general of its class. It suffices for the complete integration of the linear equation of the first order when the origin is a pole of the coefficient; and an indication has been given (~ 81) that it may serve for the expression of an integral of an equation higher than the first. The equation of the first order may be taken to be dy dy + Py = O, dx where xl~+P is a holomorphic function, s being some positive integer. Let sa, (s-1) a2 2as-1 a. p I' P = 1+ Xs +... + - - x (x), 8+-t~1 XS x~ X where I' (x) is a holomorphic function; then we easily have y = exPeI(x) = exlxP# (x), where J (x) is a holomorphic function of x, and a, a2 a. St = l+ ^... +. XS Xs-1 X It is clear that x= 0 is an essential singularity of the integral; and also that we thus have the complete primitive of the equation of the first order. It appeared, in ~ 81 and the example there discussed, that such an expression, if not in general, still in particular cases, can be an integral of an equation of higher order. As all expressions of the form e lxP' (x), * T. F., ~ 88. 262 THOM~'S NORMAL [84. where 1 is a polynomial in I, possess the same generic type of x essential singularity, we proceed to the consideration of equations that may possess integrals of this form. Such an integral is called* a normal elementary integral or (where no confusion will occur) simply normal. The quantity en, through the occurrence of which the point x = 0 is an essential singularity, is called the determiningfactor of the integral; the other part of the integral, being xPr (x) where s is holomorphic, is of the type of a regular integral, and so the quantity p is called the exponent of the integral. CONSTRUCTION OF NORMAL INTEGRALS. 85. We proceed, in the first place, to indicate Thome's method t of obtaining such normal integrals as the equation dmw da-lw dw dxm n 1 dxml ' pm- - mw = may possess. (The method gives no criteria as to the actual existence of normal integrals: and therefore, if any criteria are to be obtained for equations of order higher than the first, they must be investigated otherwise.) If a normal integral exists, it is of the form w = eLu, where f is a polynomial in -; and il is determined so that, if x possible, the equation satisfied by u may possess at least one regular integral. Let dne so that to =l, t = ', t= t' + t (p = 1, 2,...); then denw a du d&1-lu dh"\ = e nu + ntnl +....+nt + to dxn 1 dn- dx * Thom6, Crelle, t. xcv (1883), p. 75. Cayley, ib., t. c (1887), p. 286, suggested the name subregular; but the name normal is that which has generally been adopted. t Crelle, t. LXXVI (1873), p. 292. 85.] INTEGRALS 263 When these quantities, for the successive values of n, are substituted in the differential equation for w, the determining factor e" can be removed; and the differential equation for u then is dmu dm-lu du dm + ql d -li + + qm-1 +-l + qmu = 0, where m! (n - )! (m-2)! qr -tr+ p itri1 + Ptr, +... q r!(m-r)! (r-1-)!(m-r)! + (r- 2)!(n -r)!p... + (m - r + 1) p._lt + p,t, for r=1, 2,..., m. If the original equation possesses a normal integral, then, after the proper determination of Q2, the differential equation for u will possess at least one regular integral: its characteristic index cannot then be greater than rn- 1, which (after the results in the preceding chapter) is a necessary but not a sufficient condition. As ~2 is a polynomial in x-1, its form and degree being unknown, let its degree be s- 1, so that s > 2; we then have for ~2' an expression of the form a_, a, a:. _ + a. ++;. X2 w3 X S Hence in t1, the governing term (that is, the term with highest a, s. a,2 negative exponent of x) is -s; in t2, it is; and so on, so that, in ( n t,, it is;,. As in ~ 73, let wr denote the multiplicity of x= 0 as a pole of p,; then in q, the governing exponents of its respective parts are rs, P7+4(r-l)s, = 2+(r - 2) s,,... -1_l+S, Or. Thus the governing exponents in q. are, so far as they go, less than those in q,.+1 by s, and s > 2. Hence, in forming the characteristic index for the equation in ut, for the purpose of determining whether it may possess a regular integral, the governing exponent in q, is certainly greater by s than that in any other coefficient; the characteristic index is mz, the indicial function is a constant, and the equation has no regular integral. But, thus far, 12 is quite arbitrary; and it may be possible, by proper choice of its constant DETERMINATION OF [85. coefficients, to secure that a number of the terms in qm with the greatest exponents of x-1 shall disappear. If by thus utilising the governing exponent and the constants in 12', we can secure that the characteristic index of the equation in u is less than m, the indicial function ceases to be a constant and the equation may have a regular integral. In order that the indicial function may not be a constant, the governing exponent of qn-, must be less than that of qm by unity at the utmost, or that of q_-2 must be less than that of qm by two at the utmost, or (for some value of r) the governing exponent of qm-r must be less than that of qm by r at the utmost; whereas at the present moment, these diminutions are s, 2s, rs respectively, where s > 2. Hence an initial necessity is that the s- 1 terms in qm with the highest exponents of x-~ shall vanish. Now qm = tm. + Pitm-1 +... + p,_tl + pm. The s-1 terms in t, with the highest exponents of x-1 are the same as in ll', because of the form of f2' and because t/i = t',1- + - t2 _l, (but not more than those s - 1 terms are the same); hence the s-1 terms with the highest exponents of x-1, say the first s - 1 terms, in fnm +pl-&m —1 +... +-i pm-1 + Pm must vanish. 86. To render this result attainable, it is necessary that the greatest exponent must not occur in only a single term of the preceding expression, for then the term could vanish only by having a, =0; the greatest exponent must occur in at least two terms. Consequently no one of the numbers ms, +( 1)s +( 2)s, (m-), (m-2)..., Wm, may be greater than all the rest, that is, no one of the numbers 0, 1-Si - w, —28, -. m - ms, may be greater than all the rest. Of the quantities 1, ^22 i3) * 1m 1-aWI, ffW; _W3, 11~ -WM 86.] NORMAL INTEGRALS 265 let g be the greatest. Evidently g is greater than unity; for the original differential equation has not all its integrals regular, and so n > n for at least one value of n. Now s cannot be greater than g; for any such value would make all the integers in the series 0, W1-S, '2 -2s,..., rm -ms, negative except the first, that is, the first would then be greater than all the rest. Hence s <g: and s > 2, from the nature of the case. I. When g < 2, no value of s is possible; and then there is no normal integral of the type indicated. Such a case arises for the equation I 1 y/+ +py! + qy=O, when p and q are holomorphic in the domain of x = 0 and neither vanishes when x = 0. The quantity g is the greater of 1, a, that is, it is less than 2; so that there is no normal integral. Moreover, as the indicial function of the particular equation is a constant, it has no regular integral. II. When g is an integer (necessarily greater than unity), we manifestly might take s=g. For two at least of the numbers 0, WI -— S, W2-2S,..., Wm -ms, would then be equal to the greatest among them, which is zero; and then two at least of the numbers ms, Pr1+(m —)s, w2+(m-2)s,..., ym-i+s, rWm, would be equal to the greatest among them, one of these being ms. More generally, let n be the characteristic index of the original equation, so that sn + m - n, + m -, for all values of /a that are greater than n; then, adding (m-n)(s- 1) to each side of the inequality, we have rn + (m - n) s > n, + (m -,) s + (, - n)(s - 1), where, >n. In the case of all these numbers, (L -n)(s - 1) is certainly positive; so that the first s- 1 terms in our expression 266 NORMAL INTEGRALS [86. are not affected by the quantities corresponding to,~ + (m - 1w) s, and they can occur only through the quantities corresponding to X + (m - X) s, for X = 0, 1,..., n, where o, = 0, and n is the characteristic index of the original equation. We thus consider the first s - 1 terms in 'lm-nf (fn + pl2n-' + pJ2' n-2 +... + pn-1i + pn) and this holds for any value of s equal to or greater than two. As regards g, which is the greatest among the quantities ~ 1 i1, 2,2 33 3, ** - m it occurs only among the first n, in the present circumstances; for it certainly is greater than unity and if any one of the last m-n, (say - wa is the greatest of these last m - n), is greater than unity, then because =rl + m - n >, + m - u, we have Wn _ __ > ) (fJ-1 n / Pt P, that is, yn >p n /U for u is greater than n. Thus g does not occur in the last m - n of the quantities, if one or more than one of them is greater than unity; and it certainly does not occur among them, if no one of them is greater than unity. Hence g is the greatest among the quantities 21, 22, 3 *' n - Yn. It may occur several times in this set; let 1- w be the first occurrence, in passing from left to right, and - tr be the last. Take first s=g; then we have K (m - c) s = g,, +(m -r)s =mg,,+(n-X-)s<mg, ifX<c, orifX>r; 86.] THEIR DETERMINING FACTOR 267 so that the highest terms of all, being those with index mg, occur in i2/n, pK n-+ +... +Pr ln-r. If then pa-= X — +d ) (d=l,, 2,...), the equation which determines ag, the coefficient of x-g in f', is agr + cagr- +... + Cr = 0. The remaining g - 2 coefficients in 12' are given by equating to zero the coefficients of the next g - 2 terms in f2'n + plY1'- +... + pn-1' 4 p,. Each set of values of the coefficients determines a possible form of fI' and therefore a possible form of determining factor. The number of sets, different from one another, is < r. The preceding cases arise through s=g; but if g, being an integer, is greater than 2, other values of s, less than g, may be admissible. They can be selected as follows*. Mark the points 0, n; ro, n-l; m,, n-2;...; 0r, 0; in a plane referred to two rectangular axes; and taking a line through the first of them parallel to the axis of x, make it swing round that point in a clockwise direction, until it meets one or more of the other points; then make it swing in the same direction round the last of these, until it meets one or more of the remaining points; and so on, until the line passes through the last of the points. There thus will be obtained a broken line, outside which none of the marked points can lie. If a line be drawn through any of the points, say w,, n - xc, at an inclination tan-l to the negative direction of the axis of y, its distance from the origin is (1 + 2)- iW + (n - ) IA}, so that, for a given direction /u, the distance is proportional to W, + (n - K) z. It therefore follows that an appropriate value of s is given by any portion of the broken line, which is inclined at an angle tan-l' to * The method is due to Puiseux; see T. F., ~ 96. 268 NORMAL INTEGRALS [86. the negative direction of the axis of y, where / is a positive integer, > 2: the value of s being s-=/. As many values of s are admissible as there are portions of the broken line with inclinations tan-1 /, where / is a positive integer, which is > 2. For each admissible value of s, arising from a portion of the broken line, the terms in In + pli"n-1 +... + pn-1i + Pn, which correspond to the points on that portion, give the terms of highest negative power in x. If, for instance, a portion of line, having as its extremities the points corresponding to prfn'-r and ptQn'?-t, (t > r), gives a value g' (necessarily an integer, as being a value of s), then the coefficient ag satisfies an equation Crag't-r +... + Ct = 0, and the remaining g' - 2 coefficients in!' are obtained in the same manner as before. Each set of values of the coefficients determines a form of 2' and therefore also a possible determining factor; and the number of sets different from one another is t - r. And so on, with each piece of broken line that provides an admissible value of s. III. When the greatest of the quantities 1 2 2> 3, 3, is greater than 2 but is not an integer, we construct a tableau of points as in the preceding case, and draw the corresponding line. Only such values of s (if any) are admissible as arise from portions of the line, which are inclined at an angle tan-~1 to the negative part of the axis of y, I being an integer > 2. 87. In every case, where a possible form of Q' and thence a possible form of n have been obtained, we take w = e^u. SUBNORMAL INTEGRALS 269 If a normal integral of the original equation exists, the equation for u must possess a regular integral; and each regular integral of the latter determines a normal integral of the former having the determining factor el. An upper limit to the number of integrals thus obtainable is furnished by the degree of the indicial function of u; but the investigations of the last chapter shew that, when the degree of the indicial function is less than the order of the differential equation, the number of regular integrals may be less than the degree and might indeed be zero. The simplest mode of settling the matter is to take a series of the appropriate form, determined by the indicial function of the u-equation, substitute it in the differential equation, and decide whether the coefficients thence determined make the series converge. The normal integral exists or is illusory, according as the series converges or diverges. When the normal integral exists, we say that it is of grade equal to the degree of f2 as a polynomial in x-1. SUBNORMAL INTEGRALS. 88. In the preceding investigation of normal integrals, it was essential that the number s should be an integer 2: and accordingly, such values of /u, as were given by the Puiseux diagram and did not satisfy the condition, were rejected. But though they are ineligible for the construction of normal integrals, they may be subsidiary to the construction of other integrals. Let,/ denote such a quantity, given by the Puiseux diagram in the form of a positive magnitude that is not an integer: its source in the diagram makes it a rational fraction which, being expressed in its lowest terms, may be denoted by h. k. The terms which, for this quantity as representing a possible degree for f', have the highest index of x-~ in f/n + pli21n-1 +... + pn-1if + n, are those which correspond to points on the portion of the line that gives the value of A. Hence, taking f I +..., x1 270 SUBNORMAL [88. an equation is obtained by making the aggregate coefficient of this term of highest order disappear; the equation determines A. Now take a new independent variable f such that X - k and make it the independent variable for the differential equation; dl. the expression for - is d_ df2 A dx- h + so that dt; and therefore 1 = - - At-(h-k) +... h-k Thus 12 is infinite when x = 0, provided h > k, that is, for values of, that are greater than unity. Accordingly, when we proceed to consider the differential equation with 4 as the variable, values of II of the preceding form can be obtained by the earlier method: in fact, we may obtain a normal integral of the equation in its new form, the conditions being that the equation for v, which results from the substitution W = env, shall have a regular integral or regular integrals. When once the value of k is known and the transformation from x to: has been effected, the remainder of the investigation is the same as for the construction of normal integrals of the untransformed equation. Examples will be given later, shewing that such integrals do 1 exist. As they are of a normal type in a variable xk, where k is a positive integer, they may be called subnormal*. Their existence appears to have been indicated first by Fabryt. 89. We have seen that, if g denote the greatest of the quantities, 2 2, 3~3,.., * Poincare, Acta Math., t. vmII, p. 304, calls them anormales. t Sur les integrates des equations differentielles lineaires a coefficients rationels, (These, 1885, Gauthier-Villars, Paris), Section iv. INTEGRALS 271 and if the equation possesses a normal or a subnormal integral of the form ef0lz (z), then I2' is a polynomial in z-1 (or in some root of z-1) of order equal to or less than g; and therefore 12 is a polynomial in z-1 of order equal to or less than g - 1. Let g-1 =R; then R is called the rank of the differential equation for z = 0. When R is an integer, the grade of a normal integral may be equal to R: if not, it is less than R. When R is not an integer, let p denote the integer immediately less than R; the grade of a normal integral may be equal to p or may be less than p. When R is a fraction, equal to k when in its lowest terms, then a subnormal integral may exist having a determining factor el, where 1 12 is a polynomial of degree k in z l; it will still be said to be of k grade k in z, that is, of grade R. All subnormal integrals are of grade R or of grade less than R. Ex. Obtain the rank of the equation n dn-rw r=Or dzn - r=0 for z= co, the coefficients p being polynomials in z. 90. The converse proposition, due* to Poincare, is true as follows:If n normal or subnormal functions are of grade equal to or less than R, and have the origin for an essential singularity, they satisfy a linear differential equation of order n and rank not greater than R for z= 0. Any n functions satisfy a linear differential equation of order n: in the present case, let it be dew d -w d + P d +... Pw = O. dr"n dxnw= * Acta Math., t. viii (1886), p. 305: the form has been somewhat altered, so as to admit the discussion of normal and subnormal integrals together. 272 POINCAR]'S THEOREM ON [90. Let the normal and the subnormal functions be arranged in a sequence of descending grade: when so arranged, let them be wl = e~ z u1, W2-= e2Z2U,... Wn = en zn Un, so that, if Ri, R2,..., Rn be their respective grades, R > R1 > R2 Z... > -Rn-2 > Rn- >- RnNow PrA + An,r 0, where A is the fundamental determinant of the n functions, viz. dn-lw, dn-1w2 dn-lWn A dZn-l d'z-1 X dzn-1 dt-2wl dn-2w2V dn-2w dZn-2 ' dzn-2 ' " dzn-2.................................... W1, W2,.. ' q. n and An,, is obtained from A by substituting the derivatives of order n for the derivatives of order n - r in the rth row. The value of P. is Pr A 'An,r Pr = — A In order to obtain the degree of z = 0 as an infinity of P,., it will be sufficient to consider only the governing terms in A and An,r; and the degree is determined through the differences between the two sets of most important terms in the rth rows. Now if wp = e0P zcUp, we have dw = z-q (Rp+l) eaXpz7p Oq, p, where 0q,p is finite (but not zero) when z= 0. We take out of the pth column a factor egp zap, for each of the n values of p; we take out of the mth row a factor z-m (R?,+) for each of the n values of m; and then every constituent in the surviving determinants A and An,r is finite. The initial terms in 90.] NORMAL AND SUBNORMAL INTEGRALS 273 these constituents are the same for all the rows except the (r- 1)th: the difference there is that an, a2tl..., a)nl occur in A'n,r, while aln-r, a2-r,..., an-r occur in A', where A'n,r and A' are the modified determinants, and al, a2,..., a, are the coefficients of the governing terms in fi, 2,..,,n. Accordingly, if A' A +..., then A,r = A'z +..., where the other indices are higher than 0, and A, A' are constants; and therefore ^ = z-In (n-1) (R,+) eZ zp Zp A', An r - -n (n-1) (R+1) +r (R,+l) eZp Z p A'n, r the summation in the exponents being for values 1, 2,..., n of p. Hence Pr- -r(R+l A+.. Now A, being the fundamental determinant, does not vanish identically: and as z= 0 is an essential singularity, and not merely an apparent singularity, A does not vanish when z = 0; thus A is not zero. It might happen that A'=0; but in any case, if w. denote the order of z = 0 as a pole of the coefficient Pr, we have v,. r (R1+ 1). Thus the largest of the numbers -Ir r is < R1 + 1 R + I; and therefore, for z=0, the rank of the equation < R, which proves the proposition. When all the integrals are normal, which is the circumstance contemplated by Poincare, the quantities R are integers and the determinants A', A'/n, are uniform: so that the coefficients P then are uniform functions of z. The coefficients P are uniform also when the aggregate of subnormal integrals is retained: the proof of this statement is left as an exercise. NOTE. An equation, which has a number of normal integrals, is reducible; so also is an equation, which has a number of subnormal integrals. By the preceding proposition, the aggregate of the normal integrals (or of the subnormal integrals) satisfies a linear equation with uniform coefficients, say N=O0, of which they are a F. IV. 18 274 EXAMPLES OF [90. fundamental system. Denoting the original equation by P = 0, we can prove, exactly as in ~ 75, that P can be expressed in the form P = QN, where Q is an appropriate differential operator. In other words, P is reducible. The investigation of the detailed conditions, imposed upon the form of P by the possibility of such reducibility, will not be attempted here. Further, it must not be assumed (and it is not the fact) that reducible equations are limited to equations, which have regular, or normal, or subnormal integrals. Ex. 1. Consider the equation 1 1 1 y"' + PyIX qy If ry=0, where p, q, r are holomorphic functions of x that do not vanish when x= 0. To investigate the possible kinds of determining factor, we form the tableau of points 0, 3; 3, 2; 5, 1; 7, 0; and then construct the broken line. There are two pieces: one gives, =3, the other lu=2; the former joins the first two points; the last three lie on the latter. The possible expressions for a' are therefore q a+, i'=Y f 3-, =2, where a and /3 are uniquely determinate, and y is the root of a quadratic equation. Of course, the actual existence of normal integrals depends upon the actual forms of p, q, r. Ex. 2. Shew that the equation y"+ 2 pyf + 5 ' + ry= where p, q, r are holomorphic functions of x that do not vanish when x=O, possesses no normal integrals in the vicinity of x=O: but that it may possess subnormal integrals. Ex. 3. Consider the equation 3 3, 1+12X3 (1+6x3)y"'+ y'+-2y+ x2 -- =O0 which has no regular integral, because the indicial function is a constant. The numbers W, rW2, W3 are 1, 2, 6; so that g=2, and we therefore take s=2, so that x2 NORMAL INTEGRALS 275 We have to make the single (s-1) highest power of x-l vanish, in the expansion of 3s'2 3t2' 1 + 12x,Q'3 + q- -_ x (1 + 6x) 2 (1 +6.3) x6(1 + 63) in ascending powers of x; hence a3=1, so that a is a cube root of unity, and a X Accordingly, we write a y=exu; after reduction, the equation satisfied by u is found to be 3u" (a- + 6aX3) X2 (1 +6x3) 3u' (a2 + x2 + 6a2x3 + 12ax4) 3u (a2 + ax - 2x2 + 12a2x3 + 12ax4) - = 0. X4 (1 + 6X3) X.5 (1 + 6x3) The indicial equation for x=O is a2 ( -1)=0, which has a single root 0= 1; so that the u-equation possibly may possess a single regular integral which, if it exists, will belong to the exponent 1, and so will be of the form U=x (Co+C1X+C2 +...). As a matter of fact, the u-equation is satisfied by = Co (X + a22), as may easily be verified; and thus the original equation possesses a normal integral y ex (x +a2x2), where a is a cube-root of unity. But a may be any one of the three cube roots of unity; and therefore the original equation in y possesses the three normal integrals 1 a a2 ex (x + X2)), ee (x + a2X2), ex (x+ ax), where a is now an imaginary cube-root of unity. The singularities of the equation given by 1+6x3=0 are only apparent (~ 45). Ex. 4. Prove that the equation x2y"- (a+ bx2) y= has, in the vicinity of x-=-, two linearly independent normal integrals, provided a is of the form p (p + ), where p is an integer > 0; and obtain them. 18-2 276 NORMAL INTEGRALS [90. Ex. 5. Prove that each of the equations X3y" + 2xy' - y=O, x4y" + 2x3y' - (a2 + 2X2) y= 0, has, in the vicinity of x=O, two linearly independent normal integrals; and obtain them. Ex. 6. Prove that the equation y, 6-2'+=O has, in the vicinity of x= co, three linearly independent normal integrals; and obtain them. Ex. 7. Prove that the equation 4x4y" - (4 + 12x + 32) y= possesses one normal integral in the vicinity of x=0; and that one normal integral is illusory in that vicinity. Ex. 8. Shew that the equation (X + 2) xty"' + (x2 + 3x - 2) 4y "- (x + 2) x2y - (32 - 5 - 2) =0 possesses three normal integrals in the vicinity of x=0. 1 1 1 [They are xex, xe X, xe x log x.] Ex. 9. Prove that a solution of the equation y" +(a2 - y+ {b +~ +y=O X X X2 is expressed by e(n- a). [l + n, +, x (X- 1)...( + ).], e Xna 1+!Xnxs+...+ (n,)+. where n2=a2-4b, n(+1l)=a(o-+l)-c. (Math.. Trip., Part I, 1896.) HAMBURGER'S EQUATIONS. 91. The conditions, sufficient to secure that an equation, of order m and not of the Fuchsian type, shall have a regular integral, have not been set out in completely explicit form (~~ 78, 79); and consequently, the conditions sufficient to secure that such an equation shall have a normal integral have not been set out in explicit form. The foregoing examples (~ 90) afford HAMBURGER'S EQUATIONS 277 illustrations of the detailed process of settling such questions in individual instances; and the following investigation* gives the appropriate tests for a particular class of equations, which afford an illustration of the general method of proceeding. We consider the equation w = a + Az + 72 w in which a must be different from zero (~ 86) if the equation is to possess a normal integral. For any integral that occurs, z = 0 is an essential singularity. For large values of z, the integrals are regular; and a fundamental system for z = oo is composed of two regular integrals, which belong to exponents - pi and - p2 arising as roots of the quadratic equation p(p- l)=y. These two regular integrals may be denoted by zP1 (-), ZP2Pa (-2 where P1, P2 are converging power-series. As the origin is the only other singularity of the equation (and it is an essential singularity), it follows that Pi and P2 have z = 0 for an essential singularity; all other points in the plane are ordinary points for PI and P2. The expression of a uniform function having only a single essential singularity, say the origin,, and no accidental singularity, is known by Weierstrass's theorem t to be of the form P(-)e Z where P(-) is a uniform function having all the zeros of the original function (the simplest form of P being admissible), and g(-) is a holomorphic function of 1 which is finite everywhere except at z = 0. except at z = 0. * It is due to Hamburger, Crelle, t. cm (1888), pp. 238-273. t T. F., ~ 52. 278 SPECIAL EQUATIONS WITH [91. The function g may be polynomial or it may be transcendental; the discrimination depends upon the character of the origin as an essential singularity for the original function. As the present application is directed towards the determination of normal integrals, the function (-) will be taken to be a polynomial in -. If the original function has an unlimited number of assigned zeros in the plane outside any small circle round the origin, P is transcendental. When the number of zeros is limited, P(1) is a polynomial in - which can be taken in the form z ' () = k f (Z), where k is a finite positive integer, f is a polynomial in z of degree not greater than k, its degree being actually k when z = oo is not a zero. The equations to be considered are those which have integrals zPP1 (-l), zP'P2 (), as above, one (or both) of the functions P1 and P2 having only a limited number of zeros outside any small circle round the origin, with the further condition that the essential singularity at the origin is of the preceding type. Thus an integral is to be of the form W = e() zPp-f(z) e z f(z) = eQz/ (z) = earu, say, where S2 is a polynomial in -, the exponent a is a constant, and f(z) is a polynomial in z; and the differential equation for u is to have a regular integral which, except as to a factor zr, is to be a polynomial in z. Let a + a+ +m. then the equation for u is a + 1z + 7 z2 U" 2u U ( + 2 ' + u ( + n') = u 4 za 91.] NORMAL INTEGRALS 279 After the earlier explanations, it is clear that we must take m =1, a2= a. The equation for u then is 2a,2a - - 3-z U"2 ut' + U = 0, 2 z which is to have a regular integral of the type uM = zIf(z) = Z2 (Co + C1z +... + CZ1 +...), there being only a limited number of terms on the right-hand side. The indicial equation for z 0 is - 2ao + 2a-, = 0, so that -2a' Substituting the expression for u, and equating coefficients, we have, after a slight reduction, {(n~ + -) (n+ - 1)- r- } c, = {2a (n + ) +/3} c,+1 = 2a (n + 1) cn+,; and therefore (n + o) (n -- 1) - n +2al (n + 1) It is clear that, if the series with the coefficients c were to be an infinite series, it would diverge and the integral would be illusory. For this reason also, as well as by the initial condition, all the coefficients from and after some definite one, say after ck, must vanish; and therefore we must have (k + -)(k+ a - 1) = 7y, or substituting for a its value, we see that the quadratic equation t a a t t ( 2a where a2= a, must have a positive integer (or zero) for a root. This condition is sufficient to secure the significance of the series, and therefore sufficient to secure the existence of a normal integral of the equation, a4+ 3Z +7,z2 z4 280 A CLASS OF EQUATIONS [91. Clearly, there are two values of a. If for either value the condition is satisfied, there is a normal integral of the form a where a has.the value for which the condition is satisfied. The condition cannot be satisfied for both values, if the values of o are different, and arise from different values of p; for if it could, we should have a-1+=- + + 2. 2a 2a Now pl + p = 1; and therefore l + k-2 = p - -l + p - -2 = - 1, which is impossible, as neither k1 nor kI is negative. The condition can be satisfied for both values of a, if the values of a are the same, that is, if /3=0: for then the condition, that the equation (0 + 1) d= =y can have a positive integer as a root, shews that the equation - a2 + 7Z2 4 possesses two normal integrals of the form a ez z (c + clz +... + coz), a e Z z (c - c'z +... + coz~). The condition can be satisfied for both values of a, if the values of a arise through the same value of p, whether they are the same or not; and the equation then possesses two normal integrals. The limitations on the constants are given in the first of the succeeding examples. Ex. 1. Prove that the equation a +/z + -yz2 of= W possesses two normal integrals, if -=q, 4y+l=p2, a' where q is any integer, positive, negative, or zero, and p is an integer that may not vanish. (Hamburger.) 91.] HAVING NORMAL INTEGRALS 281 Ex. 2. Obtain the conditions sufficient to secure that the equation a + I bz w'+ a + fz +,/z2 + 8z3 + eZ4 w" +2 a+b +a+ + -=O Z2 z4 may have a normal integral of the foregoing type. Can it have two normal integrals? Ex. 3. Prove that the equation a b w"+-a w' + w=O z Z possesses two normal integrals, if a is an integer (positive, negative, or zero). Ex. 4. Prove that the equation w, a+z2 +yz4,WI/ - a+,3Z2 + -~- 4 -z6 possesses a normal integral if the quadratic equation ( )(^ -1 )-1 ( 2 Ja) 2 2/a) Y has a positive integer (or zero) for one of its roots for either value of,/a. What happens (i) when both its roots are integers for the same value of./a, (ii) when, for each value of ^a, the equation has a positive integer for a root? Ex. 5. Prove that the equation Wf Wl l 1 W - 2n (n+ 1) -2+4n (n+ 1) 3, + n (n+ 1) (n+3) (- 2)+a4} w=O, where n is an integer and a is any constant, has four normal integrals of the form where ( (1) is a polynomial in 1 (Halphen.) 92. In an earlier paper, Cayley* had proceeded in a different manner. If w = z"P (z), where b (z) is a holomorphic function of z not vanishing with z, we have _p _) w z + (z) = + R (z), Crelle, t. c (1887), pp. 286295; Cll. Math. Papers, vol., pp. 444452. * Crelle, t. c (1887), pp. 286-295; Coll. Math. Papers, vol. xii, pp. 444-452. 282 CAYLEY'S [92. where R (z) is a holomorphic function of z in the vicinity of the origin. Further, if w = elZPf (Z), where b (z) is a holomorphic function of z not vanishing with z, 1 and 12 is a polynomial in -, we have w =,+p+ (z) W z W Z +(Z) -= ' + * +- + R (z), say, where R(z) is holomorphic in the vicinity of the origin. Cayley transformed the equation by the substitution w' W and then proceeded to obtain, from the differential equation for y, an expansion in ascending powers of z. When once a significant expression for y has been obtained, the value of w can immediately be deduced. Applying this method to the equation,, a + /Z + yZ2 w -- w, Z4 the equation for y is at once found to be y'+ 2 = + +3 2 Hamburger's investigation shews that the integrals of the equation in w are W=zP = zplZ ) = zP2P (P,) which are valid over the whole plane but have z = 0 for an essential singularity. If an integral, say wl, has an unlimited number of zeros, the origin being its only essential singularity, then* any circle round the origin, however small, contains an unlimited * T. F., ~~ 32, 33. 92.] METHOD 283 number of these zeros: so that if, in the vicinity of the origin, the expression of w, is WI = ZP (Z), c (z) would have an unlimited number of zeros within the small circle so drawn. The expression for y is p + '(). but the function ( has an unlimited number of poles in the b (z) immediate vicinity of the origin, and so the right-hand side cannot be changed into an expression of the form Zm z2 z am.. +a + P R(z), where m is a finite integer. Accordingly, the assumed expansion is not valid in this case: and the method does not lead to significant results. But when the integral has only a limited number of zeros, so that b (z) is expressible in the form <(z)=Z/-kf(Z)eg) in the vicinity of z 0, where g (-) is a polynomial in - and f(z) z is a polynomial in z that does not vanish with z, then (Z) can (z) be changed into an expansion am a2 R -+... + * -+ R (z), and so the assumed expansion for y is valid in this case. The method therefore does then lead to a significant result*. Assuming the method applicable, and returning to the equation y' + Y2 =,a + a Z7 z3 z2 * The discrimination between the cases, and the explanation, are due to Hamburger, Crelle, t. cIII (1888), p. 242. 284 SUBNORMAL [92. we easily find ao al y = 2 + + a2 + a3z +.., 22 = 2aa0o - 2a0 = 3, 2aao + a,2- a- = 7y, and, for any value n which is greater than 2, 2 (anao + an_ a +...)+ (n - 3) an- = 0. If the constants in the equation were unconditioned, the coefficients thus determined would give a diverging series for y. But we are assuming that the method is applicable, so that the conditions for convergence are to be satisfied; and then, as W' ao a -= - +-+ a2 +..., w z2 z we have _a w = e z Zl (Co + CZ +..) where the last series converges. The method does not, however, give the tests for convergence of the series for y, at least without elaborate calculation: still less does it indicate that the convergence of the series for y is bound up with the polynomial character of the series in the expression for w. It can therefore be regarded only as a descriptive method, capable of partly indicating the form of integral when such an integral exists: manifestly, it is not so effective as Hamburger's. But the method, if thus limited in utility, has the advantage of indicating an entirely different kind of integrals of the original differential equation, which are in fact subnormal integrals, though it does not establish the existence of such integrals: for the latter purpose, other processes are necessary. It will be sufficient to consider an equation, say of the fourth order, in the form W" +plW"' +p2w" + p3W +pW 4 = 0, where the origin is a pole of p, of multiplicity w, for /u = 1, 2, 3, 4. Taking w wy, INTEGRALS 285 we have =y + y2, W w = y + 3yy + y3, y"' 4yy" + 3y' + 6y2y' + y4, w so that the equation for y is y'" + 4yy" + 3y'2 + 6y2y' + y4 + p1 (y" + 3yy' + y3) +p2(y' + y) + p3+ p =0. If this equation is satisfied by an expression of the form y = Z- (ao + a )z +...), the coefficient of the lowest power of z must vanish. Now the governing exponents for the terms in succession are -m-3, -2m-2, -2m-2, -3m-1, -4m, -uj-mT -2, -url-2m-1, - 1 -3m, - 52- m - -1, - 2- 2m, -- _53 - m, - WY4. To determine which groupings of terms will give the lowest power of z, we use a Puiseux diagram*; and in connection with each quantity 5, + km + 1, for the various values of u, k, I, mark a point (a, + 1, k) referred to two rectangular axes Ox, Oy. Through the point (0, 4) take a line parallel to the axis Ox, and make it swing in a clockwise sense until it meets one or more of the points: round the last of the points then lying in its direction, make it continue to swing until it meets some other point or points; and so on, until it passes through the point (54, 0). A broken line is thus obtained; the inclination of any portion to the negative direction of the axis Oy being tan-1 /a, the quantity p is a possible value of m, and the terms giving rise to the lowest index of z in the differential equation for y are those which correspond to the points on that portion of the line. There are as many possible values of m thus suggested as there are portions of the line. * See vol. ii of this work, ch. v, passim. 286 EXAMPLES OF [92. It is not, however, a necessity of a Puiseux diagram that only integer values of m shall thus be provided: and it does, in fact, frequently happen that rational fractional values arise. Let such an one be -, where r and s are prime to each other; and take Z= US, so that y = (a + a, +...). When the independent variable is changed from z to ', an expression for y of this type can be constructed, and it will be a formal solution of the equation; if the series for y converges, then such an integral exists, expressed in the form of a series of fractional powers, and a corresponding integral w will be deducible. Such an integral, when it exists, is a subnormal integral. It is easy to verify that the only points, which need be marked in the diagram for the purpose of obtaining the possible values of m, are those which correspond with the quantities 4m, 1 + 3m, r2 + 2m, '53 + m, 374, as in ~ 86; but fractional values of m are now admissible in every case, instead of being so only under conditions as in the former use of the diagram. Ex. 1. This indication of integrals in a series of fractional powers was applied by Cayley and Hamburger, in the memoirs already cited, to the equation* which possesses neither a regular integral nor a normal integral in the vicinity of z= 0. The only points to be marked for the Puiseux diagram are 0, 2; 3, 0; there is one portion of line, and it gives m=g. Accordingly, we take z=C2; and the equation for w then becomes d2W 1 dw /43' 4y'\ ( — - =W~4 +!2), or, writing w=iW, * This equation is used only for purposes of illustration; its integrals are regular in the vicinity of z = oo SUBNORMAL INTEGRALS 287 we have d2W +4'3'4y' d(2 = W + (2,) which is a special form of the earlier equation in ~ 91. It possesses two integrals, normal in (, if the quadratic 0(0+1)=4y'+3 has one of its roots an integer, that is, if ' = (20 - 1)(20+3), where 0 is any positive integer (or zero). To find the integrals, we have merely to adapt the solution in ~ 91, by taking a=413', /3=0, y=4y'+=-0 (0+1). Thus a=a=2/j3', o=l, and 43'( (n+ 1) cn+l={(n+l) - 0 (0+1)}) c =(n-0)(n+0+l)cn; and so, taking c= 1, we have 2p3i W-e~ ~ (i (_4_Y.i~ (O+n)! (, n=0 n!(0 n)! as a normal integral of the equation in C. Accordingly, the equation w"=3 {a+ z (20-1)(20+3)}, where 0 is a positive integer or zero, and a is a constant, has an integral z=e ~ - -112 2 n=0 4a) n! (O-n)! Manifestly, the other integral is given by - 2al-z -i i 1 +!'In n=0 4a )! (d0-n)! ' the two constituting a fundamental system. Each of them is of the type of normal integral: but the series proceed in fractional powers of the variable. It will be noted that the two values of o are the same, and that only one value of p is used; the relation is p=o-+O= +0. Ex. 2. Prove that the equation v" + v' + =0, Z 23 where X is a constant and 2/ is an odd integer, positive or negative, possesses two subnormal integrals. 288 EQUATIONS HAVING [93. EQUATIONS OF HIGHER ORDER HAVING NORMAL OR SUBNORMAL INTEGRALS. 93. There is manifestly no reason why Hamburger's method should be restricted to equations of the second order; and he has applied it to obtain the corresponding class of equations of general order, the properties of the integrals defining the class being (i) the integrals are of the regular type in the domain of z= oo; (ii) the origin is an essential singularity for each of the integrals, and at least one of the integrals must be of the normal type in the vicinity of z = 0; (iii) all the points, except z=- and z= o, are ordinary points of the integrals and the equation; (iv) the number of zeros of at least one integral, which lie outside any small circle round the origin, is limited; the second and the fourth of which are not entirely independent. Let the equation be of order n, and have its coefficients rational. The first of this set of properties requires the equation to be of the form dnw dnd-lw dw zn + zn-lp dz-Jr +... + zpn- dz + Pn = 0, where pi, 2,..., Pn are holomorphic functions of z for large values of z, and thus are expressible in series of powers of z of the form 1 1 a, + b, -+c + (=1.n). Z Z2 The third of the above set of properties requires that every value of z, except z =0, shall be an ordinary point for each of the coefficients: and by the second of the properties, z = 0 is a singularity of the equation and therefore of some of the coefficients. Accordingly, the power-series for the coefficients p, which have been taken to be rational and are limited so that every point except z = 0 is ordinary for them, are polynomials in z-. 93.] NORMAL OR SUBNORMAL INTEGRALS 289 As the integrals are regular in the vicinity of z = oo, one at least is of the form w = Q (), where Q is a series of powers of z-~, which does not vanish when z= o and converges for all values of z outside an infinitesimal circle round the origin, and where p is a root of the equation p (p-)... (p-n + 1) + ap (p- 1)... (p -n + 2) +...... + an-_p+ an = 0, the indicial equation for z =. The exponents to which the integrals belong, being regular in the vicinity of z = oc, are the roots of this equation with their signs changed; and they exist in groups or are isolated, according to the character of the roots. Let the above integral be one which, under the second of the set of properties, is a normal integral in the vicinity of z = 0, necessarily an essential singularity; in that vicinity, its expression is of the type w = enzaR (z), where R (z) is a function of z, which is holomorphic in the vicinity of z= 0 and does not vanish when z = 0, and where g is a polynomial in z-l, say 1 ^am 1 am_i al mzm m- 1 Z-l-1 z and ao is a constant. Then, in the vicinity of z = 0, we have w- am am-l al C R' - (z) W m+l m +* - z2 z R (z) =T + R1, where T is a polynomial in, constituted by E' + -, and R, is z z the holomorphic function of z given by R' (z) -R (z). But as this arises through a form of the integral, postulated for the vicinity of z= 0, while the integral is actually known to be P QF. Iv. 19~~~ F. IV, 19 290 NORMAL [93. the above form for w'/w must be deducible from this actual value. This is possible only if Q(-), which has z= 0 for an essential singularity, possesses at the utmost a limited number of zeros outside an infinitesimal circle round the origin; for if it had an unlimited number of zeros in the plane, other than z= 0, any circle round the origin, however small, would include an infinite number, and then 22 ( would be incapable of such an expansion. The requirement, that thus arises, has been anticipated by the assignment of the fourth among the set of properties of the integrals; and so we may assume Q (-) to have only a limited number of zeros. Accordingly, as in ~ 91, the form of Q (-) must be P(1-) e where P ( is a polynomial in - having as its roots all the zeros of Q (), and g (-) is a holomorphic function of -, finite everywhere except at z= 0. Let k be the number of zeros of Q; then P () is a polynomial of degree k, and so it can be represented in the form z-k G (z), where G (z) is a polynomial in z of degree k. Thus the integral is of the form zp-k G (z) e(Z) The postulated form must agree with this form; hence (-) is the polynomial!f of that form, and the holomorphic function R (z) of that form is the polynomial G (z): also = p - k. INTEGRALS 291 The expression for w'/w in ascending powers of z is thus valid, under the conditions assigned, provided B (z) is a polynomial in z. Taking T + RI = z-mr-1P1, so that P1 is a function of z, which is holomorphic in the vicinity of z = 0 and is equal to am when z = 0, we have W/ - = z-M-lPi. W Then W W / '2 d w/W zu W dz w, = Z-2m-2 (PJ2 + Zm Q) = z-2mT2P2, say. Similarly, ___ = Z-3M-3 (P13 + Zm Q2) = Z-M-3P3, say, and so on: where all the functions P2, PI,..., QI, Q,... are holomorphic functions' of z, and the first m terms in PK arise from P1K* Substituting in the equation dnw d"-l dw Z dz72 + Z'P1i dz~,' +... + ZPn-1i dz + PnW = 0, we have PI?, + Zmp1 Pn + ZlMP2Pn-2 +... + Znm Pn = 0, which must be identically satisfied. The coefficients p are polynomials in hence * KMn Z PK is expressible as a polynomial in Z, and so the highest negative power in pK is Z-Km at the utmost. Accordingly, let aK1 a aK K~n P3K =aCKO + +. K K Z ZKm for K=l,...,in. Now we have Pi = aM, + am, z +... + az m-1 + z"mT = v zmT, * If this were not the case, the assignment of a larger value of m could secure it: and so the assumption really is no limitation beyond that which is necessary for a normal integral, viz. ni must he a finite integer. 19-2 292 CONSTRUCTION OF [93. say, where T is a holomorphic function of z; and Pa = P1 + zmQ"Q_, = v + zmTml, where T,,_ is a holomorphic function of z; so that the first m terms in P1, which give all the coefficients in the exponent of the determining factor ea, are given as the first m terms of a root of the equation Vn + zmplvn-l + z2 p2 n-2 +... + ZnP = 0, when the root is expanded in ascending powers of z. When the first m terms in v are obtained, then the determining factor is known; for we have = x -m —l vdx. Moreover, after this determination, the terms involving the powers zo, z,..., zm-l in Pn zpP + pP z2mp2Pn-2 +... + znm have disappeared, so that this quantity is divisible by zm, leaving a holomorphic function of z as the quotient. 94. Having obtained the determining factor, let w = efu be substituted in the differential equation, which can now be taken in the form dnwn i n dn(w d"^ zmn+n d7w + (r"pr) Z(-r) (m+l) -r = 0. dZ r==1 dz r For this purpose, derivatives of en are required. We have e-ad dn e e dz -znL+1 let e-n2 en= e 2 z2m+2> e- en _ _ dz3 -z3m+3> 94.] NORMAL INTEGRALS 293 and so on, where v is identical with the first m terms of P1, v1 is identical with the first m terms of P2, and generally, VA is identical with the first m terms of PA+1. Now dAw _ J X! V,,K-1 dA- UK dZX =O C!( -C) zKm+K dz-K with the convention vo = v, v_ = 1; and therefore the equation for u, after dropping the factor ea, is n n-r r-K) + dn-r —K, V Y v -! K-_1 Zrmr Z(n-r-K) (m+l) a__-r-,= =~ which can be written in the form U s (, - r! d ~ "-s s ' _r= {(n-)! Z( P rm Z(n-s) (m+l) n-s 0 s==or=O ( - s) (sr -! P_ dzs -, where po = 1. The coefficient of u is n -- Vn-r-irmpr r=0 = vn-i + zm1Vn-2 + 2+ z2 Vns +...+ z(-) mpn- + VnmPn. Because the first m terms in VA_1 are the same as in PA, the first m terms in the preceding coefficient are the same as in Pn + mplPn-1 +... + mpn, and they are known to vanish, for the coefficients of zo, z1,..., zm-l were made zero to determine v; hence the preceding coefficient is divisible by zm, so that we can take Vn-, z"p- v,_ -2 +... + nmpn = zm ( + 0o + +...), where 0o is a determinate constant, because v is known. The coefficient of zm+l d is dz n-i 2 (n - r) Vn-r-2 zrmp r=0 = nn-2 + (nI - 1) Vn-3 zpi +... + 2vz(n-2) mpn-2 + Z(n-1) mn-1. The first rm terms here are the same as the first m terms in nP_1 + (n - 1) zpl P1-2 +... + 2P1 z (-2) mpn_ + z(n-1) rpn-i, that is, the same as the first, m terms in nVn-i + (n - 1) Vn-2Zmpl +... + 2Z (?1-2) mpn-2 + Z(n-l) mpn-1 294 CONDITIONS FOR EXISTENCE [94. The equation for v is vn + z"p v'n-1 + z2mp2vn-2 +... -+ Zlp = 0; and, in particular, the equation determining am, the constant term in v, is amn + - amn1 a, -2 a2,21n +.. + a,,,n = 0, giving n values of a,. 95. Let a,, denote a simple root of this equation, sometimes called the characteristic equation: then the quantity nam n- + (n - 1) amn- ai, m +... + an-, nm-m du is not zero. The coefficient therefore of z"+l dt as given above, does not vanish when z = 0: let it be no + Fi/ +.... where n0 is a determinate constant, because v is known. It follows that the equation for it, in the form as obtained, is divisible throughout by zm. Further, if it possesses (as, for the class of equations under consideration, it must possess) a regular integral, and if that regular integral belongs to the exponent a, then a is given by the indicial equation roa- + o, = 0, so that a can now be regarded as a known constant. Further, we had a = p -k, where k is a positive integer (or zero), and p is a root of the equation p(p- 1)...(p-n+l)+p(p-l)...(p - n + 2)alo + p (p - )...(p - n + 3) a2o +... + pa,,_ + ano = 0, say, of )(p)=0. Consequently, the equation I(c-o)= o, I N OF NORMAL INTEGRALS 295 regarded as an equation in k, must have at least one root equal to a positive integer or zero: if this root be denoted by K, one condition that u should be of the form U = (Z (Co + C1Z +... + CKK) (which is the form for u required by the earlier argument) is satisfied. But while the condition is necessary, it is not sufficient for the purpose. When the value of u is substituted in the equation, the latter must be identically satisfied; and so we have relations among the coefficients c. The general relation is I (o- + a) Ca + g, (a) Ca+, + g2 (a) Ca+2 +... + gm_-m (a) Ca+mnm = 0; the relations for the first few coefficients are of a simpler form. When these relations are solved, so as to give successively the ratios of c1, c2,... to Co, a formal expression for u is obtained. In this formal expression, all the coefficients cK+,, c+2,... are to vanish; that this may be the case, we must (as in ~ 79) have I (- + c) )CK = 0, I (1 + C - 1) cC_1 + g (K - ) C = 0, I (o + e -2) CK-2 + gl ( - 2) c,_ + g2 (C - 2) c = 0, and so on, being m (n - 1) relations in all. Of these, the first is known to be satisfied as above; it is the first condition for the existence of u in the specified form. The aggregate of conditions is sufficient, as well as necessary: the last of them secures that CK+1 vanishes, the last but one secures that CK+2 vanishes, and so on: the first secures that CK+,nnm vanishes; and then, in virtue of the general difference-relation among the constants c, every succeeding coefficient vanishes. Thus when the m (n - 1) conditions are satisfied, in association with a simple root of the equation amt + amnc a1l, i +... +- an, n= 0, a normal integral of the original equation exists. It may happen that the conditions are satisfied for more than one of the simple roots of the equation: then there will be a corresponding number of normal integrals of the equation. 296 NORMAL [95. The extreme case would be that in which every root of the equation amn + mn-1 al,im +... + an, nm = 0 is simple and the conditions are satisfied for each of the roots: there then would be n normal integrals. Let the n roots be denoted by 0,,..., n,, so that, if 1 +,. ar, fl.+ = I.... + -, the normal integrals will be of the form ea z Ur, where ur is a polynomial, say of degree Kr, in z. We have -r = - Kr; when these n indices -,. are associated with n quantities p, it follows that pr = (y- + K/r, for r =l,...,n. The distinct quantities pr are the roots of I(p) = 0, so that, if they are all different from one another, we have n of them; also z (a-r + Kr) =1 it (n - )a10. r=l The value of E a. cal then be obtained as follows. r=1 Construct the fundamental determinant A= Wi, W2,..., n dw, dw2 dWn dz z ' '"' dz................................... dn-lwl dn-lw2 dn-xWn dzn-1 ' dzn-1 ).). d.n-., which is equal to Ae- f dz Ae Z" that is, to alj + + I al,1,m Az-loez m zrn where A is a constant. Now if Ur = 1 + Criz +..., we have dWr reOr zr- (m+ 1) Uri, dz 95.] INTEGRALS 297 where u, is a polynomial in z which, is equal to 1 when z = 0; also d2Wr = 02enZrz0,(2m-F2) U,.2, where ur, is a polynomial in z which is equal to 1 when z =0; and so on. Thus A<ef2~~...+ zo)u1 Z'2 U2 01 2Zo, - (2rn+2) U12, 022Z0'2(2+) U2,, where 01~.. 02~. 012 +., 0 22+. As the roots 0 are unequal to one another, J? (z) does not vanish when z = 0; and it is a polynomial. We thus have a,,+. + L1 a I,,rn e z (rn -2 (,- 1)(m+1) q) (z) - Azal9 e z mz Accordingly Di+ +O l. +1 a,,,im z M ZM r=1 4D (z) = A, that is, (J? (z) reduces to its constant non-vanishing term. Thus a-, -'n(n-1)(rn+1)-a10 We saw that and thereftore Y- Kr mnh(nll) r=1 which is impossible because no one of the integers Kr is negative. It therefore follows that when the characteristic equation amn + 01Mni ailm+ ~... + an,nm = 0 298 MULTIPLE ROOTS OF [95. has all its roots distinct fromn one another, and when the quantity denoted by ao has n distinct values, associated respectively with n distinct roots of I(p)= 0, the differential equation ndil' U K dn-Kw ~nw + PKz-~p~ dz —; = 0, z znKpk dzn —= where Kr aK, s=O Z pK=~ z, 8=0 cannot have more than n - 1 normal integrals, linearly independent of one another. If, however, the quantity denoted by a hlas fewer than n distinct values, so that it could be the same for more than one of the n distinct quantities f2, the relation n E a-,.=- In(+-l)(m4+l)-aIO r=l would still hold, repetitions occurring on the left-hand side. But in that case not all the roots of the equation I (p) = O are specified, for the same value of K could be associated with the value of a- common to two integrals; and the relation E (r + A;) = 9n(n - 1) - ai no longer holds. The theorem then cannot be inferred as necessarily true: and it will appear from examples that an equation in such a case can have a number of normal integrals equal to its order. Similarly, if cr has n distinct values, and if these values are not associated with n distinct roots of I (p)= 0, the preceding theorem is not necessarily true; the differential equation can have a number of normal integrals equal to its order. 96. Next, let am denote a multiple root of the characteristic equation m n + amn-1 al,,, +... + an, nm = 0; then the quantity qr vanishes, where 77o = na n-1 + (n - 1) ar-2al, +... + an-1, nm-m. The indicial equation is o0 + 0O = 0, and a- must be a finite quantity. If 00 is not zero, the latter THE CHARACTERISTIC EQUATION 299 condition is not satisfied: and then the original equation has no normal integral to be associated with that multiple root. If 0o is zero, the preceding indicial equation is evanescent: and so further consideration is required. The differential equation for zi, on division by zm+l, becomes (01 + 02Z.) + (.) + ( Z + +...) z d dz d2u + +l (+ + Z +...)... 0, dru where the coefficient of - is of the form dzr 2 (m+) (r —o) + #1Z +...), for r= 3, 4,.... When m = 1, the indicial equation is 6^ Iq4l- + 0-+ (7 - 1)= 0 when n > 1, the indicial equation is 01 + 7 i- = 0. In either case, we can have a possible value for a. A regular integral of the equation for u, and a consequent normal integral of the original equation, exist if the appropriate conditions, corresponding to those for a simple root, are satisfied: it is manifest that they become complicated in their expression*. 97. It might happen that, in determining v, one or more roots of the equation am T + C(n-1 al, lm +... * * a, nmn = 0 is zero, while some of the remaining coefficients in v do not vanish; the implication is that (other conditions being satisfied) a normal integral exists, having a determining factor of which the exponent is a polynomial with a number of terms less than m. It might even happen that, with a zero value of am, all the associable values of the rest of the coefficients are zero, so that v = 0, and the determining factor disappears. One possibility is the existence of a regular integral, and the possibility can be settled in the particular case by the method given in Ch. VI. If, however, the conditions for a regular integral are not satisfied, then there is the * They are considered by Giinther, Crelle, t. cv (1889), pp. 1-34, in particular, pp. 10 et seq. 300 SUBNORMAL possibility of a subnormal integral of the original equation: it arises as follows. Let w = eaube substituted in the equation dnw pi dn-lw pn dz- dzn+ +*...+ w=0; then the equation for u is (by ~ 85) dzn + dz,-l +... + qnu = 0, where qn = tn + t -1 +* + n z zn and dp t - d, (e) = tp. Now f2 is to be chosen so as to diminish the multiplicity of z = 0 as a pole of qn. After the preceding hypotheses, we shall not expect to have an expression of the form a, a, a2:___ +... + z2 Z e"3+13 where mn is an integer; but after the indications in ~ 92, it is possible that 1' may be a series of fractional powers. Accordingly, assume that the multiplicity of z = 0 as an infinity of I2' is /i, so that zl'' is finite when z= 0: then in qn, we have a series of terms with infinities of orders n/ju, (n-1),+l,... (n-I1)+ m+1, (n-2)/++m+2,... (n - 2) /,+2m + 2, (n -3) m+3,... n (m + ). Construct a Puiseux tableau by marking points, referred to two axes, and having coordinates 0, n; 1, n-l;... m+1, n-; m+2, n-2;... 2m + 2, n-2;.. 97.] INTEGRALS 301 (it is easily seen to be necessary to mark only the first in each row), and construct the broken line for the tableau, as in ~ 92. If the inclination to the negative direction of the axis of y of any portion of the line is tan-' 0, then 0 is a possible value for tt. If 0 be a positive integer > 2, we have a case which has already been dealt with. If 0 = 1, there may be a corresponding integral; but it is regular, not normal. If 0 be a negative integer, 2' is not infinite for z =0, and the value is to be neglected. If 0 be a positive quantity but not an integer, it must be greater than unity to be effective; for if it were less than unity, fl would not be infinite for z = 0. Suppose, then, that 0 has a value greater than unity; as it arises out of the Puiseux diagram, it must be commensurable: when in its lowest terms, let it be 0=q where q and p are integers prime to one another, and q >p. Then take z = xP; we have an equation in u and x, and a possible determining factor en can be found such that dz and so 2 = x-(q-P) (co + c1x +...) Co C1 - +- +..., ~ q-1 q —1 ZP Z p a series of fractional powers. The investigation of the integral of the new equation in u and x, that may exist in connection with this quantity I2, is of the same character as the earlier investigations. EQUATIONS OF THE THIRD ORDER WITH NORMAL OR SUBNORMAL INTEGRALS. 98. The preceding general theory, and the methods of dealing with the cases when the equation for am has equal roots, or has zero roots, may be illustrated by the consideration of an equation 302 EQUATIONS OF [98. of the third order more clearly than by that of an equation of the second order, as in ~ 91. Taking the simplest value of m, which is unity, the equation is of the form w // 0 + 3 i, o+k / + k2Z + k z + 22 wI - 3 -22- + 4 w k3o0z3 + k lz2 + kz + 33 + 26 Wz6 which, on using the substitution kloZ +kl, dz k,, y we z = wzkoe, becomes a,,/ 2 + a~ z + a, a, 30z3 + a3 z2 + a3,z + a33 y'" + + 4 6 where the constants a are simple combinations of the constants k. The substitution adopted changes a normal integral of the one equation into a normal integral of the other, save for the very special case when it might be changed into a regular integral of the other: it therefore will be sufficient to discuss the form which is devoid of a term in y". In the present case, m 1, we take y = eZu, and a is chosen so as to make the coefficient of the lowest power in the coefficient of i equal to zero. We thus have a3 + aa22 - a33 == 0; and the equation for u then is t// - 3 // + - {a02 + (a2l + 6a)z + (a + 3a2) + - a30z2 + (a31 - aa20 - 6a)z + (a2 - aa2 - 6a2) 0, of which the indicial equation for z = 0 is (a22 + 3a2) f + a32 - aa21 - 6a2 = 0. It is clear that the equation in a will not have a triple root: if it could, we should have 22 = 0, a33 =0, a = 0, the last of which THE THIRD ORDER 303 values leads to the collapse of the process. (Account must, of course, be taken of the possibility that a22 = = a33, and this will be done later.) Meanwhile, we assume that a is either a simple root or a double root. First, let a be a simple root; then a22 + 3a2 is not zero, and the foregoing indicial equation then gives a proper value for a. If -p is the exponent to which an integral in the vicinity of z = oC belongs, p is a root of the equation f(p) = p (p - 1) (p - 2) + a20p + a30 = 0. The general investigation has shewn that this must have a root of the form p = a + K, where K is a positive integer (or zero), and that, if this condition is satisfied, the form of u is t = Z( (Co + C1z +... + cKz). We substitute this value, and compare coefficients. If gn = (a + n) (- + n - 1) ( + n - 2) + a20 (a" + n) + ao, h =- 3a'(- + n) (- + n + 1) + (a2+ 6a) (a- + n1) + a.l - a20 - 6a, cn = (a22 + 3a2) (n + 2), then the difference-equation for the coefficients c is qnc, + hnCn+i + knccn+2 = 0, for values of n > 0, together with h2_Co + ki1e = 0. As a is a simple root of its equation, a22 + 3a2 is not zero: thus all the quantities k_t, k0, kl,... are different from zero, and the preceding equations thus determine cl, c,... in succession, say in the form Cn -= Co n. In order that the integral may not become illusory, the series is to be a terminating series: it would otherwise diverge, on account of the form of gn. Let the series contain, + 1 terms; then all the coefficients cK+, cC+2... must vanish. Now cK+ vanishes if — 1 CKI-1 + h-_l CK = 0; then c,+2 vanishes if gK c = 0; 304 NORMAL INTEGRALS OF AN [98. and then all the succeeding coefficients c vanish. The latter condition gives g = 0 which, as gn =f(a +n) for all values os n, is the same as f(O + K) = 0, a known condition; and the other gives gK-1K-1 + hK-1K,= which is the new condition. When both conditions are satisfied, a normal integral exists for the equation in y. As that equatiL involves seven constants, which are thus subject to two conditic:-:, there are effectively five constants left arbitrary, subject solely a condition of inequality as regards the roots of the equation a3 + ' a22- a33 = 0; moreover, K may be any positive integer (or zero). If the corresponding conditions hold for a second simple root of this cubic equation, the number of independent constants is reduced to three, while there are two integers such as c; the differential equation for y then has two normal integrals. If all the roots of the cubic equation are simple, and f-;he corresponding conditions hold for each of them, there are three integers such as K, and there is effectively one arbitrary constant: the differential equation for y would then have three normal integrals. This, however, is impossible, if there are three diffe-re-nt values a, -', a" of a-, and three associated integers K, K, K t, ltuh that o- K, ao+ c', a-"+ KC are different roots of f(p)= 0. For then o- + K + o- + Ki + -" + " = 3. Now we have 6a2 + aia2 - a32 3a2 + a22 aa2 - (a32 + 2a22) ah aa where h = a3 + aa22- a33 = 0; 98.] EQUATION OF THE THIRD ORDER 305 hence, summing for the three roots of h, we have o-~ + + a "= 6 f a., - (a32 2a+) ah aa =6, by a well known theorem in the theory of equations. We then should have the equation c + K + // =' - 3, which is impossible as no one of the integers K, K', I" can be negative. Hence, when the equation a3 + aa2 - a33 = 0 has three distinct roots, and when there are three different values ac, a-', aof -, associated with three integers K, c', c", such that ar + c, a'+ ', 0a// K" are different roots off(p) = 0, then the differential equation, /// a2o2 + a2lz - a22,/ a30o + a31z2 + 32Z + a33 Y = y"z + Y4 f y=0 Z1 z6 cannot have more than two normal integrals. But, if the values of ao are fewer than three in number, or if the quantities a- + K are not different from one another, then the differential equation (the other conditions being satisfied) can have three normal integrals. Next, let a be a double root of the equation h = a3 + aa, - a33 = 0, so that we have 32 + a22 = 0 in order that this may be the case, the relation 27a332 + 4a223 = 0 must be satisfied. The quantity c, given by (3a2 + a22) a + a2 - aa2, - 6a2 = 0, is infinite, unless a32 - aa, - 6a2 vanishes: if this condition is not satisfied, then the regular integral for the u-equation, and consequently the associated normal integral for the y-equation, cannot exist. Hence a further condition for the existence of the normal integral is, that the equation a32 - aa2l - 6a2 = 0 be satisfied, where a is the double root: that is, 4a22 - 3a2l a33 + 2a2a32 = 0. F. IV. 20 306 EQUATION OF THE [98. Assuming this to be satisfied, the equation for u now is 3a u, a20z + a2 + 6a u, aoz + a31 - aao20- 6a 2,3 4 z z z Now 9a33 -3a = ---= CIo, say; a32 2a22 a32 a21 + 6a = -32a = 3 =C21 say; a 3a3 - c23, say; 3a33 a31- a (a20 + 6) = al - a(a20 + ( 6) = C31, say; so that the equation for u is c10 u" a2zoz + c-x u'a3oZx + c, Du = u"'" + + 2 + 021 +3 + + = 0. z2 z3 4 The indicial equation for z =0 is Clo 0 (r -1) + c2a + c31 = 0. Substituting,u = zo (Co + c z +... +n... ) in the equation, we have Du = CozO {co0 (0- 1)+ C210 + C31}, provided gnn + hnCn+l = 0, for all values of n > 0, where n= (0 + n) (O + n - 1) ( + n - 2) + a2 (0 + n) + a30, a, = ciO (0 + n + 1) (0 + n) + c,, (0 + n + 1) + c3. First, let the roots of the indicial equation be unequal, say X and,p, so that Du = CoCloO (0 - X) (0- ). Then the value of u, when 0 = X, gives an expression which formally satisfies the equation; but it has no functional significance unless the series converges. That this may happen, gn must vanish for some value of n, say KIc, when 0 = X; that is, one root of I (p) = p (p - 1) (p - 2) + a2,p + a30 = 0 must be p =+ K3, 98.] THIRD ORDER 307 where Ac is a positive integer or zero. If that condition is satisfied, then a regular integral of the u-equation and an associated normal integral of the y-equation exist. Similarly, if I (p) = 0 has another root p = + CK2, where KL is a positive integer, then the value of u, when 0= gu, has significance. It is a regular integral of the u-equation; and a corresponding normal integral of the original equation then exists. Let /3 denote the root of the cubic that is simple: then the earlier investigation shews that a corresponding normal integral may exist. If a-' be the exponent to which the regular u-integral belongs and if K3 +1 be the number of terms it contains, then the equation I(p)= 0 has a root p = r/ +- K3. But the three normal integrals, each one of which is possible, cannot coexist, if X + K1, a +- K,, a' + K3 are different roots of I(p) = 0, supposed not to have equal roots. If they could, we should have X +a ++ - + i + K2 + K C = p = 3. Now - += 1- C2 = 1 a32 C10 a22 Also o' (3/32 + a2) + a32 - 3a2 - 6/32 = 0, and /+ 2a =0, for a, a, /3 are the roots of the equation a3 + aa22 - a33 = 0; so that, a32 + 2aa2 - 24a2 a22 + 12a2 a32 + 4. a22 on reduction, after using the value of a and the relation a32 - aac2 - 6a2 = 0. Hence X + -' = 5, and therefore C1 + K2 + K3 = - 2, 20-2 308 NORMAL INTEGRALS OF AN [98. which is impossible, as no one of the integers Kc can be negative. Hence, when the roots of the indicial equation Cloa0 (o - 1) + c21- + C31 = 0 are unequal, and when I(p)=0 has not equal roots, the original equation cannot have more than two normal integrals, unless (in the preceding notation) there are equalities among the quantities X + 1, t + K2, f' + K3. If it possesses the two normal integrals associated with X and /,, it is easy to see, from the expression for h,, that, if X- / be a positive integer, it must be greater than KC2 + 1: and that, if j - X be a positive integer, it must be greater than Krc + 1. Next, let each of the roots of the indicial equation for o- be equal to: so that DU = CoCoZO ( - r)2. Thus the two quantities []=T [ are expressions that formally satisfy the equation: they have no significance unless the series converge. That this may happen, gn must vanish for some value of n, say c', when 0= r; that is, one root of the equation I () = (p ( - 1) (p - 2) + a2op + a3 = must be p = + c', where K' is a positive integer or zero. (The quantity h, never vanishes in this case and so imposes no condition.) On dropping the coefficient Co, the expression for u in general is equal to ze 9 0 Z + h-l Z2 + (-1 g '- zK + ho Aoh h,hi /o... h/-i 3 so that the two integrals are of the form v, vlogz+v1, where v = [u]0=,, and vi is an expression similar to v with different numerical coefficients, viz. the coefficient of (- 1)'rz0+ in vl is gg...gr- <r-l 1i ags 1 ahs Lhohi... hr- l s=O g, a h, a30}-=T The corresponding normal integrals are eZv, ez (v logz + v'). 98.] EQUATION OF THE THIRD ORDER 309 A third normal integral can coexist with these two in the present case in the form 2a e zu, where u belongs to the exponent a', = a32 + 4, provided I(p)= 0 a22 has a root of the form a' + C3, where KI is a positive integer (or zero). The reason why three can coexist in this case is that only two quantities r and a-' arise, and only two roots, not three roots, of I (p) = 0 are assigned. Ex. 1. Prove that, if the equation 1 4 1l 6 y"' y a2 zr y: a38z' = 2 r=O + 0 s= possesses a normal integral of the form 1e Z (CO + C+... e '~ ZZ (fCo+lZ+...+cKzK), the constants 3, a, a are given by the relations /3 + a20+a 30 = O, a (332 + a30) + 3aa2 + a31 = 0, AT (3/32 + a20) + 3a23 - 92 + aa21 +a + a22+32 =; and the equation p (p - ) (p - 2)+pa24+a36=o must have one root equal to a- +K, where K is a positive integer (or zero). Obtain the relations sufficient to secure that the series c + cz+... shall contain only K+ 1 terms. Assuming that three values of a, distinct from one another, correspond to three sets of values of a and /, prove that their sum is 9: and hence shew that, in this case, the differential equation cannot have more than two normal integrals. In what circumstances can the differential equation possess three normal integrals? Ex. 2. Obtain the constants, and the conditions of existence, of the normal integrals of the equation in the preceding example, when a30 vanishes and a20 does not vanish. How many normal integrals can the equation then have? 99. We now have to consider (i), the case in which one zero root for a occurs, so that a,3 = 0; and (ii), the case in which all the roots a are zero, so that a33 = 0, a22 = 0. Taking a33 = 0, the equation is,,, a22 + a21 + a20z2, a32 + a3lz + a3o2 y + z4 + 5 = O. 310 SPECIAL [99. Two non-zero roots are given by a2 a22 = 0; a normal integral may exist in connection with each of them. The indicial equation for z = 0 is a22 + a32 = 0; in connection with this exponent, a regular integral may exist. The investigation of the respective conditions is similar to preceding investigations. Now substitute in the equation y = eau; the equation for u is t" + 3u"/2' + ui' 3f2 + Qf" + z a + u ('3 + 3s'" + f/" + a22 + a + 20z2 Q 32+ a+31lz +3 a30z2) + 0 and by proper choice of QI, the multiplicity of z = 0 as a pole of u is to be diminished. Assume that z-~l_' is finite (but not zero) when z = 0, and form the tableau of points in a Puiseux diagram corresponding to 3,x, 2/u + 1, u + 2, /u + 4, 5, that is, insert the points 0, 3; 1,2; 2, 1; 4, 1; 5, 0. The broken line consists of two portions: one of them gives / = 2, the other gives i. = 1. The former gives the possibility of two normal integrals: the latter gives the possibility of one regular integral as above. But now let a22 =0, as well as a33= 0. The equation for a becomes a3 = 0, so that the method gives no normal integral. When we proceed to the equation for u, the coefficient of u is fl'3 + 3l'/n// + " + + a+ a20z '/ + a32 + a3z + a3z2 Z3 35 99.] CASES 311 We form the tableau of points in a Puiseux diagram corresponding to that is, we insert the points 0, 3; 1, 2; 2, 1; 3, 1; 5, 0. There is oniy a single portion of line; it gives 5 Accordingly, we change the independent variable by the relation the form of fl' is 5 4' that is, d&I 3a' 3/' /3 a _d xa + 3 2 a;3 a say. The differential equation,,, a2l + a20z /+ a32 ~ a31z ~ a3 Z 2 ly + z3 1/='Y0, with the substitution 1/ = va;2, ecomes d3V 9a21 + a;3 (9a2, -8) dlv + - {27a32 + (27a3l + 18 a2) a + (27a30 + 18a2+ ~8) x6} 0. If a determining factor exists, then (Ex. 1, ~ 98) it is of the form e X2 = xen where /83 + 27a. = 0, ac. 332 +9a21$8=O0, that is, /3=-3a,2t, a =- _ a22a.2. Substituting v - uea) 312 SPECIAL [99. and using these values of a and,8, we find the equation for u in the form ~3 8 f + ax d2U 1 33 3t R 2 a3d dx x3 dX +X6 H \/J~x~tox dx + [~ ~ +(a, ~ 63a,, + 27a31) x + {(1 2a20 + 4),8 - WI~ ~ + a (9a20 - 2) xv3 + (27a30 + 18a20 ~ 8) x1 = 0. If the equation in v is to have a normal integral, this equation in umust have a regular integral belonginDgto anl exponent a-, where it is easy to see that a~ = 3. The regular integral for m is of the form U cflXn~3. n= 0 if f = (n +3) (n + 2) (n + 1)~+(9a20- 8) (n ~ 3) +27a3~ +18a20~ 8, gn = 3ax(n + 4) (n + 3) - 6a (n + 4) + a (9a20 - 2), hn =3j3 (n + 5) (n +4) ~ (a - 9/3)(n + 5) +(12a20+ 4)I3 -6a2, lca~63a2~ +27a.1, i= 3j32 (n + 4), the difference-relation for the coefficients c is fnc~n+ gnc~n+i ~hltcn+2 + kCn~3 + lncn4 0, together with 0 +IA=O h-2c0 + -Cc1 + 12 C2 = 0, g-jco + h-lc1 + icc2 ~ 1-1c3 = 0.The conditions, necessary and sufficient to ensure that the series for u terminates with (say) the (K + 1)th term, which is the generally effective manner of securing the convergence of the series, Kc being some positive integer or zero, are fK =0) fK-1 cK-1 + gK-1CK = 0 fK-2 CK-2 + gK~2 CK-1 + hK cK = 0 K-3 cK-3 + gK-3 CK-2 + hK-3CK-1 + iceK 0 four conditions in all. CASES 313 Assuming these satisfied, we have 1/ a v= e X3 S CXn, n=O and therefore y = vxa ja32\ a21 = e Z (a32z)3 z~ cZz', n-=O a subnormal integral. If the conditions are satisfied for more than one of the cube roots of a32, then there is more than one integral of subnormal type. Moreover, the value of o- is the same for all three cube roots, and only one value of c is required: so there may be even three subnormal integrals, each containing the same number of fractional powers. In order that this analysis may lead to effective results, it is manifest that a32 should not vanish. Ex. 1. Prove that the equation + 3 / a-95z2 4-Z2 + 108z6 y possesses three subnormal integrals. Ex. 2. Discuss the integrals of the equation yt a21 + 20Z, + a31 + a32Z 3 y,+ 1 4. — O. NORMAL INTEGRALS OF EQUATIONS WITH RATIONAL COEFFICIENTS. 100. In the discussion at the beginning of this chapter, the only requirement exacted from the coefficients was as regards their character in the vicinity of the singularity considered: and a special limitation was imposed upon them, so as to constitute Hamburger's class of equations in ~~ 91-99. More generally, we may take those equations in which the coefficients are rational functions of z, not so restricted that the equations shall be of Fuchsian type; we then have dnw dn-lw po d p - + p z ++ pW = 0, 314 POINCAREJ ON [100. where po, Pi,..., p,, are polynomials in z, of degrees so, s,..., la respectively. The singularities of the equation are, of course, the roots of p = 0 and possibly z= co; owing to the form of all the other coefficients, it is natural to consider* the integrals for large values of zl. It will be assumed that the integrals are not regular in the vicinity of z = o. When a normal integral exists in that vicinity, it is of the form eaz 0, where b is a uniform function of z-1 that does not vanish when z = oo, and I2 is a polynomial in z of degree (say) m, so that the integral can be regarded as of grade m. As in ~ 85-87, the value of fi' is obtained, by making the m highest powers in the expression po 2/n + pl'n-1 +... +p acquire vanishing coefficients: and a Puiseux diagram at once indicates whether a quantity i' of such an order can be constructed. The value of m -1 is the greatest among the magnitudes 1, - 0,, (2- ~0o), (S - o),..., provided two at least of them have that greatest value, which may be denoted by h. Then for such normal integrals as exist, we have m -1 < h, when h is an integer, and m- 1 ~ [h], where [h] is the integral part of h, when h is not an integer. The integrals are of grade < h + 1, or < [h] + 1, in the respective cases; and the equation is of rank h + 1. Take the simplest general case, when the equation is of rank unity, and when, in the vicinity of z = co, it may possess n normal integrals which, accordingly, must be of grade unity. No one of the polynomials pl,..., Pn is of degree higher than po; assume the degree of pO to be Kc, and let pr = arzK + brz-l +... - kr, * See Poincar6, Amer. Journ. Math., t. vII (1885), pp. 203-258; Acta Math., t. viI (1886), pp. 295-344. 100.] NORMAL INTEGRALS 315 where some (but not all) of the coefficients a may be zero and, in particular, where it will be assumed that a0 and a, differ from zero. The determining factor for any normal integral is of the form eeZ: 0 satisfies the equation Uo (0) =a0 ao- an-1 +... + an 0 + an = 0. The preceding theory then shews that, if the roots of this equation are unequal and are denoted by 0, 0,..., 8, the normal integrals are of the form eO6zzcS1, eO2zz2,q.2,..., e0zZa' qn; the quantities o-r are given by the equations a U0 r U (r) = 0, (r= l,...), where U, (0)= bo + bl -1 +... + b_l0 + bn; and 0l, (2,..., On are uniform functions of z-l, which do not vanish or become infinite when z = c. Special relations among coefficients are necessary in order to secure the convergence of the infinite series 0; unless these conditions are satisfied, the foregoing expressions only formally satisfy the differential equation and, as integrals, they are illusory. Ex. 1. Prove that the equation x3w"' + (1 -a) x' - (1 - a2+ bx3) w=possesses three normal integrals in the vicinity of x= co, when a is a positive integer not divisible by 3; and obtain them. Ex. 2. Prove that the equation Xp W" - = W possesses three subnormal integrals in the vicinity of x= oo, when n n nn nn n n being an integer not divisible by 3; and obtain them. (Halphen.) Ex. 3. Shew that the equation y _,- 2a_ n(n+l) 1 x -TZT_1 + +(a-n)(a+n+l) y has two normal integrals in the vicinity of x=o; and, by obtaining them, verify that the points x= l, x= -1 are only apparent singularities. (Halphen.) 316 EXAMPLES [100. Ex. 4. Shew that the equation,,,n+1 / +(6n _ ) y2a y"2- y a +,y x- x possesses one integral, which is a polynomial in x, and two other integrals, normal in the vicinity of x= o. (Halphen.) Ex. 5. Prove that, if normal integrals exist for the equation Y, 2a,, 4a, +a(a -b) }+ ly x2 x3 X4 the constant a must be the product of two consecutive integers. (Halphen.) Ex. 6. Prove that, if all the singularities for finite values of z which are possessed by the integrals of the equation dnw d%2-lw -PO dn +P1 dz"n- * +PnW - 0 are poles, and if Po, P1,..., pn be polynomials in z such that the degree of po is not less than the greatest among the degrees of P,..., Pn, then the primitive of the equation can be obtained in the form Wv= Z A,,erzc/,, (z), r=1 where the constants al,..., an are determinate, and all the functions 5,..., pn are rational functions of z. (Halphen.) Ex. 7. Apply the preceding theorem in Ex. 6 to obtain the primitive of the equation (i) "- ( + 1) +a =, where n is an integer; also the primitive of the equation 1 +-n2,I /1-n2 ) (ii) w'+ 2W-( + W= where n is an integer prime to 3. (Halphen.) Ex. 8. Similarly obtain the primitive of the equation x2 (x2 y" - 2x3y'-6 (4 + 2- 1) y =0, in the form X2y= AeX W6 {(x3 + 3x) V6 - 7x2 - 3} + Be - ',6 {(x2 + 3x) V6 + 7X2 + 3}. (Math. Trip., Part II, 1895.) LAPLACE'S DEFINITE INTEGRAL 317 POINCARn'S DEVELOPMENT OF LAPLACE'S DEFINITE-INTEGRAL SOLUTION. 101. Several instances, both general and particular, have occurred in the preceding investigations in which formal solutions, expressed as power-series, have been obtained for linear differential equations and have been rejected because the power-series diverged. These instances have occurred, either directly, in association with an original equation, or indirectly, in association with a subsidiary equation, when an attempt was made to obtain regular integrals of an equation, some at least of whose integrals were not regular; and they have arisen when an attempt has been made to obtain normal integrals of an equation, which is of the requisite form but the coefficients of which do not satisfy the latent appropriate conditions. In such instances, the expressions obtained for formal solutions do not possess functional significance. But Poincare has shewn that it is possible to assign a different kind of significance to such solutions in a number of cases. In particular, there is a theorem*, due to Laplace, according to which a solution of the given differential equation with rational coefficients can be obtained in the form of a definite integral; this solution has been associatedt by Poincare with the preceding results in ~ 100 relating to normal integrals. For this purpose, let w = fe Tdt, where the contour of the integral (taken to be independent of z) will subsequently be settled, and T is a function of t the form of which is to be obtained. If this is to be a solution of our equation, we must have f(potn + plt- +... + pn) etz Tdt = 0: or, if U0 = a0tn + alt-1 +... + a~n UT = botn+btn1-+...+ bn,,,,....,....... Uk = ko tn + kl t;-l +... + k-n * See my Treatise on Differential Equations, ~ 140. f In the memoirs quoted in the footnote on p. 314. The following exposition is based partly upon these memoirs, partly upon Picard's Cours d'Analyse, t. in, ch. xIv. 318 LAPLACE)S [101. the necessary condition is fUozk~ Uizk-+...+ Uk) etzTdt=0. Let d dr-i for r= 1, 2,.)...Ik. Then frTUk-retzdt = [etz V] + -ir (TUk-r) etzdt, dtrk-r ezt for each of the k values of r; and the value of [etz Vr] depends upon the contour of the definite integral. Using this result, the above condition becomes k d dk= 0 L1 etzvj + etz U- (TUk) + + (- )dk (TUo)}dt =0, "dt dtklU j which will be satisfied, if T be a solution of the equation d dk TUk- d (TUk-1) +... + (- I)k (TU) =0 dt ~~~~~dtk and if the contour of the integral be such that Lr=i etz Vr] = 0. The equation for T is U0 dT + oU, dk-'T d dt dtk-Il U,~ n + In (kU, dU, -U,) u'c) d + lk, k 1) _k - 1) +... = 0, dt2 ( 1)dt +2dtk~-2 so that its singularities are the roots of U0 = 0, that is, are the points O1, 02,..., O,,,, and possibly infinity.- Writing the equation in the form dkT dki di~ j- PI -IT d-2T 0 dt-k + tk dtk-' + P dtk_ 0, bo CO the value of PI when t is infinite is,that of P2 is —,and so ao ao' on. Further, the quantity C Vr involves derivatives of T up to r=1 order i - 1 inclusive. This equation for T has its integrals regular in the vicinity of each of its singularities O1, 02,..., O,: their actual form will be DEFINITE INTEGRAL 319 considered later. Let TP denote the most general integral of the equation for T in the vicinity of 0,; then, assuming that the conditions connected with the limits of the definite integral can be satisfied, we have an integral of the original differential equation in the form w= fetzsdt, and this result is true for s= 1, 2,..., n. Now 8s is certainly significant, because it is a linear combination of k regular integrals of the equation for T; hence we have a system of n integrals of the original differential equation. 102. This system of n significant integrals can be transformed into the system of n normal integrals, when the latter exist. They can be associated with the formal expression of the n normal integrals, when the latter are illusory. A preliminary proposition, relating to the given differential equation, must first be established*. In the first place, let it be assumed that all the constants in the equation for T are real, and that T and t are restricted to real values. That equation can be replaced by the system dT- T dt-1 dT1 crt _ T= dt -T............ dt= -P Tk- P2Tk-2 - -Pk T. When we substitute Tr,= re-, (r= 0 1,..., k-), with the conventions that T = T and (0 =,, the modified system is d- = X6 + o), d~t d r-= XOr + OHr+, dc- =Pk-Pk, - P -... -P2, k2-(P- ) ek-. * It is due to Liapounoff (1892); see Picard, Cours d'Analyse, t. II, p. 363, note. 320 LIAPOUNOFF'S [102. Hence d (2+ O +...12 + 0-= X ( 02 + +. + O k2)+(x -) P) 62k + 00, +0 + +-... + 0 k-20k- Pk(00(Hk-i. -. P.2k-2-k-A - Take a real quantity to, smaller than the least real root of U0 = 0; as t ranges along the axis of real quantities between - ox and t0, all the quantities P1, P2,..., Pk remain finite. Hence, by taking a sufficiently large value of X, the quadratic form on the righthand side can be made positive for that range of values of t; and therefore, as t increases from - oo to t0, the quantity (H)2 + 612 +... + 2k__1 steadily increases in value. Consequently, when t decreases from to to - oo, the quantity 2 + 912 +... + 02-1 steadily decreases in value. As to is not a singularity of the equation, the values of (0), (O,..., 0_, for any integral that exists at to are finite there; their initial values are finite, and therefore each of the quantities 10[, 101,..., \0~-1 remains finite and decreases steadily, as t decreases from to to - oo. Hence the quantities qText, Text,..., T, lext all remain finite within that range, that is, no one of them can become infinite, for a value of X sufficiently large* to make the quadratic form positive. Next, suppose that the constants are complex, so that T, T1,... can have complex values; but let t still be real. Then we write Tr= Or+ ifr, * For the tests, see Williamson's Differential Calculus, 3rd ed., p. 408. In the case of k =4, the conditions are: X>0, X2_ >0, x (X2 ~) >o, so (t2 - )2)2 + P32 + u42t to teP3 (1 - P2 - th) +te tt (1 - P2 + P4)> so that it is sufficient to take X greater than the greatest positive value which makes the left-hand side in the last inequality vanish. THEOREM 321 for all values of r, where b and *fr are real; the system of equations takes the form dk1= Pl~k-1 -ql~k-I - P20~k-2 - 2*- - -P& - qk*k dt where b0=~, ~~ and P = p + iq,. We now have 2/c equations; on substituting 0,= Dr e-t 'k = Tr ext, they give the modified set dt dto dt Hnedtk-i -q D- - (p'L - ), +... k qA -pPI' 2dt ~, P2 k-2 X $ (~IDr2 ~'TPr2) + (X -pl) (4)2kl + 'I'2k-,) + bilinear terms. r=1 As before, by choosing a sufficiently large value of X, the righthand side can be made always positive. Then, by taking a value t0, smaller than the least real root of U, = 0, and by making t decrease from t0 to - 00, so that all the quantities p and q are finite, it follows that, for such a variation of t, k-1 k-i -) cJ,2 + $:qr r=i r=1 steadily decreases, and therefore that each of the magnitudes (Dr + Prj remains finite within the range from t, to - 00. Hence each of the quantities Text, Tl e't,..., Tk-16et F. IV. 21 322 LIAPOUNOFF'S [102. remains finite within the range of t from to to - oo, for a value of X sufficiently large to make the quadratic form positive. Lastly, let the constants be complex, so that T, T,... can have complex values; and now let t be complex in such a way that, in the variation from to towards - oo, where t = to + -re, a remains unaltered. The independent variable now is 7, a real quantity, varying from 0 to - o; and the preceding argument applies. A finite number X can be found such that each of the quantities TeXT, Ti e',..., T,_1 eX remains finite within the range of t. But eXa = eX(t-to) (cosa-i sin a) hence a finite quantity X' can be chosen, so that each of the quantities Tex't, Telx't,..., Tk-l et remains finite within the range of t from to towards - o. In the first and the second cases, let =X+ o-, where a is any real positive quantity that is not infinitesimal; and in the third case, let / = X' + -ee-i, where a- is any real positive quantity that is not infinitesimal. Then, because e(-^ )t and e(k- '), in the respective cases tend to zero, as t becomes infinite in its assigned range, it follows that a quantity /z of finite modulus can be obtained, such that Teit, TleLt,..., Tk-lejt all become zero when t becomes infinite in its assigned range. This is true, afortiori, when, is replaced by another quantity of the same argument and greater modulus. THEOREM 323 It also is true when any one (or any number) of the quantities T should happen to be multiplied by a polynomial in t. For all that is necessary is to take a value, + p, where p has the same argument as /; then eptP, where P is a polynomial in t, is zero in the limit, when t is infinite in its assigned range. Thus a quantity, can be chosen so that TPelt, T1 Pl eit,..., TkI Pkl et, where P, PI,..., P,_1 are polynomials, all become zero when t becomes infinite in its assigned range from to, which is not a singularity of the equation, to - oo. 103. This result is now to be applied to the equation which determines T. Let t= 0O be any one of the roots of U = 0, and consider a fundamental system of integrals in that vicinity. If P _ 1 ] + (k - 1) -a - 1, at t=02. at t=or the indicial equation for 0, is ( (+-1)... ( - + 2) ( - p) = 0. Suppose that p is not an integer. The integrals which belong to the exponents 0, 1,..., k- 2 are holomorphic functions of t- 0r in the vicinity of 0, (Ex. 12, ~ 40); and the integer which belongs to p is of the form (t - e.)P P (t - Or) where P is a holomorphic function of its argument. The contour of integration has yet to be settled. In connection with the value 0, we draw a straight line from that point towards - co, either parallel to the axis of real quantities by preference, or not deviating far from that parallel, choosing the direction so that the line does not pass through, or infinitesimally near, any of the other roots of U0 = 0; and we draw a circle with Or as centre, of such a radius that no one of those other roots lies within or upon the circumference. The path of t is made to be (i) in the line from - o towards 0, as far as the circumference of 21-2 324 DISCUSSION OF [103. the circle, (ii) then the complete circumference of the circle, described positively, (iii) then in the line from the circumference back towards - oo. So far as concerns the conditions imposed upon T by the relation r k 0 [ etzV] =0 r=l at the limits, we have only to take the values at the two extremities t = - o. Now Vr is a linear function of T, T1,..., Tk_l, the coefficients of these quantities in that linear function being polynomials in z and t; hence, taking z as equal to the quantity / of the preceding investigation, or as equal to any other quantity of the same argument as u and with a greater modulus, we have k E etz V = 0 r-l at each of the two infinities for t; and so the conditions at the limits are satisfied. In these circumstances, the complete primitive of the equation for T is = A (t - r) P (t - 0) + Q(t - ), where Q is a holomorphic function of t - r, involving n-1 arbitrary constants linearly. The corresponding integral of the original equation then arises in the form fetz dt, taken round the chosen contour. 104. We proceed to discuss this integral for large values of zzl. Let a be the radius of the circle in the contour, so that the series P and Q converge for values of t such that It - 0rI a. For simplicity of statement, we shall assume* that the duplicated rectilinear part of the contour passes parallel to the axis of real quantities from t= -a to t= - oo. From the nature of the integral T, we know that a finite positive quantity X exists, such that the value of eat T * The alternative would be merely to take t = 0r tl"e with a suitable constant value of a, and then make t" vary from - a to - o. 104.] THE DEFINITE INTEGRAL 325 remains finite, as t decreases from Or-a to - o. Let 8 denote the maximum value within this range; then et T ( a, for all the values of t, and then Or - a fO - a etzTdt < 8 e (z-)tdt J -s0 J -00 [e (Z-) t] r - Let z have the same argument* as X, and have a modulus greater than 1XI, that is, with the present hypothesis, let z be positive; then the part corresponding to the lower limit is zero, and we have eOr-at ( etz T dt < e — (Z-A) (Oe-a) J -oo Z-X for values of z that have the same argument as X and have a modulus greater than 1X; and 8 is a finite quantity. Similarly, if, after t has described the circle, 8' denote the maximum value of eltT for Or -a>t > - oc, then the second description of the linear part of the contour gives an integral, such that etz T dt < e(z-A) (Or- a) XJ XZ - A for similar values of z; and 8' is a finite quantity. If, then, these two parts of the integral be denoted by I' and I"' respectively, we have zqe-Zr I' < - e-az-A (r-a) where a is a positive quantity; hence for any constant quantity q, however large, we have Limit (zqe-zrI') = 0, when z tends to an infinitely large positive value. Similarly, in the same circumstances, we have Limit (ze-OrI"') = O. * This form of statement is suited also for the variation of t indicated in the preceding note. 326 POINCARE'S DISCUSSION [104. Now consider the integral round the circular part of the contour. As Q (t -,.) is a holomorphic function over the whole of the circle, we have etzQ(t- Or)dt=O, taken round the circle; and therefore the portion of the integral etT dt contributed by this part is I", where I"= (t -,) etP (t - Or) dt, on taking A = 1. The function P is holomorphic everywhere within and on the circumference, so that we may take P (t- Or) = Co + cl (t - Or) +... + Cm (t - Or)m + Rm, where IRI can be made as small as we please by sufficiently increasing m; for if g be the radius of convergence of P (t-,.), so that g > a, and if M denote the greatest value of P (t - Or)\ within or on the circumference of a circle of radius c, where g > c > a, then* M \< CP' and iRm < n 't -, i m+ + It-O~. It-t. -2 }-O om+1 C C2 for values of t such that It-.l a < c. The value of the integral taken round the circumference can be obtained as follows. Draw an infinitesimal circle with,. as centre, and make a section in the plane from the circumference of this circle to that of the outer circle of radius a along the linear direction in which t decreases towards - oo. The subject of integration is holomorphic over the area of this slit ring: and therefore the integral taken round the complete boundary is zero. Let * T. F., ~ 22. 104.] OF LAPLACE'S INTEGRAL 327 J' denote the value along the upper side of the slit, J" the value along the lower side, K the value round the small circle which is described negatively; so that J' = r (t - Or)P etzP (t - 0r) dt, r-a O~r -- a J"= e-"2'1ip (t - Or)P etzP (t - Or) dt J r - e- 2ipJ' and, if the real part of p be greater than - 1, then* K= 0. Hence, beginning at the point on the outer circumference which is on the lower edge of the slit, we have I" +J' +K +J"O, that is, I" = (e-ip 1) J'. Let u denote the integral u=J (t- O-r) etz dt, and consider the value of ie for large values of z. Let t - r = - = reri; then ra Jo U = e'ii+Z f rTKe-ZTdr. Taking real positive values of z, write rz = y, so that, as z is to have very large values, the upper limit for u with the new variable is effectively + o; thus *00 ' = e7riK+Z0?r z- (K1 yK e-~ dy = (- 1) ezO z-(K+1) r (K + 1). * T. F., ~ 24. 328 LAPLACE)S SOLUTION [104.. Also, if v denote the integral f (t - 0a)P etz R dt, then v=(- 1)PezrfTP e-ZTRmd = (- 1)P ezOr Z- (P+i) yP e R,m dy. Further, yP - -m d < yP e`-Y dy f00yPeYuRndyY <M\ (~l\~1 if 1 — Cc a ( which, when the real part of p + 1 is positive, can be made less than any assigned finite quantity as m increases without limit, because a < c. Using these results, we have 0O,. in ' Or,-a81 VnCc (t - r)" + Rm} (t - Or)P etz dt = (J a)pezZr-ZP~1) r (- 1)"Z-CF(p + a + 1), a=O when m is made as large as we please, and the real part of p is greater than - 1. Hence I" is a constant multiple of this quantity. 105. If now w, denote the integral of the original equation, we have Wr = ftz I'dt -.1 - 1, -I I"', so that W,e-ZBP4 = e-ZBP+I' + e-zOrzP~1I"' + ezzP~'ll I For very large values of z, the first term on the right-hand side tends to the value zero; so also does the second term. The third is a constant multiple of $ (l1)"zz'caF(p a 1). a=O AND NORMAL INTEGRALS 329 Hence, dropping the constant factor, we have Wr = ezEz-P-1 2 (- 1)az-ca r (p + a + 1), a=0 for very large values of z. If the coefficients, of which ca r(p+a+l) is the type, constitute a converging series, then this expression has a functional significance. If they constitute a diverging series, the result is illusory from the functional point of view. Now we have P+l= w_ Ut —.Or; and therefore the preceding integral, when it exists, is of the and therefore the preceding integral, when it exists, is of the form wr = ezrze 2 (- l)az-aCa r (a - c0r) a=O When the series converges, this expression agrees with the form in ~ 100, which is Wr = eZGrzor r, where br is a holomorphic function of z-1 for large values of z. It thus appears that, when Laplace's solution of the equation, originally obtained as a definite integral, can be expressed explicitly as a function of z, which is valid for large values of z, it becomes a normal integral of the equation. This normal integral has arisen through the consideration of the root 0r of the equation U0 = 0. When the corresponding conditions are satisfied for any other root of that equation, there is a normal integral associated with that root. Hence, when n normal integrals exist, they can be associated with the roots of the equation U0 = 0, which comprise all the finite singularities of the equation in T. Note. It has been assumed that p is not an integer. When p is an integer, logarithms may enter into the expression of the primitive of the equation for T, and they must enter if p has any one of the values 0, 1,..., k - 2. There is a corresponding investigation, which leads from the definite integral to the explicit expression as a normal integral. When the normal integral exists, 330 EXAMPLES [105. it can always be obtained by the process in ~ 100. If logarithms enter into the expression of eau, they enter into the expression of u in the usual mode of constructing the regular integrals of the equation satisfied by u. Ex. 1. The preceding method of obtaining the normal integral gives a test as to the convergence of the series in its expression. If the infinite series IE (-1)z Ca r (p+a + 1) a=converges, which must be the case if the expression for the developed definite integral is not to prove illusory, its radius of convergence r is given by the relation* 1 Lim car,(p+a+l)a =1. a=oo r But, from Stirling's theorem for the approximation to the value of r(n), when n is infinitely large, we have 1 Lim r (p+a+ l) =; a=0o hence 1 Lim I c, =O. a=oo Thus the series co + C1 (t-Or) + C2 (t- )2 +... must converge over the whole of the t-plane; and therefore the integral Tr is of the form (t - 0r.)P d (t), where fi (t) is holomorphic over the whole plane: a result due to Poincare. Ex. 2. Prove that, if the condition in Ex. 1 is satisfied, a normal integral certainly exists. (Poincare.) Ex. 3. Consider Bessel's equation x2w" + xw' + (X2-n2) W = 0, for large values of Ix. The integrals in the vicinity of x= oo may be normal-they are not regular-and, if normal, must be of grade unity. Accordingly, let O = e0x 6; w=eO u; then the equation for u is X2U" + (x+ 202) u' + {x2 (1 + 02) + Ox- n2} u= -0. We take 02+ 1=0, * T. F., ~ 26. 105.] BESSEL'S EQUATION 331 and then seek for a regular integral (if any) of the equation x2u" +(X+ 20X2) u'+ (Ox - n2) u-O If an integral, regular in the vicinity of x = co, can exist, it is of the form Z XP-V1 M=O Substituting, and making the coefficients in the resulting equation vanish, we have co (20p+O>) = 0, and, for all values of m, cm ((p - rn?)2 - t2) + Ocm + 1 (2p - 2m - 1) = 0. The former gives P= and the latter then gives cm - {(m \)2- n2}{(r(n -!)2 12}{.()2 n2} cm =Y cg Hence, taking co=1, and 0 = i, a formal solution of the original equation is M- 12 21,n2 u {x(m -)2/'-n 2}.{Q)2 - n2} {(1)2.-n2 1 n=0 in! (2im)m and takipg 9=? co= 1, another formal solution is w eix {2 2 I)2n2) 1 W2=x -i (m-;6e1,)(B m= - m! 2ix)m If 2n is an odd integer, positive or negative, both series terminate; and the formal solutions constitute two normal integrals of the equation. It is not difficult to obtain an expression given by Lommel* for J,, in a form that is the equivalent of V2 n,,i-n-1W, f1 +l 2 -If 2n is not an odd integer, both series diverge; and the formal solutions are then illusory as functional solutions. When Laplace's method of solution is adopted, so as to give an integral of the form w=fetx Tdt, the equation for T is d 12 n2T- (tT) +' (t2 3 1) Tj 0 dt dt2 that is, (t2 + 1) T" + 3tT'+(1 -n2) T= -0 On writing t= i- 2iv, where v is a new independent variable, the equation for T is dTV v M(l t. v) +n,3v)t i ( 87 n2) 7=015 * Math. Ann., t. Iv (1871), p. 115. 332 BESSEL'S EQUATION AND [105. This is the differential equation of a hypergeometric function, whose (Gaussian) elements are given by Y=2, a+/3+1=3, a3==l-n2 The contour of the integral consists of (i) a circle round i as centre with radius less than 2 (so as to exclude -i, the other finite singularity of the equation in T), and then (ii) a duplicated line from a point in the circumference passing in the direction of a diameter continued towards - o. The argument of t and the argument of x must be such that the real part of xt is negative. In order to construct the integral, we need the complete primitive of the T-equation in the vicinity of v=O: it is T=AF(a, t, y, v)+Bv- F(a-y+l, /3-y+l, 2-7y, v), where A and B are arbitrary constants. The part multiplying A, being a holomorphic function, merely contributes a zero term to w; and we need therefore substitute only the other part. Manifestly, we may write B=1. Now F(a-y+l, /3-y+l, 2-y, v)=F(a-~, 3-~, -,, = CmVm, m=0 where (a - ) (a+ )...(+(am- ) ( - ) (+ + ~)...( +m - ) n1. 2m+1. -2 2..... But (a+p) (3 +p) =p2 +2p+1 - 2=(p + 1)2-n2; also 2m+ 1 '2..... =II ( + 2) (- 21); 2 2 so that C (m - {(t)2 - n2}.. ((m - _)2 n2 cm -2 2 m2( )(-. m II (m. ) 2 Taking this value of cm, we substitute T=v- ~ C m m=O in the definite integral. In the preceding notation, we have or=i, p=-, Or=-(p+i)=-2, n (,m - r) = n (m + ); so that, when the solution w= etx(v -i 2 c,,n)dt, m=0 where t=i-2iv, is expanded into explicit form, it becomes a constant multiple of exix- {(-1)m -m - 2) (m +- ) 1=o t(- 2i) DEFINITE INTEGRALS 333 But e n(m + )= I(')- } ( ) - }'"(M- )2- n(- ~; so that, after substituting for c and rejecting the constant factor (-), so that, after substituting for Cm and rejecting the constant factor II (the integral becomes a constant multiple of,eix_ E {(-)2_ n2} {()2-n2}{...{(m- 1)2 -2} 1 m=o m (2ix) which agrees formally with the expression earlier obtained. The corresponding integral, associated with the primitive of the T-equation in the vicinity of t= -i as a singularity, can be similarly deduced*. Ex. 4. Shew that the equation *y" + (ax+ bl) y' + (a2+ b2) =0, where al2 - 4a2 is not zero, can be transformed to xw" +(X1+X2 + 2) w' + {x + i (X1 - X2)} t = O. Assuming X1,, X 1+X2 not to be integers, prove that the latter equation is satisfied by = f(t - i) (t+i) 2 extdt, for an appropriate contour independent of x; and deduce the normal series which formally satisfy the equation. (Horn.) DOUBLE-LOOP INTEGRALS. 106. Before proceeding further with the investigation in ~ 101-105, which is concerned partly with the precise determination of a definite integral satisfying the linear differential equation, we shall interrupt the argument, in order to mention another application of definite integrals to the solution of certain classes of linear equations. It is due to Jordant and to Pochhammer$, who appear to have devised it independently of one * In connection with the solution of Bessel's equation by means of definite integrals, papers by Hankel, Math. Ann., t. I (1869), pp. 467-501; Weber, ib., t. xxxvII (1890), pp. 404-416; Macdonald, Proc. Lond. Math. Soc., t. xxIX (1898), pp. 110-115, ib., t. xxx (1899), pp. 165-179; and the treatise by Graf u. Gubler, Einleitung in die Theorie der Besselschen Funktionen, (Bern), t. I (1898), t. II (1900); may be consulted. J Cours d'Analyse, 2e ed., t. II (1896), pp. 240-276; it had appeared in the earlier edition of this work. + Math. Ann., t. xxxv (1890), pp. 470-494, 495-526; ib., t. xxxvII (1890), pp. 500-511. 334 DOUBLE-LOOP [106. another. A brief sketch is all that will be given here: for details and for applications, reference may be made to the sources just quoted, and to a memoir by Hobson*, who gives an extensive application of the method to harmonic analysis. As indicated by Jordan, the method is most directly useful in connection with an equation of the form (w) = Q dw, - Q (l) a (a + 1) Q ( ) dn-w dzidfl dZ +n- 2 Q"(Z) d- fl. dn-(lw dn-2w -R (,) d-1 + (a + 1) R' (z) d - 0 _ = 0, where Q (z) and zR (z) are polynomials, one of degree n, the other of degree < n in z, R (z) also being a polynomial. For simplicity, we shall assume Q (z) to be of degree n. Consider an integral W = f T (t - z)a+n-1 dt, where T is a function of t alone: this function of t has to be determined, as well as the path of integration. We have a W - (- 1) (a+ n - 1) (a + n - 2)... (a + 1) [a (t - z)a {Q ()+(t-z)+ ) +Q (Z) + 2 - Q " (z+... + (t - z)M {R(z) + (t - )2 R11 ) +(Z) +...}]Tdt = f[a(t - )a-1 Q () + (t - z) R (t)] Tdt, the summation being possible because Q and R are polynomials of the specified degrees. The integral will be capable of simplification, if the integrand is a perfect differential; accordingly, we choose T so that TR (t) = {TQ(t)}, which gives 1 J ( dt 1 q- a * Phil. Trans., 1896 (A), pp. 443-531. 106.] INTEGRALS 335 The preceding integral then becomes fdV, where v=(t - z) TQ (t) R (t) = (t - z)a e a(t). Hence the original differential equation will be satisfied if fdV=0; and this will be the case, if the path of integration is either (i) a closed contour such that the initial and the final values of V are the same: or (ii) a line, not a closed contour, such that V vanishes at each extremity*. Each such distinct path of integration gives an integral. It is proved by Jordan that there is a path of the first kind, for each root of Q; and that, when there is a multiple root of Q, paths of the second kind are to be used. Again, restricting Q(z) for the sake of simplicity, we assume that each of its n zeros is simple; let them be a., a2,..., a. As the polynomial R (z) is of degree less than n, we have (t) n 7r Q (t) r=1 t - ar' where ry,..., yn are constants; and then V=(t- z)a H (t - ar),.. r=l To obtain the paths desired, take any initial point in the plane; from it, draw loopst round the points a,..., an, z, and denote these by Al, A2,..., A,, Z. Take any determination of T (t z)+n —, * A third possibility would arise, if the path were a line such that V has the same value at its extremities: but this case is of very restricted occurrence. + T. F., ~ 90. 336 DOUBLE-LOOP INTEGRALS [106. that is, of n I = (t - Z)a+n-l1 I (t - ar)rr=l which is the subject of integration in W, as an initial value; and let the values of W, for the various loops A,,..., An, Z with this as the initial value, be denoted by W (a,),..., W (an), W (z) respectively. An integral of the original differential equation will be obtained, if the path of integration gives to V a final value the same as its initial value. Such a path can be made up of ArAsA,-1As-, that is, first the loop Ar, then the loop A,, then the loop Ar reversed, then the loop As reversed. Let W(ar, as) denote the value of the integral for this path; then W(ar, as) is a solution of the differential equation. Taking the above initial value (say Io) for I, we have W(a,, as) = W (a,) + e2Wivr W (a,) - e2riys W(a) - W(as) = {1 - e2"iY8} W(ar) - {1 - e2r}l W (as); for after the description of Ar, the initial value of I is e27rrIo for the description of As; it is e2ri(r+Ys) J for the description of Ar~, and it is e2niTs Io for the description of A,-1. It is clear that W(ar, as)=- W(as, ar), {1 - e2riTt} IW (as, ar) = { - e2riYr} W (as, at) + {1 - e2ri} W (at, a,); and therefore all these values of the integrals, for the various appropriate paths, can be expressed linearly in terms of any n of the quantities W (ar, a), in particular, in terms of W(z, a), (z,a),..., W (z,an). Each such quantity is an integral of the original equation; and we therefore have n integrals of that equation. Note. For the special cases when a or any of the constants y is an integer; for the cases when Q (t) has multiple roots; and for the cases when R (t) is of degree n- 1, while Q (t) is of degree less than n- 1; reference may be made to the authorities previously cited. As already stated, all that is given here is merely a brief indication of the method of double-loop integrals. 106.] EXAMPLES 337 Ex. 1. Consider the equation of the quarter-period in elliptic functions, viz. d2W dwv Z (z- 1) d (2z- 1) d 12z00. Here we have Q (z) =Z (Z- 1), a= —, n=2, thus R(z)__ 2 Q (z) 2 z-1' so that _1 Yo 2= Yi) and 1 )t(t-l):=tI(tl) t (t-1 Accordingly, we have W=ft (t - 1) -(t - z)- dt, and the path of integration has to be settled. We have Y(O)==2fdW, where at marks the initial point of the loops. Hence W(O, 1)=2W(O)-2(1)=4fdW2, W(O, z)=2W(O)-2W(z)=4fdW; and thus tvo integrals of the equation are given by fdW, dJW. The comparison with the known results is immediate. Exv. 2. Integrate in the same way the equation d2W dwv (1 ) - -2) 2az ++bw==O, dz2 dz where a and b are constants. (This is another form of the equation Z)d2W _dw (1 - 2)dW 2(m +1)z' +(n - m) (n + m + 1)wqv= 0, d'Z2 dz discussed by Hobson (1. c.) for unrestricted values of the constants m and n.) F. IV. 22 338 ASYMPTOTIC [106. Ex. 3. Prove that when the equation d2w d — = w { 1n (n+l) (u) +/h}, where A is a constant, is subjected to the transformation *= p (U), the transformed equation (which is of Fuchsian type, ~ 54) can, under a certain condition, be treated by the foregoing method: and assuming the condition to be satisfied, obtain the integral. Ex. 4. Apply the method to the equation d + (n + 1) d+zw= apply it also to the equation of the hypergeometric series. (Jordan; Pochhammer.) Ex. 5. Apply the method to solve the equation (1 -z2) w"- 2zw'- = 0, for real values of z such that - l <z < 1. Shew that the equation is transformed into itself by the relations (z-1)(Z-1)=4, w(z+1)'= W(Z+1)l; and deduce the solution for real values of z such that 1 < z < o. (Math. Trip., Part II, 1900.) POINCARI'S ASYMPTOTIC REPRESENTATIONS OF AN INTEGRAL. 107. After this digression, we resume the consideration of the investigations in ~~ 101-105. In those cases when the infinite series in a normal integral diverges, the normal integral has been rejected as illusory from the functional point of view. There are, however, cases belonging to a general class which, while certainly illusory as functions of the variable, are still of considerable use in another aspect: they are asymptotic representations of the integral, to use Poincare's phrase*. A diverging series of the form C + C 2 + n c x x++-+... + +..., X X2 X" * Acta Math., t. VIII, p. 296. 107.] REPRESENTATIONS 339 is said to represent a function J(x) asymptotically when, if Sn denote the sum of the first n + 1 terms, the quantity x11 {J(x) - Sn} tends towards zero when x increases indefinitely: so that, when x is sufficiently large, we have xn {J(x)- Sn] < e, where I e is a small quantity. The error, committed in taking Sn as the value of J, is less than Xn which is much smaller than Cn X7n that is, the error in taking Sn as the value is much smaller than in taking Sn_. (The definition, though stated only for large values of x, applies also to the vicinity of any point in the finite part of the plane, mutatis mutandis.) The asymptotic representation is, however, not effective for all values of the argument of the independent variable. If xn fJ(x) - Sn tended uniformly to zero for all infinitely large values of x, the function J(x) would be holomorphic, and the series would converge: the permissible values of the argument of the independent variable are therefore restricted. It is manifest from the nature of the case that, when such a series is an asymptotic representation of a function, the series can be used for the numerical calculation of the approximate value of the function for large values of x with a permissible argument: the error at any stage is much less than the magnitude of the term last included. Without entering upon any discussion of the question why a diverging series, which is functionally invalid, can yet, when it is an asymptotic representation of a function, be of utility for the numerical calculation of the function, it is proper to mention one conspicuous example of the use of such series, as found in their application to dynamical astronomy*. The normal series, derived from the solution of the equation as represented accurately by the definite integrals, are proved by Poincare to give this type of asymptotic representation of the * In particular, see Poincar6, Mecanique Celeste, t. xi. 22-2 340 NORMAL INTEGRALS AND [107. solution. For, denoting the solution by w, and the sum of the first m 4 1 terms of the series ( l)az-c r(p + a+ 1) a=O by S,,, we have zm {we-z0 ZP+ -} = zl+ m (t - O,)Pez (t- o dt. - - = rn f Now = / (t - Or)m+ 1 cm+l 1 C where it- 0,.a a < c, and K <1. Then, as before, we have zp+l+' m Rm (t- 0.)Pez (t-o) dt - M la Z4147, +p (-) =_ Cf Ji 1 - 0++(t - 0) +P ez(t-) dt) CM I - 0~* c which is a multiple of p' Uf -?z P+1+Mrl T++p e-zTdr _ tr C by a quantity independent of z. When we take so that, as z is to have large values, the limits of y effectively are 0 to + o, the last definite integral is a multiple of I f:. yP+'+me- dy. z I - - c This definite integral is finite. Denoting its value by I, we have z'n(we-xZzP+1 - S_ ), where a is a quantity independent of z, and I is finite. Hence, when z is sufficiently large, we have zm (we-zorzP+ - S) <, 107.] ASYMPTOTIC EXPANSIONS 341 where e is a small quantity; and so we can say that Sm asymptotically represents we-ZrsP+1, or we can say that the normal series is an asymptotic representation of the actual integral, the representation being valid (on the hypotheses adopted earlier) for large positive real values of z. Note. For further discussion of these asymptotic expansions in connection with linear differential equations, reference may be made to Poincare's memoir*, which initiated the idea. Among other memoirs, in which the subject is developed and new applications are made, special mention should be made of thoset by Kneser, and thosell by Horn. Picard's chapter:: on the subject may also be consulted with advantage: and a corresponding discussion on integration by definite integrals is given by Jordan~. Ex. 1. Shew that the complete primitive of the differential equation d 2 /2 a2 \ — 2 +W a2+1 +2+.. =0, in the vicinity of x= oc, can be asymptotically represented by (ao+ + 2+..) cos +( + + + +...) sin X, where X= ax + - -1 log x, and ao, 80 are arbitrary constants. (Kneser.) Ex. 2. In the differential equation dx ( dC) +(B ) += 0, k2 is an arbitrary parameter, A, B, C are real functions of x and (with their derivatives) are holomorphic when a x: b; moreover, A and B are positive. Prove that an integral of the equation, determined by initial values that are independent of k, is a holomorphic transcendental function of k; and shew that, for large values of c, its asymptotic expansion is of the form = (+ +- '... )cos + ( +...) sin ki, where C0, (1, )2'..., wv are functions of x. (Horn.) * Acta Math., t. vIi (1886), pp. 295-344. t Crelle, t. cxvi (1896), pp. 178-212; ib., t. cxvII (1897), pp. 72-103; ib., t. cxx (1899), pp. 267-275; Math. Ann., t. XLIX (1897), pp. 383-399. I| Math. Ann., t. XLIX (1897), pp. 452-472, 473-496; ib., t. L (1898), pp. 525 -556; ib., t. LI (1899), pp. 346-368; ib., t. LII (1899), pp. 271 —292, 340-362. + Cours d'Analyse, t. II, ch. xiv. ~ Cours d'Analyse, t. II, ch. I, ~ iv. 342 RANK [107. Ex. 3. Shew that the equation d2y dc2 + (k2 - k cos 2x) y= has a solution of the form 00 2 kll, (A,, cos kx + B,, sin kx),?n=0 where Am, B,, are rational functions of k, and that it has an asymptotic solution of the form (0o+ +...) cos kx(+ (- + k4+...) sin kx; and indicate the relation of the solutions to one another. (Poincare: Horn.) EQUATIONS OF RANK GREATER THAN UNITY REPLACED BY EQUATIONS OF RANK UNITY. 108. When the differential equation dnw dn-lw Po dn + pi d;z +i- +. p, uW = 0 possesses, in the vicinity of z= oo, normal integrals which are of grade m, then, denoting the degree of the polynomial p, by so, it follows (as in ~ 85) that the degree a,. of the polynomial pr is such that. < o 0 + r (f - 1), the sign of equality holding for some at least of the degrees. Also, if el be the determining factor of any such integral, then 2' is the aggregate of the first m terms in the expansion, in descending powers of z, of a root of the equation poZ' +plZ1'-1 +... + pi = 0. The existence of the normal integral then depends upon the possession of regular integrals by the linear equation in u, where w = e ic. In the case where m= 1, the method of Laplace certainly gives the integrals of the differential equation, even when the normal series diverge; but it is not applicable, when m is greater than unity. Poincare, however, devised a method by which the given equation is associated with an equation of grade unity: Laplace's method is applicable to the new equation, so that its primitive is 108.] OF EQUATIONS 343 known: and from this primitive, an integral of the original equation can be obtained by means of one quadrature. The new equation is of order nm; and the investigation leads to an expression for 1 dw w dz ' which, when it exists, can be obtained more directly by Cayley's process (~ 92). Poincare's method is as follows. Let the given equation be supposed to possess n normal integrals of grade m, say, in the form el, (z)q l (z), e (z 02 (z),..., eQ (z) On(Z); let these be denoted by f (z), f2 (z),..., f (z). 27ri Let a denote a primitive rmth root of unity, say e7m; and consider, in connection with any integral f(z) of the original equation, a product m-i y= n f (a'z). r=0 Then y satisfies an equation of order nnt, which possesses nn normal integrals fa (Zfb (,)fc (a2Z)... fk (am —z), where a, b, c,..., k are the numbers 1, 2,..., n or some of them, any number of repetitions being permitted; and these normal integrals are of grade m. Let nm = N, and let the equation for y be Q d + QN-1 dZ- +... + Q. Y = o, where, if QN be of degree 0 in z, then the degree of QNr in general is equal to + r (m-1), because of the grade of the normal integrals. Owing to the source of the quantity y, which clearly is not changed if z be replaced by zaY, s being any integer, it follows that the equation for y must remain substantially unchanged, when this change of variable is made; hence Q_-r (zaC) Ca-(N-r) QN-r (Z) where X is independent of r. 344 RANK CHI. NGED [108. Now let the variable be changed fr-om z to x, where zT =mx; then, because dKy - I dK-ay Z(-)M for all values of Kc, the coefficients CII being numerical, the equation for y takes the form N d-.:~ 1-q 0, where q R 'q = Cqv., N (-N-q)m- (NL-r) QNL-r. r=0 The degree of 11Nq in z, as it is determined by the highest terms in Qy, q,is 0 + q (m - 1) + (N- q) m - (N- q = 0+N~m ) w~~hich is independent of q; so that the degree* of all the coefficients A is the same. Further, we have q -R (Z~S) =: (S (q mNVrN QN-1 (zca~ q for the power of a is A as (N-q) in -(L~m)s(NV-q)1 thus -R,,q (Za-s) = XliN-9 (Z). Hence the equation is substantially nnaltered, when z is replaced by za8 in the coefficients BR; hence, multiplying by a power of z, say zK, where K + 0+ NQ t) 0 (mod in), 1? becomes a uniform function of x, when we substitute,n X *Some might have vanishing coefficients in particular cases: the argument deals with the general case. BY TRANSFORMATION 345 The new equation is therefore an equation in the independent variable x such that all its coefficients are uniform.- They all are of the same degree, so that it is of rank unity; it has normal integrals, and some of its integrals may be subnormal. Laplace's method can be applied to this equation; and we then have a solution in the form of a definite integral. The way in which this definite integral is used, in order to bring us nearer a solution of the original equation, is as follows. Let =f (za8), (s=, 1,..., ), and let Y = Wo0 W... Wm1. This has to be differentiated N(=nm) times, derivatives of wo, Wu,..., wnm- of order n being replaced, whenever they occur, by their values in terms of derivatives of lower order, as given by the differential equations which they satisfy; and, from the N+ 1 dy d-uy s equations involving y, dx... dxy, the products dawo dbw, dkw,,_. dza ' dzb.''' dZk where a, b,..., k each can have the values 0, 1,..., mn-1, are eliminated. The result is the equation for y. The N equations dy d-'y involving y, d'... dxN- can be regarded as giving these N products of the type dawo dbw, dkw,,_dza ' dzb.'"' dzk each in terms of derivatives of y and the variables. Let two such be W0 WU... 1, _ 1U-1- = Y, dw, d ~Wl... Wn-l, = c; then 1 dwo = w, dz y Assuming y known, as an integral of its own equation, the value of w, is derivable by a quadrature. If y, first obtained as a 346 POINCAR}'S METHOD [108. definite integral, can be evaluated into a functionally valid normal integral, it is of the form y = ea Y. The function () is linear in y and the derivatives of y, so that, when we substitute the value of y, we have ) = eax t, where P is free from exponentials and then 1 dwo T Wo dz Y' which can be expressed as a series in terms of z. The exponent to which it belongs is easily seen to be an integer, owing to the form of d?; thus I dwo aa am,2i -- dz = aozm-l z- +... +,a_ + a +.... Wo dz z z2 But if y cannot be evaluated into a functionally valid normal integral, there may be insuperable difficulty in dealing with the quantity-. In instances, where the actual expression of a normal integral (if it exists) is desired, the process is manifestly cumbrous: as it does not lead to explicit tests for the existence of normal integrals, the simpler plan is to adopt the process indicated in ~ 85 —88, which gives either a normal integral or an asymptotic expression for an integral in the form of a normal series. For further consideration of Poincare's method, reference may be made to his memoir, already quoted, and to a memoir by Horn*, who discusses in some detail the case, when the linear equation is of the second order and of rank p. Ex. 1. In the case of an equation of the second order which is of rank 2, say d2wo dw dX2 +Ao (x) ++ A1 () w=0, shew that, if wt -- (x), and if zl= q (- x), which will satisfy the equation d2W, dw \ 2 ~- q (- A) 1 (-(-x) W1-'O, * Acta Math., t. xxiii (1900), pp. 171-201. EXAMPLES 347 dx2 + Bo (x) dw- + B, (x) wL = 0, then a variable y, where y^/= 20Wv^1 d generally satisfies an equation of the fourth order, and that d is expressible uniquely in terms of y. If, however, the invariants of the two equations are equal, so that 1 dBo i2 dAo iA2 B 1-2d 4 0-Al-2 - - 0 1 dw shew that y satisfies an equation of the third order, and that - d is the root W dx of a quadratic equation, the coefficients of which are expressible in terms of y. (Horn.) Ex. 2. Discuss the equation y,=-(x3 + 1) y, for large values of x. (Poincare.) Ex. 3. Shew that, in the vicinity of x= o, the equation y"=(x2 +a)y possesses a normal integral of the second grade, when a is an odd positive integer. Ex. 4. Obtain the normal integrals of the equations (i) X2y=(X4+ ) y, (ii) x2y" = 2.x (1 + bx) y + (4 - b22 - 2bx - -) y, in the vicinity of x = o. CHAPTER VIII. INFINITE DETERMINANTS, AND THEIR APPLICATION TO THE SOLUTION OF LINEAR EQUATIONS. 109. IN the investigations of the present chapter, infinite determinants occur. These are not discussed, as a rule, in books on determinants; a brief exposition of their properties will therefore be given here, but only to the extent required for the purposes of this chapter. Their first occurrence in connection with linear differential equations is in a memoir* by G. W. Hill: the convergence of Hill's determinant was first establishedt by Poincare. Later, von Koch shewed+ that the characteristic method in Hill's work is applicable to linear differential equations generally; with this aim, he expounded the principal properties of infinite determinants~. The following account is based upon von Koch's memoirs just quoted, and upon a memoir II by Cazzaniga. Let a doubly-infinite aggregate of quantities be denoted by ai, k, where i, k acquire all integer values between - oo and + oo; the quantities may be real or complex, and they may be uniform functions of a real or a complex variable. They are set in an * First published in 1877; republished Acta Math., t. vmII (1886), pp. 1-36. t Bull. de la Soc. Math. de France, t. xiv (1886), pp. 77-90. + Acta Math., t. xv (1891), pp. 53-63; ib., t. xvI (1892-3), pp. 217-295. ~ For further discussion of their properties and their applications to linear differential equations, see a memoir by the same writer, Acta Math., t. xxiv (1901), pp. 89-122. 11 Annali di Matematica, Ser. 2a, t. xxvi (1897), pp. 143-218. Other memoirs by Cazzaniga, dealing with the same subject, are to be found in that journal, Ser. 3%, t. i (1898), pp. 83-94, Ser. 3a, t. II (1899), pp. 229-238. INFINITE DETERMINANTS 349 array, so that all the quantities with their first suffix the same occur in a line, the values of k increasing from left to right, and all the quantities with their second suffix the same occur in a column, the values of i increasing from top to bottom. We then have an infinite determinant, which may be represented in the form [a k] - Construct the determinant Din,n, where Dn,n-= [Ci,k] a n; then if, as mn and n increase indefinitely and without limit, Dm,n tends to a unique definite value D, we regard the infinite determinant as converging to the value D. In all other cases, the infinite determinant diverges. To secure this convergence to a unique definite value D, it is sufficient that, when any arbitrary small quantity 8 has been assigned, positive integers M and N can be found, such that jDn.+p,n+q — Dm,,n < 8, for all values of mi greater than M, for all values of n greater than N, and for all positive integers p and q. The aggregate of all the quantities for which i = k, that is, of the quantities..., a-_,_-, a0,0, a,,,, as they occur in their place in the determinant, is called the principal diagonal, sometimes briefly the diagonal; and a constituent of reference in the diagonal, naturally chosen in the first instance to be a,,O, is called the origin. Let aj,j= + Aj,j, aj, = Aj,, (j = ); then the infinite determinant converges, if the doubly-infinite series o0 00 E X iAi,kl -oO -oo converges, all values of i and k between - oo and + co occurring in the summation. To prove this, let i=m k=m _ i=n k A=m Pm,n = I +; s A,, Pn= I i+ E |Ai,,4, -n -n- -n — n 350 CONVERGENCE OF [109. and consider Dmn= [ai,k] }Let P,,n be expanded; by omitting suitable terms and changing the signs of others, we obtain D~,n. Hence, taking D,,n, making all the terms positive, and adding certain other positive terms, we obtain Pm,,. Similarly,, we can pass from D,,+p,nq to Pm+p,n+q. Now take Dn+p,n+q-Dm,n; make all the terms positive, and add certain other positive terms, and we have Pm+p,n+q -Pm,n; hence I Dm+p, n+q - Dn,n i < P n+p, n+q - Pr, n. But, because of the convergence of the series o00 00 X E Ai,kI, -00 -co the product Pm,n converges when in and n increase without limit; hence, assuming any arbitrary positive quantity 8, however small, integers M and N can be determined such that Pm+p,n+q - Pm,n < 3, for all values of m greater than M, for all values of n greater than N, and for all positive integers p and q. Consequently, for the same integers, we have IDm+p,n+q- Dmn, n < 8; and therefore the infinite determinant converges. Such a determinant is said* to be of the normal form. All the determinants with which we have to deal are of this type. Next, the origin may be changed in the diagonal without affecting the value of the determinant. All the conditions for the convergence of the determinant with the new origin are satisfied; let its value be D', and let D be the value with the old origin. Then taking any small positive quantity 8, we can determine integers M and N such that ID-Dm,nl<, D) '-D-Dm,,1n <8, * von Koch, Acta Math., t. xvI, p. 221. 109.] INFINITE DETERMINANTS 351 for all values of m greater than M and all values of n greater than N, the determinant D',,,l being the same as Dmn,,,so that, if ao,0 be the new origin, mr = m - 0, n1 = n + 0. Manifestly, Dm,n can be chosen so as to include the new origin. Hence ID- D' = D - Dmn D- (D, - D\ } < _D-Din, nI + o D-D'l,,, a < 28, so that, in the limit when 8 is made infinitesimal, D = D'. Similarly, the value of the determinant changes its sign when two lines are interchanged, and also when two columns are interchanged: so that, if two lines be the same, or if two columns be the same, the determinant vanishes. Further, if the determinant be changed, so that the lines (in their proper order) become columns and the columns (in their proper order) become lines, the principal diagonal being unchanged, the value of the determinant remains unaltered. If, in any line in a determinant of normal form, each of the constituents be multiplied by any quantity t, the value of the determinant is multiplied by /u; likewise for any column, and for any number of lines and columns, provided that the product of all the factors (when unlimited in number) converges. Further, if all the constituents in any line of a converging normal determinant be replaced by a set of quantities of modulus not greater than any assigned finite quantity, the new determinant converges. In the determinant D, let the line ao,k (the constituents occurring for values of k) be changed, so that a0,k is replaced by Xk, where xk < A, A being finite; and let D', D',,,, for the new determinant correspond to D, D,,,. For comparison with D',mn, construct a product Pmn,n, where in. n i -n A i having all values from -n to +m, except i =0. Then, when D',,, is expanded, there occurs in Pm,n a term corresponding to 352 PROPERTIES OF CONVERGING [109. every term in D'm,n, the latter having some one factor xp that does not occur in Pmn; hence term in D'n,n\ ~ A iterm in Pm,n n Now some of the terms in D)'n,n are negative, while all the terms in Pm,n are positive; and terms arise in Pm,,,, the terms corresponding to which do not occur in D',,,. Hence ID L,, < A I Pn,n, Similarly, ID'm+pD, n+q - Di, n 1 < A Pm+p, n+q - Pin, n < A3, where 8 can be chosen as small as we please, because 00 ( 00 A Ti I+ E2 IAl,, k is a converging product. The result, which is due to Poincare, is thus established. PROPERTIES OF CONVERGING INFINITE DETERMINANTS. 110. The development of an infinite determinant can be deduced from the preceding properties. We have Din, n= a —n, -_, a —n, -n+l,., ) a-n, m a-n+l, -n ) a-n+l,-n —,..5 a-n+i, m....... _................................. am, -n, am, -n+i, ~.., am, m -E + (a_ n,-n a_n,+1,-1+1. am,m) n - m,n) say. In this expanded form, let a,i = 1 +Ai,, ai,k =Ai,k, (i:+ k); and let every term in the new expression be changed, so as to have a positive sign and so that each factor is replaced by its modulus. The resulting expression is greater than |Em,n[; and every term that occurs in it is contained in Pm,n, where Pmn= I+ 2E |Ai,kl -n -n INFINITE DETERMINANTS 353 Also, Pm,, contains other terms, all of which are positive; thus lwm,n| < Pmn n Similarly, _ _ Simi, m+p, n+q - m, n < Prm+p, n+q - P, n for all positive integers p and q. But Pm,,,, with indefinite increase of m and n, is a converging product; hence m,,,, in the same limiting circumstances, converges absolutely. Thus the usual method of development of a finite determinant holds in the case of an infinite converging determinant of the normal form, and we have D [ai,k]Z-= A... ~a-2, p2a_, p, ao,po al, q, a2, q2 * * (- I1).+ (p2-2) + (p-1) + (po-o) + (q-1) + (q2-2)... the sum being extended over all the permutations *..., Pi, P, P o q, q2,... of the integers...-2, -1, 0, 1, 2,.... Writing ai,= 1 + Ai,i, ai,k = Ai,k, (i + k), for all values of i and k, we at once have the expansion D=1+2Aii+2 A i,i, Ai,j +:, A,, Ai,j, Ai,k +... Aj,i, Aj, j Aj,, Aj, j, Aj,k Ak,i, Ak,j, Ak, k the summations being for all integer values from - o to + co such that <j <.... 111. It follows from the preceding expansion of a converging determinant D of normal form that, when a constituent aik enters into any term of the expanded form, no other constituent from the line i or from the column k enters into that term. Taking the aggregate of terms (each with its proper sign) into which aik enters, their sum may be denoted by ai, kik; and the determinant may be represented in the form 00 D- E ai,k ai,k, k= — x F. IV. 23 354 MINORS [111. or in the form 00 D = E ai, k ati,k. i= -00 The quantity aik is called the minor of ai,k, and sometimes it is denoted by (;) It can be derived fiom D by suppressing the line i and the column k, or, what is the equivalent in value, by replacing ai,k by 1, and every other constituent in the line i or in the column k or in both by 0, and then multiplying by (- )i-k. Manifestly, we have /i\ as ~i, k - ' - a, ' It is an immediate corollary that 00 0= 2 aj,k ai,k, k= - (i j) o I[ k= - o for the right-hand side in the first is equivalent to D with the line i replaced by the line j, so that the latter is duplicated; and in the second, the right-hand side is equivalent to D with the line j replaced by the line i, so that the latter is duplicated. More generally, if, in the lines a1, Ia2,..~., a, and in the columns 1, 3 21,.2, is') we replace all the terms by 0, except a0,,g,, aa,,,2,..., aa,,,, each of which we replace by 1, and then multiply by (- 1)(a-l- )+( + x,2-2) +-...+ (or-r),) the result is the coefficient of aa~,~,.., aal,Pr aaLr, il.., Caar,or 111.] OF FINITE ORDER 355 in D. It manifestly is a minor of order r; and it is denoted by t a, a2,..., (18 22 **@2 2-) Clearly all the minors of any finite order are determinants of normal form, converging absolutely. If D is not zero, some at least of the minors of constituents in any line must be different from zero, and some of the minors of constituents in any column also must be different from zero. Similar results, when D is not zero, hold for the minors of any order r of finite determinants, which are constructed out of r selected lines and any r columns, or out of r selected columns and any r lines. Further, the minor.-r, -,+4l,..., 1,..., s r, -r+1,..., 0, 1,..., tends to the value unity, as r and s increase. To prove this, let Qs,r= H {1 + IAp,,}, where the product is for all the values of p, and the summation is for all the values of q, that are excluded from the ranges p=- r to +s, q=-r to +s. Expanding the minor, and changing every term so that its sign is positive and each factor in the term is replaced by its modulus, we have a new expression every term of which is contained in the expanded form of Qs,,.; and Qs,, contains other terms. Further, the expanded minor contains the term + 1 as does Qs,,., and all other terms involve the quantities A; hence -r, -tr + 1,, s But the product II X+ E IAp, ql - converges; and therefore,when any small positive quantity is converges; and therefore, when any small positive quantity 8 is assigned, integers -r and s can be determined such that Qs,r-1 <8. 23-2 356 EXPANSION OF [111. Taking these as the integers defining the minor, we have +(-a, -..+, ), _ - -, ) -- I < 6, r, -r+ 1,, so that r, -r+l,..., s 1-3< <; + + Moreover, as integers s', r' are chosen, greater than s and r and gradually increasing, the quantity Qs',r- 1 decreases; and thus the minor tends to the value unity as r and s increase. One or two properties of minors may be noted. We have (k, I) k, I) (1, k) (1 k); for the changes from one of these expressions to another are equivalent to an interchange of two lines or an interchange of two columns, each of which changes the sign of the determinant. Similarly for minors of any order. Again, expanding ai, by reference to constituents of a column, we have f(/i)\ ik aj/z, ); \K/ j,X'SU3,l\f^) 1/ and expanding it by reference to constituents of a line, we have -fi\ k ai, I \ when q is neither k nor 1, because it is a minor of the first order Similarly, I / j, i\ l a(, j\ /i\ I \Fc, L, Further when q is neither k nor 1, because it is a minor of the first order with two columns the same; also Zah, ' =0, h ]c, 111.] INFINITE DETERMINANTS 357 when h is neither i nor j, because it is a minor of the first order with two lines the same; and h or q 'q ( v where h is neither i nor j, and q is neither k nor 1, because it is a minor of the first order with two columns the same and two lines the same. Similarly for minors of higher order. The similarity in properties between finite determinants and converging infinite determinants of normal form is not exhausted by the preceding set: in particular, infinite determinants can be multiplied, and determinants framed from minors of an infinite determinant are connected with their complementary in the original, exactly as for finite determinants. The simpler of these properties are contained in the following examples. Ex. 1. If A=[ai, k], B=[bi, k] are converging determinants of normal type, and if j=oo Ci,k= ai,j bk, j, -00 for all values of i and k, then c= [c, k] }-o} is a converging determinant of normal type, and AB=C. Ex. 2. If ai, denote the minor of ai, k in the determinant A =[ai,k], }-}' then "/ILI ~) ai lr * - -k 25. D k P') aik 7 ~..., ai k ai^lc.-X^ ai.kr with the preceding notation for minors of order r. Ex. 3. In connection with the determinant A =[a, } } 358 INFINITE DETERMINANTS AS [I1. prove that i Vi, '2 + i, i2 __ (W l ) 2) )2A k, k/2 V1k2, k k2, k k, k) l,2 Vk \2l~k22 VkVkil k2J Vk )kl) k2 ]k1lk and, more generally, that "i 0, (1 i,2 5 ~ *.r _ _, il '2... *. r A '*0 /n+l, j kn+2, k /,, where, in the typical term, kn, kn+, k+2,..., kn-1 preserve the same cyclical order as ko k, k 2,..., k,. In the first of these, the right-hand side vanishes if k is equal to kl or k2; in the second, it vanishes if i is equal to i1 or i2; in the third, it vanishes if ko is equal to any one of the quantities kl, k2,..., k,.; and so in other cases. 112. The infinite determinants which arise in the-discussion of linear differential equations have, as their constituents, functions of a parameter p. The preceding results are still valid, if the condition that 00 o0 Ai, j (p) -oo - o is an absolutely converging series is satisfied; in particular, the determinant converges absolutely, and its value may be denoted by D (p). The parameter may be made to vary; and then it is important that the convergence of D(p) should be not merely absolute, but also uniform, in order that it may be differentiated. Suppose that, in any region in the p-plane, all the functions Ai j(p) are regular functions of p, such that the series 00 0G Aij(p) -Xo -oo converges uniformly and absolutely. For all values of p within that region, any small quantity 8 can be assigned, and then integers 11 and N exist, such that for all integers m > M, and integers -n < - V, Wn - 0 By analysis that follows the earlier analysis practically step by step, we then infer that, for all integers m > M, n > N, and for all positive integers p and q, and for all values of p within the region indicated, we have Dm+p,n+q (p) Dm-, (p)1 < 28; 112.] FUNCTIONS OF A PARAMETER 359 so that D(p) converges uniformly. Hence, within the domain considered, D(p) is a regular analytic function of p. The expansions of D (p) in terms of its constituents have been proved to converge absolutely, by comparison with the expansions of Pm,,, where pmnTI1+2 Ai,~. PMm, n I- + E;|A, k I A As the series O 0 E A,,k(p) -00 - X converges uniformly and absolutely, Pm,, is a product that converges uniformly with indefinite increase of m and n. The corresponding modifications in the investigation lead to the conclusion, that the expanded form of D(p) converges uniformly as well as absolutely. Moreover*, this expanded form can be differentiated, and its derivatives are the derivatives of D (p). In particular, we have aD E aD aa,,k aaia k p 9ak -pap Thus if D vanish for a value p' of p, and if all the first minors of D vanish for that value, we have =, k 0, while a-ahl is not infinite; the first derivative of the uniform ap function D vanishes, and therefore p' is at least a double zero of D. In that case, we have a2J _ D a2ai$, k s av i a aa, aaj, ap2 kap2 aZ+ 8ai p ap (71, 1 Dap ap Hence, if all the second minors of D vanish for that value of p, we have aoD =0; ap* The proof is similar to those given for preceding propositions; see von Koch's memoir, Acta Math., t. xvi, p. 243. 360 INFINITE SYSTEMS [112. and so p' is at least a triple zero of D. And generally, if all minors of all orders up to r-1 inclusive vanish, but not all minors of order r, when p = p', then p' is a root of D in multiplicity r; and D is then said to be of characteristic r. The quantity r cannot increase indefinitely, for we have seen that minors of sufficiently high order tend to the value unity, so that the general vanishing of all minors of the same order is possible only for finite orders. But it need hardly be pointed out that the converses of these results are not necessarily true: thus p = p' might be a double root of D, while not all the first minors of D would vanish. 113. The purpose, for which infinite determinants are to be used in this place, is in connection with the solution of an unlimited number of equations, linear in an unlimited number of constants. Let 1c=oo Ui- ai, k Xk, - 00 and suppose that the infinite determinant D, where D = [ai,k] k- converges uniformly; it is required to find the ratios of the quantities x to one another which satisfy the equations Ui = 0, (i = - 0 to + 0 ), the quantities x being themselves finite, so that we have ]Xk X, where X is finite. We know that j / converges absolutely; its value is D when j = k, and is 0 when j is different from k. Moreover, the series Ea, k is an absolutely k converging series, and hence for values of x considered, we have Sai, kxk Xa i, <k U, k 113.] OF EQUATIONS 361 where U is finite. Hence, by one of the propositions already established, the quantity S, defined by the equation S=:5jk aj, xj, also converges absolutely, so that S= () u= xj am, j, i j ' NA- = xkD, for all the other terms give a zero coefficient for x. Hence, if ui = 0 for all values of i, and if we are to have values of Xk different from zero, then D=0, which is a necessary condition. We shall assume this condition to be satisfied. If some at least of the first minors are different from zero, then the equation (7) ui =0O shews that any one of the quantities u, which it contains, is then linearly expressible in terms of the others, and so the corresponding equation u =0 is not an independent equation. Let u0 then be omitted on this ground; we have ~/ t S I- ai^ xq, ki \(, 1 i q k, I ) where on each side the summation is for all values of i except i = O. The coefficient of Xq on the right-hand side is Ev (k a q.\ This is zero, if q is different from both k and I; it is =0o, ^ai,l=-,a0,k, if q =; and it is =i \(', ) aik = C,, = 2;(gk ls,-a, 362 INFINITE SYSTEMS [113. ifq=k. Thus k, g )ui —,'k X1 + a o, lXk. But all the quantities ui vanish; hence -,k Xl + ao, 1 Xk = 0. We thus have XI = a0,I, for all values of 1; and ~ is any finite quantity, for only the ratios of the quantities x are determinate. Similarly, if D be of characteristic r, so that the minors of lowest order which do not all vanish are of order r, let al, a2,..., ar / 1, 2,,/3 be such a minor different from zero. We then have = 1,, ] — a1, n,, an+l, *, ]r) mv 1n ***n /371-1a, qn Xq.* r m q \P l...., n-1i P, Pn+l? ** * Pr/ Thus the coefficient of Xq is 1.al,, an-i, m, afn+l,.., atr mwe \P n1,,-.-,,1 n n+l, ),3 P "r When q is equal to any one of the integers,/l, *..2, *ir, this coefficient is equal to a minor of order r- 1 and so vanishes. When q is not equal to any one of those integers, the coefficient is equal to a determinant with two columns the same, and it is therefore evanescent. Hence S=0O, and therefore I, }a,q _ Ir (,(1/al,..., an-l, m, an+1,..., a,) Pi,.*., p. \/, l, ]n-1, n, n+il,.**, / ' where, on the right-hand side, m must not be equal to any one of the integers a,,..., ar. It thus appears that there are r relations among the quantities u; and that, in particular, each of the quantities ua,, ua,..., 'Ha, is linearly expressible in terms of the 113.] OF EQUATIONS 363 remaining quantities u. Accordingly, we assume these r quantities 'a omitted from consideration. Denoting by a any integer other than a,..., a, and by / any integer other than 01..., /,., we have aP, al, Xr\ P\ 2...3 all S. U0, =:, r. t', Y X ~ ~ C7, Ctl) a2 0(?' g - il a2 )... Ct), Xg, - (71 5C(2) ~l "92 (2a0, a23,...,ar3\ 1? - (X1/3 2 /3/3 X~ in the same way as for the simpler case; hence, as all the quantities it vanish, we have /32, /32,.~., /3 k/31, /3,2..., /31-1, /3, /3+..., 3.1. so that all the quantities xg are linearly expressible in terms of 'r such quantities. For further properties of infinite determinants, reference may be made to the memoirs quoted at the beginning of ~ 109. APPLICATION TO DIFFERENTIAL EQUATIONS. 11-4. When the differential equation is given in the form dn W dnZ-1 ~W d W Q &-d t.. + Qn-1 + Q &,W 0 0 dxh dn 1T dz the substitution fQidCz W= we leads to an equation of order na in w, which is devoid of the term dn-iw involving dzn-l' The coefficients of the new equation are linearly expressible in terms of Q2, Q3,., Qn1, Q, and the expressions involve derivatives of Q, up to order n - 1 inclusive and integral powers of Q,. We may therefore take the differential equation in the form dnw d'1-2 w dw P (w) dW + P2 d + +.+ Pl1dz + PnW = 0. 364 INFINITE DETERMINANTS APPLIED TO [114. We assume that, in the vicinity of z = 0, it possesses no synectic integral, no regular integral, no normal integral, and no subnormal integral. The point z = 0 is then a singularity of the coefficients; and, if it be only an accidental singularity (of order higher than s for Ps, in the case of some value or values of s), the conditions for the existence of a normal integral or a subnormal integral are not satisfied. We assume the coefficients P still to be uniform functions of z, and we shall suppose that their singularities are isolated points. Let an annulus, given by R < Izl < R', be such that its area is free from singularities, no assumption being made as to the behaviour of the coefficients P within the circle of radius R; then it is known* that each such coefficient can be expanded in a Laurent series 00oo Pr = c c.,, (r = 2, 3,..., n), 00 which converges uniformly and unconditionally within the annulus. Without loss of generality, it may be assumed that R < 1 <R': for, otherwise, we should take a new variable Z= z (RR')', and the limiting radii R and R' of the annulus for Z then satisfy the conditions R <1 < R. Further, owing to the character of the convergence of Pr, we have dPr ~ dz = E C~,2Z, dd ( dPr 0 - dz z dz / _o.,z and so on; all these series converge uniformly and unconditionally within the annulus. Hence also 00 E R () Cr,,zZ - 00 * T. F., ~ 28. LINEAR DIFFERENTIAL EQUATIONS 365 similarly converges within the annulus, where R (p) is any polynomial in A/; and therefore, taking the circle zl =j1, every point of which lies within the annulus, the series IR () c,W l 00 converges. 115. From the general investigations in Chapter II, it follows that the equation certainly possesses an integral of the form = zP (z), where p is any one of the values of -.log o, the quantity o being a root of the fundamental equation associated with an irreducible (but otherwise simple) closed circuit in the annulus; and the quantity b is a uniform function of z. As the integral is not regular, the number of negative powers of z in b is unlimited; and so we may write y = a ZP+m. -_00 In order to have an adequate expression of the integral, the quantity p must be obtained; the value of a-m a0, for rn= + 1, + 2,..., + oo, must be constructed; and the resulting series must converge for values of z within the annulus. We first consider the formal construction of the expression for the integral. Let (p) =p (p - 1)...(p -n + 1)+ c,,_, p(p-l )...(p - n + 3) + C3,-3 P (p - 1)... (p - n + 4).. + Cn-_],-n+i, p + n,-n,; Cr.. = (P + ). * * (p + - n + 3) C2,,r-,,-2 +(p + n )...(P + )/ - n + 4) C3,r-_-3 + **... + (p + P) Cn-1, r-,-n+l + Cn, r ----n; and write o00 Gm (p)= (p m) an, + Z' Cm,. a, - 00 where, in the last summation, the values of p are from - o to + oo, with p =m excepted. Then we have P (y) = G, (p) zp+,'-n -00 366 366 ~~INTEGRAL IN THE FORM OF[15 [1,15. so that y is an integral of the differential equation if Gin (p) = 0, for all values of m fromn - oo to ~ oo, there being no assumption that the negative infinity is the same numerically as the positive infinity. Let cm, 0 -for all values of 4 other than / = m; and introduce a quantity *JInl with the convention =m1 then Gm (p)zzq(m +p)y *M a where the summation now is for all values of F.t. We then require the infinite determinant (P)= [#in,J1A I., 'k*-2,-I) "k-2,0, "k-2,1, *J-2,2,. * -I, -2 1,* I,~ 0 k,~ * I, '-I, 2.. the necessary and sufficient condition of the convergence of which is the convergence of the double series for all'values of mn and at between -- oo and + oo except m =4 116. In order to establish the convergence, we first transform the expression of U~.Let Ft m-X; then we may take =(P+M -X)"(p+M -X -1)... (p:m - X-p+l1) =(p+rn')P + a', (p + n)P ~ ap,2 (p m)P+ * a A LAURENT SERIES36 367 where %,,8' is a polynomial in X of degree r. Using this for all the terms in c.Awe have Can,~ A, (X) (p + M)-2 + A3 ()(p + M)n-3 +... + An, (X, where Ar (X):::On-2, r C-,A2+On3 - C3,X. -3 +. +Cr,,kr. Accordingly, we have ran, 9 =, 10 (p +rn)4-2 ______3 MA4(p+m) mXA (P + M) m A (p + M) Now the series I R (X) Cr, A converges for every value 2,:3,..., n of r, where 1? (X) is any polynomial in X. Hence n-r A=oo A ~~~p=2X=-GO every term of which (for the various values of p) converges, because anV8-+ is a polynomial in X of degree s -p + 2, and therefore the whole of the right-hand side is a converging series. Accordingly, we may write A= - 00 and then each of the quantities JH8 is finite. We thus have m b(p + n)2 m (p + M)n ~ M Ix lit ~~b(p + m) =0, M 368 INFINITE DETERMINANT [116. each of which is of degree n, we know that all the series (p + rn)" +(p + m) converge absolutely, for the values / = 2, 3,..., n. Moreover, the sum of each such series is a function of p: and then, if p varies in a region no point of which is at an infinitesimal distance from any of the roots of b (p + m), the convergence of the series is uniform. Accordingly, the double series E Efrm,, K rn,u converges uniformly and unconditionally; and therefore the infinite determinant f (p) converges uniformly and unconditionally, provided p does not approach infinitesimally near any root of any of the equations p (p + m) = 0. Clearly, i (p) is a uniform function of p, for such values of p. Further, we have An,p. (p) f (P + n) = COn, (p), and therefore m+l,,,+, (p) (p + rn + 1) = Cm+l,,+1 (p) = c,,(p+ 1) = mrn,, (p + ) + (p +1 + m), so that m+1,,+l (p) = Kmn,. (p + 1). Construct the infinite determinant 2 (p + 1), and then replace each constituent rl,, (p + 1) by 4m+-,^.+1 (p); the result is to give the modification of 12 (p), which arises by moving each column one place to the right and by depressing each row one place, in other words, by taking 1,, (p) in the diagonal as the origin instead of r0, 0 (p). But such a change makes no difference in a determinant which converges absolutely; we therefore have f(p + 1)= I (p), or the infinite determinant 12 is a periodic function of p. Lastly, by making p infinitely large in such a manner, that it does not approach infinitesimally near any of the roots of any of the equations b (p + m) =0, 116.] MODIFIED 369 (which roots for different values of m differ only by real integers, so that if we take p = u + iv, where i and v are real, it will be sufficient to take v large), we reduce to zero every constituent that lies off the diagonal of 2 (p). As every constituent in the diagonal is unity, and every constituent off the diagonal is zero, it follows (from the law of expansion of an absolutely converging determinant) that Lim 2 (p)= 1, p=OO provided p tends to its infinite value in the manner indicated. MODIFICATION OF' THE INFINITE DETERMINANT 12 (p). 117. It is convenient also to consider another infinite determinant associated with f2 (p). The equation Gm (p) =0 was taken in the form 00 (m + p) -m,, a, = 0: - 00 and the infinite determinant 12(p) was composed of the constituents m'Em,. If an infinite determinant were composed of constituents p(m+p)rm,,, then the row determined by the integer would have a common factor (mn4 + p); and thus there would be an infinitude of factors, the product of which either should converge or should be made to converge. Let,, p2,..., p be the roots of b (p) = 0, so that (p) = (p - P) (P - p)... (P - P), and therefore (p +m) = mn (1 + -P1) (1 + P -2)... (1 + P To change this into a form suitable for an infinite converging product, we multiply by P - P P - P2 P-P hm(p)= P-e m.e m.....e m with the convention ho(p)=1. As h (p) remains finite and is not zero for finite values of p, we may replace the equation Gm (p) = 0 by hm (p) Gm () = 0. F. IV. 24 370 MODIFICATION OF THE [117. Now let Xm, m (p) = hm () + (p + m) =II P - P P - Pa = {(i + -P)e -m, o=1 D\ ' e n for all values of m except m = 0, and o0, = n (p-pa); cr=l also let Xm,, () = hm (p) ( (,+ + p) %m,, = m (p) C,,,. Then the equations between the constants a have the form 0o X, x = =0. -00o In association with these equations, we consider the infinite determinant D (p)= [Xm,] } — o0)..., X-2,-2, X-2,-1, X-2,0, X-2,1, X-2,2,..*.. -1-2 -1,-1 X —l, X —1, X-,, X-, 1, X-*, *, X.., XO,-2, Xo,-1, Xo, o Xo, 1, X0,2, '*..., X1, —1, X1,0, 1,1, X1,2, *, * *.. X2, -2, X2,-1, X2,0, X2,1, X2,2, **..................................................... Taking the diagonal to be...* X-,-2, X-i, —, Xo, % X, 1, X2,2,..., we require to establish, (i), the convergence of the series X S m, ) summed for all values of m and Mu, except rn = /, from - oo to + o, and (ii), the convergence of the series S (Xmm-), -00 in order to know that the infinite determinant D (p) converges. We consider first the double series 6X~mx. Let n P - Pa k( (p)= ml, (p) = n e m, (m + 0). 7a= INFINITE DETERMINANT 371 The quantities p,, p,..., pn are finite; hence, so long as p remains within a finite region that does not lie at infinity, there is a finite quantity K which is larger than any value of | km (p) for values of p within that region. Hence, as Xn, 4 = hm (p) Cm, f - km ( ) Gm/ we have I <A K I Cm, AL when?n is not zero. When m is zero, we have X0, = ho (p) Co0, = Co,/. Proceeding exactly as with the series SZm,/. in ~ 116, summing for all values of m other than zero, and for all values of /L other than m = /,, we have S cl? U IZZ, I E21 E IP + I - 1-1..g + m ml n, + I.Z Hnq/ n m m m~ m MIn mmn every term of which is finite, and therefore mm is finite. Also jCo,,l 111 Hpn-2 + I31 lp-3l +... +, /A which is finite, so that X Co,pj converges. Hence the double series Xm,, summed for all values of m and /u between - oo and + oo except m =, converges. Moreover, all the series, which occur in the superior limits in the inequalities, converge uniformly within the region of p considered; hence the double series converges uniformly. The establishment of the convergence of the series — 00 E (Xm,m-1) 24-2 372 CONVERGENCE OF [117. is simple. We know, by Weierstrass's theorem, that the series I Xm,m -o00 converges uniformly and unconditionally; so that, if 0m,m = Xm,m - 1, the infinite product 00 H (1 + Omm) -00 converges uniformly and unconditionally; I (1 + I O mI ) converges. But and thereforet H (i + m,mn) <1 + lOm,,]; hence O I0m,ml converges uniformly, that is, the series X (Xm,nm1) - oo converges uniformly and unconditionally. The convergence can also be established as follows. Let P - Pa. u.= )1 +P e m and choose a finite positive integer p, such that, for values of p under consideration, we have IP-P'I<P, where p' is any one of quantities Pi, P2,..., Pn. The sum of the terms E (Xm,m-1) -P is finite, and may be omitted without affecting the convergence: and we consider the sum of the remaining terms, for which we have I| ml>p We have u=e a, where T= - (P - )2 (p _ p-)3 m2 'F., + T. F., ~ 50. ~ T. F., ~ 49. THE DETERMINANT 373 and therefore 2 IT[<p-pP-{l I+ P -P+p-p2 -} 2e |r < I P-p_ 12 -- + P p 22 + < I P -12 1 I P-Pr m Now for all the values of m under consideration 1 - P -PoI ) I P ~ PI P I -- i ->1i- 7> 1 rnm I t1m I p+1 p+l' and therefore so that we may take where Hence p+l I\p-pl12 T P< 2 2 P - P 12.2m2 -Pa)2, _I L (p_ p XM, m = IIVu = e T, p+l1 n = e -l% (P - p)2 =e 2' '=1 Now 2 /ra (p- pa)2 is finite for all the values of p under consideration, and (r=l it is finite for all values of m if I,, involves m; let M denote the greatest value of its modulus. Again, for any quantity 0, we have leo-11 +1 -+.. eol-1, so that or writing we have and therefore 1 3rV= (p + 1) 2}/, IXm,m-l|<em - -1, IXmn-1 <eI - N N2 1 _VS 1 < +, + ~ +...; m 2! m4! m3" 1 N2 1 N3 1 O + T2! 2 W4 + * * *; 1 N2 ( 1 \2 + 3 1 3 1 2 + '; (2) + 7,<r2 + 2 V2 (2 N3 f\32 3 3 2-! 3) 3(! (3) shewing that the series converges. < -1, < e (XN 1 ) Y (Xm, n-1 ) 374 EVALUATION OF THE [117. The infinite determinant D (p) thus converges uniformly and unconditionally for all values of p in the finite part of its plane. Its relation to &i (p), which converges similarly for values of p that are not infinitesimally near any of the roots of any of the equations b (p + m) = 0, is at once derivable from its mode of construction from /2 (p). The row of quantities Xm,, (p) in D (p) for the same value of m is derived from the row of quantities m,j, in /2 (p) for that value of m, through multiplication of the latter by hm (p) b (m + p). Hence D (p)= n (p) n hm (p) / (n + p) = n(p) (p), where -00 m=c00 P- n ='5= I l+( e n(p-p (), 11 [ I{( + P - a)- e m I J (p m= - 00 r=l m =l and II' implies multiplication for all values of m between + oo and - o except m = 0. Also (p-p,) n I+-P e =b si -; (p-) II' {(1+ P P) - PO Sin (p- p) r and therefore (p)= n - {si (p p) Now D(p) has been proved to be finite (that is, to be not infinite) for all finite values of p; and manifestly, from its form, it is a uniform function of p, so that it is a holomorphic function of p everywhere in the finite part of the plane. Further, 2I (p) is a uniform function of p; and it has been proved to be not infinite for values of p, which are not infinitesimally near any one of the roots of any of the equations f (p + m) = 0, the aggregate of all these roots being pI + m, m,..., pn +, (= -oo to + o ). Hence, owing to the relation D (p) = l (p) II (p), 117.] DETERMINANT 375 it follows that these roots are poles of B (p). Take a line in the p-plane inclined at a finite angle to the axis of real quantities, choosing the inclination so that it does not pass through any of the points pe + m for all values of a- and m; let it cut the axis of real quantities in a point f. Take the point f+ 1 on that axis, and through it draw a line parallel to the former, thus selecting an infinite strip in the p-plane. Since (p + ) = n (p), the uniform function 1 (p) undergoes all its variations in that strip: and within the strip, we have Lim fI (p)= 1. p=OO Owing to the nature of the poles of 1f (p), the strip contains n of them, which may be regarded as the irreducible poles: suppose that they are p,,,p2..., pn Within the strip, p = co is an ordinary point of the simply-periodic function D (p); it follows* that the number of its irreducible zeros is also n, account of possible multiplicity being taken; let these be p,', p',..., pn'. Hence I (p) =\ A sin {(p - p') 7r} sin {(p - p') 7r}... sin {(p - Pn') 7r} n (p)=A sin {(p - p ) 7r} sin {(p - p ) 7r}... sin {(p - pn ) r}' taking account of the holomorphic character of D(p) for finite values of p, and of the relation D (p) = a (p) n (p). Here, A is independent of p. To determine A, we use the property Lim p (p) = 1, p=OO which holds for p = a + iv, in the limit when v is infinite, whether positive or negative. Taking v positive and infinite, we have Aeir (Ipa - p,) = 1; and taking v negative and infinite, we have Ae - it (2pR'- Zp,) = 1. * T. F., ~ 113. 376 FORMATION OF [117. Hence Xp,'- Sp is an integer; if it is not zero, we can make it zero by substituting, for the quantities p', values congruent with them. Assuming this done, we have Sp' = Spa, A = 1, A=l, so that n ( \ sin t(p - p') 7r..." sin ~(p - p)7r} sin {(p - pi ) 7}... sin {(p - p, ) 7r} and therefore D (p) = i sin (p- p/)Vr}I D(p) = H. o=l 7r Moreover, the quantities pi, p,..., p, are the roots of s (p)= 0, so that Spa =1n (n-1); hence Sp2'= n (n- 1). 118. Next, we consider the expression -00 /0 Y= E Zk; we proceed to prove that this series converges for all values of z within the annulus. It manifestly arises from D (p), on replacing Xo, k in D (p) by Zk; we shall therefore assume that Y is transformed into this modified shape of D(p). When the determinant is in this shape, we multiply the column associated with m by z-, and the row associated with m by zm; these operations, combined, do not change Xm,, and they do not alter the value of the determinant. Let this combined pair of operations be carried out for all the values of m from - c to + o; the result is to give a determinant, which is equal to Y and has XP, q Zfor its constituent in the same place that %p,q occupies in D (p). Hence, as for D(p), so Y converges uniformly and unconditionally for values of p within the p-region selected, and uniformly and unconditionally for values of z within the annulus, if the doubly-infinite series 2%P, ZP q, (P+q), 118.] AN INTEGRAL 377 converges uniformly and unconditionally within those regions, and if (Xq,-1) converges uniformly and unconditionally. The latter condition is known to be satisfied, owing to the convergence of D (p). It remains therefore to consider the convergence of the double series. With the notation of ~~ 115-117, we have Xm,,uZmy = hm (p) Cm,,zA = hm (p) [(p + m)n-2 A2 (X) zA + (p + m)n-3 A (X) zA +... + An(X) zX]. Now A (X) z = a_, C2, A-2 zA + n-3,,r-1 C3, A-3 Z +... + Cr, A-r Z. Owing to the definition of the coefficients in the original differential equation, the series 00 > Cs,h —s ZA-s A= -o converges uniformly and unconditionally, for values of z within the annulus R< Iz < R'; and therefore the series X A,(X)zA - oo converges uniformly and unconditionally for the same range. Denoting this by Jr, we have A=c0 Je= ) Ar(x)z^; -0 and IJrl is not infinite for any of the values of z. Again, as (~ 117) X0,o = CO,a, and Xmz km (P) mn, when m is not zero, we have Xm -Zm = km (P) m ^, Z- - 77/n 378 CONSTRUCTION OF [118. Proceeding with the double series Z Xi,,,zm-,, exactly as with the double series oXm.i, omitting for the present the terms corresponding to m = 0, and remembering that the summation is for all values of m other than m = /, we have <xi[jl, Zlp + - n "-2 lp + M1-3 < K[ +IJP +* + I X every group of terms in which is finite, so that X: 1%m,,u z y/ I is finite. Also, taking account of the terms omitted for the value m = O, we have,, \ Z-\ < |IJ2 Ipn-21 + J3 p3 -31+... + IJn which is finite, so that converges. Hence:i XM, -, summed for all values of m and fL between - oo and + oo except m = /L, converges unconditionally. Moreover, all the series which occur in the superior limits in the inequalities converge uniformly, both for the values of z considered and the retained range of p; hence the double series converges uniformly and unconditionally. The proposition is therefore established for E (~.k A similar investigation shews that the series k=, b ay wa, f or any value of r, the numbers a and / being any whatever, converges uniformly and unconditionally for values of z within the annulus, and for values of p in the range that has been retained. 119.] INTEGRALS 379 CONSTRUCTION OF IRREGULAR INTEGRALS. 119. These results may now be used, by a generalisation of the method of Frobenius in Chapter III, to construct expressions for the integrals of the equation P dn dn-2w 'W 0 P (w) + P2 d.tnw~+ p-o d dz"Writing y = am zp'm, 00 and adopting the notation of ~ 115, we have P (y) = Gm (p) ZP+n-n = Gi (p) Zp+i-n, if Gin(p) = 0, for all values of m between - co and + cc, except m = i. The last equations are equivalent to hm(p) Gm (p) O, that is, to Y, %m, a, = Ur = 0, - 00 for all the values 0 + 1, ~ 2,. of m, except rn = i. Let Gi (p) ~Xi, 1, a, u 00 We have 300 I its = ak D (p), that is, U~ Li = ak D (p), for all the values of le. Hence, writing a A2 a0 = we have u= A D (p), and ak = A 380 IRREGULAR [119. Thus the quantity y, where y=A l Zp,+, - oo satisfies the equation P (y) = AP+i-n D (p). The determinant D (p) is of normal form; the series for y converges uniformly and unconditionally, alike for values of z within the annulus R < lzI < R', and for values of p within the finite region contemplated. 120. Let p =p' be an irreducible simple root of D (p)= 0. Then the first minors of constituents in any line cannot vanish simultaneously for p = p'; for aD k= as, k ap -0 ap and the left-hand side does not vanish for p = p'. Selecting minors of constituents in the line i, we have yl=A Z ( P _) - oo\0 p=p and P (yl) = AzP'+i- D (p') = 0: that is, y1 is an integral of the equation. Similarly for any other irreducible simple root of D (p) = 0. 121. Next, let p = p' be an irreducible multiple root of D (p) = 0 of multiplicity a. Firstly, suppose that some of the first minors of D (p) do not vanish for p = p'; let some of these non-vanishing minors be minors of constituents in the line i. Then, in the vicinity of p = p', we have y=A: ()zpQ+k, as a quantity satisfying the equation P (y) = AzP+i- (p - p') R (p - p') where R (p - p') does not vanish when p = p'. It therefore follows that p ( = (p - P')C_ (z, p, p'), ^ \aplL INTEGRALS 381 so that, if, < - - 1, we have - p /=' = o, and therefore /a^ Y, = (\app=p ) is an integral of the equation. Hence, corresponding to the irreducible root p' of multiplicity a, there are integrals Yo = X (k) Zpk, yl = ap (I +)]Zpk + yolog =l + yo logz, 2 = Laa2 (p] ZP+ + 27 log z yo (log z) = 7 + 271 log z + yo (log z)2,........................................................................... (-1)-2 (lgg+(a- 1)2(a -2) y.-1 = 1-i + (a - 1) cr-2 log z + l e3 (log )2 +... 1.2.. + (o - 1) W (log z)a-2 + Yo (log z)-1, when, in each of these expressions on the right-hand side, we take p =. 122. Next, still taking p= p' to be an irreducible root of D (p)= 0 of multiplicity a, suppose that, of the minors of successive orders, those of order r are the first set which do not all vanish for p = p'. Let the lowest multiplicity of p' for first minors be ai, for second minors be -2, and so on up to minors of order r - 1, the lowest multiplicity for which is denoted by Or_1. Then, owing to the composition of D in relation to first minors, to the composition of first minors in relation to second minors, and so on, we have - > o- > 0-2 >... > 7r-1_. There are two ways of proceeding, according as r < a, or r = a-. First, let r < a. With the preceding notation, we have y = A _I ^ok, — 00 I P (y) = AzP+'-" D (p). 382 SUB-GROUPS OF [122. After the explanations given in the construction of these expressions, we know that p = p' is a root of multiplicity o- for some of the minors in the expression for y. As before, in ~ 121, the quantities ay ap - ly Y) ap 0 ap O-,) when in each of these we take p = p', are such that P (a,- =0,;ap*/ p=p, for X = 0, 1,..., - 1. But owing to the fact that p = p' is a root of all the minors (c) of multiplicity a1, all the quantities ay aa-ly Y, ap."a'' pal-1 vanish when p = p'. Hence the non-evanescent integrals which survive are a-T y ao-,+1y a0-1y ap^il' api+fl "' apwhen p = p'. They have the form y,,=A );; iZ+k, y~, = A _E + a + P+k('r + 1) y,, logz = w1,2 + (o-1 + 1) y,l log z, co a-1+2 li y1,3 = A E ap+2 1 ZP + ( —, + 2) 1, 2log z (a-1i 2) (a-,+ 1) +( 2 )('+ yl, (log z)2, and so on: their number being O' - 0-1. Next, p = p' is a root of least multiplicity a, for some of the minors of the constituents of any line i: and there must be at least two such minors. For D (p) = kai, k; IRREGULAR INTEGRALS 383 if p = p' is a root of multiplicity - + 1 for all the minors but (k), then, as it is of multiplicity > e r- + 1 for D (p), it would be of multiplicity o7 + 1 for ( ). Similarly for any other line. Once more substituting y = am zP+ -00 in P (y), we have P (y) = E G (p) p+m-n — 00 = Gi (p) p+i-n + Gj (p) p+-n, provided Gp (p) =, for all integer values of p from - o to + oo except p =i, p =j. The last equations are equivalent to hp (p) Gp (p)= 0, that is, to /A=00 E Xp, al, =O, 00 for all integer values of p except i and j. Consider quantities ao of the form for all values of 0, the quantities A and B being arbitrary. With these expressions for ao, we have Xp,, a,,=A E X,, 2+BY, +XPg. -00 / Each of the sums on the right-hand sides vanishes, when p is not equal to either i or j: and thus the preceding expressions satisfy the equations hp (p) Gp (p) = o, for all integer values ofp except i and j. Further, hi () Gi (p)= I Xi, ^a,^ - _ 00 4 A; Xi, + By, X? -A ( jl) -B[jk \~~~ ~~~ ii,\,' 384 SUB-GROUPS OF [122. and, similarly, hj (p) Gj(p)=-A +B Using these values, we have ~, {A ( )+ B( )}zP+m as the expression for y; and it satisfies the relation P (y)= G, (p) zP~'-+ +Gj (p) p~j-n As the right-hand side of the last equation has p = p' as a root of multiplicity a-,, the quantities hi (p) and hj (p) having no zero for finite values of p, it followvs that for X = 0, 1,., - 1. Therefore all the quantities when p = p', satisfy the equation P (w) = 0. Owing to the form of y above obtained, which has p p= as a root of multiplicity c~, all the quantities vanish when p =p'. Therefore the surviving integrals are 1 =, 1 2 a0_2+2q +(-+2)l, oz (0-2 + 2) (a-2 + I) y21 (log, z)2, and so on: their number being 0-1 - a-2. 122.] IRREGULAR INTEGRALS 385 Similarly for the next sub-group. With the same notation as before, we have P (y) = G (p) p+i-n + Gj (p) P+j-n + Gh () +h-n, provided Gp(p)= O, for all values of p, other than i, j, h, from - oo to oo. The analogy of the preceding case suggests a= A, j, m) + C, h for all values of 0, where A, B, C are any quantities. With these expressions for ao, we have 2p,a =AS(; 0,, )x} +BZ 5 fh) ) Xpj, +h hi,' -X \, l, n / p, k, 1, m) P, + i Each of the three sums on the right-hand side vanishes, when p is not equal to either i orj or h: so that the preceding expressions for as satisfy the equation Gp(p)= 0, for all values of p other than i or j or h. Further, -00 (ih(p) IXi, m a,,k A A ( &(' m + B (m k + C (k I()' Gj (p) = A (,' ) + B (, h) + C (, )' G(p)=A (A +B( + C(I Thus P (y)= (, P), where ) (z, p) is a linear combination of minors of the second order; and y= 2 aozP+O, - 00 the coefficients ao being linear combinations of minors of the third order. F. IV. 25 386 SUB-GROUPS [122. As ~ (z, p) has p = p' as a root of multiplicity 02, it follows that Pkp /D= o,= 0, for = 0, 1,..., a - 1; so that all the quantities ay ar2-ly Y' ap' *" a2p_-l1 when p = p', satisfy the equation P (w) = O. Owing to the form of the coefficients ae in y, each of which has p=p/ as a root of multiplicity c3, all the quantities ay a3-ly Y, ap' "' ap3 —1' vanish when p = p'; and we therefore are left with the integrals a3y Y3, 1=a ao-+ly Y3,2 = ap+, = W3,2 + (03 + 1) y3,1 log z, Y3.3 p+ = 33 + (o3 + 2).3,2 log z + 2 (og )2 and so on: their number being 0-2 - 0-3. Proceeding in this manner, we obtain successive sub-groups of integrals; the total number in the whole group is (- O1) + (01 - 0-2) + (a2 - 3) + * + (o-r-2 - O'r-) + o'r-1 which is the multiplicity of p = p' as a root of D (p) = 0. 123. Two cases, both limiting, call for special mention. It is manifest that, if o- -- > 1, the first sub-group contains integrals whose expressions involve logarithms; likewise for the second sub-group, if o- - 0 > 1; and so on. If, then, all the integrals belonging to the multiple root p = p' of D (p) = 0 are to be free from logarithms, we must have 0 — l= 1, O1 —'2= 1,..., 123.] OF INTEGRALS 387 and therefore which thus is a limiting case of the preceding investigation. An intimation was given that, when r = a, a different method of proceeding is possible. As a matter of fact, the property of the infinite system of linear relations, established in ~ 113, leads at once to the result. Let /ai, CT2, *, t ar\ be one of the non-vanishing minors of order r belonging to D (p); then,^ i, al, (_,2, *2,.A i, 2.. -1, Otp, OapIl,... atA )31 /3 2, * ry p=1 181 i 182) -X * p-1, i 1).. *I )dry and the quantities a,,, a,,..., a,, are bound by no relations, so that they are arbitrary constants. The integral determined by these coefficients is oo it manifestly is a linear combination, with arbitrary coefficients a,,,..., a,,, of r integrals which are, in fact, the group of integrals above indicated. The other limiting case occurs when r = 1: all the ao integrals belong to a single sub-group. In that case, there exists at least one minor of the first order which does not vanish when p = p' the condition is both necessary and sufficient. 124. We thus have a set of a- integrals, belonging to an irreducible root p' of D(p)=O which is of multiplicity -. Similarly for any other irreducible root of D (p) = 0; hence, when all the irreducible roots are taken, we have a system of n integrals. We proceed to prove that this system of integrals is fundamental. For, in the first place, it follows (from the lemma in ~ 27) that the integrals in any sub-group are linearly independent, on account of the powers of logz which they contain. Next, there can be no relation of the form C, yi + C2y2a,1 +... + CrYr, = 0, 25-2 388 FUNDAMENTAL SYSTEM [124. with non-vanishing coefficients 0. If such an one could exist, the coefficient of every power of z in the aggregate expression on the left-hand side must vanish. Writing j, j, A, h P, P2, P3'... Pr) /C, 11 Ml q 21q3 q we have 00 laas ae\ PISt) Ys~i (DT ao P1~ where ao=A s-A_ PP:,,::) + S2,.,P + 0, q2, qS qI,, qS +A8, q(2:PI '..qs-1, 0) and the quantities As,,, AS,2,..., A8, are at our disposal. Let these last be chosen so that A8,1, A8,2 =... = A, 8- = 0, A,8 = 1. Then the coefficient of 2p'+qt in ys,I is zero if t < s, and it is different from zero if t = s: let it be denoted by [y,,I]qt. The above relation being supposed to hold, select the coefficients of 2p'~q1 ZP'+q2,... p'9+,. in turn. As they vanish, we have C, [yi, iq, 4- C0 [y2,2]q, +... + Cr [yr,jq1 = 0, from the coefficient of zp'+ql; every term vanishes except the first, and [sli]q, is not zero; hence C,= 0. The vanishing of the coefficient of Zp'+q2 then gives 02 [Y2, i1k, + 03 [Y3,1 ]lq +.+ Cr [Yr, j]qr = 0; every term after the first vanishes, and [y2,ilq, does not vanish; hence 03= 0. And so on; every one of the coefficients C vanishes; and thus no relation of the form Cy1,1~I C02Y2,1+... + Cryr, I= O can exist. 124.] OF INTEGRALS 389 Next, there can be no linear relation among the a- members of a group. For, in any expression Cs, t Ys, t, the coefficient of the highest power of log z is of the form E Cs, t ys, I and this can vanish, only if the coefficients Cs,t are evanescent; hence Cst ys,t can vanish, only if the coefficients Cst are evanescent. Lastly, there can be no linear relation among the members of different groups. For let Y(p', z), Y (p", z),... denote the most general integrals of the groups belonging to the irreducible roots p', p",... respectively, of D (p) = 0. Let z describe a contour enclosing the origin; then Y (p', z) acquires a factor e2rP', Y(p", z) acquires a factor e2RiP", and so on. Thus, if there were a relation a Y (p', z) + Y (p z) +... = 0, then ae27iP' Y(p', z) + Je27p" Y (p", z) +... = 0; and similarly, after K descriptions of the contour, ae27ip'K Y (p', z) + Ie2rip"K Y (p", z) +.... 0 for as many values of the integer K as we please. Now p', p",... are the irreducible roots of D (p) = 0; no two of them are equal, and no two can differ by an integer. Hence the preceding relations can be satisfied, only if a=0, f=0,...; in other words, no linear relation among the n integrals can exist. They therefore form a fundamental system. THE EQUATION D (p)=0 IS THE FUNDAMENTAL EQUATION OF THE SINGULARITY. 125. Consider the effect which the description of a closed contour, round the origin and lying wholly in the annulus, exercises upon this fundamental system. Let 'r=e2ip', "= e2p",...; ___ 390 FUNDAMENTAL EQUATION [125. and let y' denote, at the completion of the contour, the value of the integral which initially is y. We have Y11/ = 'yll, yi2' = 1'12 + a-1 OyL (log z + 27ri) = ac2,y, + 0'y12, y13I = a31Y1 + a(32y12 + 0/y13,..................,....................... y21 = 0y21, y2 = /321 11 + 0 y22, and so on. Hence the fundamental equation (Chap. II) is 0 = 0, where >= 0'-0, 0, 0,..., 0,... 21, 0-,. a1 Of'0 0 0,... a31i, a32, 06 -0,., 0 ' ***********************************.............,.......,................. 0, 0, 0,... 0'-, 0,,...,,.................................. - /2, 0 -, O,..., 0,......,.....,0,...,...,........,.......,,............. 0.....O................................., 0, 0, 0,..., I -,.................................. 0 O..., 0'21, 0 - 0,... =(0' - 0) (0" - "..., where r' is the number of integrals in the group belonging to the root p' of D (p) = 0 of multiplicity a-'; -" is the number in the group belonging to the root p"; and so on. Now it was proved that D I n D(p)= - -I sin {(p- p') r. But I _ (p + p,') ri (e2rip _ e2rip,') sin (p- ) - p,' +) r (e2= e2 ') = - (P+ P+P)i (0-_ ), if = e2?rip Hence 1 e D (p) = e (0- 0). (27wi)'4 a-I 125.] OF THE SINGULARITY Also (~ 117) Ep'= n (n- 1), so that e- rizPa = e - 2n (n - ) i; and therefore e- npwi D (p) = (2 )n (0 -0 (0 -As the quantity e-np7i has no zero for finite values of p, it thus appears that, so far as roots are concerned, D (p) = 0 and e = 0 are effectively the same equation, when the relation between p and 0 is taken into account. Also, so far as roots are concerned, f (p)=0 is effectively the same as D (p) 0; hence any one of the three equations =o0, D(p)=O, a(p)=O, may be used for the determination of 0 and the associated quantity p. It is known that 3 = 0 is an equation remaining invariantfive for all modifications of the fundamental system: and, for the form of equation adopted in ~ 114, the term in ) independent of 0 is equal to unity (~ 14). This property in the present case is verified by means of the values of the quantities 0', 0",...; for (- 0')' (- 0")0"... = (- 1)n e2ripa' = (_ l)n en (n- l)ri = (_ l)n. The remaining coefficients in (9 are known (~ 14) to be the invariants of the equation, whatever fundamental system be chosen. Replacing e by 12 (p) for purposes of this discussion, we have n( _ sin {(p - p) 7r... sin {(p - pn) 7r} sin {(p - p, ) }... in {(p - p,) 7r}' Now sin {(p - p') 7r} = ei(p) - 0' sin (p - pr )7r} 0 - 0r' where = e2 2rir, r = e 2ripr'; so that, as Spr = Spr', and therefore II.= 1 r'i- e2wiin(n-l) =, r=l r=l 392 FUNDAMENTAL [125. we have n( (0-01)(0-02)..(0-0) O"-... +(-1)n Hence, when i2 (p) is expanded in descending powers of 0, the term in 0~ is unity; and when it is expanded in ascending powers of 0, the term in 0~ is likewise unity. When the quantities 01, 02,..., 0 are unequal, then 12 (p) can be expressed in the form (P) + M 'fi n(p)=l+^ -:l 0 - 0a, On account of the character of 12 (p), when expanded in ascending powers of 0, we have Ma/ = 0, cr=l so that there are n-1 independent quantities Ma', and these are equivalent to the n-1 invariants. The equation may also be expressed in the form f(p)=+1 S M. cot {(p-p)7r}, cr=l where M,= 27riMa, and therefore n Ma = O. 7=1 Corresponding expansions occur in the case when equalities occur among the quantities pl, 2,..., pn. 126. The integrals, which have been obtained, are valid within the annulus represented by R!zl R'; the inner circle may enclose any number of singularities of the equation, and the outer circle may exclude any number of other singularities of the equation. But care must be exercised in particular cases. If for instance, the only singularity within the inner circle is the origin, and the integrals are regular in the vicinity of the origin, then in the expression of any integral, such as E a2tn zP+m, 126.] EQUATION 393 there can be only a finite number of terms with negative values of m: the method, which is based upon the supposed existence of an unlimited number of such terms, is no longer applicable. If the only singularity outside the outer circle is z = o, and if the integrals are regular in the vicinity of z = oo, then in the expression of any integral, such as amzp+m, there can be only a finite number of terms with positive values of m: the method again ceases to be applicable. In such cases, the best procedure is to construct a fundamental system which shall include the regular integrals: this is the customary procedure for, e.g., Bessel's equation, the integrals of which have z = o for an essential singularity and are regular near z = 0. The method, which uses infinite determinants, is best reserved for equations which have their integrals non-regular in the vicinity of every singularity: it is nugatory when applied to Bessel's equation. Ex. 1. Consider the equation dz2 3++ =o. It is clear that the point z=O0 is an essential singularity, there being no integral regular in its vicinity, when a is different from 0: and that z= 0 is likewise an essential singularity, when y is different from 0. We shall assume that both a and y are non-vanishing quantities. Let Z-x=(-), a= (ayl; the equation becomes dx +,3 + x- +, =0. With the notation of the preceding paragraphs, we have (A(p)= p (p - 1) + b = (p- P) (P - P2); Cm, IL =- C2, m- - 21 so that Cr, r-b, Cr,r+1a, Cr,r-l=a Cr,,=0, when u< r-1, and when p> r+1; also #r,r 1, rr-1= (p ),r +) 1, a whe (p + r)' 4'r,,, =O, when, < r - 1, and when / > r+ 1. 394 EXAMPLES [126. Hence the value of Q2 (p) is.............,........ o o...oo........................................................................... a a.o(p 2)_',0 O... ***'~', o, o, 0,o.... ( - 2)'' 5(p —)2)' a a cj7 ' (p- (p)1) a a...,0o. o, o,,( ), I,,, 0,o,... 0..., O O 0, 0, 0, )..~, o, o,,, o, o ~ ' 0 (p+2)' 1 (p +2)'........................................................................................................ The general investigation shews that, when pi and P2 are unequal (which will be assumed), Q (p) = 1 + M1r cot (p — pi) qr + M2rr cot (p - P2) r, with the condition Jl+ J12==0; that is, we have 1 (p)= 1 + rirf[cot {(p - P) rr} - cot {(p - P2) 7r}], where M is independent of p. Taking the determinantal form for Q (p), and expanding according to the law established in ~ 110, we have Q (p)= 1 +a2, +a4 + a66 +..., where odd powers of a do not occur because the combinations which they multiply all vanish. Also m= oo 1 1 -oo (p+m) c (p + m + l)' 7.=oo. p=o0 1 4=- -o 2 -=d so p=on. Hence we ha and so on. Hence we have M=a21V+a41+aol +..., and N2k is the coefficient of - (that is, the residue of p= pi) in M2k. P-Pi To find 12, we notice that the only terms in M2, which have p=pi for a pole, are those given by m =0, m= - 1, these being 1 1 / (p)> (p+l),/ (p - 1) / (p) 126.] EXAMPLES 395 Now (P) = (P - P) (P - P2); hence Pl-P2 (P l+l)+ <(l-) 1 1 -1 P1-P2 1 - (Pl - P2)2 4b(p-P2) Again, writing D (p+m)=4) (p+m)> (p+m+l), we have M4 (p+ m) (p+p)' Consider n=o 2 n=-oo (p+n)J it contains the terms n=oo 2 -o.4.. (p +n) ' which do not occur in /4; it contains terms oo 1 2 -_ + (p +n) (p+m+) ' which do not occur in M4; and it contains the terms 1 > ln+l 4 (p + m) ~ (p +p) twice over, once in the form 1 p>m+l 1 (p+m), (p +P)' and once in the form 1 n >p+l ( (P + m) F (p +p) Hence _ _oo 1^ n2=oo 2 oo +2 + 24, {n (p ) = J (p + n).... (p +In,) (p +nm+ 1) 4 so that '4 —'1 ('p -I" ~..) -~ M (p+ 1) — 2 i — 2 -o (p+'n)} -E (p+m)+ 4(p+m+l)' The first term on the right-hand side is 2 2 = 'V22 7r2 [COt (p - p) 7r - cot (p - p2) 7]; the residue of this function for p =p is = - A22 7- cot (P - P2) 7T - 7 cot {(p1-p2) 7} 1662(1 - 4b) 396 EXAMPLES [126. In the second term on the right-hand side, the residue for p = pi can arise only for the values n=O, n= - 1; thus it is 1 (pi) +," (pi-l) "-3 (Pi1) +2 ~43 (p1-1) _ 1 (+2p,-2p2)(2+pi -P2) 1 (1 -2p +2p2) (2 —P+P2) (P1- P2)3 (1 P1- P2)3 2 (Pi- P2)3- P1 + P2)3 1 - 6b + 4b2 8b3 (1 - 4b) (p - p2) after reduction. Similarly, from the third term, the residue is 3-8b 1-8b 8b2 (3 + 4b) (PI - P2) 8b2 (1 - 4b) (Pl- P2) Hence - cot {(Pi - P2) 7} 3- 8b - 52b2 + 16b3 " I4 16b2(1 - 4b) 8b3 (3 + 4b) (1 -4b)(p1 - 2)' after reduction. Other coefficients could be calculated in a similar manner: but it is clear that even N6 would involve considerable numerical calculations, and it is difficult to see how the general term could thus be obtained. But the method of approximation may be effective in particular applications. Thus, in Hill's discussion* of the motion of the lunar perigee, the convergence is very rapid; and comparatively few terms need be taken in order to obtain an approximation of advanced accuracy. When this is the case, the values of p' for the integrals are given by ~ (p)=0, that is, cos 2p7r = - cos {(Pi - P2) 7r} - 27rM sin {(Pl - P2) 7r}; and two irreducible values of p' chosen are to be such that Pl +P2 = Pl + P2 = 1 The expressions for the integrals are to be obtained. Denoting still by p either of the quantities Pt' and P2', the relations between the coefficients are ( a,-) a +, + ar+ a 0= q5 (p+r) d /(p+r) a+' and considering in particular the row 0, we know that the constants a are proportional to the minors of the constituents in that row in the determinant Q (p). Thus ao:()=aK: ( for all positive and negative values of K so that, if we take ao= 1, * See the memoir already quoted in ~ 109. 126.] EXAMPLES 397 we have (o) aK = O) and our solution is _00 for the effective expression of which it is sufficient to find the first minors, as the series is known to converge within the annulus. In order to obtain 0 from Q (p), we replace)' ( +1 0 iP - '1)' 4)'(p+l) by zeros; it will therefore be necessary to do this in the expanded form. We thus have I =l+a2MO,2+a4 Mo,4+ X where a2 a2 =a22+2a ((p) (p - 1) O(p- 1) (p) Similarly for Mo,4 from M4; and so on. In order to obtain ( ) from 2 (p), we replace a-) in the -1 column by unity; the quantities 1 and a ( ) in that column by zeros; and the q (P -2) a quantities 1 and ) in the 0 line by zero. We then easily find 0 - (p) -( O:_ - )q5 (p )a2M-l'2+a4 — l'1 + '" where a2 a2 a2 a2 = 02po, f + (P (p _ 1) q, (p - 2); and similarly for the others. In the same way, we have +) a+2M,2 +a4M, 4 +..., where a2,2 - a2A[ a2 a2H, 2 =a20, 2 +f (p + 1) oq (p + 2) and so for the others. 398 MODES OF CONSTRUCTING [126. Lastly, for negative values of p less than -1 and for positive values greater than +1, we have =a2Mp,,2+a4'+, 4..., where 1 1 P, 2 =o, 2 + + 2 2 (p+p) (p+p+l) + (p+p) (p+p-1); and so for the others. After the remarks made in relation to the formal development of Q2 (p), it is manifest that these expressions for the integrals are mainly useful for approximate numerical expansions: they cannot at present be held to constitute a complete formulation of the integrals. Ex. 2. In his classical memoir, already quoted, Hill considers the equation 1 d2w dt2- a + al cos 2t a2 cos 4t -..., i dt2 the coefficients al, a2,... being considerably smaller than a,(. The memoir will well repay perusal, both for the analysis (account being taken of the lacuna as to convergence supplied by Poincare), and for the numerical approximations. It will be noticed that the effectiveness of the method is largely influenced by the data as to the smallness of a,, a2,..., when compared with ao. Ex. 3. Discuss the equation in Ex. 1, when b —, so that pI=p2. Ex. 4. Given an infinite system of differential equations of the form dxm I-m=: a(, =,, (2e, = 2..., O ), dt n=l where the coefficients am, are regular functions of t within a region I t I R, such that la,,,, < SAn in this region, where Sm, An (for m, n=l,..., o ) are such that the series SAl + S2A2+...+ SnAn +... converges. Shew that, if a set of constants c1, c,... be chosen, so that the series clA1+-c2A2+... — +nAn,... converges absolutely, then a system of integrals of the equations is uniquely determined by the condition that Xm=cm, when t=O, for all values of m. (von Koch.) OTHER MODES OF CONSTRUCTING THE FUNDAMENTAL EQUATION FOR IRREGULAR INTEGRALS. 127. The preceding method, so far as it is completed, leads to the determination of the fundamental equation for a closed circuit round the origin, the circuit lying entirely in the annulus; 127.] THE FUNDAMENTAL EQUATION 399 and it leads also to the determination of the integrals. Other methods have been proposed by Fuchs*, Hamburgert, Poincare+, and Mittag-Leffler~, some of them referring solely to the construction of the fundamental equation. But all of them seem less direct than the preceding method, due to Hill and von Koch; and they are not less devoid of difficulties in the construction of the complete formal expression of the integrals. Ex. 1. A modification of Hamburger's method, applied to the equation dx2+ a+ b a\ = already discussed in Ex. 1, ~ 126, may give some indication of his process. Changing the variable from x to t, where =eit we have the equation in the form d2y dy - dy(b + 2a cost), dt'2 C = or writing ye- 2it the equationll for Y is d2 JY dt =Y(c+2a cos t), where c= b -. Let x describe a circle round the origin, say of radius unity; then on the completion of the circle, t has increased its value by 2rr. Let y=f(x), y=g (x) be two linearly independent integrals; and when x describes its circle, let these become [f(x)], [g(x)], respectively, so that we have [f (x)] = allf (x) + a12 (x), [g (x)] = a2 f (X)+ a22g (x). The fundamental equation for the circuit is all-o), a12 0, a21, a22 - o) * Crelle, t. LXXV (1873), pp. 177-223. I Crelle, t. LXXXIII (1877), pp. 185-209. In connection with this memoir, reference should be made to two papers by Giinther, Crelle, t. cvi (1890), pp. 330 -336, ib., t. cvII (1891), pp. 298-318. + Acta Math., t. IV (1884), pp. 201-312. In connection with this memoir, reference should be made to Vogt, Ann. de l'Ec. Norm., Ser. 3e, t. VI (1889), Suppl., pp. 3-71. ~ Acta Math., t. xv (1891), pp. 1-32. II In this form, it is a special case of Hill's equation: see Ex. 2, ~ 126. 400 EXAMPLE [127. that is, by Poincare's theorem (~ 14), 02 - (a11 + a22) w + 1 = 0,) so that ajj+a22 is the one invariant for the circuit. Let F(t) =e tf(eit), C (t)=e- -ig (e't) then F~(t + 2 7,) = - a,1 F (t) a,2 G (0)~ G(t+2v)= -a22 F(t)-a2 GC (t) The fundamental equation is independent of the choice of the linearly independent system, and it is unchanged when any particular selection is made. Accordingly, let the integrals be chosen so that F(t)=1, F' (t)=0, G(t)=0, G' (t)=1, when t=0; then, using the foregoing equations, we have F(27r)= -al, CG'(27r)= -a22; and therefore a,, 1+a= a F (27r) - G'(27T), which accordingly gives the value of the invariant, when the values of F(27r) and G' (27r) are known. To obtain these, let qt=sin2It, so that u increases from 0 to 1, as t increases from 0 to 27r. The equation becomes d2 Y dYY qt (I- U) +I (I- 2u 4 Y(c + 2a - 16au + 16aU2) -T \ -ff du and this remains unaltered when we change u into 1-u. Two linearly independent integrals, constituting a fundamental system in the vicinity of U =0, are given by Y,, t" Y2= YI C00+15+ a n=0 n=O where a =1, co=I; also an is the value of b, when p=O, and c, is the value of bn when p =, the quantities bn being given by the equations 60=1, (p+ 1) (p +) b1= p2 + 4c + 8a, (p + 2) (p + ) b2 = {(p+ 1)+ 4c+8a b - 64a, and, for values of n >3, (n + p) (n + p - 1) b, = {(n + p - 1)2 + 4c + 8a} bn-1 - 64ab,,2 + 64ab,,3. Similarly, a fundamental system in the vicinity of u= 1 is given by LZ, = 2 an (I -U)n, Z2E 2 Cn(lcZ ) n + n=O ~~~n=Q 127.] EXAMPLE 401 The integral F(t), defined by the initial conditions F t)1, F' (t) =O, when t=O, is given by F (t) = Y1. The integral G (t), defined by the initial conditions G t)=, G' (t) =1, when t= 0, is given by G ()4Y2. To obtain expressions for F(27w), G' (27w), consider values of u, which lie in the vicinity of u = 1 and are less than 1. By the ordinary theory of linear equations, we have Y1 = A Z +BZ2, Y2=0CZ, + DZ2. First, let U=~ so that I1- u=~1; then we have F(7r)=AF(r) +'BG (7r), G (r) =4CF(v)+-DG (r). Next, differentiate with regard to t and then take u I~ 1- u we have Moreover, F (t) G' (t) -F' (t) G (t)=constant by taking the initial values; hence F (7) G' (7) -F' (g) G (7>r These relations give A =F (7) G' (w) +F'(7r) (7)= -D, B- 2F(r) F' (w7), 4C=2 (7r)C' (7r). Now G (2w7 - r) =4 Y2 (2w7 - T) {4 CZ, (r) + DZ2T), so that - G'(2r) =4[ 1dr df2] -D, and F (2 r-,r) =Y1 (2 7r-T) =AZ, (r) + BZ2 () so that F(2r)=~A. Hence F (2r) +CG' (2r) = A - D = 2F (w) G' (r) + 2F' (7) G (7). P. IV. 26 402 EXAMPLES [127. Now, when t is 7r, the value of u is -, so that oo an 00 na F (7,)=2-, G (or)=2 2 i (n+ )c. n-O 2n+ 1 and therefore a +a22= - F(27r)- G' (27r) I 9 an - (M+)Nm, na " CM, = L[n==o 2n n-0 2n +1 + nO 2 +1 m] 21 E: (inm+ n+ ) Anc,. - n=O m=O 2m+n+l which is the invariant of the fundamental equation. This gives a formal expression, the only operations required being in the direct construction of an and cm, and no one of these operations is inverse; but the result is less suited to numerical approximation than is the method of infinite determinants in the case when a is small. We shall return later (~~ 137 —139) to a different discussion of this equation. Ex. 2. Apply the preceding method to Hill's equation 1 d2hw t -- a= al cos 2t a2 cos 4t -..., in the case when al, a,... are not small compared with ao. Ex. 3. Discuss, in the same manner as in Ex. 1, the equation dZ3 ( 3 y\ dq+( a++ sy=0o. In particular, obtain expressions for the invariants of the fundamental equation for z=0. CHAPTER IX. EQUATIONS WITH UNIFORM PERIODIC COEFFICIENTS. 128. ALL the equations which hitherto have been considered have had uniform functions of the variable for the coefficients of the derivatives; and the only particular class of uniform functions, that has been specially adopted with a view to detailed discussion of the properties of the equation, is constituted by those which are rational. Many of the properties, however, which have been established in the preceding chapters, hold for uniform functions whose form, in the vicinity of a singularity, is similar to that of a rational function when expressed as a power-series in such a vicinity. Among the classes of uniform functions, other than rational functions, there are two characterised by a set of specific properties: viz. simply-periodic functions, and doubly-periodic functions; and accordingly, it seems desirable to consider equations having coefficients of this type. The present chapter will be devoted to the discussion of equations the coefficients in which are uniform periodic functions. EQUATIONS WITH SIMPLY-PERIODIC COEFFICIENTS. We begin with the case in which the coefficients have only a single period; and we take the equation in the form d"w dm-lw dm +P dm- + + pw = where pi,..., pm are uniform functions of z, are periodic in o, and have no essential singularity for finite values of z. Let a 26-2 404 4EQUATIONS HAVING [128. fundamental system of integrals in the domain of any point be denoted by fi (2), f (0, ~.., fm (z, which therefore are linearly independent. A change of z into z + o leaves the differential equation unaltered: hence i (Z + o), f2(Z + W),..., f (Z + ) are integrals of the equation. That they are linearly independent, and therefore constitute a fundamental system (it may be in a new domain), is easily seen; for SCrfr ( + 0) satisfies the equation for all values of z, and by making z pass from any position Z + o to Z without meeting any singularity, the integral changes from Scrfr (Z + w) to crfr.(Z). If, then, values of c could be found such that the equation OCrfr (z + 0) = 0 is satisfied identically (and not merely for special zeros of the function on the left-hand side), then we should have WCrfr (Z)= 0, also identically. The latter is impossible, because the integrals f (z),..., fm (z) constitute a fundamental system; and therefore the former is impossible. Thus fi (z + w),..., fm (z + Cw) constitute a fundamental system. Suppose now that the domain, in which the original fundamental system exists, and the domain, in which the deduced fundamental system exists, have some region in common that is not infinitesimal; and consider the integrals within this common region. As f, (z + o),..., fm (z + co) are integrals, and as f (z),..., f,(z) are a fundamental system, we have equations of the form fi (z + o) = a, fi(z) +... + a,, fm (z))................................................, fm (z + C) = a,,,f (z)+... + a,,mf (z) where the coefficients a are constants; their determinant is not zero, because the set of integrals on the left-hand side constitutes a fundamental system. UNIFORM PERIODIC COEFFICIENTS 405 Consider any other integral in this region: it is of the form F(z) = lif0 (z) + c2 f2 (z) +... + KCmfm (Z), where cK, K2,...> /m are constants; and so m m m F(z + o)=X an,, j (z) + S aCr,f2 (Z ) +... + E a, r,?fm (z). r=l r=l r=l In order that F (z) may be characterised by the property F (z+ co)= OF(z), where 0 is some constant, the coefficients K must be chosen so that n E arpKr= 0Kp, (p 1 2,...) M), r=l a set of n equations linear and homogeneous in the coefficients K; and therefore 0 must satisfy the equation A (0) = a,- 0, a1,..., am =0, a21, a22-0,..., a2m........,..................... am2, am2, *, mm- 0 an equation involving the coefficients a, and so apparently depending upon the choice of the fundamental system fi,..., fm. But, as with the corresponding equation for a set of integrals near a singularity (~ 14), we prove that this equation is independent of the choice of the fundamental system, so that the coefficients of the powers of 0 are invariants. The proof follows the lines of Hamburger's proof for the earlier proposition. Let another fundamental system g, (z),..., g (z), existing in the region under consideration, be such that gr (Z + o) = b gi (z) +... + b..gmm (z), (r = 1,..., m), the determinant of the coefficients b being different from zero. The equation, to be satisfied by the multiplier 0 of F(z), is B()= bl, bl2,., bim =0. b2, b22-, b,..... b 2....,.......m - bm,, bM2,..., bmm - 0 406 406 ~~FUNDAMENTAL EQUATION[18 [128. As the integrals f are a fundamental system in the region, in which the integrals g exist, we have gs )csifAi(Z) +..+ csmfm (Z), (S=1.,m), where the determinant of the coefficients C~t, say 0, is not zero. Thus, as bagI~ (z) +... + brmgm = gr (z + W) =Crij; (Z +co) +. +Crmfrn(Z + c), we have t==4 __s=1 t=1 This homogeneous linear relation among the linearly independent integrals f must be an identity; and therefore I brs Cst = I?s s 8=1 s=1 = art, say. Then CB(0)= ct21-c,10, af12- C120,..=A (0)0C, a(21 - C21 0, a22 - c2260, so that, as C is not zero, we have B (0) =A (0), and the equation is invariantive. We therefore call it the fundamental equation for the period w. Let A (z) denote the determinant A(Z) = dm-1f, dm'lf2 d mlf dzm-1 dzm-1 'dzmdf 1 df2 df m dz dz dz A,.. fin then, as in ~9, we have JPI (x) dx Hence A(z + ()= (zo)e J~ P (x) dx 128.] FOR THE PERIOD 407 so that [s+@ A (Z + w) Pi (x) dx A (Z) where we may assume the integration to take place along a path that does not approach infinitesimally near the singularities of p, if any. Now, as pi is a uniform function, simply-periodic in o, it is known* that pi is expressible in the form a= oo 2rrzi pi(Z)= A,e, -oo within such a region as encloses the path of integration; and the series is a converging series. But rz+ 27rxia e X dx=0, if the integer a is distinct from zero; hence A (z + ) eA A (Z) But, substituting in A (z + o) the expressions for fi (z + c),..., fm(z + o) and their derivatives, in terms of f(z),..., fm(z) and their derivatives, we have A (Z + ) A(z) Z/\ -^all. 12;, aimX a21, a22, a.., m am,i am 2 m.., a mm which is the non-vanishing constant term in A (0); and thus A (0) = eco +... + (- 1)( 0. In particular, when p, is zero, so that the differential equation dm-lw contains no term in dz_, we have A, = 0; and then A () = 1 +... + (- 1)mom. 129. The generic character of the integrals depends upon the nature of the roots of the fundamental equation. * T. F., ~ 112. 408 ROOTS OF THE [129. If the m roots of the fundamental equation are different from one another, and if they are denoted by 01, 0,, 0..,, then a fundamental system of integrals exists, such that F, (z + ow) = eF, (z) = 1,...,m ). Consider any simple root 0r of the equation A (0) = 0. Then not all the minors of A (0) of the first order can vanish for = 0r; hence m - 1 of the equations S aspKs.= OCp, (p= 1, 2,..., m), s=1 determine ratios of the m quantities K, and consequently determine a function F, (z) having a multiplier 0O. This holds for each of the m different roots: and thus m different functions F(z) are determined. These m functions are linearly independent of one another. If there were an equation 71 F(z+ () + 72 () +... + 7mFm () = 0, which is satisfied identically, then also ty1F, (z o+ ) + 72F2 (z + 0) +... + 7mFm (z- + ()= 0, that is, O1'YI'F (z) + 22 () +... + Ommm (z) = 0. Similarly, O2lFl () + o22yF2 ) +... + Om2,mFm (z) = 0; and so on, up to 01 -lFyll (z) + 02 r-ly (z) +... + mm-1,yFm (z) = 0. Now the determinant!010, 021, 3,..., 7, -l does not vanish, because the quantities 0 are unequal: hence 71F (z)=0, 37F (z)=0,..., 7mFn (Z) = 0, so that the constants 7 all vanish. The m functions F therefore constitute a fundamental system. 130. Next, let ~ be a root of A (0)=0 of multiplicity 1, where / > 1. The equations m 2 aspcs= 0, (p= 1...), s=1 130.] FUNDAMENTAL EQUATION 409 are consistent with one another, though not necessarily independent of one another: any m-1 of them are satisfied by ratios of the quantities Kc, which are finite and may contain arbitrary elements. Giving any particular values to the last, we have an integral, say <>I (z), defined by means of these quantities: it is such that,1 (Z + a) =, (Z), and it is a linear combination of f (z),..., fm (). Taking any one of the integrals which occur in the expression of this linear combination, say f, (z), we modify the fundamental system so as to replace f, (z) by c>P (z). Let the equations for the increase of the argument by o in the modified fundamental system be ~1 (Z + a) = rQ1 (z), f. (Z + 0)) = Cr? (z) + Cr2f2 () +... + Crm fm (), ( = 2,..., ) then the fundamental equation is —, 0, 0,... =0, C21, C22 -0, C23, *. C31, C32, C33-0 *..,.... ***,,..o,.. o............ which, owing to its invariantive character, is A (0) = 0, and therefore has?- for a root of multiplicity p. Consequently, the equation A1(0) = 22 - 0, 23,... =0 C32, C33 -....................... has ^z for a root of multiplicity / - 1; and therefore the equations (C22- I 2 + c23 IC3 +... + C2nm = 0,......................,............................ Cm2 k~2' 3 Cm3 K3' + - *. + (Cmm - ) Kim = 0, are consistent with one another, and any m - 2 are satisfied by ratios of the quantities K', which are finite and may contain arbitrary elements. Giving any particular values to the latter, and writing 4'2 (z) = K2 f (z) + K3 3 (Z) +... + Km'fm (z), we have (2 (z + )o) = X211 (Z) + 'I2 (z), 410 FUNDAMENTAL SYSTEM [130. where m X21= Kr Cl CrI, r=2 so that X\2 is a constant, which may be zero. The quantity 4)> (z) is an integral of the differential equation: we use it to replace some one of the integrals in its expression, say f2(z), in the fundamental system, so that the latter then is constituted by "DI (Z), ) (Z), f3 (Z),.(z, f. M (z). Proceeding similarly from stage to stage, we infer that, associated with a root S- of multiplicity /t of the fundamental equation, there exists a set of pu integrals such that pl (Z + a)= q$~ (Z), 42 (z + ) = X,211 (z) + 12~ (z), )3 (z + 3) = X31 I (z) + X323 () + 1) (Z),..................o.................... o........ o. 4, (Z + w) = X.111 (z) + X,.2 (z) +... + X,- 1 (_-1 (z) + j.~ (a), where the coefficients X are constants. Similarly, if the roots of the equation A (0)= 0 are i-,,..., '5 of multiplicities )p,...,,u respectively, so that pA +... + -n = m, the fundamental system can be chosen so that it arranges itself in n sets, each set being associated with one root of the fundamental equation and having properties of the same nature as the set associated with the preceding root of multiplicity W. A function, characterised by the property F(z+ o)=F(z), is strictly periodic, and sometimes it is said to be periodic of the first kind. A function, characterised by the property F (z + ) = OF (z), where 0 is a constant different from unity, is pseudo-periodic, and sometimes it is said to be periodic of the second kind, 0 being called its multiplier. A function, characterised by the property F (z + o) = exZ+~ F (z), where X and p are constants, is also pseudo-periodic, and sometimes it is said to be periodic of the third kind. 130.] OF INTEGRALS 411 With these definitions, the preceding result can be enunciated as follows*:A linear differential equation, the coefficients of which are simply-periodic in a period c, possesses integrals which are periodic of the second kind: and the number of such integrals is at least as great as the number of distinct roots of the fundamental equation for the period. Ex. 1. Prove that, if the equation d2w dw+ d- +PI (Z) dz +P2 (Z) W =0 possesses an integral which is periodic of the third kind with a multiplier e~z+I, then i1 (2 z+ ) =p1 () - 2X, P2 (z+o-0) =-P2 (Z)- Xl (z)+X2. Hence integrate the equation d2 8rr2z dw 167r4 dz2 ao2 dz + 2 shewing that Xco=47r2. (Craig.) Ex. 2. Shew that, if the coefficients in the equation d2w dw dz2+P1 (z) d +P2 (z) w= have the form 2Xz i (Z) = (Z) + —, Xz X2 P2 (Z) = + (z+) + c +Z2 where ( and + are periodic of the first kind, then the equation certainly possesses one integral that is periodic of the third kind. (Craig.) 131. On the basis of these properties, we can take one step towards the analytical expression of the integrals. The integral (<P (z) is a periodic function of the second kind. As regards the integral 2 (z), we have (D2 (Z + O) _>2 (z) X21 so that ()2 (Z + -() _ 21 (Z +.) 2 (Z) X21 Z (le, n' xI 13..21 1 (Z + W)) S 0 1 (z) rw * Floquet, Ann. de 'VEc. Norm., Ser. 2e, t. xII (1883), p. 55. 412 PERIODIC INTEGRALS [131. so that the function on the right-hand side is a periodic function of the first kind, say * (z). Therefore (I? (Z) = (D21 (Z) + z(II (Z), where I',, (z) is a constant multiple of 4, (z), and the constant factor may be zero; and (21 (z), = * (z) (I (z), is a periodic function of the second kind, with the same multiplier as 1, (z). As regards the integral I1, (z), we have 43(Z w) 4?3(Z) X32 4)2 (Z) X31 (Di (z + q0, 5 P' D,(zj lD3 (z) X32 (X21 31 1(z) (A) Now, if (& 3 (z) X32 ____ z2 + X32.X21 - 21hAX31 IB (z) - )- Z*j (z) 2 2 C _ =, (Z) 2 c2o A) we have 0 (Z + (0) = 0 (Z), so that 0 (z) is periodic of the first kind. Hence (I3 (z) = 441 (z) + Z132 (z) + Z24331 where lD.1 (z) = 0 (z) I, (z), and therefore is a periodic function of the second kind with the same multiplier as 1,; where (132 (z) is a linear combination of (')21 (z) and (i (z), and thus is periodic of the second kind with the same multiplier as 'i, (z); and D,, (z) is a constant multiple of (I (z), in which the constant factor, viz. XC32-X21 may be zero, and certainly is zero if P22 (z) disappears from I2 (z) owing to the vanishing of its constant factor. Proceeding in this way stage by stage, we obtain expressions for the integrals in succession; and we find (Ic (Z) = 4I1ri (z) + ZFD, (z) ~ Z2 (D'.3 (Z) +... + Zr11lcJr (Z), where (1?rr (Z) - (1) Xrr —i Xr-1, r2... 3221( so that it is a constant multiple of (, (z), the constant factor being capable of vanishing; and all the functions rI~, (Z), (Iy (z),... ) (Di, I.(z) are periodic functions of the second kind with the OF THE SECOND KIND 413 same multiplier as >i (z), and are expressible as linear combinations of 1, 4>21),,31,..., Pr-,l. This holds for the values r = 1, 2,..., is. Similarly for any other set of integrals, associated with any multiple root of the fundamental equation of the period. It may, however, happen that some one of the coefficients X, s-l vanishes, so that, for all values of r > s, the term in iD, (z) disappears. The alternative result is that a linear combination of the functions Q>s (z), (s-i (z),..., ), (z) can be constructed which is periodic of the second kind. This linear combination can be used to replace 4s (z), and thus may be the initial member of another set of integrals in the group associated with the multiplier W. The proof of this statement is simple. Assume that Xs,s-1 vanishes, and that no one of the coefficients Xr,r- for values of r ~ s vanishes; and construct the linear combination Cs (s (Z + )) + _ss-1_S-1 (Z + )) +... + 4 l242 (Z + )), choosing the coefficients K so that the term in >D, disappears and that the remaining terms are {r {KsDc (Z) + I1 ( + S- - ( ). + K2 + 2 (Z)}. To satisfy these conditions, we must have 0 = KSX\S + cK8 —1,1 +... + K4X41 331 + 23X3 + 221, 0 = KCSX82 + KSc-1XS-1,2 +... + /K442 + K3X,32, 0 = Ks X83 + KS-1X5-1,3 4+... + K4X43,.................................... o = KcsX,S-2 + KS-1. —1,S-2-.Transfer the terms in KS to the left-hand side: the determinant of the coefficients K on the remaining right-hand side is + XS-1, S-2XS-2,S-3... 43X32X21 which by the initial hypothesis does not vanish. Some of the coefficients Xs,, S2,..., X,,s-2 are different from zero, for (Ds (z) is not a periodic function of the second kind; hence there are finite non-zero values for the ratios of KS-1,..., K2 to K. When these values are inserted, let s (Z) = csq z (Z) +... 4+ fK2c (); then s (z + w) = w3s (), 414 SETS OF [131. so that T (z) is periodic of the second kind, with the common multiplier 5-; it can replace b>s (z) in the fundamental system, and then can, like (D (z), be the initial member of another set within the group of the same type as (< (z),..., <Q-i (z). The statement is thus proved. 132. Any set, such as (, (z),..., (<-_ (z) in the preceding group of integrals, whether s =, or be less than /L, can be replaced by an equivalent set of simpler form. Let the equation be written O=Pw (= + P d +... pm) = (Dm + plDm-l +... + pm) W, so that P = Dm + pDm-l +... + pml D + p. Also let P1 = mDm- + (m - 1) pDm-2 +... +pm-1 AD ' a2P P2 =aD and, generally, let asP S M.DS Let the integral of the set containing the highest power of z, say zr-l, be expressed in the form W = ZrI lO1 + (r-1 ) Zr-2 02 + + (r-1 ) ~ + (r- 1)! 1 +,, the binomial factors being inserted for simplicity. Then, as P ( vK) = zKP () + + KZK-P (#) + KC (- 1) K-2P (*) +..., we have o= P(w) =- z- P (s/) + (r -1) zr-2 (sb) + ~(r - 1) (r - 2) zr-3P ( 4) +... + (r - 1) {r-2P (k2) + (r - 2) zr-3P, (2) +.. + (r -1)(r- 2) {zr-P (,)+ + +..................... INTEGRALS 415 which can be satisfied identically, only if 0=P(,b1), 0 = P (,2) + P (,0), 0 = P (0) + 2P1 (0) + P2 (01),.................,.......,.0 The first of these conditions shews that w = )1 is an integral of the equation. The second shews that W = 02 + 201 is an integral; the third that w = b3 + 2zc2 + Z201 is an integral. And generally, if w denote (r- 1)!,'-~,/, +(r-l),'-, +...+(_ 1)! (-+'...+ (r - l) 'r-, +,,r, o being an integral of the equation, then each of the quantities O1 dW 2! (r- 3)! a2 (_ - 1) I (r - Ar) a-UT w r-1 a~' (r- l)! a2 *' (r-l)! ar- ' is an integral of the equation, when 5 is replaced by z after differentiation. Accordingly, the group of r integrals in the set are linearly equivalent to Itl = 01) 2 = 02 + Z1, T3 = (3+ 2zb2 + '201, U4 = 04 + 3zb3 + 3Z202 + Z301,..............................,. = r + (r - 1) Z,_r- +... + (r - 1) zr-202 + Zr-, and any linear combination of these is an integral of the differential equation; all the quantities b which occur in them are periodic of the second kind, having the same multiplier. Similarly for any other set; and thus the m integrals of the equation will be constituted by sets of r1, r2,..., rn integrals of the preceding types, where r + r2 +... + rn = m, and the system contains n periodic functions of the second kind. 416 GROUP OF INTEGRALS ASSOCIATED [133. GROUP OF INTEGRALS ASSOCIATED WITH A MULTIPLE ROOT OF THE FUNDAMENTAL EQUATION OF THE PERIOD. 133. These results can also be obtained by using the properties of the elementary divisors of the quantity A (0), when it is expressed in its determinantal form. Let the elementary divisors associated with the root '- be (0 _-)-, (0- _)1,2*..., (0 - ~)T-2 A — (0_- )-, so that, as in ~ 15, the highest power of 0- i common to all the first minors of A (0) is (0-?)1, the highest power common to all the second minors of A (0) is ( 0- )2, and so on; and the minors of order T (and therefore of degree m - in the coefficients) of A (0) are the earliest in successively increasing orders not to vanish simultaneously when 0 $=. As in the earlier case discussed in ~ 15, 16, we have - x - 4 - p- 2 ' p2 - p-3 > *.. >T -1. Proceeding on lines precisely similar to those followed in ~ 23 for the arrangement, in sub-groups, of the group of integrals associated with a multiple root of the fundamental equation belonging to the singularity, we obtain a corresponding result in the present case, as follows:The group of p integrals associated with the root - of multiplicity u, belonging to the fundamental equation for the period co, can be arranged in T sub-groups, where 7 is the number of elementary divisors of A (0) which are powers of 0 - S-. If the X members of any one of these sub-groups be denoted by g (z), g (z),... g, g(z), these integrals of the differential equation satisfy the characteristic equations gl (z + w) -= rg (z) g2 (Z + 0) = Sg2 (z) + gl (z) g3 (Z + (c) = Sg3 (Z) + g2 (Z) gA (z + c) = ~gA (z) + gA-1 (Z) Taking all these sub-groups together, the number of first equations which occur in them is equal to the number of the sub-groups, that is, the number of the elementary divisors of A (0) connected 133.] WITH A MULTIPLE ROOT 417 with 0-; the number of second equations which occur is the same as the number of those indices of the elementary divisors connected with 0- - that are not less than 2; the number of third equations is the same as the number of those indices that are not less than 3; and so on, the number of equations in the first sub-group being x- r1. The analogy with the Hamburger sub-groups in Chapter II is complete. COROLLARY. The total number of integrals of the second kind, defined as satisfying a relation of the form g (z + w) = Og (z), where 0 is a constant, is the total number of elementary divisors of A (0) associated with all the roots of A (0) = 0; a theorem more exact than Floquet's (~ 130). For the total number of such integrals, in the group associated with a multiple root of A (0)= 0, is equal to the number of elementary divisors of A (0) associated with that root: and the total number of groups is equal to the number of distinct roots of A (0)= 0. 134. Some approach to the analytical expressions of the functions, satisfying the equations characteristic of the sub-group, can be made, as in ~ 23. Let and introduce a difference-symbol V, such that* VF (z) = F (z + co)-F (z) for any function F; also let G(z)= Xx + l Xx- +( 2 1) '-2+.....+ (X 1 ) - + 'X1, where the functions Xi, X2,..., X are periodic functions of z, with a period co, and A(-1) (X-1)! r r! (x-1-r)! * For these difference-symbols in general, see a memoir by Casorati, Ann. di Mat., Ser. 2a, t. x (1882), pp. 10-45. P. IV. 27 4~18 GROUP OF INTEGRALS ASSOCIATED [134. Then if we take z hx-n 12)= n V,,1VG W,) for all values of n, we have z hx-, (z + )) = ~rW ~3rn1l"n (z + W) z = 13rw Yn+'Vin {VF (z) + F(z)} - hX-n-I (z) + '3rhkn (Z), holding for all values of n. These are the characteristic equations of the sub-group; and we therefore can write z gX-n (Z) = 1 ';7~1Vr G (z), with the above notations, for n - 0, 1,..., X - 1. These X integrals are a linearly independent set out of the fundamental system; the systemn will remain fundamental, if gw g2,.., gx are replaced by X other functions, linearly equivalent to them and linearly independent of one another. This modification can be effected in the same way as the corresponding modification was effected in ~ 24, viz, by introducing a set of functions, associated with G and defined by the relations GI (z) = XI, G3(z)= X2 + Xl~, G, (z) - X, + 2X,~ + X1~ 2 G= Gx(z)=Xx+(X -1)x-l+ + ( X2 /x 2+ XIWx the functions X being periodic functions of z, with the period o. Constructing the expressions VG, V2G,..., Vx-1G, we find VG = cl,/J~ — + C~1, fG2 + + C1,,k-2G2 + C1, -1G1, V2G = C2,1GAx2 + C2,2GX-3 ~.+. + C2X2GI, '7X-2 G= cx- 2,1G2 + CIX-2,2G1, Vx-1= cx-,, IGI, where the constants c are non-vanishing numbers, the exact values of which are not needed for the present purpose. 134.] WITH A MULTIPLE ROOT 419 It follows, from the last of the equations, that G, is a constant z multiple of VA-1G, and therefore that S G1 is a constant multiple z of g# (z); we replace g, (z) in the fundamental system by 1"w G1. It follows, from the last two of the equations, that G2 is a linear z combination of V^-2G and V^-1G, and therefore that 0G2 is a linear combination of g2 (z) and g, (z). As g1 (z) has been replaced z in the fundamental system, we now replace g2 (z) by w G2; and the system remains fundamental. And so on, for the integrals in succession. Proceeding thus, we obtain X integrals of the form z z z WU G, (Z), G2 (Z), (..., w GA (Z). Further, these integrals are linearly independent, and so they are linearly equivalent to g1 (z), g2 (z),..., gx (z). For if any relation, linear and homogeneous among these quantities, were to exist with non-vanishing coefficients, we should, on substitution for G,1 G2..., G. in terms of G, VG, V2G,..., VX-1G, obtain a relation, linear and homogeneous among the quantities g (z),..., go (z) with non-vanishing coefficients. Such a relation does not exist. Accordingly, the X integrals z z z GI (Z), c2 (Z),..., GW GA (z) can be taken as constituting the required sub-group of integrals. We now are in a position to enunciate the following result, defining the group of integrals associated with a multiple root? of the fundamental equation of the period:When a root? of the fundamental equation A (0)= 0 is of multiplicity A, there is a group of p, integrals associated with that root; the group can be arranged in a number of sub-groups, their number being equal to the number of elementary divisors of A (0) which are powers of?- 0; the number of integrals in the first sub-group is equal to the number of those elementary divisors; the number in the second sub-group is equal to the number of the exponents of those divisors which are equal to or greater than 2; the number in the third sub-group is equal to the number of the 27-2 420 GROUP OF INTEGRALS [134. exponents of those divisors which are equal to or greater than 3; and a sub-group, which contains X integrals, is equivalent to the X linearly independent quantities z z z w G, (Z G (z),..., W G (Z), where '" + ( 2 r(^~) = %r~+ (r 7 l~?( (r3 1) X 2-~ + X.'.., for r= 1, 2,..., X: the quantities Xi,..., Xx are periodic functions of z, but they are not necessarily uniform: ~ denotes Z, and (r -r-2~ ~ ~ ~ ~() </r- t! (r-1 -c)! NOTE. By taking Xn = 0-"bn, for n= 1,..., X, and writing orGr (z) = Gr (z), the integrals become z z z rw G, (z), w G,2 (z),..., G (z), where Gr (Z = Or + (1 )r- + r-22 +. \2 *+ (r1) 2 zr-2 + 'l1r —l the functions Q having the same character as the functions X. 135. There is a theorem of the nature of a converse to the foregoing proposition, which is analogous to Fuchs's theorem proved in ~ 25-28. The theorem, which manifestly is important as regards the reducibility of a given equation, is as follows:If an expression for a quantity u is given in the form z u = S ( + (b-1, + fn-z22 +... + 42 n -2 + (ln-1}, where? is a constant, all the functions 01,..., /)n are periodic in w, and ' denotes -, then u satisfies a homogeneous linear differential 135.] CONSTRUCTION OF UNIFORM INTE(:RALS 421 equation of order n, the coefficients of which are uniform periodic functions of z, having the period co; moreover, atu u 2 an-1u as I ) **" an-1 yb., ~,' '-l are integrals of the same equation and, taken together with u, they constitute a fundamental system for the equation of order n. The course of the proof is so similar to the proof of the corresponding theorem as established in ~~ 26-28 that it need not be set out here*. It can be divided into three sections; in the first, it is proved that a,..., _a satisfy such an equation, if u satisfies it; in the second, it is proved that these must form a fundamental system, for no homogeneous linear relation with nonevanescent coefficients can exist among them; in the third, it is shewn that the linear equation, which has these quantities for its fundamental system, has uniform periodic functions of z with period o for its coefficients. The details of the proof are left to the student. MODE OF OBTAINING INTEGRALS THAT ARE UNIFORM. 136. The further determination of the analytical expressions of the integrals, on the basis of the properties already established, is not possible in the general case. Thus the functions Xi,..., XX occurring in the sub-group specially considered in ~ 134, are periodic functions of the second kind with a multiplier -. If we take new functions 1r (z),..., f, (z), such that xr (Z) = ~ % (Z) = elog ( (z(r), these new functions are periodic of the first kind. But further properties of the functions must be given if there is to be any further determination of their form. When we limit ourselves to the consideration of those equations whose integrals are uniform functions, (criteria are determined * Some of the analysis of ~ 132 is useful in establishing the theorem. 422 EQUATIONS HAVING [136. independently by considering the integrals in the vicinity of the singularities), some further progress can be made; but, of course, the assumption that the integrals are of this character must be justified by appropriate limitations upon the forms of the coefficients p,,..., pm in the original differential equation. In such cases, every quantity such as *,(z) is a uniform simply-periodic function of the first kind; it can therefore* be expressed in the form of a Laurent-Fourier series such as -00 2riKz A,e. K= -00 Such a form of expression does not lead, however, towards the determination of the criteria for securing such a result or any other result of a corresponding kind for any other assumption. In particular examples, we adopt a different method of practical procedure. In order to determine some of the functional properties of the integrals, it frequently is expedient to change the variable so that, if possible, the transformed equation belongs to one or other of the classes of equations considered in preceding chapters. Thus if the coefficients pi,..., p, which are uniform periodic 2rzr 27rz functions of z, occur as rational functions of sin -- and cos then, introducing a new variable t, where 27rzi t=e, we obtain a linear equation, the coefficients of which are rational functions of t. Some characteristic properties of the integrals of the equation in the latter form can be obtained by earlier processes; it may even be possible to determine the fundamental system of integrals. The preceding transformation is, however, not the only one that can be used with advantage; and it often happens that the special form of a particular equation suggests a special transformation which is effective. In particular, if the coefficients in the equation are alternately odd and even functions, * T. F., ~ 112. 136.] SIMPLY-PERIODIC COEFFICIENTS 423 such that pi, ps, p5,... are odd, and p,, p4,... are even, then we may take 7rz. 7Z t =cos -, or sin -, as a new independent variable: it is easy to prove that the transformed equation has uniform functions of t for its coefficients. Also, some indication is occasionally given as to a choice between these two transformations; for example, if an irreducible pole of the original equation is z = 0, we should choose rz t = sin(.) as the transformation, and consider the integrals in the vicinity of t = 0; whereas the other would be chosen, if an irreducible pole of the equation is z = co. Another transformation, that sometimes can prove effective, is rX% t = tan; any uniform function of z, periodic in c, can be expressed as a uniform function of t; and the differential equation is transformed into one which has uniform functions of t for its coefficients. Ex. 1. Consider the equation d2qw dw Z d- +2a- cot z + (b+ c cot2z) w=, where a, b, c are constants. Writing w sina z=y, we have the equation d2y dz2 +(/3+ cot2 z) y=0, where /3=a+b, y=c+a-a2. As the equation is periodic in 1r, and as z=0 is a singularity, we take a transformation t=sin z; and the equation is ( 1-t2) dy t +y +a- =. The indicial equation for t=0 is p (p- 1)+y=0. If y=2an tp+ 4249 EXAMPLES [136. satisfies the equation, we have {(n+p) (n+p - l)+yja,= {(n+p - 2)1 -.3+Yj a-21 so that a- A2+2n (p - 2) +- 4 - p - 3p an2 +n(2p-1) a-2. The form of a,,, in terms of a.- 2 shews that the series for y converges for values of t < 1. If the two roots of the indicial equation are Pi and P2, and f(t, Pi), f (t, P2) be the two values of y, the primitive of the original equation is,w = sin- a z {Af (sin z, l) +Bf (sin Z, P2)). Ex. 2. Consider the equation d~y d2y sln2z=ay, where a is a constant. Taking ilp = cot 2, we find the transformed equation for z to be d- {1-2) d y=0 which is Legendre's equation and so its primitive is known. Ex. 3. Obtain integrals of the equations d2w dw (i) a+ - cot z- w cosec2 z=O; d2w dw (ii) dz2 +4 cosec 2z + 2wsec2 z=O; (ii.\ d2w ( 2 1\dw,(21\z '~dzW k(sin z cos z JdzCOS2z sin z cosz) U Lx. 4. One integral, f(z), of the equation d2wu dZIe 4 (2 - sin z) d + 2 (3 sin z + 2 cosz) - 6) A+(5 - 3 cos z - sin z) w=O dz2 dz satisfies the relation f(z+27r) +e2Tf(z)=0; find the general solution. (Math. Tripos, Part ii, 1896.) Ex. 5. Shew that the equation Ild2w 2 -W~ ~ $u' dz'2~ sin'z+/ has an integral sin (z - zj) ez cot Z, sin z where z, has an appropriate constant value; and obtain the primitive. (M. Elliott.) LIAPOUNOFF'S INVESTIGATION 425 Ex. 6. Obtain an integral of the equation I d2W 6 z dX2 sin2z where h is a constant, in the form sin (z - zj) sin (z - Z2) ez (cot z, + cot Z2) sin2 Z where z1 and z2 are appropriate determinate constants: and obtain the primitive. (M. Elliott.) Ex. 7. Integrate the equation d2W fnn(n+i) k W, dz2 =lsin2Z+ where n is an integer, and h is a constant. (M. Elliott.) 137. A somewhat different form of the theory is developed by Liapounoff*, whose investigation deals with a more general equation, given by d2W -d + ~wp (z) = 0, where 1a is a parameter, and p (z) is a uniform periodic function of period w. Let f(z) and p (z) be two integrals of the differential equation, respectively determined by the initial conditions f (0) I g (0)= 01 f'~~~ =O) Oj} Then we have relations of the form f (z + W) = a4 (Z) ~ iP (4~ P (z + o) = 'yf(Z) + cp (z)i and the equation for determining the multipliers is (n - a) (fl - 8)- /37 =0, that is, ~n2 _ (a + 8 ) fj + I 0, as in ~ 127, Ex. 1. Clearly, we have f(o)==a, ' (0)=1; C Gomptes Rendus, t. cxxiii (1896), pp. 1248-1252; ib., t. cxxviii (1899), pp. 910-913, 1085-1088. 426 LIAPOUNOFF'S [137. so that, if we write A = -{ f(o)+ b' (@)}, the equation is 21- 2A + 1 = 0. Writing p=A +(A21)2, and assuming that A2 - 1 does not vanish, we obtain two integrals in the form z z pwF(z), p F(), where F, (z), F2 (z) are functions of z, periodic in w; and thus the complete primitive of the equation can be obtained. The actual expressions for F (z) and F2(z) can be constructed as in the preceding sections; and the value of p depends upon that of A. When /a = 0, the primitive of the original equation is w= C + Dz, shewing that the equation for determining the multipliers is (a-1)2 =0; and then A = 1. Hence, when p is not zero, and when A is expanded in powers of u, it is inferred that A is of the form A = 1 -.Ai +,C2A - /A3A +.... When A,, A2, A,... are known, the two values of 12, which satisfy the equation 2-2 - 2Af + 1 = 0, can be regarded as known, and the primitive of the differential equation can be obtained. For the purpose of obtaining the value of A, which is A = {f(o) + ' (ao)}, where the integrals f(z) and b (z) are defined by the initial conditions, we assume both f(z) and c (z) expanded in powers of u. Let f() = Uo 4 /Ui + U + 2 +...; then, in order that it may satisfy the equation dw wp d + /LWp (z) = 0, dz2 137.1 EQUATIONS 427 we have d2u d2U1 ~ u"p (z) = 0, dz2 d2u2 + u~p (z) = 0, dz 2 and so on. From the first, we have Io= ao ~ b0z; from the second, we have ul= a, + b1z -f dyf YIt (x) p (x) dx; from the third, we have U= a2 + b2Z-~ dyf Yu, (x) p (x) dx; and so on. Now accordingly, fO=,f()O a0 + /ka, + Jk2a2 +..=1 Taking account of the fact that ~k is parametric, we have ao =1, a.= 0 for s = 1 2,...,I b = 0 for s 0, 1, 2,...; and thus we have = 1, l= -f dyf p (x) dx, U2= -f dyful (x) p (x) dx, and so on. The value off f(z) is given by f z=L+u+2 U2+ Similarly for 0b (z), which is determined by the conditions its value is given by 00=,0()1 428 LIAPOUNOFF' [137 where VI= - ffy (x) dx, V2= -f dyfp (x) VI (x) dx, and so on. We require the quantities f (w) and 0' (w): let them be denoted by where u, = f dyf p(x) dx, VI = -f xp (x) dx, and so for the others. Substituting the valne of A in the form we have 2A,= U1 - V11 fdyfp (x) dx ~xpr(x) dx fdyfp (x) dx + yp(y) dy. But d yfp (x) dx} fp(x) dx +yp(y); integrating between the limits 0 and o, we have 2A I= COfr(x) dx. Next, we have 2A2=- U2- V"2 - dyfui(x) p(x) dx, -fp (x) vI(x) dx. To transform these definite integrals, we write fp(x) dx =P (x), P (o) fl, 137.] METHOD 429 so that uix) dtfp (O)dO P- P(t) dt, =, W f dt Op (0) dO fdtf{P (9) -P(t)} dO. We have.d u (Y)fp 1(y) dy} = Ui (y)p (y)- P2 (y); therefore fU1(x) p(x) dx u, (y) P(y) +fP2 (x) dx =fP (x) {P (x) - P (y)} dx; and thus the first integral in the expression for 2A2 is equal to fdyf{P (y) -rP(x)1 P(x) dx. Similarly, we have d {vi (y) p(y) dy} =v1 (y) p (y) _ yP2 (y) + P (~y)fP (x) dx; therefore fvi(x) p(x) dx= v (co)&2 +fyP2 (y) dy fdyfPQy) P(x) dx — f2f yP (y) dy + {2 dy fY P (x) dx + fP2(y) dy -fdyf P (y) P (x) dx. The first and third terms on the right-hand side together are dy Il-P(~ P (y) dx, 430 LIAPOUNOFF'S EQUATIONS [137. so that v (x)p (x) = - dy { - P (y)} {P (y) - P (.)} dx, fo f o ' o which gives a transformation of the second integral in 2A2. Combining the results of transformation for the two integrals in 2A2, we have 2A2 = dy fy - P (y) + P ()} {P (y)- P (x) dx. Similarly, it may be proved that the value of 2A8 is dz dyf {a- P (z) + P (x)} {P (z) -P (y)}{P (y)-P (x) dx, and so on: so that the value of A, and therefore the value of P, is known. The investigation is continued by Liapounoff, especially for the purpose of discussing the values of a/ which satisfy the equation A2- _ 1 =0; and the results appear to be of importance in the discussion of the stability of motion. The reader is referred to the notes by Liapounoff already cited (p. 425, note); other references to more detailed investigations are there given. Ex. 1. Establish by induction, or otherwise, the general law for the coefficients A, viz. 2An= dxi dx... @dx, where = { - P (x) + P ()} {P (xI) - P (x2)} {P (2) - P (x3)}... {P(xn-) - P (x,)}. Ex. 2. Shew that, if the periodic function p (x) always is positive, then all the coefficients A are positive; and prove that m!in! Am+n<(m+ AAn. Hence shew that, when p (x) is positive and satisfies the inequality,U o p (x) dx: 4, then A so that o then A2< 1, so that |p|= 1. 137.] EQUATION OF THE ELLIPTIC CYLINDER 431 Ex. 3. Prove that, if the periodic function p (x) be real and odd, so that the series for A contains only even powers of i, then A2= -2 dx 1(PI P2)2 dx2, Jo o A4- 4 f dx- f' dx2 fdx P)2 (P- -P4)2 dx4, A 4 AdJ 1 Jo Jo2 J3 and so on, where P, denotes P (r). Prove also that, if j p(x)dx=P, a the constant a being determined so that f Pdx=0, and if o,c0 P2dx 4, then A2< 1. Ex. 4. Discuss the values of / which are roots of the equation A2=l. (All these results are due to Liapounoff.) DISCUSSION OF THE EQUATION OF THE ELLIPTIC CYLINDER. 138. One of the most important equations of the class, which has been considered in ~ 137, is the equation d2w d ~- + (a + c cos 2z) w = 0, commonly called the equation of the elliptic cylinder; it is of frequent occurrence in mathematical physics and astronomical dynamics. It forms the subject of many investigations*. It is known (~ 55) to be a transformation of the limiting form of an equation of Fuchsian type. Moreover, it has already (~ 127, Ex. 1) been partially discussed in connection with another equation and for another purpose. In this place, it will be brought into relation with the preceding general theory. Let new independent variables u or v be introduced, such that u = cos2 z, v =1 - = sin2 z. * Heine, Handbuch der Kugelfunctionen, t. I, pp. 404-415; Lindemann, Math. Ann., t. xxII (1883), pp. 117-123; Tisserand, Mecanique Celeste, t. II, ch. I, at the end of which other references are given. 432 EQUATION OF THE [138. The equation becomes d2W dw (I1-u) dit + -1(1 - du2u) + '(a-c +2cu) w=0, when ua is the independent variable; and it becomes V 2WV+d2+1 - 2v) dw+ (a + c -2v) w=O0, when v is the independent variable. Accordingly, if w =f (, c) is an integral of the equation, another integral is provided by w =f (v, - c). The indicial equation for ua = 0 is p (p - 2D=0; if w =:~aup~p be the integral, the scale of relation between the coefficients a. is with the relations (p ~ 1) (p + 1) a, p2 _-1 j(a - c). When p =0, let al,= 0(p, c); when p=, let ap ='~r(p, c). Then two integrals of the equation are $UPO0(p, c), x1= UV'~ u~(p, c), P=O p=O with the, convention It is clear that, when z is 1 7r, so that uis0, dz x=,dxz _ Moreover, as the equation in w is satisfied by x0, and x~, we have dx, dx1 C XdXO du ={u (1 -u)ji 138.] ELLIPTIC CYLINDER 433 But du = - 2 sin z cos zdz = - 2 {m (1 - u)J}dz, so that Xi x,- - J2C. dx dz When z-=7r, the left-hand side is equal to 1: hence C=-, and therefore, for all values of z, we have dxo dx, dz - dz 1 Two other integrals of the equation are given by yo = VP 0 (p, -c), y v+ (p, c); p=O p=O they are such that, when z is 0, and therefore v is 0, dy o dyl yo = 1, d O, y- O dy- 1, and, for all values of z, dy d- ddyo Yo dz- ydz Now when z is real, both u and v are real and lie between 0 and 1; and, in particular, when z = ~qr, then u = v = ~. For such values, Xo, x1, yo, yi, coexist; and so we have relations of the form yo = axo + 1X) yl =, 7xo + 8xi' where a, 3, y, 8 are constants. Hence yo (2-) = Xo (O) + x, (0), - Yo (2)= XO (1) + x1 (2), where,/ (1)\ = fdo -dxoz) and so for the others. Hence a = - Yo ( ) + ' ( y ) xi (). /3= yoCDxO'(4K+yo'(Dxo(). Similarly = yl) ( X ) + y/ (g) 0() 28 F. IV. 434 EQUATION OF THE [138. and it is easy to verify that ac8 - by= 1. Moreover, we have o0 = 4y0- 3ylI x =- 7yo + ay') 139. The integrals x, and xI are valid in the domain of u = 0; the integrals Yo and y, are valid in the domain of v = 0, that is, of u= 1. Lindemann proceeds, as follows, to obtain uniform integrals valid over the whole of the finite part of the plane. After a small closed circuit of u round its origin, x0 returns to its initial value and x1 changes its sign; hence Yo becomes ax - /x,, and y1 becomes yx0 - 8x1. After a small closed circuit of u round the point 1, the integral yo returns to its initial value and yj changes its sign. Consider a quantity a7, where X = Ayo2 + Byy2, as a function of u. It remains unchanged when u moves round the point 1. Its two values in the vicinity of u = 0 are (A22 + By2) x2 + (A/2 + B82) x12 + 2 (Aa/3 + By8) x0x1, (Aa2 + By2) Xo2 + (A/32 + B$2) x12 - 2 (Aa/3 + ByS) x0x, which are the same if Aa/3 + By8 = 0: hence the function is uniform in the vicinity of u = 0 if this condition is satisfied, that is, the function is uniform over the whole plane. The condition is satisfied if we take A =-y, B= a/3; and then ] = a/3yl2- Y/0o2. Moreover, in the region of existence common to yo, y1, x0, e, we have a<y12 - 8y2 = /8X12 - ayx02. Hence defining the function qJ in the domain of z = 0 by its value in terms of y, and y,, and defining it in the domain of z = 1 by its value in terms of x, and x1, we have a function F(u) = F(u) = Fo z)= (2), * Math. Ann., t. xxII (1883), pp. 117-123. ELLIPTIC CYLINDER 435 say, which is regular in the vicinity of u = 0, regular in the vicinity of u = 1, and therefore is regular over the whole finite part of the z-plane. Now let Y, = yi(a + yo (y8)Y o; Yo -y, (a)2 - y0 (ry8) then IdY1 dY0,I dy1 dyo Yod -- _ dY, = _ 2 \(as~/3 5 Y o d - yl dz - 2 (a/3,88). Also, YYo= (z), and therefore d Y d Yo dz- + Y z Hence 1 dY1 1 V(z) (a!y8)2 Y, dz -2 D (Z), (Z)' 1 dYo (z) (/ry) Yodz 2= >(,) ~(Z); and therefore Y,= K { (z)} e J K(z) = K { z)} e 1 () M f (dz YO= K' {) (z)}t e ( where 1M= (afy8)1, KK' =1. These integrals of the original differential equation are valid over the whole of the finite part of the plane. Accordingly, we may take two integrals -fdz G (z) -{^ J'(z~)}I- e I(z) G, (z) = ^ (ze { e Gas i) { (h)}i e the a as integrals, which are valid over the plane and have z- oo for their sole essential singularity. We now proceed to shew that they are uniform over the plane. Substituting in the original differential equation, we have (a + c cos 2z) )2 - V 1)/2 + I(I// + M 2 = 0; 28 2 436 LINDEMANN'S [139. so that, as M in general is not zero, any root of I = 0 is a simple root. Let k denote such a root: then M= 1 " (k). Now let z describe a simple closed contour, including k and no other root of (< = 0, and passing through no root of ' = 0. Then, at the end of the contour, {(I (z)}l has changed its sign. As for the exponential factors in G (z) and G1 (z), they are multiplied by f dz e - (Z) respectively, the integral being taken round the contour, that is, they are multiplied by 27ri e2 (k) that is, by -1. Thus G(z) and G (z) are unaffected by the contour; they are therefore uniform in the vicinity. Moreover, in the immediate vicinity of k, we have () (z) (z - k) ' (k) +..., so that G (z) [{ (k)} (Z- +...] e- og(z-k)+P(z-k) = {Vf (Jk)}2 eP (z - k) Q ( ), G (z)= {f' (k)Z} (z- k) e - P (-k) Q (z-k), so that k is a simple root for one of the integrals and it is not a root for the other. Similarly, in the vicinity of any other root of <) =0; hence G and G1 are uniform over the whole plane. Now take any path from z to z + vr, for 7r is the period for the original equation. We have 1 (z) = F (cos2 z), where F is uniform; hence (Z + T) = (Z {< + r))' = (-()I {& (Z),2 where r is 0 or 1, depending upon the path from 0 to rt. The effect upon the exponential factor of G (z) is to multiply it by e 2+V dz 139.] METHOD 437 We know that D(z) is regular over the whole plane, that it is periodic in 7r, and that it has only simple roots; hence, taking a path between z and z + r, that nowhere is near a root, we can 1 expand in the form n=00 -(-)-= E Cne2nzi, (z) - valid everywhere in the range of integration. Then M.z+7r dz e z - (Z) e- -MCo 7 and, consequently, if p =(- 1)re- M0o r, then G (z + r) = G (z). Similarly G1 (z + r) = - G (). Hence G and G1 are the two periodic functions of the second kind, which are integrals of the original equation*; and they have been proved to be uniform functions, regular everywhere in the finite part of the plane. Ex. Shew that the equation z (1 d+(I- z2) d + (az+ b) w=O has two particular integrals the product of which is a single-valued transcendental function F (z); and shew that the integrals are Y = {F(z)}. exp. f { (1-z)}F (z)] 2= {F (z)}. exp. - C {z (1-z)} LJ{z(l-z)}iF(z) where C is a determinate constant. In what circumstances are these two particular integrals coincident? (Math. Tripos, Part ii, 1898.) 140. The multipliers p and - are thus the roots of the equation n2 - I + 1 =, * This inclusion of Lindemann's special result within the general theory is due to Stieltjes, Astr. Nachr., t. cix (1884), pp. 147, 148. 438 INVARIANT OF THE EQUATION [140. where the invariant I of the period o is (- 1)r (eMCo" + e-MCor). Another expression for this invariant, consequently leading to another mode of obtaining these multipliers, has already been given in Ex. 1, ~ 127. Both processes are dependent upon the determination of simple special solutions of the original differential equation. Another method of proceeding is as follows. Let L -= eprih, so that eih (z+r) = -e ihz; so that, if G (z) = eih (z), then, as G (z) is a uniform function of z, regular over the whole plane, ~ (z) is a uniform periodic function of the first kind, regular over the whole plane; and 7r is the period. Hence we have n —oo (H (Z) = n xne2nzi, -o00 and therefore G(z)= KX e((2n^)zi - 00 Now in the vicinity of z = 0, the integral y, is even and y, is odd: hence G (z) contains both odd and even parts. The form of the differential equation shews that, if f(z) is an integral, then f(-z) also is an integral; hence, as G (z) exists over the finite part of the plane, G (-z) also is an integral. Hence, taking H(z)= G (z) + G(- z)} cos a + i {G (z) - G(-z)} sin a n= 00 = Kncos {(2n + h)z + a}, -oo where a is an arbitrary constant, it follows that H(z) is an integral of the original equation, which exists for all finite values of z. Substituting in the differential equation, and noting that cos 2z cos {(2n + h) z + a} = - cos {(2n - 2 + A) z + a} + 2 cos {(2n + 2 + h) z + a}, we have - o0 -o [{a -(2n +h)2}Kf ~ +c (K_, +n+i)] cos {(2n + -h)z~ c4}= 0, 140.] OF THE ELLIPTIC CYLINDER 439 as an equation which must be identically satisfied; hence {a - (2n + h)2} Kn + -C (ACn-1 + K-1+l) = 0, for all values of n from - oo to + 0. The mode of dealing with this infinite set of equations by means of infinite determinants has been indicated in a preceding chapter, and much of the analysis of the first example in ~ 126 is directly applicable here: so we shall not further discuss this mode of obtaining h and the ratios of the coefficients K. There is, however, another method of obtaining these quantities: it is due to Lindstedt* and is specially adapted to the differential equation under consideration, for purposes of approximation when c is conveniently small. Writing ac = 2 (2n + h)2-2a, we have Kn ten KCn-1 an C c in /n+ C =- R |n /n_1 On C / n+i an Kn C C2 c2 Xfa anan+l an+lCan+2 ad inf. 1- 1- 1 -Owing to the form of - for increasing values of r, it is easy to ar prove that this infinite continued fraction converges, for all values of n. We therefore have c C2 C2 E__ al a1 1 a23. ad inf. cK 1-1-1 -Similarly c c2 c2 C -- 1= a_ — n a —n-1 a —X- 1 a —2... ad inf., C_n+~ 1 - 1- 1 - * Mem. de l'Acad. St Petersbourg, t. xxxI (1883), No. 4. 440 LINDSTEDT'S [140. which is a converging continued fraction; and, in particular, c C2 C2 K-1 aL1 a-la-2 a-A-3...ad inf. But, from the fundamental difference-equation, -C, + K-1 h2- a of 22 K0 C therefore C2 C2 C2 C2 C2 C2 Cta t C C fC Ct~ - a-CC ( 2C 3 1=a~a1 a~ca2 a2~c2 ~c~ct-l or,, az a transcendental equation to determine h, which of course is equivalent to the corresponding equation arising out of the vanishing of the infinite determiniant D (p). Denoting the first continued fraction by - and the second by q q so that these values may be regarded as convergents of infinite order, we easily find 2 o00 2 00 2 2 2 00 00 00 C2 C2 C2 r=2 s=r+2 t=s+2 (%rC4r.i CtsCts+1raft+i a 00 2 00 2 2 -Q ~ C C C r=1 12r ar+1 r=1 s 1r+2 Ctrar,_t fsCas+1 00 00 C 2 C2 C2 r=1 s=r+2 t=s~2 Orar+l Of as+i Otati t+ the values of p' and q' are derivable from the expressions in p and q respectively, by changing a, into a-, (for all values of it) wherever a,, occurs. The equation manifestly lends itself easily to successive approximations. Thus, if we neglect C4 and higher powers, we have C2 '2 1 = C+ C a0a1l a a-I which, to this order of approximation, gives C2 h2=a + 1 -The calculation of the coefficients can similarly be effected. 140.] METHOD 441 Ex. 1. Prove that, up to sixth powers of c inclusive, _ 1 c2 c4 15a2 — 35a + 8 16 a (1- a) 1024 a2 (1- a)3 (4-a) c6 105a5- 1155a4+ 3815a3-4705a2 +1652a-288 16384 a3 (1-a)5 (4-a)2 (9-a) (In astronomical applications, a is usually not an integer, and c is small compared with a.) (Poincare, Tisserand.) Ex. 2. Taking Ko= 1, and writing hz + a=Z, prove that, up to c3 inclusive,.c C3 3a+4q2+15q+16 } TT/N=caiiZ~ r1+ I 1 1024 q (I + # (2 + q) (I - q) c c3 q3-4q2+15q - 16 - cos(Z-2z) H7 (,)= cosZ+ (Z+ 2z) + - q 1024 q(1 - q)3(2 - )(+)} cos (Z- 2z) (Sc)2 f cos (Z+ 4z) cos (Z- 4z) 2 (1+q)(2+q) +(1-q)(2-q)J + (3 cos (Z+6z) + cos (Z- 6z) 3! (1 + q) (2 +) (3 +q) (1- q) (2 - q) (3 - g) where q2= a. (Poincare, Tisserand.) Ex. 3. In the investigation of ~ 138, the quantity M is supposed to be different from zero. When M is zero, the integrals G (z) and G, (z) are effectively the same; and neither of them is uniform, so that the remainder of the investigation does not apply. Discuss the case when M=O. (Heine.) EQUATIONS WITH UNIFORM DOUBLY-PERIODIC COEFFICIENTS. 141. We proceed now to the consideration of linear equations, the coefficients in which are uniform doubly-periodic functions of the independent variable. Let the equation be dmw din-lw dz-~ + plzd_ +... + *+pmw = 0) where pl,..., pm are uniform functions of z, which have no essential singularity in the finite part of the plane and are doubly-periodic in periods w and o', such that the ratio of o' to o 442 DOUBLY-PERIODIC [141. is not purely real. A fundamental system of integrals exists in the domain of any finite value of z, and may be denoted by fl(Z, Z 0), -..., f, (), which accordingly are linearly independent of one another. The differential equation is unaltered when z + w is written for z: hence fA (Z + ), 2 ( + ),..., fr (Z + ) are integrals of the equation and, as in ~ 128, they constitute a fundamental system of integrals. Similarly, as the differential equation is unaltered when z + )' is written for z, f, ( + z <'), 2(z + w'),..., f (Z + o') constitute a fundamental system of integrals. Choosing therefore a region common to the domains of these three fundamental systems (a choice that always can be made because the singularities of the integrals are isolated points, finite in number within any limited portion of the plane), we have relations of the form fi (z + = all i (z) +... + am 1m (Z ) fm (z+w)=arn/f(z)+...+ammf/(z))............................................... fm (Z + w ) = ay; (z) +... + amf,,(z) I and fi (z + o) = bll f (z) +... + bm fm ()................................................, fm (Z + (') = bmlfi (Z)+ +.. + bmfm (z) valid within the region chosen. The coefficients a are constant, and their determinant is not zero; the coefficients b also are constant, and their determinant also is not zero. The two sets of relations may be represented in the form f(z + co)= Sf (z), f(z + c') = S'f(z), where S and S' denote the linear substitutions in the relations. The coefficients in the two substitutions are not entirely independent of one another. We manifestly have f/r {( + ) ) + O} =fr {(Z + W') + }, for all values of r. The symbolic expression of this property is f (z + ) + c} = S'f( + o) = S'Sf (), f t(z + W') + } = Sf(z + co')= SS'f(z), 141.] COEFFICIENTS 443 so that SS'= S'S, or the linear substitutions are interchangeable. The explicit expression of relations between the constants is obtainable from the equation brfi (Z + ) + b,2f2 (z + w) +... + brmfn (z ~ w) =fr (Z + o + )') = arif (z + o') + a2f2 (z + o') +... + armfm (z + co'), by substituting for the functions f(z + 0) in the left-hand side and the functions f(z +w') in the right-hand side. The result must be an identity, for otherwise there would be a linear relation between the members of the fundamental system f (z),..., f (z); hence, comparing the coefficients of f (z) on the two sides after substitution, we have b.c a4, + b,.a2S +... + br, ams = a,., bl +~ ar2b, +... + a, brms = Ors, say. This holds for the m2 equations that arise from the values r, s = 1,..., m. Of the m2 equations, only m2 - m are independent of one another, a statement the verification of which (alike in general, and for the special values m = 2, m = 3) is left to the reader: it can also be inferred from some equations which will be obtained immediately. The number of the relations is less important, than their existence and their form, for the establishment of Picard's theorem relating to integrals with the characteristic property of doubly-periodic functions of the second kind. Consider a linear combination of the members of the fundamental system in the form F(z) = Xlf () X2f2 ( z)+... + Xfm (z), where X, will be taken as equal to unity when it is not bound to be zero; and let the constants X2,...,, be chosen so that, if possible, the relation F (z+ co)= OF(z) is satisfied, 0 being some constant. To this end, we must have X1 0 = X all + X a2l + 4X3a31 +... + Xmai al, X2 0 = Xlal2 + X2a2 + X3a 2 +... + Xmam2.................................X............ Xm,0 = Xilal. + X2a2m + X3a3Mr +... + X), amil, 444 FUNDAMENTAL EQUATIONS [141. and therefore l (0) = alln-, a2l,..., ami =0, a12, a 22-,..., am2................................ aim, a2m,..., amm - 0 the equation satisfied by 0. As in the case of the single period in ~ 128, it may be proved that this equation is independent of the original choice of the fundamental system of integrals f (z),..., f,, (z). The coefficients of the various powers of 0 are therefore invariants, and the equation is called the fundamental equation for the period o. Now let XI'= b Xib + X2b2, + Xb +3 +... + mbm /2 = Xlb12 + X2b22 + X3b32 +... + Xmbm2............,............ o....................... /tm = Aibim + Xa2bm + X3bsm +-... + Xmbmm. Multiply the earlier equations, which define the quantities X and lead to the equation Q (0) = 0, by blr, b2,.,..., bm respectively, and add: then [tLr = Xi (all bir + a12 b2r... + am bmr) + X2 (a21 b,. + a22 b2r.... + a2m bmr) +.......................................... + Xm (amb br + abr + + amnbmr) = Xi (bll air + bi2 ar +...+ bim amr) + X2 (b2i air + b2, a2 +-.. + b2m amr)............................................ + Xm (b, mair + bm2 a2r +... + bmmarmr) = X1arO' + a2r2 +3 + a3r3 -... + amrPmThis holds for all values of r; and thus we have X10 = Xal, +- a2a + - a31 +... +- am,, 0 = Xa2 +~ a22 + a32 +. + a,,2...........,......................................,, rwm,_0 = tl+2 P+3 * + m r 6'm — ki aim + am + + 6'2 0' a am... amm FOR THE PERIODS45 445 When these are compared with the earlier equations, we have uniquely 19 =Xr, for all values of r; and therefore the same values of X2,..., Xm,, that enable the equations connected with the period w to be satisfied, lead to the equations Xi 0' = Xi b1i + XV21 + N3 b31 ~. + Xm bmi, X2 0' = Xib12 + V2b22 + X3b32 +-.. + Xmbm2,Y Xm0' Xibim +X2b2M+ XAbm+-.. + Xmbmm,m Hence F(Z + W) X1f1 (Z +o/)~X2f2 (Z+aCO) +.+XmJm (Z + &i') / 0' X~f1 (Z) ~ X2f2 (Z) +.. + Xm}'m (z)} = 'F (Z), on using the preceding equations. Moreover, this multiplier 0' satisfies the equation f~' (O) = b1 - O', b21. bmin = 0.,1 2 - 0',..., This equation, like &2 (0) =0, is independent of the initial choice of the fundamental system of integrals f, (z),..,fm (z), the proof being similar to that -in ~ 128. The coefficients of the various powers of 0' are therefore invariants; and the equation is called the lundamnental equation for the period ol'. The term independent of 0 in fl (0), and the term independent of 0' in f2' (0'), can be obtained simply. Let A~ (z) denote the determinant z~()-dm-1f, dmn-lf2 dn lf df, df dfm dz 'dz '' dz f, f2 f.. 446 FUNDAMENTAL EQUATIONS [141. then, as in ~ 9, we have fz A (z) = A () eJ () Hence rz+w I p, (x) dx A (z + o)= A (o) eJ zo so that rz+w A (Z + =) i pl(x)dx A (Z) and similarly z+W/ A(z + w) e P1 (x) dx A (z) where we manifestly may assume that the path of integration does not approach infinitesimally near the singularities of p,. Now p1 is a uniform doubly-periodic function with no essential singularity in the finite part of the plane; if, therefore, a,,..., a, denote its irreducible poles, and if (z) denote the usual Weierstrassian function in the same periods w and o' as pi, we have* n b dA/ \ - a,.) p= C+ E Ar,'(z-ar)+ > Br -) r=l r=1 dz + dc (z-a)., l dz2 with the condition ZA,r= O. Now [z+(, / (z + c - a,) n p, (x) dx= Co + A, log - - ^ ^+(-^)-^^o- (z -= a(,-.) - d c,. {~ (z + ( - ar) - (z - a.)) _... nZ n~n r=1 = C(+ S A, {iT + 1 + 2j(z-ar)}+ S 2^r r=l r=1l = Co - 2 S Ar,a,. + 27 Z B, = D, r=l r=l * T. F., ~ 129. OF THE PERIODS 447 say; and similarly,Z+o/ n n p, (x) dx = Cw' - 27r' Arar + 2g' Z Br = D, - z r=l r=l say. But, substituting in A (z + co) the expressions for f, (z + co),.., (z + co) and their derivatives in terms of fi (z),..., f (z) and their derivatives, we have A (z + o) a,,, a12... aim a (z) a21, a22 *., a2m........................ aml, am2,..., atmm which is the non-vanishing constant term in 2f (0). Thus n (0) = eD +.. +(-1)m0m; and similarly (0') = eD'+ +... + (- 1) 10' In particular, when p, is zero, so that the differential equation dm-lw has no term in dz_-, we have D = 0, D' =; and then 1 (0) = 1+... + (- 1)~Om, fn' (0')= 1 +... + (-1 )M0'. INTEGRALS WHICH ARE DOUBLY-PERIODIC FUNCTIONS OF THE SECOND KIND. 142. Let 0 be a root of the equation {I (0)= 0. Then quantities X2,..., Xm exist such that the equations leading to 1 (0) =0 are satisfied; and a quantity 0' is obtained, when the values of X2,..., Xm are substituted in its expression. It thus follows that there is an integral F(z) of the differential equation such that F(z + o) = 0F(z), F (z + w') = 0'F (z), where 0 and 0' are constants. Such a function is called* doublyperiodic of the second kind: and therefore it follows that a linear differential equation, which has uniform doubly-periodic functions for its coefficients, possesses an integral which is a doubly-periodic function of the second kind: a result first given by Picard. * T. F., ~ 136. 448 PICARD'S [142. When 0 is a simple root of the equation 2 (0)=O, then X\2..., Xm are uniquely determinate: and 9' is uniquely determinate. When 0 is a multiple root of its equation, quantities 2,..., Xm exist satisfying the associated equations but they are not uniquely determinate: and assigned values of X2,..., I determine 0'. Similarly for O' as a root of the equation f' (') = 0. Combining these results, we have the theorem*: A linear differential equation, having doubly-periodic functions for its coefficients, possesses at least as many integrals which are doubly-periodic functions of the second kind as either of the equations f (0) = 0, f' (9') = 0 has distinct roots. By using the elementary divisors of 2 (0) = 0, we can obtain a more exact estimate of the number of integrals which are periodic functions of the second kind, associated with a multiple root. Let 90 be a root of 2 (0) = 0 of multiplicity X,, and let nl be the number of different elementary divisors of /2 (0) which are powers of 0 - 01, so that the minors of 2 (0) of order n1 are the first in successively increasing order which do not vanish simultaneously when 0=01. Then (~ 133) the number of integrals, which satisfy the equation I(z + ) = 011 (z), is precisely equal to n,. * These equations appear to have been considered first by Picard in general; see Comptes Rendus, t. xc (1880), pp. 128 —131, 293-295; Crelle, t. xc (1880), pp. 281-302. Their properties were further developed by Floquet, Comptes Rendus, t. xcviII (1884), pp. 82-85, Ann. de l'Ec. Norm. Sup., 3me Ser., t. i (1884), pp. 181-238, which should be consulted in connection with many of the following investigations. A proof of Picard's theorem, different from that in the text, is given by Barnes, Messenger of Mathematics, t. xxvII (1897), pp. 16, 17. Investigations of a different kind, leading to equations the primitives of which are expressible in terms of doubly-periodic functions, are carried out in Halphen's memoir "Sur la reduction des equations differentielles lin6aires aux formes integrables," Mem. des Sav. Etrang., t. xxvIIi (1882), No. 1, 301 pp.; particularly, chapters ii and Ix. The most important equation of the type under consideration is the general form of Lame's equation. It had been considered by Hermite, previous to Picard's investigations; and it has formed the subject of many memoirs, references to some of which will be found in my Theory of Functions, ~~ 137-141. 142.] - THEOREM 449 Moreover, in that case, ni of the equations in ~ 141 for determining the quantities X are dependent upon the remaining m -n1. Let the last m -nI be a set of independent equations, determining ~.,X,,, in terms Of X1, 'X2,..., X,,,; and suppose that the expressions are 'X -:k8l XI ~ k'82'X + k83 X3 +..+ k8n, Xnj Y for s= n, ~1, n. 2,..., m. Then F (z) = Xif1 (z) ~ X2f2 (Z) +..+ X~nfM, (Z) = X1g1 ()+ X2Y2 (Z) +..+ Xn, gn, (z), where g, (z) =fr. (Z) + '~ korfs (z), for r = 1, 2,..., ni; and each of the functions g,,..., is such that g, (Z + co) = 1g,.j (Z). As regards the possible multiplier O1' for the other period, we have 01' =Xib1l + X2b21 + X3b31 + +. ~ mbmi =X1B, + X2B2 +.. ~X,81Bnj, say, where Br =bri- + - l-sr bsi; and the effect upon F (z) of the increase of argument by the period w' is given by F (,-+o) 01' Fz) Now 01' is not zero, for it is a root of SQ' (0') = 0 which has no zero root; and therefore not all the quantities B,, B2,..., Bn, can vanish. Let Bi, B2,..., B. be those which do not vanish; then wehave Xl~l ( +(t) + 2.2 ( + )) ++Xnlgnl (Z + CO) = (X1B1 + X2B2 +... + x B,,) {X~g1 (Z) ~ X2g2 (Z) +... + X211gn, (z). As some one of the quantities XI, X2,..., X,2, is not zero (for, thus far, all these quantities are arbitrary), we shall take X1 = 1. In order that this equation may hold, we assign definite values to X2 AS; we write gI (Z) + )2g2 (z) +..+ X~g, (z) = G (z), F. IV. 29 450 DOUBLY-PERIODIC INTEGRALS [142. and then, as s+,..., Xn, can remain arbitrary, we have G (z + co') =,'G (z), gr (Z + o') = O'gr (z), for r=s+l,..., n,. Moreover, on account of the composition of G (z), we have G (z + w) = OG (z), and we had 9g. (Z + o) = gr (z). Accordingly, the number of integrals, which are doubly-periodic functions of the second kind and are associated with the multiple root 01 of the fundamental equation 1 (0) = O, is 1 + n - s, where nl is the number of elementary divisors of -Q (0) which are powers of 0 - 0, and s is the number of quantities In bri + kbsrbs s=nl+1 which do not vanish, so that 0 < s < nj. 143. We now can indicate the total number of integrals, which are doubly-periodic functions of the second kind. Let 0, be a root of multiplicity Xi of n1 (0)= 0, and let it give rise to n, elementary divisors of 2 (0) which are powers of 0- 0-; and let s, be the number of quantities lIt b,.r + E ks,.bs s=nl+l s = Bn +1 in the preceding investigation which do not vanish, so that 0<s <ni< i. Let 02, 03,... be other multiple roots; and let X,, n2, s,; Xn, n3, s3;... be the numbers for them, corresponding to XI, n,, si for 01; so that X\ +2 X + X...=. Then the number of integrals, which are doubly-periodic functions of the second kind, is E (1 +,.-s,). r=l 143.] OF THE SECOND KIND 451 In particular, if the roots of l (0) = 0 be all distinct from one another, a fundamental system can be composed of m integrals, each of which is a doubly-periodic function of the second kind; the constant multipliers are the m roots of f2(0)=0, and the corresponding quantities 0' derived from them, these quantities 0' themselves satisfying the equation 2' (0') = 0. Moreover, the relation between the equations satisfied by 0 and X,., X, and the equations satisfied by 0' and X\,..., Xm, is reciprocal; for each set can be constructed from the other as in ~ 141. Hence, if either of the equations 2 (0) = 0 and f2' (0')= 0 has all its roots distinct from one another, there is no necessity to take account of possible multiplicity of the roots of the other, so far as the present purpose is concerned: the implication merely is that one of the two multipliers has the same value for several of the integrals. Further, if 0 and 0' are two associated multipliers, each of them arising as repeated roots of their respective equations, we shall suppose, for the same reason as in the preceding case, that the construction of the doubly-periodic functions of the second kind is initially associated with that one of the two equations which has the repeated root in the smaller multiplicity. MULTIPLE ROOTS OF THE FUNDAMENTAL EQUATIONS AND ASSOCIATED INTEGRALS. 144. We have now to consider the form of the integrals associated with a multiple root of S2(0) =0, the fundamental equation for the period o; and we assume that the corresponding root of 1' (0') = 0 is also multiple, to at least as great an order of multiplicity. Denoting this root by 0, and the corresponding root of 1f' (') = 0 by 0', we know that there certainly is one integral, which is doubly-periodic of the second kind and has multipliers 0 and 0'; let it be denoted by 01, so that 1 ( + o) = 00b (z),, (z + o') = 6 (z). Considering the integrals first in relation to the period co, we know (~ 134) that the number of them associated with the multiple root 0 is equal to the order of multiplicity of 0: and 29-2 452 INTEGRALS ASSOCIATED [144. further, that this group of integrals is linearly equivalent to sub-groups of integrals of the form i1t= =1, q b2 2 + Z01, U b3 + 2z,2 + Z201, U4 = 0 + 3zb3o + 32202 + Z31, the aggregate number of integrals in the various sub-groups is equal to the order of the multiplicity of 6, and each of the functions k is such that J (z + o) = 60 (2). In these integrals, b2 can have any added constant multiple of c/h; also f3 can have any linear combination of constant multiples of b2 and sb; and so on. All the functions 0, so changed, still have the multiplier 0 for the period a. Now u1 has the multiplier 0' for the period w'. The simplest case arises when some other integral of the group, say u,., also has this multiplier 0' for the period w': for then all the intervening integrals have this multiplier for the period 0'. What is necessary to secure this result is that, first, 02 (Z + 0)) + (Z + a)') 01 (Z + c)') = 0' 102 (z) + z10 (z)}, that is, QP2 (2 + Ol) + 01'01 (Z + co) = 0'b2 (Z), and therefore 02 (Z + WI, 2 (Z) +0 Secondly, we must have 0b3 (Z + 0)) + 2 (2 + 0) jb'2 (2 + 0' ~ (z + 016)2 Cbl (2 + co') = 0' {S3 (2)+ 2z1)2 (z) + Z201 (z)J, which, in connection with the preceding equations, is satisfied if 13 (2 + co) + 260/'2 (Z + CO')+ +'b (z + co) 0'c1)3(z), that is, if 13 (2 + 0') )2,ck(2 + 60) ) 2 (Z 01 (z + CO") 01 (z + CO') 01 (Z) 14~4.] WITH MULTIPLE ROOTS 4~53 Similarly, we must have /4 (Z W'), 3 (Z (z ) / 02 (Z +w') /3 4 (z) + U/ + 3Co/ + W' - 01 (Z + 3C0 (1 (Z + 31 2(Z _ _(z) and so on. Let f'(z) denote the usual Weierstrassian '-function, with periods o and w'; and let iq, q' denote the increments of '(z) for an increase of z by the respective periods, so that we have r7WI - 'q'w = ~ 2njri, the sign being the same as that of the real part of w' + ico. Then, if a function it (z) be defined by the equation we have u (Z + O)= m (z), u (Z + co') = t (Z) + o'. Then we have '2 (Z + ('A) (/)2 (z) 06 (Z + 01 + ~ = (z) that is, the function on the right-hand side is periodic in w'. Moreover, 02 and 0b have the same multiplier for w, and u (z) is periodic in o; hence the function on the right-hand side is periodic in w also. It thus is a doubly-periodic function of the first kind; denoting it by #2, we have f2 = r1'2 - U011 so that 01#2 is a doubly-periodic function of the second kind, with the same multipliers as 0b, viz. 9 and 0'. Similarly, we have 03( 2u (z + w') 02 Z + + U (Z + W)122 01 (Z + ')1 ( + _3 (Z) (2 (z) - _~ + 2 u (z) + 4 Z (z)}2, 01 (Z) 01 (Z) so that the function on the right-hand side manifestly is periodic in co'; and it is periodic in w on account of the properties of ut and 454 454 ~~~INTEGRALS ASSOCIATED[14 [144. the functions b.Denoting this doubly-periodic function of the first kind by J3 we have 03 = #3- 2uoi#2~m 2 01. And so on in succession. The group of integrals, in the case suggested, can be represented in the form X2 - U1 X, - 2aUX2 + U2 b1, X4 - OUX3 + 3ua x2 - 1 where the functions 0b1, X2) X3' X4' are doubly-periodic functions of the second kind with the multipliers 6 and 0', and U (Z) ~= 2w ~ Z - (Z 145. Returning now to the less simple case, when not more than one of the integrals associated with the corresponding multiple roots can be assumed to be doubly-periodic of the second kind, we know that one integral certainly exists in the form of a doubly-periodic function of the second kind with the multipliers 6 and 6'. Denoting it by 0b1 (z), we use it to replace some one of the integrals, say f~(z), in the fundamental system, which then becomnes We have f' (z + W)= 0f,i (z), f2 (Z + W)C21i01 (Z)+ C22f2(Z+.+ C2M fM(Z), fM (Z + W) = CMl 01 (Z) ~ c2M2f2 (Z) +... + clamfrn (Z).The fundamental equation for the period co is f2(x) = 0- x, 0, 0,., 0 =0; C21, C022 - X, C23.. C2m CMi, CM2 I CM3, CM., n - X and so 6 is a root of fl, (X 022 -...,23C2m =0, Cm2, Cm3,...I1 Cmm -X of multiplicity less by one than its multiplicity for f~ (x) = 0. 145.] WITH MULTIPLE ROOTS 455 Similarly, we have relations of the form f, (z + W') = 0',0 (z), f2 (z + w') = d21 0, (z) + d22 f2 () +... + dm fn (),...................................................o fm (z ')= d (z + d f ( + ) = f (Z) + 2 () + + dmfm (z) the multiplier 0' being a root of fn' (x) = d22-, d23,..., d2m = 0,.........,,,,,.............. dm2, dm3,... drmm- of multiplicity less by one than its multiplicity for Q' (x)= 0. The coefficients cj and di must satisfy conditions in order to have fK {(z + 01) + O} =fK {(z + o) + o'}, for all values of K: these conditions are dr, cis + d'2c2s. +. + drcms r= cr dds + crd2s +...* * * crm d say, with the limitations cll=0, di,=0'; cs=0, di,=0, if s>1. Owing to the fact that 0 is a root of fQ (x)=0, quantities K3,..., Km exist such that OKCs = C2s + C3s K3 +... + CmsKm, (S = 3,..., m), C = C122 - C32 K3 +... + Cm2 Km Let r 2= d 32K+ 3 +. + dmz2 Km r- = d2r+d3r 3+... +dm,.K, (r =3,..., m); then 6r, = d2r (c22 + C32 K3.+ + Cm2 KCm) + d3r (c23 + C33 K3 + *. + Cm3 Kim)................................. + dmnr (c2m + C3m3~ + * * + Cmm Kn) = d22 C21 + d23 C3r +... + d2an Cmr + K3 (d32 c2r + d33 C3r +... + d3m Cmnr) 4.................................... + Km (dm2C21' + dm3c3r +... + dm Cmr) C= 2r- + c3r 03 + C4rT4 +-... + Cmr 0m; 456 456 ~~~MULTIPLE ROOTS AND[1. [145. and therefore r 03 O C+ ~~~- 2r+C~~~~~~~~~~~~~~~r~~~~~~C~~~~~~~r~~-+... 0m,M2 holding for all values of r. Comparing with the earlier equations in c, we have for all values of r; and thus the second set of equations is '~r= d22 +d32 K3~.. ~dm2 icn fK,.=drd + d,,K3 +.. dmrl~m, (r=3..,i). Eliminating the quantities K, we have so that 0' is the value of '~r. Now consider the integral X2 (Z) =f2 (z) + K3f3 (z) +.. ~ KMfrn1 (Z). We have x2 (Z~4-) f2 Z+(z 0)~+ K3f3 (Z + 60) +... ~/4nf"(Z~W () = (c21 + c3C'31 +... + /cm CM) 0b' (Z) ~ 0X2 (Z) = a20b1(Z)~0yO2 (Z), say, and = (d2l + K3d31 +... + Kcmdma) 0f1 (z) + 0IX2 (Z) = b2p1 (Z) + O'X2 (Z),I say. When a2 and b2 vanish, X2 is doubly-periodic of the second kind: but in the general case, a2 and b2 are distinct from zero. The property X2 J( + W) + (O'1 X2 {(z + Co) + o}1 leads to no relation between a2 and b2. If the multiplicity of 0 as a root of &i~ (0) = 0 and that of 0' as a root of &2' (0') = 0 be greater than 2, so that 0 and 0' are multiple roots of {~1 (x) = 0, f2,' (x) = 0, we proceed as above. The newly obtained integral X2 is used to modify the fundamental system by replacing f2 say, so that the system consists of ASSOCIATED INTEGRALS 45'7 Then, in the same way as above, it is proved that an integral X, exists such that X, (z ~ O) = a.s6O + c3X2 + 6X3, X3 (+to)= b341 - d3X2 + 0'X3. Since x. {(Z ~ to) + ' X3 {(Z + to)+ WO, we find, on substitution, b2C3= a2d3, so that we may take c3=Xa2, d3= xb2, where X is any parameter. This parameter may clearly be absorbed into X3 by taking X3. X, and also into a, and b3 by division. Thus our integrals 0,, X2, X3 are such that X2(Z +to)= a2q1 ~ X y3 (z + t )= a3b1 + a2X2 + OX3, 0y (z + to') = b101, X2 (Z + 10 = b2ol + O'X2, X3 (z + O)') = b3p, + b2X2 + O'yX. And so on, until a number of integrals is obtained equal to the lesser of the orders of multiplicity of 0 and 0'. Thus the next integral is X, (say), where X4 (Z + at) = a4b1+ (a3+ Xa2) X2+ ~ a2X3 + 0X4 X4 (Z + to') = b4c + (b8 + xb2) X2 + b2X3 + 01X4 146. From these descriptive forms, we can proceed one stage towards the construction of an analytical form of the integrals. For this purpose, we introduce (as in ~ 144) the functions U (z) Z-~(z) v +z)=~ rrc) O /z 27() 3 ri w 2- i where the doubtful sign is the same as that of the real part of t'~ito; we have U (z +co)= it (Z) + v to)= (x)1 u(Z + to')= U(z)+ ii V(z ~ )V Then the various integrals can be expressed as non-homogeneous polynomials in u (z) and v (z), the coefficients of which are doubly 458 ANALYTICAL FORM OF INTEGRALS [146. periodic functions of the second kind, with 0 and 0' for multipliers. In particular, the integrals have the form 1 (z) = F (z), X2 (Z)=1() = + 11F (z), x (z) = F3 (z) IF (+ () + F (z), X4 (Z) == F4 () + I1F3 (z) + J2F2 (z) + 6I3F1 (z), and so on. The functions F are doubly-periodic of the second kind with factors 0 and 0'; I1 is a polynomial of the first degree in u (z) and v(z); I2 and J2 are polynomials of the second degree in the same quantities, having J12 as the aggregate of their terms of the second degree; Is is a polynomial of the third degree in the same quantities, having I3 as the aggregate of its terms of the third degree; and so on. To prove this, we note in the first place that l (z) is a doublyperiodic function of the second kind with the multipliers 0 and 0'. As for X2 (z), we have X2 + w) X2 () a2 01 (z + ) 1(Z) 0 X2 (z + W) X 2 (z) b2 01 (Z + ~') 01 (Z) 0 If therefore we take the function p21 (z), where a b2 p21 (Z) = 2(Zj + 2 U (Z)\ we have 0X(Z + (Z + 1 ). -p21(z) =x. (z + ')-P21 ( z + O'); (~1f(* -{ - /-21 (z + Ol); and therefore X2 (z) X2(Z) _P21 () 01(Z P is a doubly-periodic function of the first kind. Let F2(z) denote the product of this function and 01(z); then F2(z) is doublyperiodic of the second kind with multipliers 0 and 0', and we have X2 (z) = F2 (z) + p21 (z) 01 (z). 146.] ASSOCIATED WITH MULTIPLE ROOT 459 Similarly, if P31 (z) = {2 () - (Z) + { () - - (Z), we find X3 () X (z) X3 p. (Z) + P21 (z) + 31 (z) to be a doubly-periodic function of the first kind. Let F3(z) denote the product of this function and 1 (z); then F3 (z) is doubly-periodic of the second kind with multipliers 0 and 0', and we have X3 (Z) = F3 (z) + p21 (z) F2 (z) + {P212 (Z) - p31 (Z)} 1 (). Similarly, after reduction, X4 (z) = F4 (z) + p21 (Z) 3() ( z) -+ P4 (z)} F2 (Z) + {6P 213 () - P42 (Z)} 01 (Z), where F4 (z) is a doubly-periodic function of the second kind with multipliers 0 and 0', p41 (z) is a polynomial in u and v of the first degree, and p2 (z) is a polynomial (not homogeneous) in u and v of the second degree. And so on, in general: the theorem is thus established. CONSTRUCTION OF INTEGRALS THAT ARE UNIFORM. 147. Further progress in the effective determination of the analytical forms of the integrals on the basis of the foregoing properties is not possible in the general case. When particular classes of limitations are imposed upon the coefficients in the original differential equation, such progress might be possible: but it frequently happens that some more special method leads more directly to the solution. The simplest case is that in which the equation possesses a uniform integral, or in which the equation has several uniform integrals: but, of course, the preceding investigations in ~ 141 -146 apply to all equations of the type considered, whether they have uniform integrals or not. When all the integrals are uniform (and this can be determined independently by considering their forms in the vicinity of the singularities), then the 460 EQUATIONS HAVING [147. doubly-periodic functions of the second kind arising in the preceding investigation are uniform functions of z; and a general method of constructing such functions is known*. Instead, however, of using the preceding results, it sometimes is more convenient and more direct to infer the irreducible singularities of the integrals from the differential equation itself. These are used to construct an appropriate uniform doubly-periodic function of the second kind; the remaining quantities needed for the precise determination of the integral are then inferred by substituting the expression in the differential equation. Ex. 1. Consider the equation t d3w dw d — 3{2p (z) + a) +W=0, with the usual notation for the Weierstrassian elliptic functions; a and 3 are constants. The only irreducible singularity that an integral can have is z=0. The indicial equation for z= 0 is n (n- 1) (n- 2)-6n=0, the roots of which are -1, 0, 4; and the expansions that respectively correspond to the roots are easily proved to be w1 — +1 az + 112 +2 (410 g - a2) 3+... W2 1 + _L_ 'z3 + I/ a~z5 +..., W3=z4 +- aZ6 +.... Thus no logarithms are involved; every integral is a uniform function of z, being of the form A w1 + Bw2 + Cw3; and at least one integral of the equation is thus a uniform doubly-periodic function of the second kind. We proceed to its construction. This doubly-periodic function of the second kind cannot be devoid of poles, if it is to involve the first of the above integrals in its expression. (If it were devoid of poles, it would also: be devoid of zeros in the finite part of the plane: and then (I.c.) it could only be an exponential of the form eKz, which is manifestly not a solution of our equation.) It has one irreducible pole; it therefore has one irreducible zero in the finite part of the plane. Let the latter be denoted by - a, which at present is unknown. We now consider ~ the elementary function o-(z+a) e\z a- () a- (a) * T. F., ~~ 137-139. t It is a modified form of an equation given by Picard, Crelle, t. xc, p. 290. + T. F., ~ 139. ~ T. F., I.c. 147.] UNIFORM PERIODIC INTEGRALS 461 which has z=0 for an irreducible simple pole, and - a for an irreducible simple zero; its expansion begins with z-l, and the function must therefore agree with the integral above obtained in the vicinity of z=0. (The constants X and a determine, or are determined by, the multipliers of the periodic function; but at present these are unknown, and so X and a must be determined in another manner.) To expand the above function in powers of z, we have cr(z +a) a' (a) z2 a" (a) Z3 a"' (a) a o- ( (a) 2! - (a) + 3! o-(a) =l+ + zl (-2 - )+ I z3 (C3-3 - Je Y)+ the Weierstrassian functions on the right-hand side being functions of a. Also ( 1 4,4 -- () =ze Tg2 -;. and therefore w= a(z+a) eXZ a- (z) a (a) =- (X+)+ 1 + {(X +)2- + Z2{(X+ )- 3 (x + o- } 24 73 + 24 {( + C)4- 6 (X+ ()-4 (X + C) 3, 32+ ~2} +... This is to satisfy the differential equation, so that it must be of the form Awl + Bw2 + Cw3. Clearly A = 1, B=X +, for this purpose: the value of C would be needed for the complete expression, but we merely require X and a at present. Comparing the coefficients, we thus have A=l, B=X+(, Aa = (X)+- )2 - 1 A/= (X + ()3- 3 (X + -() l - so that X and a are determined by the equations (X + ()2 - (X +~ ) - 3 (X +: ) ~0 - O' = We have (a-2g ' where a is determined by the relation ' + (3a2 — g2) 2 - a3g=3 0 The function on the left-hand side is a doubly-periodic function of the first kind: it has a single irreducible pole, which is at z=0 and is of multiplicity three. Hence it has three irreducible zeros, say al, a2, a3; and their sum is congruent to 0, so that we may take a + a2 +a,= 0. 462 EXAMPLES [147. In general, al, a2, a3 are unequal, because a and 3 are general constants; the discussion of the critical conditions, that lead to equalities between al, a2, a3, and of the consequent modifications in the complete primitive, is left as an exercise. Let l' (a,)+~+ X?=- ((ar) (-=1, 2, 3); then Wr a (z +ar) eXrZ o- (z) o- (ar) is an integral of the equation for each of the three values of r. The primitive of the equation is w= (z a,) X z r=1 o- (Z) a (a,t) where AD, A2, A3 are arbitrary constants. Ex. 2. Obtain the relations which express the integrals wl, w2, W3 of the equation in the preceding example in terms of Wl, W2, W3; and determine the multipliers of the integrals. Ex. 3. Obtain the primitive of the equation (P' + P2) w" - (_P' + PI" - Yd3) w' + (p'2- g42g)' - pI ) W=O, in the form w= Aes (Z) + B (z). Ex. 4. Verify that the primitive of the equation d2y k sn x cn x d 2y d2 dx2 dn x dx is y = A cos (n am x) + B sin (n am x). (Gylden.) Ex. 5. Prove that, if I be an odd function and J be an even function, both doubly-periodic in the same periods, the integrals of the equation d2w dw -d- + Iz+ Jw =O dz2 dz can be expressed in terms of dp (z). Hence (or otherwise) integrate the equation W" + Pe'2 _g" V'- I QIo = o. Ex. 6. Determine the relations among the constants (if any) in the equation w"' + (a - 3p) w' + (y +, /3 - 2p') w =, in order that every integral of the equation should be uniform; and assuming the relations satisfied, shew that the equation has three integrals of the form r (z+a) Az a (z) or (a) 147.] EXAMPLES 463 Ex. 7. Shew that the equation y"' + (a - 3k2 sn2 x) y' + (3 + yk2 sn2x- 3k2 sn x cn x dn x) y = has an integral of the form H (x.+ o) ex {x-Z ()} 0 (x) provided 3a+y2 = 3 (1 +k2); and that it then has three integrals of that form. Obtain these integrals. (Mittag-Leffler.) Ex. 8. Obtain the integral of the equation dx3 + (h - 6k2 sn2 x) dY+ hly= in the form H (x + ) x{ - z (,)} "~ (x) e where the constants X and c are given by the equations h- (1 +k2) + 3 (X2 _ ks2 n2 o)=0, 2X3- 6Xk2 sn2 o + 2X (1 + k2) - 4k2 sn co en Co dn o - h = 0. Verify that, in general, three distinct integrals are thus obtained. (Picard.) Ex. 9. Prove that the equation dx4 +(a 12k2 sn x2 + d + (y + k2 sn2 )y= has an integral of the form H (x + o) ex{_-Z(W y" (x) provided 2a+=8 (1 +k2); and that, if this relation be satisfied, it has four such integrals. Obtain them. (Mittag-Leffler.) Ex. 10. Verify that the equation (sn2x-sn2a) dy -2 sn n dn -+2 1 - 2 (1+ k2)sn2a+3k2sn4 a=l has an integral of the form (x + co) ex~_-Z(w), 0(x) provided sn co n co dn co sn4 a (2k2 sn2 a-1 - k2) -— sn scn asn2Co,sn2 CO = sn2 a sn2 sn = 3k2sn4a - 2 ( + k2)sn2a+ l; and obtain the primitive. Hence integrate the equation d2), d- =u (6k2 sn2 X + h), wher h s acontan. (ermte. where h is a constant. (Hermite.) 464 LAM 'S [147. Ex. 11. Discuss the equation d4w d2w dw 74 +{a - 12 } (z)} d2+^ z + y +{+ ()} w=O, for those cases when every integral is a uniform function of z. Ex. 12. Shew that there are three sets of values of the constants a and b, for which the equation 3dY2 l / (x)+7" (x) + a (x) + b y admits as an integral a uniform doubly-periodic function; and obtain the integral. (Math. Tripos, Part ii, 1897.) Ex. 13. Prove that the equation y"" - 2n (n + 1) y" (z) - 2n (n + 1) y' p' (z) +n (n + 1) ( 3) (?- 2) V" () + } 0 where a is an arbitrary constant and n is a positive integer, has a uniform function of z for its complete primitive. (Halphen.) Ex. 14. Construct the equation which has w = a + a2-F (z) + a3 ' (z) a4 " (z)}/ (z) for its complete primitive and, for a properly determined value of f(z), is d3w devoid of the term in -3. Likewise construct the equation which has = {a +a2 sn z + a cn + a4 dn }f(z) for its complete primitive, with the corresponding determination of f(z) to d3w remove the term in --. In each case, the quantities a,, a2, a, a4 are to be regarded as arbitrary constants. (Halphen.) Ex. 15. Prove that the primitive of the equation w-"'- 'n2W' P (z) - -n (n+3) (4n-3) iv' (z)=O is a uniform function of z, when n is an integer multiple of 3; and discuss the primitive, when the integer n is prime to 3. (Halphen.) LAME'S EQUATION. 148. One of the most important instances, in which a differential equation with uniform doubly-periodic coefficients has a uniform doubly-periodic function of the second kind for its integral, is Lame's equation or, rather, the more general form of 148.] EQUATION 465 Lamd's equation as discussed in the investigations of Hermite, Halphen, and others. The form used by Hermite* is 1 d2w - d = n (n + 1) k2sn2 + B, w dz2 where n is a positive integer and B is a general constant; the form used by Halphent is 1 d2W d =d Z (n+ l)(z)+B, with the same significance for n and B. We shall use the latter form of equation: it is selected for convenience and for its slightly greater generality owing to the functional independence of the periods. The mode of discussion is the same for the two forms. As we are concerned with the application of the general theory+, rather than with the special properties of the functions defined by Lame's equation, only an outline of the solution of the equation will be given here. The detailed developments, and references to further memoirs, will be found in the authorities just quoted. It may be not without interest to indicate how this form of equation arises from the equation a v a2V a2V. -+ ~- 0, ax2 ay2 a2 characteristic of the potential in free space. When orthogonal curvilinear coordinates a, /3, y, as defined by three orthogonal surfaces a(w, y, z)=a, (x, y, z)=, = a, (x, y, z) =, y, are used, then the equation becomes a a\ a B a a C av Da \BC Da ) + a8 CA Do3) a 8y AB 87y) * "Sur quelques applications des fonctions elliptiques," a separate reprint (1885) from the Comptes Rendus. t Traite des fonctions elliptiques, t. iI, ch. xII. + That is, the theory of the uniform doubly-periodic functions of the second kind which are integrals of the differential equation. It has been proved (~ 54) that, by an appropriate transformation, the equation can be changed so as to be of Fuchsian type. F. IV. 30 466 SOURCE OF [148. where A2= \~/ + ~(/ay + \aZ/, A2= (2Y)2 + (8)2 + (8) 2 confocal with a given ellipsoid; and let X, /, v be the roots of the equation 2 a Y2 a)3 aa z2 B + + a.2 + b2+ c2+O=. a cubic in 0. Then* we take a2 + (a) -e + X = p( (a - e2, c2 + X = p (a) -- e, a2+t= (I) - ei, b2+ p= -(/3)- e2, c2+ = (a3)- e3, a2 + v = (y) - e,, b2 + v = (Y) - e2, C2 + V = P ( - e3. a +v=p(7)-el, b +v=p(/)-e2, C2+u=p(7)-e3. Now p2 (a) A=2 =a2 /aX\2 (ax2 )A - + ~ +() =4p 2 where pA (a2 + X)2 (b2 + )2 (2 + X)2 (X- h) (X-V) (a2 + X) (b2 + X) )(2 + X) _ 4 (-,) (X - ) Y- 2(a) ' hence A2 1-= 1 ( - ~) (X - v)' Similarly 1 C 1 B2 = C = (k- X) (i- V)' (_- x)(v - ); so that the equation for the potential becomes a ( - V) a a, (V, - } X) * Greenhill, Proc. Lonld. MiLath. Soc., t. xvIII (1887), p. 275. 148.] LAMP'S EQUATION 467 or, what is the same thing, a2V a2/ aV.2V -v()- ()} + () - ((c) )} ) + {P (a) - (/3)} a = 0 For the purposes contemplated in the transformation, the quantity V is the product of a function of a, a function of /3, and a function of y, or is an aggregate of such products; and it is a uniform function of its variables. Hence, writing V =f(a) g (/3) h (y), where j, g, h denote uniform functions of their arguments, we have I dcf d-g {g (/)-e (Y)}j fa + ({ (Y) -P (a)} da' + - '=0. + k (a)- (i)} ~ d = o. Thus 1 d2w - - = A p (z) + B, w dz2 where w =f when z = a, w =g when z = 3, w = h when z = y, and A, B are constants independent of a, /3, y: they must be such as will, if possible, make w a uniform function of its argument. The only possible singularities of w are z = 0 and points congruent with z = 0; hence, after the earlier investigations, we consider the irreducible point z = 0. The form of the equation shews that it will be an infinity of w; and thus it must be a pole, say of order n, where n is a positive integer. Thus we have, in the vicinity of the pole, a0 a 17 w = n + n-l + R (z), d2w n1 (n + 1) aO n (n +1) wdz) an+2 n ns o z s where R (z) and R, (z) are regular functions of z, such that R_ (z= 1 + powers of z. R(,) Hence, in the vicinity of z = 0, we have 1 d22W n (n + 1) w d2 - n - (1 + powers of z), uw dz2 z2 30-2 468 INTEGRATION OF [148. and therefore A =n(n + 1), a limitation upon the form of the constant A. But there is no limitation upon B, necessary for the existence of integrals of the type indicated; and therefore the differential equation may be taken in the form as stated. To obtain Hermite's form, we write f{)6 elS - e 2 e2 - e3 g (z)- e3= en - e I y = z (el - e,), k2 -e sn2y ' el - e3 as usual, and then take y = x + iK'; the equation becomes I d2W d2 = n + - 1) k2 snx + B", where B" is a constant. 149. The method of solution of the equation is based upon the knowledge that there is at least one integral in the form of a doubly-periodic function of the second kind: the limitations, that have been imposed upon the equation, secure that this function is uniform. Moreover, the integral has only one irreducible pole, viz. at z= 0, and the pole is of order n. There are two modes of using these results in order to construct the integral. By one of them, we use* the further property that a uniform doubly-periodic function of the second kind has as many irreducible zeros as it has irreducible poles, account being taken of the orders of the points in each category. Accordingly, in the present instance, the integral has n irreducible zeros: let them be - a, - a2,..., - a,. Consider the uniform function = (z a+ a) (z + a2).. +o z (+n) on (z) which is doubly-periodic of the second kind; its (single) irreducible pole is of order n and is at z = 0; and it possesses the * T. F., ~~ 139, 141. 149.] LAMJ'S EQUATION 469 necessary n (unknown) irreducible zeros, so that it is of a suitable form. We have 1 dw n 1 dw = P - ^(z) + (z+ a,.). w d? r=1 In order to simplify the right-hand side, it is convenient to take n p=X- S (a,); r=l and so 1 dw - -l = X + s { (z-q-a)- + a (a,) - n$(z). w dz r=1=1 Hence 1 d2w 1 (zdW\2 n w dz2 W2 Kdz) r=l But 1 dw _ _ _ _ _ _ _' I d- + - - (ar) - (z) w dz 2 r (a,.)- f (z) ' and therefore 1dw2 = g J_(ar) _(Z)2 2 Wdz] — 4 1r=l g(ar)- J (Z) +~ n ~ (ar) /)- g(*,) -j)-' (z) + 1 2 r=l 1s= (a,.) - (z) ' (a,) - (z) + x '(a)- () + X2. =l p (ar) - (z) The first term on the right-hand side is equal to I {(z + a.) + ) (a,.) + d (z)}. - '=1 To modify the second term, where the summation is for pairs of unequal values of r and s, we have ' (a,)- ' (z) ' (a) - g' (z) (aor) - (Z) ' g (a,) - P (z) 4V3 (z) - g2Z (z) - g3- o' (ar) (' (as) - d' (z) {' (ta) + g' (as)} {g (ar.)-go (Z)} {g (a.)- (z) = 4 IV (z) + g (a,)+ ' (as) lr ( (a.) ' ~3- ' ( *G () } ' after easy reductions, where L _ p (a,) + (as). '~ ((ar)- (a) 470 LAM~'S EQUATION [149. and thus the second term in the expression for2 (- ) becomes 2 2w2dz n (n - 1) p (z) + 2 (n - 1) p (a,.) + - -(a,)- () ' LI,, r=l r. l P -(z)- r (a,) s=1 where the summation is for all values of s from 1 to n except only s = r. Then 1 d2w n u d2 n ( + 1) P( ) + (2 n- 1) > p (a)+ X2 W d Z2 2 -" I 2 -? - -r 1~d S ( )- ' (x) {C' L - 2X} ~ i (z) - p(ar) s=1 Comparing this result with the differential equation under consideration, we naturally take ' L,, = 2X, s=1 for all values of r, that is, ' (a,) + ' (a2) ' (a + ' (a) 2 g (a,)- (a2) + (a,)- g (a3) p' (a2) + p' (al) +' (a2) + p' (a3) + P (02)- lp (al) (2) - p (a3) (an)- p (al) ' (an)- p (a,) and (2n -1) E g(a,.)+X2 =B. r=1 Adding the n former equations together, we have 0 = 2nX, so that X vanishes. Hence if the n quantities ac, a2,..., a, are determined by the equations p' (a) + pi (a2) + (a,) + ~' (a3) + +..., p (al) - o (a2) O (al) - ) (a3) V' (a2) + ' (a ) )' (a2) + p' (a3) + 0 gp (a2)- (al) p (a2)-P p (fas) (which are equivalent to only n-1 independent equations, because the sum of the n left-hand sides is zero) and by n (2n- 1) 2 g (a,.)= B, 2=1 149.] INTEGRATED 471 then Fz (Z + a,) o ( + a2)... * (z + a,) -z ~l(a. an (z) is an integral of Lame's equation 1 d2w w ad2 = (n + ) () + B. The equation remains unchanged when -z is written for z; hence F(-z) is also an integral. Save in the case when the constants a are such that F(z) and F(-z) are effectively the same function, we have two independent integrals of the equation, which therefore is completely solved. 150. Another method of arranging the necessary analysis is as follows. Consider the equation 1 d2w F w dz2 where F(z) is a doubly-periodic function; by Picard's theorem (~ 142), an integral (say w1) is known to exist in the form of a doubly-periodic function of the second kind. If then we write 1 dw' the quantity is a doubly-periodic function of the first kind; and the quantity v is a doubly-periodic function of the first kind; and it satisfies the equation dv dv + v2 = F(z). The irreducible poles of wi, in their proper order, are known from the singularities of the original equation; let them be n in number, account being taken of multiplicity. Then each of them is a pole of v, of the first order; and the sum of their residues for v is - n. The number* of irreducible zeros of w, is also n, account being taken of multiplicity; each of them is a pole of v, of the first order, and the sum of their residues for v is + n. We therefore construct a uniform doubly-periodic function of the first kind, having these poles, all simple, viz. the known poles arising through the singularities of F, and the unknown poles * T. F., ~ 139. 472 EQUATIONS HAVING [150 arising through the zeros of w1, taking care to have -n and + n for the respective sums of the residues. The general expression for such a function is known*: when substituted as a trial function in the above equation, comparison of the results leads to a determination of the constants. As an illustration, consider the equation 1 d2w - dz- = 2g2 (z) + B. w dz2 The irreducible pole of (z), viz., z = O, is the only irreducible pole of w, and it is of the first degree. Accordingly, it is a simple pole of v, with a residue - 1. Further, there is (by the preceding argument) only one other pole of v: it is simple, and has a residue + 1. As v is a doubly-periodic function of the first kind, an appropriate expression is v = ( -c) - C (z) + k = (z - c)- (z) + ~(c) + b, say; and b, c have to be determined by substituting in the equation dv dz + v2= 2 (z) + B. Now dv = go (z - c) + P (z) = 2 (z) + e (c) - 4 { (z) - ' (C)}2 and by the addition-theorem for the '-function, we have v-b+141 (Z) + p' (C) v = b + 2 g (z) + ' (c) 2 P (Z)- f (C)W Substituting, we have 2 (z) + p (c) + b + b (z) + = 2g (z)+ B, p (z) - go (c) which must be satisfied identically. Accordingly, b=0, (c)=B; and thus, with a known value of c, v= (z - c) - (z + (c), * T. F., ~ 138. 150.] 160.] ~DOUBLY-PERIODIC COEFFICIENTS47 473 so that r- (Z) There are two values of c, equal and opposite: the construction of the primitive is imimediate. Ex. 1. Shew that two independent integrals of the equation 1 d2wV in the case when B =el, are given by {9p (z) - ell (9d(z) - ei} { z+w + elz4 and obtain the integrals in the cases, when B =e2, and B =e3, respectively. Ev. 2. Obtain the primitive of the equation 1dy2k2 sn2x-a, (where a is constant), in the form H= ( H~+a)_-zZ(a) +BH~x-a) ezZ(a) yA J(X) Go where dn2 a=a-k2. Discuss the solution in the three particular cases a=1I+k2,I 1, k2. (ilermite.) Ex-. 3. Shew that a (z +a,) o-(z +a2) a-(z +a3) ej(j-~a)+a) o3 (Z) satisfies the equation _1 dz2w if 9- (a,), P (a2), ~9(a3) are the roots of the cubic equation 463 - 12b02 +(24b2 -g2) - 60b3 +4g2b-g3=O; and deduce the primitive. (llalphen.) Ex. 4. Shew that the primitive of the equation d2W can be expressed in finite form for appropriate values of the constant B in the following cases: 1. When it is an even integer, = 2m, then either w = arnkm (z) +am l% P91- 1(Z) +...ao or wv= [{9d (z) - e~} (9~(z) - e~j11{crni ml (Z) +... -+Co}b where e., e1A are any two of the three constants el, e2, e3: 474 EXAMPLES [150. IIL When n is an odd integer, =2mn - 1, then either w={z)( ) 2 ex}3{C?n- 1 ()+.... +coj, or w= 'd' (z) {b,, - 2. 2 (Z) ++... + bo} where el, is any one of the three constants el, e2, e3. IDetermine the number of solutions of the specified kind in each of the cases indicated. (Crawford.) Ex. 5. Shew that an integral of the equation I d2WV k2 2A (n+ 1) = n (n1,+ s n2 12X+i W dz2 sn2r where lb is a constant and m, n are integers, can he expressed in the form m+n W 1 (Z - Z1) 01 (Z - Z2)... 01 (Z - Zin + ne ~ U. {O, (x)}'" {O (2)}m in the usual notation of the theta-functions, z1, zD..., I Zin + being appropriate constants. Obtain the primitive. (M. Elliott.) Ex. 6. Obtain the primitive of the equation d2W wU d2 4w 2 (z) _{4 (z) - 2g2 9d (z) - 3g3} in the form ={(~' (z)}t {A eS2(z)+B&}e - S). Ex. 7. Shew that there are two values of k0, for which the equation 1 d2w w z = k+ok1 V(z) - 2mg )' (Z) +Im2 gd (Z), where mn is a constant, possesses an integral of the form w -e az+n(z)n (z b) W=e T (Z) and, for each such value, obtain the primitive. (Benoit.) Ex. 8. Shew that there are n + 1 values of k0, for which the equati on 1 d12w w dz2 =ko~k1,k (z) -in (n +1) ga' (Z) + 1n22 g4" (z), where m is a constant and n a positive integer, possesses an integral of the form H (z - br) r=1 e - az4-mr(z) Orn (Z) Prove also that, if the right-hand side of the differential equation be increased by P (z), where 'P is a doubly-periodic function of the first kind having all its poles simple, a corresponding theorem holds as regards the integral, if ko be properly determined. (Benoit.) 150.] ALTERNATIVE PROCESS 475 Ex. 9. Integrate the equation 1 d2w _ 1 (a + 1) +' (/'+ 1) dn' M " + ("n + 1) k2 cn2 x (l w dz2 sn2 cn2 dn2x+ where /A, /', /', n are positive integers. (Darboux.) 151. The other mode of utilising the known properties of the integral, when it is a uniform doubly-periodic function of the second kind, is to obtain the actual expansion of the integral in the vicinity of its irreducible pole and thence to construct its functional expression in terms of the elementary function a (z + a) ez a- (z) where a and X are initially unknown constants. Some indication of the process is given in Ex. 1, ~ 147; but a slightly different form will be adopted for the present purpose. We take the elementary function in the form z = (z + a) ez{P- (a)} o (z) o- (a) where p and a are now to be regarded as the constants to be determined. The expansion of this function in the vicinity of its irreducible pole at z = 0 is G (z) = + p + I {p2 -P (a)} z + {p3 -3pp (a)- -' (a)} Z2 + - {p4 - 6 p2g (a)- 4p2' (a) - 3g2 (a) + 5g2} 3 +.... If, in the same vicinity, an integral of the differential equation exists in the form u= (_)n- a, + 1(- 2( 1n 2) an-1 + zn n... + + a0 + positive powers, z then we may take d~n-G dn-2G w = a d + an- dzn +... + al G, where a comparison of expansions serves to determine the constants a and p. The integral thus is known. 476 EXAMPLE [151. An illustration will render the details clearer. In the case when n = 2, the equation is 1 d2w w d 2 = 6p (z) +B. Let w -+ a + a, + a2Z + a.z 2 + 2 2 be substituted in the equation; we find ao= O, a2 =O, a, = 0,... a,=-IB, a3=I B2 -3g2,... so that W -'-~B + (-B2 _ 3 Z22+ z2 62 24 402) Manifestly, the form to take is dG dz' and then comparing the two expansions, we have { p2g (a)} -I B, E 6R p3_- 3pgd (a) - VY(a)=O. These equations give B3 + 2y7g3 gd (a) 9B2- 27g2' ' B - 6B- (a) The former in general leads to two irreducible values of a; the latter uniquely determines p for each of these values of a. Denoting the two values of a by a and - a, and writing a- ( + a) e1 {6 ) (a)} a (z) a) (a) cr(z -aC) -Xi3'()-c = (2) a (a) the primitive of the differential equation is d+, MdG2 w=L dz dz 151.] EXAMPLES 477 Ex. 1. Discuss the integral of the equation 1 d2W when a, as obtained in the preceding solution, has the values 0,,co', co" respectively. Ex. 2. Prove that an integral of the equation 1 d2,W 12 P(z) +15b is given by dZ2 where a Z) (z +a)ezp-W o- (z) o- (a) the constants p and a being given by the equations l2b k.) (a) + 20b2 +' ~32=(0 1 ~ A)3, __- kd' (a) AP-2P(a) - 4b' IDeduce the primitive. CHAPTER X. EQUATIONS HAVING ALGEBRAIC COEFFICIENTS. 152. THE differential equations, considered in the preceding chapters, have had uniform functions of the independent variable for their coefficients. We now proceed to consider (but only briefly) some equations without this limitation: one of the most important classes is constituted by those which have algebraic functions of the variable as their coefficients. For this purpose, let y denote an algebraic function of the independent variable x, defined by the equation * (X, y)= 0, where F is a polynomial in x and y, and the equation is of genus p. With this algebraic equation we associate the proper Riemann surface of connectivity 2p + 1. We assume that the linear differential equation has uniform functions of x and y for its coefficients, so that each of these is a uniform function of position on the surface: and we write the equation in the form dmu dint-h dm-2u dxm al (x, ) dx1 + a2 (x, Y) d- +- + am (x, y) u =0. Let a, (x, y) dx W -= Wue J the exponential in the factor of u on the right-hand side being an Abelian integral; then the equation for w is dmw dm-2w d-3w, dxtM P, ) + (X, (, y) d-3 +' + -P (x, y) = 0, 152.] ALGEBRAIC COEFFICIENTS 479 devoid of the derivative of order m-1; all the coefficients P2,..., Pm are algebraic functions of x, and are uniform functions of x and y. This is the form of equation which will be discussed. Let (x0, y,) denote any position on the surface, which is not a singular point on the surface and in the vicinity of which each of the coefficients P is regular. Then an integral exists, which is regular everywhere over a domain in the surface, and is uniquely determined by the assignment of arbitrary values to w and to its first m - 1 derivatives at (x0, yo). In fact, all the results relating to the synectic integrals of an equation with uniform coefficients hold for the present equation in the domain of (x0, y,). Next, let account be taken of the singularities of the equation = 0 and of the associated surface. As these affect all the coefficients of all differential equations of the class considered, and thus afford no relative discrimination among the functions defined by those equations, we shall assume them simplified as much as possible before proceeding to consider the properties of the functions. Accordingly, we shall suppose that, if the equation r = 0 (or the Riemann surface associated with it) possesses a complicated singularity, it is resolved* into its simplest form by means of birational transformations, so that we may write x-e = g (, qr), y-f= h (:, t7), where g and h are uniform functions which, in connection with =0, admit of uniform expressions for: and V in terms of x- e and y -f, and are such that = 0, r = 0 is an ordinary position on the transformed Riemann surface. The positions on the surface, that have to be considered in connection with the differential equation, are now ordinary positions: and therefore, in dealing with the theory of the equation, no generality is lost if we assume that the singularities of the equation r = 0 and of the Riemann surface are ordinary positions for the integrals. (Of course, in any particular example, it may happen that a multiple point on the curve f = 0, or a branch-point of the associated surface, is definitely a singularity of the equation. In order to discuss the nature of the integrals in the vicinity of such a point, we taket x-e=0q, y-f= PS(E), * T. F., ~ 252. tf. F., ~ 97. 480 FUNDAMENTAL SYSTEM [152. where p and q are integers, and S is a holomorphic function of its argument that does not vanish when %=0; and then we investigate the character of the integrals in the vicinity of '= 0.) Lastly, let (a, b) denote a position on the Riemann surface (being a pair of values given by the differential equation) such that the coefficients of the equation are not regular in the immediate vicinity of (a, b); after the preceding explanations, we may assume that y - b is a holomorphic function of x - a in the immediate vicinity of the position. The character of the integrals in that region is determined, after substitution of y- b in terms of x - a, in association with an indicial equation; and the general processes of the theory, in the case of differential equations with uniform coefficients, are applicable to the integrals in the vicinity of (a, b). As in that earlier theory, we have a fundamental system of integrals existing at any ordinary position on the surface, the system being composed of mn linearly independent members. Continuation of these integrals is possible: and by taking all admissible paths from one ordinary position to any other ordinary position (care being taken to avoid the actual singularities), and assuming an arbitrary set of initial values at the first point, we shall obtain all possible integrals at the second point. Similarly, by taking all possible closed paths on the Riemann surface, which begin at an ordinary point (x0, y0) and return to it, we obtain new integrals at the end of the path: and each of these integrals is linearly expressible in terms of the members of the initial fundamental system. A FUNDAMENTAL SYSTEM OF INTEGRALS, AND THE FUNDAMENTAL EQUATION. 153. Let w,, w2,..., w, denote a fundamental system at an ordinary position (x0, yo); and let the variable of position describe a closed path on the surface returning to (xo, yo), this closed path being chosen so as to include the singularity (a, b) but no other singularity of the differential equation. Suppose that the effect upon the fundamental system, caused by this variation of the variable of position, is to change it into OF INTEGRALS 481 wI', W2',..., WmM': then, as in the case of uniform coefficients, the latter set also constitute a fundamental system, and the two systems are related by the equations m W = A A, ( = 1..., ), A=l where the determinant of the coefficients a is different from zero. This determinant is (as in ~ 14) equal to unity. For let A denote the determinant of the fundamental system dm-lwl dm-lw2t dm-lwM dxn-l' dn-l' dxm-1 dm-2 Wl dm-2 u2 dmn-2wM dxm-2? dXn-2 ' " dnm-2 WI, W2,..., and let A' denote the same determinant in relation to the fundamental system w'; then, if A denote the determinant of the coefficients ax,, we have A' = AA. Now, because the term involving the (m - 1)th derivative of w is absent from the differential equation, we have, as in ~ 14, A = C, where C is a constant. Let the function A, which is equal to C in the vicinity of (x,, yo), be traced along the closed path which the variable of position describes on its return to (xo, yo); it is steadily constant, and its final value is A', so that A =C; and therefore we have A=1. Further, as in the cases when the coefficients are uniform functions of the independent variable, it is possible to choose a linear combination v of the members of the fundamental system such that, if v' denote the value of v obtained by making the variable of position describe the aforesaid closed path, we have vI = Ov. r. iv. 31~~~~~~~~ F. IV. 31 482 FUNDAMENTAL EQUATION [153. The multiplier 0 is a root of the equation A (0) = -0, a, 0.., ai, a21, a22-0 ) * *., 2m aml, LmM2 a ** C Ma mm 0 = 1 + 1,0 + 120 +... +,M_-10- + (_ 1)mm = 0. This equation is independent of the choice of the fundamental system, so that its coefficients may be regarded as invariants of the linear substitution, which the fundamental system undergoes in the description of the closed path round (a, b). 154. If some, or if all, of the integrals in the vicinity of (a, b) are regular in the sense of ~ 29, then an indicial equation for the singularity exists; and if p be a root of this equation for an integral with a multiplier 0, then 0 = e27ip. If no one of the integrals is regular, there is no valid indicial equation. In the first case, the general character of an integral is determined by the value of p: and the explicit form is obtained by substituting an expression of the appropriate character so as to determine the coefficients. In the second case, various methods* for obtaining the value of 0 have been suggested, by Fuchst, Hamburger+, and Poincare~; the most general is the method of infinite determinants, due to Hill and von Koch, and expounded in Chapter vIII. Without entering upon details, it may be said briefly that many of the properties of linear differential equations having algebraic coefficients can be treated by processes that, except as to greater complexity in the mere analysis, are the same as for equations with uniform coefficients. It therefore seems unnecessary to discuss them at any length, as they would lead to what is substantially a repetition of a discussion already effected for less complicated equations. * See ~ 127. t Crelle, t. Lxxv (1873), pp. 177-223. + Crelle, t. LXXXIII (1877), pp. 185-210. ~ Acta Math., t. iv (1884), pp. 208 et seq. 154.] EXAMPLE 483 A systematic discussion of equations having algebraic coefficients and development of many of their characteristic properties will be found in a series of memoirs by Thome*. Ex. 1. Consider the equation d2w a +- w-O= 0 dx2 (ax + by + c)2 where the variable y is defined by the relation 2 +y2= 1, and a, a, b, c are constants. The position at infinity is a singularity of the differential equation in each of the two sheets of the Riemann surface. The integrals are regular in that vicinity in one sheet, and the exponents to which they belong are the roots of ( ( 1l +l) +b)= 0, (a + bi)2 provided a+bi is not zero; but, if a+bi=0, the integrals are irregular at infinity in that sheet. Similarly, they are regular in the vicinity of infinity in the other sheet, and the exponents to which they belong are the roots of (+ 1)+ (<x bi- = 0, (a - bo)2 provided a-bi is not zero; but, if a- bi=O, the integrals are irregular at infinity in that sheet. The other singularities of the equation are given by ax +by +c=O} x2+y2-_ 1 =0 When these are distinct from one another, let them be denoted by x=cos 0, y=sin 0; x=cos ), y=sinq. The integrals are regular in the vicinity of each position; and the respective indicial equations are P (P- 1) a = 0 -(a - b cot )2 -P (P- l)+ a 0. (a - b cot 0)2 When the two singularities coincide, let the common position be denoted by x=cos +, y = sin +; and then a- b cot 4= 0. In the vicinity, we have x = cos, + +,,2 3 cos y = sin +- cot -i-n + si n..., sin3 + sin6 + * Crelle, t. cxv (1895), pp. 33-52, 119-149; ib., t. cxix (1898), pp. 131-147; ib., t. cxxi (1900), pp. 1-39; ib., t. cxxII (1900), pp. 1-29. 31-2 484 EXAMPLES OF EQUATIONS HAVING [154. so that the equation is d2w 4asin6{ +cos -+... }-2 d-2+ b2 4 1+ sin2 + - w The integrals are not regular; but the equation may have one normal integral, and can even have two normal integrals, of the type 2i sin3 / aa e b~ af(~), where f is a polynomial in $. The forms, and the conditions necessary to significance, can be obtained as in ~~ 85-87. Ex. 2. Discuss in the same way the singularities of the same differential equation, when the irrational quantity y is given by the respective relations (i) +y3 = l (ii) y2=4x3-g2x-g3. Ex. 3. Let u, and u2 denote a fundamental system of the equation in Ex. 1, for y=( - x2); and let v1 and v2 denote a fundamental system of the same equation for y= - (1 - x2)5. Shew that the linear equation of the fourth order, which has ut, u2, v1, V2 as its integrals, has rational functions of x for its coefficients; and obtain them. Ex. 4. The equation dw d=Wo (x, y), where, (x, y) = 0, has its primitive in the form w = Kef (, y) dx It is natural to inquire whether an equation d2wo dw =, j (x, Y) can have an integral of the type w=ef (x, y)dx where w (x, y) is a rational function of x and y. A general method for such an inquiry has been given by Appell*, though it is not carried to a complete issue as regards detail: it will be sufficiently illustrated by means of the equation d2w ax + 3y dx2 y (ax+ by)2 where x2+y2= 1, it being required to find under what conditions, if any, the equation can have an integral of the form w=efw(x, y)dx * Ann. de l'Ec. Norm. Sup., 2me S6r., t. xII (1883), pp. 8-46. 154.] ALGEBRAIC COEFFICIENTS 485 where w (x, y) is a rational function of x and y. Since 1 dw w dx ' we have d+ ax +y dx y (ax + by)2 We assume that each of the quantities a+~3i, a bi, is different from zero. By adopting the method in the preceding Ex. 1, the integrals of the equation in w are easily seen to be regular in the vicinity of x= oo, so that they have the form w=XXR (I) where R ) is a holomorphic function for large values of x, not vanishing when x=co; and thus '==- + +..., X X+L in the vicinity of x= oo. Substituting in the equation for w, we have x_ a+oi i(a +bi)2 Now the infinities of w are included among the points (i) x=c, which has just been considered; there are two possible values of X in each sheet: (ii) /=0, with x=l, x= -1, which are the branch-points of the surface: (iii) ax+ by=0, in each sheet. Moreover, the zeros of w are unknown from the differential equation: but they must be considered, because each of them gives a pole of w. Let such an one be (iv) x=f, the number of such points being unknown. All these points, whether infinities of w or zeros of w, can be singularities of w. As regards the branch-points (ii), we may take y= r, x=l-7,2+..., in the vicinity of 1, 0, where 7 is small; and then 1 ds 2 a + w a2 - ] dq+ a"r so far as the governing term in w is concerned. If this be A.4 1o )n 486 EQUATION OF SECOND ORDER [154. where n > 0, then n+2=2n, A2+nA=O. Thus n= 2; and we can have A = -2, or A =0, as possible values. Similarly for the vicinity of - 1, 0. Next, at the two points (iii), where ax +by=O, we have x y 1 b -_ c.(a 2+b2) say x=sin +, y=cos +, a tan += -b. Then, in the vicinity, we take x=sin +,, y=cos + - tan ~ +..., so that ax + by= a (a -- b tan #) +... a2 + b2 -= a.... a Thus the equation is dz _ (ab - 3a) a 1 d- +r2= - _ _ d~S; (a2 + b2z)2 62X so far as the governing term in w is concerned. If this governing term be ao r (a2 + b2) 1 zr a ax + by we have ~2_ _ a (ab- 3a) (a2 + b2)2 Thus there are two possible values of ar at each of the two points. Lastly, as regards a point such as x=f in the set (iv), it is easy to see that, if the governing term in w be B (ad then 2n=n+1, B2=nB; that is, n=l, and either B=1, B=0, are possible values. This holds for every such point x=f and in each sheet. Our required function w (x, y), if it exists, is to be a rational function of x and y, and we have obtained all the singularities that, in any circumstances, it might possess. We accordingly must take some combination of the possible infinities, which are x = G, with any of the values of X, x= + 1, y=0, with either A = - 2, or A =0, ax + by = 0, with any of the values of r, x=f, with B=1, or B=0. 154.] HAVING ALGEBRAIC COEFFICIENTS 487 A possible form is clearly C ax + by' where C is a constant. We have (if this be admissible) C a + bi from the first of the possible infinities: we take A = 0 from the second: then a2+ b2 C-= a — a from the third: and we take B=0 from the fourth. Hence we must have a- bi a for some possible values of X and of o-: that is, 4ai-443U 2 a-bi f( 4a 1+1- 1 l-_ (ab -+ —) (1_ 4ai -4} a - bi ( + b2)2 the signs being at our disposal. This leads to a single value of f, viz. a2 a and the condition is satisfied by taking the negative sign on both sides. We then have a C=; so that, with the above value of 3, an integral of the equation d2w ax +ay dx2 y Y(ax + by)2 is given by b (ax+by) w=e b adJv Actual evaluation of the integral in the exponential can easily be effected. Of course, it would have been possible to discuss the particular equation by taking 2t - t2 Xl+t2 Y=l+t2' with t as the new independent variable; for the algebraic relation is of genus zero, and therefore* the variables can be expressed as rational functions of a new parameter. The new form of equation would then have uniform coefficients. But the foregoing method, that has been adopted, is possible for an equation +f (x, y) =0 of any genus. * T. F., ~ 247. 488 INTRODUCTION OF [155. ASSOCIATION WITH AUTOMORPHIC FUNCTIONS. 155. It is manifest that some of the complexity in the analysis associated with the construction of integrals, either in general or in the vicinity of particular points, would be removed, if the equation could be changed so that, in its new form, its coefficients are uniform functions of the independent variable. This change would be secured, if both the variables x and y in the relation (X, y)=0 were expressed as uniform functions of a new variable z. Now it is known* that, when the genus of this relation is zero, both x and y can be expressed as rational functions of a new variable z, which itself is a rational function of x and y: moreover, the expressions contain (explicitly or implicitly) three arbitrary parameters, which may be used to simplify the form of the resulting equation. Againt, when the genus of the relation is unity, both x and y can be expressed as uniform doubly-periodic functions of a new variable z, while g (z) and g' (z) are rational functions of x and y; moreover, the expressions contain (explicitly or implicitly) one arbitrary parameter, which again may be used to simplify the form of the resulting equation. And, in each case, definite processes are known by which the formal expressions of x and y, in terms of the new variable, can actually be obtained. When the genus of the algebraical relation k (x, y)= O is greater than unity, a corresponding transformation is possible by means of automorphic functions: not merely so, but such a transformation can be effected in an unlimited number of ways. Further, it is possible to choose transformations that simplify the properties of the integrals of the differential equations to which they are applied. But, down to the present time, the instances in which the complete formal expressions of x and y have been obtained, and the application to the differential equations has been made, are comparatively rare. The results that have been * T. F., ~ 247. + T. F., ~ 248. 155.] AUTOMORPHIC FUNCTIONS 489 established are of the nature of existence-theorems. It is true that indications for the construction of formal expressions are given; but the detailed analysis required to carry out the indications is of so elaborate a character that it may fairly be said to be incomplete. The subject presents great, if difficult, opportunities for research in its present stage. A brief account, based mainly on the work* of Poincare, is all that will be given here. References to the investigations of Klein and others in the region of automorphic functions will be found elsewheret. The main properties of infinite discontinuous groups and of functions, which are automorphic for the substitutions of the groups, will be regarded as known. It is convenient to associate with any group a region of variation of the variable which is a fundamental region; and for the sake of simplicity in the following explanations, it will be assumed that this region is such that, when the substitutions are applied to it in turn, the whole plane is covered once, and once only. Further, also for the sake of simplicity, it will be assumed that the axis of real quantities in the plane is conserved by the substitutions of the group. There are corresponding investigations, which establish the results when these assumptions are not made; but, as already indicated, the results are mainly of the nature of existence-theorems and cannot be regarded as possessing any final form, so that the kind of consideration adduced will be sufficiently illustrated by dealing with the simplest cases. In order to deal with the most general cases, it is necessary to utilise the theory of automorphic functions in all its generality; yet the subject still is merely in a stage of growth, being far from its complete development+. 156. It is known ~ that, if x and y be two uniform functions of a variable z, which are automorphic for an infinite * This work is best expounded in his five valuable memoirs in Acta Mathienzatica, t. i (1882), pp. 1-62, 193-294, ib., t. II (1883), pp. 49-92, ib., t. iv (1884), pp. 201-312, ib., t. v (1884), pp. 209-278. t T. F., chapters xxI, xxin. + The most consecutive account of the subject is to be found in Fricke und Klein's Vorlesungen ii. d. Theorie d. automorphen Functionen (Leipzig, Teubner; vol. i, 1897; vol. ii, part i, 1901). ~ T. F., ~ 309. 490 AUTOMORPHIC [156. discontinuous group of substitutions effected on z, then some algebraic relation (x, y) =0 subsists between them. Conversely, if this algebraic equation be given, it is desirable to express the variables x and y as uniform automorphic functions of a new variable z. For this purpose, we note that for general values of x, the variable y is a uniform analytic function* of x; but there are special values of x, being the branch-points, at and near which y ceases to be uniform. Now suppose that x can be expressed as a uniform automorphic function of z, say =f(z), the fundamental polygon being such that the branch-point values of x correspond to its corners (or to some of them), which include all the essential singularities of the uniform function f(z). Then, when substitution is made in the above relation, it becomes an equation defining y as a function of z; so long as z varies within the polygonal region, y does not approach those values where it ceases to be uniform, for they are given only by the corners of the polygon. Hence y becomes a uniform functiont of z; and as x is automorphic for the group of the polygon, it is at once seen that y also is automorphic for that group. Further, suppose that at the same time there is given a linear differential equation of any order, in which the coefficients are rational functions of x and y. In addition to the branch-points which may be singularities of the equation, it may have a limited number of other singularities. Let such a singularity be x = a, y = b, where of course Or (a, b) = 0: for the moment, the question of the regularity of the integrals in the vicinity is not raised. If the polygon is constructed, so that x = a corresponds to one of its corners which is an essential singularity of the group, then that corner is an essential singularity of f(z). Hence, when the differential equation is transformed so that z becomes the independent variable, the original singularities no longer occur so long as z is restricted to variation within the fundamental polygon: they can occur only for the special values at the corresponding corners. If, further, the function f(z) is such that no special * T. F., ~ 97. t Another method of obtaining this result is indicated in ~ 160. 156.] FUNCTIONS 491 singularities for values of z are introduced for values of x that are ordinary points of the equation, which will be the case if f'(z) does not vanish within the polygon, then all the values of z within the polygon are ordinary points of the equation, and all the integrals are synectic everywhere within the polygon. The singularities have been transferred to the boundary of the z-region; and thus the variables x and y, as well as all the integrals of the given linear differential equation which has rational functions of x and y for its coefficients, can be expressed as uniform functions of z within the region of its variation. AUTOMORPHIC FUNCTIONS AND CONFORMAL REPRESENTATION. 157. The relation between the variable z and the function x=f(z) can be considered in two different ways, the analytical expression of the significance being the same for the two ways. In the first place, the relation can be regarded as one of conformal representation. Assuming for the sake of simplicity that all the singular values of x are real, consider the problem* of representing the upper half of the x-plane bounded by the axis of real quantities conformally upon a polygon in the z-plane, bounded by circular arcs and having m sides: this conformal representation is known to be possible. If its expression be x=f(Z) then f'(z) must not become zero or infinite anywhere within the polygon, that is, for any finite values of x; for otherwise, the magnification would be zero or infinite there, a result that is excluded save at possible singularities on the boundary. It is manifest that the representation remains substantially the same, if the z-plane be subjected to any homographic transformation a'~+ b' c' + d'" where a'd' - b'c' = 1; for this will merely change the polygon bounded by circular arcs into another polygon similarly bounded. * 1'. F., ~ 271. 492 CONFORMAL REPRESENTATION AND [157. Hence, in constructing the function for the conformal representation, account must be taken of this possibility; and therefore, as {z, xI = {, aX), where {z, x} is the Schwarzian derivative, we construct this function {z, x}. We have* 1 - a2 A{Z,2 (x-a)2 +} =1 ( + (x), say, where the summation on the right-hand side extends over all the singular values a of x; the internal angle of the z-polygon at the corner homologous with a is a7r, and the coefficients Ao are real quantities. If o is an ordinary value of x, so that no angular point of the polygon is its homologue, then 0 = SAo, O = Ao + (1 -a2), 0 = Ea2Ao + a (1 - a2). If oo is a singular value of x, which has an angular point of the polygon as its homologue, with the internal angle equal to Kcr, then EAo =0, oaA0 = - a2) + - (1 _- 2), the summations being over all the finite singular values of x. The number of constants is sufficient for the representation. In the case when cc is not the homnologue of an angular point of the polygon, we have m constants a, m constants a, and m constants A, subjected to three relations as above; as all these constants are real, there are 3m - 3 independent constants. But, if a"X + b" = "X + d"' where a"d" - b"c" = 1 and the constants a", b", c", d" are real, then the upper half of the x-plane is transformed into itself; hence the m constants a are effectively only m- 3 in number, and thus the constants in I(x) are equivalent to 3m -6 independent constants, which can be used to make a solution determ * The whole investigation is due to Schwarz: see T. F., ~ 271. 157.] AUTOMORPHIC FUNCTIONS 493 inate. On the other hand, to determine the polygon, 3m constants are needed, viz. two coordinates for each of the m corners and a radius for each arc: but these are subject to a reduction by 6, for the representation is determinate subject to a transformation a'l+ b' c' + d" where a'd'- b'c' = 1, and the constants a', b', c', d' are complex, so that there are six real parameters undetermined. The number of available constants is therefore sufficient for the number of conditions that must be satisfied. In the case when oo is the homologue of an angular point, we have m - 1 constants a, m constants a, and m constants A,, subjected to two relations as above; as all these constants are real, they are equivalent to 3m - 3 independent constants. The remainder of the argument is the same as before; and we infer that the number of constants is sufficient to satisfy the number of conditions for the conformal representation. It need hardly be pointed out that, thus far, the polygon bounded by circular arcs is any polygon whatever; it has been taken arbitrarily, and it does not necessarily satisfy the conditions of being a fundamental region suited for the construction of automorphic functions. 158. That polygons can be drawn in the z-plane, suited to the construction of automorphic finctions in connection with a given algebraic relation t (x, y) = 0, may be seen as follows. For simplicity, let the polygon be of the first family*, and let it have 2n edges arranged in n conjugate pairs; and suppose that q is the number of cycles of its corners, each cycle being closed. The genus p of the group is given by 2p=n+ 1- q. When the surface included by the polygon is deformed and stretched in such a manner that conjugate edges are made to coincide by the coincidence of homologous points, then for each cycle in the polygon there is a single position on the closed * T. F., ~~ 292, 293. 494 FUNDAMENTAL REGION [158. surface obtained by the deformation. This closed surface corresponds* to the Riemann surface for the equation, (x, y) = o. which also is of genus p; and thus there are q positions on the surface, each associated with one of the q cycles. Each such position requires a couple of real parameters to define it; and thus we have 2q real parameters. Equations, which are birationally transformable into one another, are not regarded as independent: and therefore the effective number of constants in r (x, y) = 0 to be taken into account is 3p - 3, being the numbert of class-moduli which are invariantive under birational transformation. Each of these is complex, so that the number of real parameters thus arising is 6p - 6. We therefore have to provide for 6p - 6 + 2q real parameters, by means of the polygon. In order that the polygon may be properly associated with a Fuchsian group, it must satisfy certain conditions. Its sides must be arcs of circles, the centres of which lie in the axis of real quantities. As it has 2n sides, we therefore require 2n centres on that axis and 2n radii, making 4n real quantities in all; but three of the centres may be taken arbitrarily, for the polygon now under consideration is substantially unaffected by a transformation / az + b Z' cz d) ' where a, b, c, d are real; so that the total number of real quantities necessary is effectively 4n - 3. They are, however, not sufficient of themselves to specify an appropriate polygon: for conjugate sides must be congruent, a property that imposes one condition for each pair of edges, and therefore n conditions in all: and the sum of the angles in a cycle must be a submultiple of 27r, so that q conditions in all are thus imposed. Hence the total number of real quantities necessary is = 4n - 3- n - q =3n.- 3 - q = 6p - 6 + 2q, in effect, the same as the number of real parameters given. * T. F., ~ 310. t T. F., ~ 246. 159.] FUCHSIAN EQUATIONS 495 AUTOMORPHIC FUNCTIONS AND LINEAR EQUATIONS OF THE SECOND ORDER: FUCHSIAN EQUATIONS. 159. In the second place, the variable z, and the automorphic functions x and y, can be associated with a linear differential equation of the second order. Let (dx\ _dx\1 \dz \dz so that V2 z= — V1 then it is easy to verify that 1 dl _ 1 d2v2= {x, z} Vl dX2 v2 2 -d 2 X2 where {x, z} is the Schwarzian derivative of x with regard to z, and x'= dx/dz. It is a known property* that, if x is an automorphic function of z, then the function {x, z} X'2 is automorphic for the same group; hence it can be expressed rationally in terms of x and y, where ' (x, y) = 0. Denoting its value by - 21, where I is a rational function of x and y, which may be a rational function of x alone, we have v1 and v2 as linearly independent integrals of the equation d'2v d2 + Iv = 0; dX2 the quantity z is the quotient of the two integrals. The analytical relation is effectively the same as before; for if {Z, x} = 21, we knowJ that z is the quotient of two integrals of 1 d2v - -+1=0. v dX2 * T. F., ~ 311. t Treatise on Differential Equations, ~ 61. 496 AUTOMORPHIC FUNCTIONS AND [159. Moreover, x, Z = - X2, x}; so that the results agree in form. The difference is that, regarding the relation as a problem of conformal representation, we have been able to calculate the value of I in greater detail than in the alternative mode of regarding the relation: but the considerations adduced in connection with the differential equation have been of only the most general character, and have not permitted any discussion of the form of I. When an equation of the form d2v d- + Iv = 0 is given, where I is a rational function of x, or a rational function of two variables x and y, connected by an algebraic equation * (x, y) = 0, it may happen that x and y are uniform functions of z, the quotient of two integrals of the differential equation. But these uniform functions are not necessarily, nor even generally, automorphic for a group of substitutions of z. Judging from the result of the consideration of the question as a problem of conformal representation, we should be led to expect that the constants, which survive in I after the conditions for uniformity are satisfied, might be determinable so that the uniform functions of z are automorphic. When this determination is effected, the equation is called* Fuchsian by Poincare, if the group be Fuchsian. 160. We proceed to consider more particularly the properties of the equation d2v -- + Iv = 0, dx2 in relation to the quotient of its integrals. Let x = a, y = b be a singularity of the equation, where r (a, b) = 0; and let Limit [(x - a)2I]a = p, so that the indicial equation for a is n(n-1) + p =0. * Acta Math., t. iv, p. 223. 160.] FUCHSIAN EQUATIONS 497 Let n1 and n2 be its roots, when they are unequal; then two integrals of the equation are of the form v = (x - a)n, +..., v, = (x- a)n +...; and so Z = 2 = (X a)n2 —a +... V1 If a7r be the internal angle of the z-polygon at the angular point homologous with a, we must have n2 - -= a, and therefore 1 - 4p = a2, that is, p=K(l- a2), so that 1 - a2 I-i1 i- ~ (x- )2 + the remaining terms being of index higher than - 2. This is valid, if a is not zero. When a is zero, so that n, = n and therefore p = a, the integrals of the equation are VI = (x - a)p +..., v2 = (x - a)[{1 +1...} log (x - a) + powers of x - a], and so, in the immediate vicinity of a, we have z = -2 log (x - a) + powers; V1 and then I- +4..+ (x-a)2 the remaining terms again being of index higher than - 2. The quantity a, in terms of which the leading fraction in I is expressed, depends upon the character of the singularity at (a, b). If the latter denote a singular combination of values for the equation, (x, y) = 0, then it is known* that the variables x and y can be expressed in the form x-a=,~q, y-b='pS('), * T. F., ~ 97. F. IV. 32 498 FUCHSIAN [160. where S(') is a regular function of, which does not vanish when g= 0, and the expressions are valid in the immediate vicinity of the position. Let r be the least common multiple of p and q, and write 1 q 1 a=-, = = z +...; then in that vicinity, we have (x - a)- = z +..., y- b = PP' S(z), so that both x and y are uniform functions of z in the vicinity. The commonest instance occurs, when (a, b) is a simple branchpoint; we then have p, q== 1, 2, so that a=. If (a, b) be a singularity of some given differential equation of any order, say dmw m- 1 dnW dwrcn - on_~( 'y) dae ) where * (x, y) = 0, three cases arise. Firstly, let all the integrals be free from logarithms, and let all the exponents to which the members of a fundamental system of integrals (supposed regular) belong be commensurable: then they are integer multiples of a quantity k-1, and we take a = (x-a)" = z.... In that case, any integral is of the form w = (x - a) R (x - a) M =(x - a)k R (x - a) = z R (z), so that the integrals of the equation, as well as the variables x and y, become uniform functions of z in the vicinity of z = 0. Secondly, let the integrals (still supposed regular) of the fundamental system belong to exponents some of which at least are not commensurable quantities. We take z = log (x - a) + powers; 160.] EQUATIONS 499 and then an integral of the form (x - a) R (x - a), becomes eIZ Rz (z), i.e., a uniform function of z, valid for large values of lzl: and this uniformity is maintained whether,/ is commensurable or not. Thirdly, let x = a be an essential singularity of one or more of the integrals, supposed irregular there. As in the last case, we take z = log (x - a) + powers; the integral may or may not become uniform for large values. of Izl. In the last two cases, if the expression for x in terms of z, say x =f(Z), be automorphic, then z = o is an essential singularity of the function f(z); and then, when z varies within the polygonal region, x does not approach the value a for which the integrals of the equation cease to be regular. Within the region, the integrals are uniform. It is to be noted that the relation, adopted in the second case and the third case, would be effective in the first case also, so far as securing uniformity; but the converse does not hold. The relation which, as seen above, corresponds to the vicinity of an angular point of the polygon where the sides touch, is the most generally applicable of all: the form of relation, corresponding to the first case, is applicable only under the somewhat restricted conditions of that case. 161. These conditions and limitations affect the quantity I in the equation d2v dx + Iv = 0 for they determine the leading coefficient in its expansion near any of its poles; but, in general, they do not determine I completely. On the other hand, we so far have only secured the uniformity in character of the functional expression of x in terms of z: the automorphic property of the functional expression has not been secured. The latter is effected by the proper assignment of the remaining parameters in I. 32-2 500 CONSTRUCTION OF [161. As a special instance, take the case in which the genus of the group and of the permanent equation is zero; so that, if the polygon has 2n edges, the number of cycles q is given by q=n +1. Taking the angular points in order as A2, A2,..., A2, and making the sides A1A2 l A2 A2 3 3 l An-iAn ) An An+i5 AA2n) A2n A 2n-i, ) An+3An+2' An+Q2An+i to be conjugate pairs, the necessary n + 1 cycles are Al; A2, A2n; A3, A2n-;...; An, An+o2; An+l. To define the polygon of 2n circular arcs, which have their centres on the axis of real quantities, we require the 4n coordinates of the angular points; but these effectively are only 4n - 3 quantities, because the z-plane is determinate, subject only to a transformation / az+b\ tz' cz+d) ' where a, b, c, d are real. In each cycle, the sum of the angles is a submultiple of 2Sr: so that n + 1 conditions are thus imposed. Again, the edges in a conjugate pair must be congruent; so that n further conditions are thus imposed. Accordingly, there remain 2n - 4 real independent constants to determine the polygon. The polygon thus determined defines a Fuchsian function; as the genus is zero, every function can be expressed rationally in terms of x, so that the equation for v (leading to z, as the quotient of two integrals) is d2v dx + Iv = 0, where I is a rational function of x. Corresponding to the n + 1 cycles, there are n + 1 values of x; let these be a, a2,..., an, o0. Let ac7r be the sum of the internal angles of the z-polygon corresponding to ar, so that ar is the reciprocal of an integer; and take a,,+1 to be the quantity a for oo. Then in the vicinity of a, we have 1 -a.r2 ( - ar)2+ 161.] FUCHSIAN EQUATIONS 501 for each of the values of r. Thus, if we write A = (x - a,) (x - a,)... (x - a,), and remember that I is a rational function of x, we have _ (x) 1= A2 where G (ar Jx=ar for r= 1,..., n. In order to satisfy the condition for x oo, G(x) must be of order 2n -2, and G (x)= ~ (1 - a2n+) x2n- +.... The number of coefficients in G (x) is 2n - 1; but the coefficient of the highest power is known, and there are n relations among the rest, owing to the conditions at a,,..., an; hence there remain n-2 coefficients independent of one another. Each of these is complex in general, so that they are effectively equivalent to 2n-4 real constants. Assuming that the quantities a,,..., an are known, it is to be expected that the 2n - 4 conditions for the polygon determine these 2n - 4 real constants. In the simplest case, we have n = 2; and we may take a, = 0, 2 = 1, so that 1- a 2 1 - a2 p x2 4 (X -1)2 x x-1 The conditions for x= o00 give p + c=0, ( - a2) +i(1 - a 2) + (1 - a32), where al, a2, a3 are the reciprocals of integers; the quantity I then is the invariant of the hypergeometric series. 162. As another illustration, which may be treated somewhat differently, consider the equation y = x (1 - x)(1 - x), where c is a real constant less than unity; and write ac = 1, 502 EXAMPLE OF A [162. so that a is a real constant greater than unity. Here, the points x= O, 1, a, oc are each of them singular; and the value of a is g for each of them. Consequently, = (1-1) 1 (1-1) (1 - i) + A B. C — 4_ + 4 -+ + X2 (Xx-1)2 (x-a)2 x x-1 x-a and the conditions for x = oc give A +B+C=O, -19- + B + Ca = to3 One constant in I is left undetermined by these conditions; thus -3_ _ 3 3X X 16 + 16.+ T 6x +X X2 (x- 1)2 (x - a)2 (x- 1)(x-a)' say, where X is the undetermined constant. It is possible to determine X, so that x is a Fuchsian function of z, where z is the quotient of two solutions of the equation d2v 2 + Iv = 0. dX2 As regards this Fuchsian function, its polygon may be obtained simply as follows. We take four points A, B, C, D in the z-plane to be the homologues of 0, 1, a, oo; owing to the value of a, which is I in each case, the internal angles of the polygon must each be -r. We make the edges AB, CD conjugate, and likewise the edges BC, DA; and then there is a single cycle, ADCB, the sum of the angles in which is 27r. With the former notation, we thus have q= 1, n = 2; so that 2p = 2 1 - 1 = 2, and therefore p = 1, as should be the case. Further, the sum of the angles of a curvilinear triangle, entirely on one side of the real axis, is less than rT, when the centres of the circular arcs lie on the real axis: so that, if our polygon be curvilinear, the sum of its angles would be less than 27r (for it could be made up of two triangles), whereas the sum is actually 27r. Hence the polygon can only be a rectangle, and the Fuchsian functions are doublyperiodic. We therefore take x=sn2z, y=snz cnz dnz, 162.] FUCHSIAN EQUATION 503 as is manifestly permissible; and then, dz 1 1 dxr 2y - 2c'~x (x - 1) (x - a))2 which leads to ri_ 1] 1 3-a-1 {Z. x}8 ~ + ( _- 1)2 + ( a)2 --21, so that we have X=- (a +l). This value of X renders x (and so y) a Fuchsian function of the quotient of two solutions of the equation d2v d + Iv=0. dx2 As regards the integrals of this equation, the indicial equation of x = 0 is p(p-1)+ ~: =0, so that p =, p = 3. Denoting by v, and v2 the integrals that belong to ~ and 4 respectively, we have v=x { jl+2-x+..., 1 - 1+2-x+... and therefore = ~2 + s - -4 -... V2 &a = 4( - 1 + c) a4 +* 1x- x-t... = sn2, after the earlier analysis. Fl, =( c1)i{l+2x+(x-1)+ }l I-a '"' X -1 a' " 504 EXAMPLE OF A [162. and then, taking V2 we find X- I=?12+ 3 8 + 14 +.. 3 —a = 4' + I a - 4 +... = I2+ 1 - 2c 4 Now x -1 = - en2 ~ - (1 - c)n 2 dn2 (- K) =-(1-c)(-K)2 + (1-2c) (l-c)(-K)4-..., so that =(c- 1)4(- K). Hence V= (C -1)t ^-K so that, as Vr2= Av2 + Bvl, V =- Cv2 + Dv,, where AD- BC= 1, because dVT7 dV, V2 - - - = constant, dx dx we have V2=(c - 1) (V2 - Kv)} r, =(c - 1) —vI Again, in the vicinity of x = a, we find integrals Ui= (Q-a) 1+ 2 X- I (x ) (- {+ a (a - 1) U2= (x - a)'l+23a (x_-)+...}; and then, taking U2 fc2- g' 162.] FUCHSIAN EQUATION 505 we find x-a=-+2 a(a- 1) '2 +.. =2+ 1 (2-c)c +.... ~ 1-c Also 1 - a = sn2 -- c — dn2 1 -c sn2(- K - iK') c cn2(- K-iK') _- -c (_ K-iK')2 + 1 -c) (-c) ( _ K-iK)4 +... C C so that 2= ( - iK'). Proceeding as before, this leads to the relations U2= ( V2-(K + iK') v} ) u (1-C UI- - Vl Lastly, for large values of x, we have W =X{1- ( +a)+... W,=x l- (l+)-+ I...; and then, taking W2 3 WI we find = 2 -(1 + a) 4+.... Now 1 1 x sn2 ' = c sn2 ( - iK') = c(- iK')2 -C2 (- iK')4(1 + a)+..., 506 INTEGRALS EXPRESSED AS [162. so that c3 = - -iK'). Proceeding as before, this leads to the relations W,= c (v, - iK'v,) Wi= c-i )v The relations, in fact, have enabled us to construct the expressions for each fundamental system in terms of the first and, therefore by inference, in terms of every other. Ex. 1. Discuss in the same way the Fuchsian differential equation ld2v. f 1a x- - v dX2+ 16 (Xex- - e2 + ( X - e )2 (x - el) (x - e2) (x- e3) connected with the equation y2= 4x3 _ g2- g3' Ex. 2. Shew that, if x= P (log z), where p denotes Weierstrass's elliptic function, [ 1 1 I 1 {z, X L8(X - el)2 + (x - )2 (+ -e3 - ( el) (x-) (x-e3 ) and discuss the significance of the integral relation in regard to its pseudoautomorphic character for the equation y2 = 43 -g2x-g3. Ex. 3. A fundamental polygon in the z-plane is composed of two semicircles, one upon a diameter in the real axis for values of z corresponding to values of x equal to 0 and 1, the other upon a similar diameter for values of x equal to 1 and a, (where a> 1), and of two straight lines drawn, through points corresponding to 0 and a, perpendicular to the axis of real quantities. Prove that the subsidiary equation of the second order, for the construction of x as an automorphic function of the quotient of two of its integrals, is ld2v I 1 1 ] lu x~=_-~ ~+i-i~+ - + x (x - 1)(x - a)' v dX2 4 [j2 + (X_1)2 + (x-a)2] X (X-1) (X-a) where the constant p is to be properly determined. AUTOMORPHIC FUNCTIONS USED TO MAKE THE INTEGRALS OF ANY EQUATION UNIFORM. 163. If, for any given equation, there is only one singularity, it can be made to lie at the origin. In order to obtain a variable z, in terms of which the integrals of the given equation can be expressed uniformly, we construct an 163.] UNIFORM FUNCTIONS 507 equation of the second order which has x= 0 for a singularity, of such a form that the indicial equation for x = 0 has equal roots (~ 160). This auxiliary equation may have other singularities, but otherwise it may be kept as simple as possible. Such an equation is d2V X = — V; dx2 x2 the indicial equation for x = 0 is 0 (0-1)=\, so that X =- if it has equal roots. Thus the equation is d2v v dx2+4 2~0o Two integrals are given by V = X-, v2 = x logx; thus V2 z - = log x, V1 which is the new independent variable. An equation of the kind indicated is (~ 45, Ex. 6) d2u /3 1 \du /1 dx — + +- + + 2X242 dr u -24 =O when the variable is changed from x to z, where x=e3, the equation becomes d +(2+U e-) d+(e-z- e-2 0. &~ +(2+~-e) T+ (e e2') The integrals are synectic for all finite values of z. 164. When a given differential equation has two singularities, a homographic transformation can be applied so as to fix them at =0O, x=1. To obtain a variable z in terms of which the integrals of the given equation can be expressed uniformly, we construct an equation of the second order, having 0 and 1 as its singularities and such that the respective indicial equations have repeated roots. An appropriate equation is d2v a + o3x dx2 X.2 (X _ 1)2 508 EXPRESSION OF INTEGRALS [164. The indicial equation for x = 0 is P (P-1)=a, so that a=- 1; the indicial equation for x = 1 is (p -1)=a +3, so that a + /- -, and therefore / = 0, so that the equation is d2v __ __ 0. d2 + 21) vO. dx2 x2 (X _ 1)2 One integral is easily found to be v1 = (x- 1)i; and then z, the quotient of another integral by vI, is given by dz C -1 dx v2 (x-1) ' on particularising the constant C, which may be arbitrary. Thus ez x -- ez - 1 gives a new variable z, such that the integrals of the given differential equation are uniform functions of z. Thus let the equation be dzy 2x+a dy b dx+ (x- 1) dx x2(x- 1)2 which has x=0 and x=1 for real singularities: it is easy to verify that x=oo is not a singularity but only an ordinary point for every integral. When the equation is transformed so that z is the independent variable, it becomes d2y (a+1) dY+b the integrals of which clearly are uniform functions of z. 165. When a given differential equation has three singularities, a homographic transformation can be used so as to fix them at x= 0, 1, oo. We may proceed in two ways. It may be possible to choose, as the fundamental region in the z-plane, a triangle, having circular arcs for its sides, and having Xrr, /7rr, 7rr for its internal angles at points which are the homologues of 0, oo, 1 respectively: 165.] AS UNIFORM FUNCTIONS 509 X, u, v being the reciprocals of integers. Then the subsidiary equation may be taken in the form I d2v (1-2) (1-V2) 2- + 2-1 v dx2 2 ' (- 1)2 4 (x —1) which is the normal form of the equation of the hypergeometric series with parameters a, y3, 7, where =(1-_ )2, 2 =(a-/)2, V2=(-a-/3)2. The variable z may be taken as the quotient of two solutions of the subsidiary equation; and so _ F (a+1-, /3+1-y, 2-, x) F(a, /, rx) It is known* that x, thus defined, is a uniform automorphic function of z. This transformation will render uniform the integrals of a differential equation, which has no singularities except at 0, 1, so, provided the integrals are regular in the vicinity of those singularities and belong to indices which are integer multiples of X, v, Ad respectively. If these conditions are not satisfied, in particular, if the singularities are essential for the integrals, then we proceed by an alternative method. We take a subsidiary equation having 0, 1, so for singularities, such that the indicial equation for each of them has equal roots. Let it be d2v a' + c'x + y,'X2 dx2 x2 (x - 1)2 V where a', /', m' are to be chosen so that the indicial equation for each of the singularities has equal roots. These equations are ( - 1)=a', a (-l)= a'/ + '+', (7+ 1)= ', so that a'=-4, /3'=, ' and thus the equation is d2v 1 - x + X2 dx +V x (X - 1)2 * T. F., ~ 275. 510 GENERAL APPLICATION OF [165. The coefficient of v is the invariant of a hypergeometric equation, of which the parameters are a=/3=-, 7=1; so that z, the quotient of two integrals v, is also the quotient of two integrals of the equation d2w dw x (1 - x) dw+ (I - 2x) x -w = 0. This is the equation of the quarter-periods in elliptic functions: so that K (x) -K' (x) This relation effectively defines x as a modular function* of z: the fundamental region is a curvilinear triangle. The function exists over the whole z-plane: the axis of real quantities is a line of essential singularity. Any differential equation, having x = 0, 1, oo for all its singularities no matter what their character may be, can be transformed by the preceding relation so that z is the independent variable: its integrals are then expressible as functions of z which are uniform over the whole of the z-plane, their essential singularities lying on the axis of real quantities. Ex. A differential equation has only three singularities at x=a, b, c, such that the roots of the indicial equations of those points are integer multiples of a, /3, y respectively, where a, 3, y are reciprocals of integers. Shew that a variable, in terms of which the integrals can be expressed as uniform functions, is given by taking the quotient of two Riemann P-functions with the appropriate singularities and indices. AUTOMORPHIC FUNCTIONS APPLIED TO GENERAL LINEAR EQUATIONS OF ANY ORDER. 166. At the beginning of the preceding explanations and discussions, it was assumed (~ 157) that all the singular values of x are real. The assumption was then made for the sake of simplicity: it can be proved to be unnecessary. * T. F., ~ 303. + Poincare, Acta Math., t. iv, pp. 246-250. 166.] AUTOMORPHIC FUNCTIONS 511 Firstly, let the singularities be constituted by al, a2,..., am, all of which are real, and by c, which will be supposed complex. With these we shall associate Co, the conjugate of c; and we write () = ( - c) (X- Co), a quadratic polynomial with real coefficients. Then all the quantities o, c (al), c (a2),..., Qj(am), ( c+ Ico) are real. Construct a fundamental region in the z-plane, such that the foregoing m + 2 quantities are the homologues of the corners; and let X = F(z) be the relation that gives the conformal representation of the region upon half the X-plane, so that F(z) is a Fuchsian function of z. Consider the variable x, as defined by the equation (x) = F(z). So long as z remains within the fundamental region, x is a uniform function of z; it could cease to be so, only if that is, if x= oc + Co, and then we should have F(z)= (-c t-2o), which is not possible for values of z within the region. Also, dx. d- is not zero for any value of z within the region; for then we should have F' (z)= 0, which would make a zero magnification between the X-plane and the z-region: this we know to be impossible for internal z-points. This uniform function x, whose derivative does not vanish within the polygon, cannot acquire either of the values c or co within the polygon, for then we should have F ()= 0, 512 AUTOMORPHIC FUNCTIONS AND [166. which is possible only at a corner. Nor can it acquire any of the values a1, a,..., an for points within the z-polygon: for at any such value, we have F(z) = (a), which again is possible only at a corner. Now since X = F(z) is a relation that conformally represents the half X-plane upon a z-polygon bounded by circular arcs (this polygon being otherwise apt for the construction of autororphic functions), we have (~ 157) {z, X} = 2# (X), where I (X) is a rational function of X. But for any variables X and x, we have [tXz, =[, X}}z X +, }; and therefore, in the present case, {z, x} = 2 (2x- - Co)2 (2 _ CX- CoX + CCo) - X(2x - c - CO)2 = 2r (X), say, where T(x) is a rational function of x. Hence z is the quotient of two integrals of the equation d2v dx + vT (x)= O. Now x is known to be a uniform function of z; it is therefore a Fuchsian function of z. And we have proved that, for values of z within the polygon, x cannot acquire any of the real values al, a2,..., am or either of the complex values c, Co, and, further, dx that d- does not vanish. Secondly, to extend this result to the case, when x is not to acquire any one of any number of complex values for z-points within the polygon, we adopt an inductive proof; we assume the result to hold when there are q-1 pairs of conjugate complex values, and shall then prove it to hold when there are q pairs. It has been proved to hold, (i), when there are no complex values and, (ii), when there is a pair of conjugate complex values: it thus will be proved to hold generally. 166.] LINEAR EQUATIONS IN GENERAL 513 Suppose, then, that the given x-singularities are made up of a number m of real values al, a2,..., an, and of a number of complex values. Let the latter be increased in number by associating with each complex value its conjugate complex, whenever that conjugate does not occur in the aggregate; and let the increased aggregate be denoted by c, cl; C2, C2;...; Cq, Cq; arranged in conjugate pairs. Write q 4 (x) = I (X - Cr) (x - Cr), r=l which is a polynomial of degree 2q with real coefficients. The equation do (x) _ dx of degree 2q-1 with real coefficients, certainly possesses one real root; its other roots, when not real, can be arranged in conjugate pairs, the number of pairs not being greater than q - 1. Let its roots be denoted by bl, b2,..., b2qan aggregate which contains not more than q-1 conjugate pairs. In the series of quantities 0; < (a1),..., ( (a,m); ) (b),..., (b2qi); there are certainly m + 2 real quantities; and there are not more than q - 1 conjugate pairs of complex quantities. According to our hypothesis, a Fuchsian function G (z) can be constructed, such that the foregoing 2 + 2 + 2 (q - 1) quantities are the homologues of the corners of an appropriate fundamental region, and G' (z) does not vanish within the region. Then, proceeding on the same lines as in the simpler case, we consider a variable x, defined by the relation () = G (z). So long as z remains within the fundamental region, x is a uniform function of z; it could cease to be so, only if +' (Z) =0, that is, if x = b, b2,..., or b2q_-, and then we should have G (z)= (b), b (b2),..., or ( (b2q-1), F. IV. 33 514 FUCHSIAN EQUATIONS HAVING [166. which is not possible for values of z within the region. Also, dx does not vanish for values of z within the region; for otherwise dz we should have G' (z) = for such values, and this is known not to be the case. Further, x, being a uniform function of z whose derivative does not vanish for values within the polygon, cannot acquire any of the values c,. or c,', for r = 1,..., q, within the polygon; if it could, we should have ( (x)= 0 there, and then F (z) = 0, which is possible only at a corner. Nor can it acquire any of the values a,,..., a, for values of z within the polygon: if it could, we should have F (z) = (a,), (a2),..., or (am), which again is possible only at a corner. Now since Y, = G(z), is an automorphic function, it follows* that (d Y>-2 which is equal to - z, Y}, also is an automorphic function. Consider the upper half of the Y-plane. So far as the equation Y= G (z) is concerned, certain points on the upper side of the axis of real quantities are exceptional, not more than q-1 in number; these can be considered as excluded, and cuts drawn from them to singular points on the real axis. We then can regard this simply-connected and resolved half-plane as conformally represented upon the polygon by the equation Y= G (z); hence t {z, Y}= 20(Y), where 0(Y) is a rational function of Y. But (x) = Y, where b (x) is a polynomial; hence {Z }-{Z, } = {Yd Y{ 1Z) XI = 1z I dx; + I Y, xI =20[ ( P)] [d (]) +{2(x) x} = 2~ (x), * T. F., ~ 311. t T. F., ~ 271. 166.] ASSIGNED SINGULARITIES 515 say, where (x) is a rational function of x. Hence z is the quotient of two integrals of the equation dx2 + wO (x)= O. Now x is known to be a uniform function of z. It is therefore a Fuchsian function of z, which acquires the particular assigned values only at the corners of the fundamental region and nowhere within the region; its derivative does not vanish anywhere within the region. The statement is thus established. 167. The preceding explanations, outlines of proofs, and analysis, will give an indication of the kind of result to be obtained, and the kind of application to differential equations to be made. It will be recognised that such proofs as have been adduced are not entirely complete: thus, when a number of real constants is to be determined by the same number of equations, whether algebraical or transcendental, it would be necessary to shew that the constants, if determined in the precise number, are real. As, however, it was stated at the beginning of these sections that only an introductory sketch of the theory would be given, there will be no attempt to complete the preceding proofs: we shall be content with referring the student, for the long and complicated processes needed to establish even the existence of certain results without evaluating their exact form, to the classical memoirs by Poincare, and to the treatise by Fricke and Klein, which have already been quoted*. It may be convenient to recount the most important and central results of Poincare's investigations, which have any application to the theory of linear differential equations. Let dqw q-1 dKw -z =Y.O (X Y) XK dxq K=0 be a linear equation of order q, having rational functions of x and y for its coefficients, where y is defined in terms of x by the algebraic equation (x y) =0; * A memoir by E. T. Whittaker, "On the connexion of algebraic functions with automorphic functions," Phil. Trans. (1899), pp. 1-32, may also be consulted. 33-2 516 POINCARI'S THEOREMS [167. this equation in w will be called the main equation. Let d2v dx= vO (x, y) be another equation, in which 0 (x, y) is a rational function of x and y; it will be called the subsidiary equation, and its elements are entirely at our disposal. Let x = a,, y = b,, be a singularity of the main equation. If all the integrals are regular at this singularity, if they are free from logarithms, and if they belong to exponents, which are commensurable quantities (no two being equal), let k-1 (where k is an integer) be a quantity such that the exponents are integer multiples of k-1. We make x= a,, y = b/, a singularity of the subsidiary equation. In the case of the indicated hypothesis as to the integrals of the main equation, we make the difference of the two roots of the indicial equation of the subsidiary equation equal to k-1. In every other case, we make those two roots equal. This is to be effected for each of the singularities of the main equation. Thus the subsidiary equation is made to possess all the singularities of the main equation. It may have other singularities also; for each of them, the difference of the two roots of the corresponding indicial equation is made either zero or the reciprocal of an integer, at our own choice. By these conditions, the coefficient 0 (x, y) will be partly determinate: but a number of parameters will remain undetermined. The effect of these conditions is, by the analysis of ~ 160, to make x and y uniform functions of z, where z is the quotient of two linearly independent integrals of the subsidiary equation; and no further conditions for this purpose need be imposed upon the parameters, which may therefore be used to secure other properties of the uniform functions. The various forms of 0, corresponding to the various determinations of the parameters, determine a corresponding number of differential equations; all of these are said to belong to the same type, which thus is characterised by the singularities and their indicial equations. Poincare has proved a number of propositions connected with the results that can be obtained by the appropriate assignment 167.] ON AUTOMORPHIC FUNCTIONS 517 of values to these parameters. Of these, the most important are:I. It is possible to assign a unique set of values in such a way as to secure that x and y are Fuchsian functions of z, existing only within a fundamental circle. II. It is possible to assign sets of values, unlimited in number, in such a way in each case as to secure that x and y are Kleinian functions of z, existing over only part of the z-plane. III. It is possible to assign a unique set of values in such a way as to secure that x and y are Fuchsian functions or Kleinian functions of z, existing over the whole of the z-plane. There are limiting cases when the Fuchsian function becomes doubly-periodic, or simply-periodic, or rational. POINCAR]'S THEOREM THAT ANY LINEAR EQUATION CAN BE INTEGRATED BY MEANS OF FUCHSIAN AND ZETAFUCHSIAN FUNCTIONS. 168. Consider now the integrals of the main differential equation, when they are expressed in terms of the variable z. We shall assume that x and y have been determined as Fuchsian functions of z, existing only within the fundamental circle. Near an ordinary point xo, Yo, any integral w is a holomorphic function of x - x0; near such a point, x is a holomorphic function of z-z0; so that w, when expressed as a function of z, is a holomorphic function of z. In the vicinity of a singularity (a, b), there are two cases to consider. If all the exponents to which the integrals belong are commensurable quantities, so that they are integer multiples of some proper fraction k-1, where k is an integer, and if the integrals are free from logarithms, then every integral is of the form w = (x - a) S (x- a), 518 POINCARI'S THEOREM AS TO THE [168. where S is a holomorphic function of x- a. As in ~ 160, we have 1 z - c = (x - a)k T (x- a), so that x - a = (z - c)kR (z- c), where T and R are holomorphic functions. Hence w = (z - c)0 G (z- c), where the function G is holomorphic in the vicinity of c. Thus w is a uniform function of z; if /a is positive, then c is an ordinary point; if u is negative, it is a pole. In all other cases, whether the integrals involve logarithms, or the exponents to which they belong are not all commensurable, or the singularity is one where some of the integrals, or even all the integrals, are irregular, the roots of the indicial equation for the subsidiary equation are equal. In consequence, the two circular arcs of any polygon touch, and thus the angular point is on the fundamental circle. As we consider the values of z within the fundamental circle, the character of the integral, when expressed as a function of z, does not arise for the point of the kind under consideration. It thus appears that, when z is restricted to lie within the fundamental circle of the Fuchsian functions which are the representative expressions of x and y, any integral of the main equation is a uniform function of z. When this uniform function has poles, it can be represented in the form G (z) G (z)' where the zeros of GI (z) are the poles of the integral in unchanged multiplicity, and both G(z) and G, (z) are holomorphic functions of z within the fundamental circle. When the uniform function representing the integral has no poles, it can be expressed in the form H(z), where the function H(z) is holomorphic everywhere within the fundamental circle. Hence we have Poincare's theorem* that the integrals of a linear differential equation with algebraic coefficients can be expressed as uniform functions of an appropriately chosen variable. * Acta Mlath., t. iv, p. 311. 169.] INTEGRALS OF LINEAR EQUATIONS 519 169. The characteristic property of these uniform functions can be obtained as follows. Taking the equation in the form dqw q 2 dKw — Xl, Yd tf(x, 1) = 0, dxQ K=O dc = where it is supposed that the term (if any) which involved dq — has been removed from the equation by the usual substitution (~ 152), we denote by 01, 0,..., 0q a fundamental system of integrals in the vicinity of any singularity (a,, b,). Let a closed path on the Riemann surface, associated with the permanent equation, be described round the singularity; then, when the path is completed, the members of the fundamental system have acquired values 01', 02',..., 0q' such that On' a =~ 8+ ay 0 02 +... a () 0, (n = 1, 2,... q, 1, n, 2, n q, ~' 2 where the coefficients a(-) are constants such that their determinant is unity, because the derivative of order next to the highest is absent from the differential equation. Now x and y are Fuchsian functions of z, existing only within the fundamental circle in the z-plane; hence, when the path on the Riemann surface, which cannot be made evanescent, is completed, x and y return to their initial values, and z has described some path which is not evanescent. It follows, from the nature of the functions; that the end of the z-path is a point in another polygon, homologous with the initial position, so that the final position of z is of the form 7yZ + Sy The integrals 01, 02,..., 0q are uniform functions of z; let them be denoted by 4b (z), P2 (z),..., q (z). Moreover, 0n, is the value of 0, at the conclusion of the path; thus ni, _ ( a f + 13p so that the integrals in the fundamental system consist of a set of uniform functions of z, which are characterised by the property ny +-) = a (z) + 2,) (z) +... a () q (z), (n=l,...,q). 520 Z ETAFUCHSIAN [169. Corresponding to the substitution of the Fuchsian group, we have a linear substitution S, in the quantities b0, 1, 2,.., bq: the aggregate of these linear substitutions S, forms a group, which is isomorphic with the Fuchsian group. Functions of this pseudo-automorphic character are called* Zetafuchsian by Poincare: and thus we can say that linear differential equations can be integrated by means of Fuchsian and Zetafuchsian functions which are uniform. It is, however, necessary to obtain explicit expressions for the functions b, in order that the equation may be regarded as integrated. This is effected (I.c.) by Poincare as follows. Let represent the substitution inverse to S,^ so that the quantities A()n are the minors of the determinant of S,. Take any q arbitrary rational functions of z, say H, (z), H (z),..., Hq (z); and by means of them, in association with the Fuchsian group, construct p infinite series, defined by the equations ()= Ot (i.i Z + (+3 1+ z(Z) YE.=1, A -Up z ~ __+ 8; (,iz __+ 8i)2r' for the q values 1,..., q of /J; the quantity m is a positive integer; and the summation with regard to i is over all the substitutions of the Fuchsian group. This integer m is at our disposal: by choosing it sufficiently large, and by limiting the rational functions H, so that no one of the quantities Hy aiz + li\ \7iz + 8i is infinite on the fundamental circle, all the series can be made absolutely converging: but we do not stay to establish this resultt. Assuming this convergence, and writing ^iz + /i S ( + 8, St (Sk (z)) = sp (s), * Acta Math., t. v, p. 227. t It can be established on the same lines as the convergence of Poincare's Thetafuchsian series: T. F., ~~ 304, 305. 169.] FUNCTIONS 521 so that., for any value of k and all the values of i, we get all the values of p for the group, we have C kZ +13k) q (z+/P)(Y-k~t+8 2m p v=i 1~V pZ~~ 7pZ+~, p But Yak) I a~~@=)v=1n- (P) H y(p (p+ ) n /13fl _ I= innVVYz s 7Z 8)M Owing to the properties of the isomorphic groups, we have Si Sk= Sp, and therefore that is, n=1,n n,v 4 and therefore ~tZ+O -k + k )2mn I aM 4n (z). 'Yk Z + 8k ~ n=i a Now let O (z) represent a Thetafuchsian series*, with the parametric integer m, and possessing the foregoing Fuchsian group: then, for each substitution of the group, we have We introduce functions Z1, Z2,..,Zq, defined by the relations They satisfy the conditions ZnaZ+, - a ()Z, (z) + a ()Z2 (z) +... ~ a ()Zq (z); and therefore we may take or the q functions Z, which are Zetafuchsian functions, constitute a system of integrals of the differential equation. 170. As regards the Zetafuchsian functions thus constructed, it will be noted that the rational functions H1,..., II., which -*T. F., ~ 305. 522 PROPERTIES OF A [170. enter into their construction, are arbitrary; so that an infinite number of Zetafuchsian functions can be formed, admitting a Fuchsian group G and the linear group (say G) isomorphic with G. Further, the Thetafuchsian series { (z) with the parametric integer m is any whatever; but, as x=J (z) =f ~(, + k we have dx k /Z a, + /3kA 1__ dz =f (z) vK k (7z + k)2' so that we may take (z) (dx) P (x, y), where P (x, y) is any uniform function of x and y. The simplest case occurs when P (x, y)= 1. Again, we have (Zn z + ) ) a k),(z) + a(k Z2 (z) +.. a+ a Zq (Z); and therefore 1 Z k (\z + = a(k) dZ ( () ) dZ (7ykz + k)2Z kZ + 8J a d + 2, r n dz qn + dz ' so that dZ1 dZ2 dZq 1 y/k + Z/3) (kc) dz (lc) dz (k) dz + A (z~+ n, +a = w dr,+ arL + + an d kyz + e) dz dz dz that is, d Z (k + 3k _ (k) d.Zl + a(() dZ q dxZn a +a - a +a -d * adx kZ + 84- 1/ n dx 2'n dn7q, n dx' Hence dZ, dZ2 dZq dx ' dx ' ' dx ' are a Zetafuchsian system, admitting the Fuchsian group G and the isomorphic linear group G. The same property is possessed for all the derivatives of any order of the system Z1, Z2,..., Zq with regard to x. 170.] ZETAFUCHSIAN SYSTEM 523 This property is used by Poincare to obtain the most general expression for a Zetafuchsian system, admitting the groups G and G. Let it be Ti, T2,..., Tq; and construct the matrix dZ, dq-'Z, 1' dx ' " dxq-l' 1 dZ2 dq-lZ2 Zd, T' dx ' ' dx- '.............,................... o dZq dq-'Z, zq' d'"" q-l' Tq q' dxt ' ') dxq-l' q Denote by (- 1)-lAAi the determinant obtained by cutting out the ath column from the matrix: then, by a known property of determinants, we have dZn dq-_Zn cn+ n+ +A+qn AoZn + Al d-n +... + AqQ dxq_. +,qTn-O, dx A^ 4... q-iA=dxq-0, for all values of n. Hence oT - z 1\A dZn_ Aq-_ dq-Zn Inl - n A q dx ' A, dxq-1 When z is subjected to any transformation of.the group G, the quantities in any column in the matrix are subjected to the corresponding linear transformation of the group G; so that each of the q + determinants A0, A1,..., Aq is multiplied by the determinant of the linear transformation. Hence Ar. Aq is unaltered, that is, it is automorphic for the substitution of the group G; and therefore, as this property is possessed for each substitution, Ar + Aq is automorphic for the group G. Consequently, Ar Aq is a rational function of x and y, say ^Al- r, (r= 0I,,..., q-l); and therefore dZ dq-_Zn T = FoZ + F d +... + F Z for n =1, 2,..., q. This is Poincare's expression for the most general Zetafuchsian system, admitting the Fuchsian group G and the isomorphic linear group G. 524 CONCLUDING [170. We can immediately verify that Z1,..., Zq satisfy a linear differential equation, having coefficients that are rational in x and y. For dqZ, dqZ2 dqZq dxq' dxq ) ''. dxq' are a Zetafuchsian system, admitting the Fuchsian group G and the isomorphic linear group G; and therefore rational functions b o0, 'l,..., q- exist, such that d 7Zn dZnb dq-1Zn dxq = O + 01 d + q- dxq-1 holding for all values of n. Thus Z1,..., Zq are integrals of the linear differential equation dqZ dZ d _-_Z &d = fOZ + 01 dx +... + q-i d-'-l Similarly, T,..., Tq are integrals of a linear differential equation also of order q, having rational functions of x and y for its coefficients, and characterised by the same groups G and G as characterise the equation satisfied by Z1,..., Zq. CONCLUDING REMARKS. 171. The Zetafuchsian and Thetafuchsian functions thus used occur, for the most part, in the form of series of a particular kind; as they were first devised by Poincare, his name is frequently associated with them. The main aim in constructing them was to obtain functions which should exhibit, simply and clearly, the organic character of automorphism under the substitutions of the groups; and they are avowedly intended* to be distinct in nature from series adapted to numerical calculation, such as series in powers of z. Unless both these properties, viz. the exhibition of the organic character of the function and its adaptability to numerical calculation, are possessed by the functions involved, it is manifest that they are not in the most useful form. It is unlikely that the best development of the general theory can be effected, until * Acta Math., t. v, p. 211. 171.] REMARKS 525 functions have been obtained in a form that possesses both the properties indicated. In this connection, Klein* quotes a parallel instance from the theory of elliptic functions, viz. the series of the form (rm + m'w')-l, usedt by Eisenstein, which exhibit the characteristic automorphic property of the modular functions, but are not adapted to numerical calculation. Their deficiency in this respect has been met by the possession of the theta-functions and the sigmafunctions. The generalisation of the Jacobian theta-function and the Weierstrassian sigma-function, required for automorphic functions, has not yet been attained. We thus return to the statement made at the beginning of the foregoing sketch of Poincare's theory of linear differential equations with algebraic coefficients. The explicit analysis connected with the theory of automorphic functions has not yet acquired sufficiently comprehensive forms upon which to work; and therefore its application to linear differential equations, as to any other subject, can be only partial and imperfect in its present stage. The theory of automorphic functions in general presents great possibilities of research: the gradual realisation of these possibilities will be followed by corresponding developments in many regions of analysis. * Vorlesungen ii. lineare Dibferentialgleichungen d. zweiten Ordnung, (Gottingen, 1894), p. 496. See also Fricke und Klein, Theorie der automorphen Functionen, t. n, p. 155. t For references, see T. F., ~ 56. INDEX TO PART III. (The numbers refer to the pages in this volume. The Table of Contents at the beginning of the volume may be consulted.) Adjoint equation, Lagrange's, 253; and original equation are reciprocally adjoint, 253; is reducible if original equation is reducible, 254; composition of, 254; properties of, in relation to the number of regular integrals, 257. Algebraic coefficients, equations having, Chapter x; character of integrals of, in vicinity of branch-point, 479, and in vicinity of a singularity, 480; mode of constructing integrals of, 483; Appell's class of, 484; associated with automorphic functions, 488 (see automorphic functions). Algebraic equation, roots of, satisfy a linear equation with rational coefficients, 46, 174; connected with differential resolvents, 49. Algebraic integrals, equations having, 45, 165, Chapter v; connected with theory of finite groups, 175; connected with theory of covariants, 175; equations of second order having, 176 et seq.; equations of third order having, 191 et seq.; equations of fourth order having, 201; construction of, 184, 198; and homogeneous forms, 202. Analytical form of group of integrals associated with multiple root of fundamental equation of a singularity, 66; likewise for multiple root of fundamental equation for a period orperiods, 416, 454. Annulus, integral converging in any (see fundamental equation, irregular integral, Laurent series). anormales, 270. Apparent singularity, 117; conditions for, 119. Appell, 209, 484. Asymptotic expansions, Poincare's theory of, 338; represent normal integrals when functionally illusory, 340. ausserwesentlich, 117. Automorphic functions, and differential equations having algebraic coefficients, 488; and conformal representation, 491; associated with linear equations of second order, 495; constructed for a special case, 500; when there is one singularity, 506; when there are two singularities, 508; when there are three, 509, 510; in general, 510 et seq. Barnes, 448. begleitender bilinearer Differentialausdruck, 254. Benoit, 474. Bessel's equation, 1, 13, 34, 100, 101, 126, 164, 330, 333, 393. Bilinear concomitant of two reciprocally adjoint equations, 254. Bocher, 161, 169. Bocher's theorem on equations of Fuchsian type with five singularities, 161. Boole, 229. Boulanger, 195, 197, 198. Brioschi, 206, 208, 218. Casorati, 55, 60, 417. Cauchy, 11, 20. Cauchy's theorem used to establish existence of synectic integral of a linear equation, 11. Cayley, 94, 113, 121, 182, 216, 233, 246, 262, 281, 282, 286. Cayley's method for normal integrals, 281; for subnormal integrals, 284. Cazzaniga, 348. 528 INDEX TO PART III Cels, 254. Characteristic equation belonging to a singularity, 40. Characteristic equation for determining factor of normal integrals, 294; effect of simple root of, 294; effect of multiple root, 298. Characteristic function of an equation, 226. Characteristic index, defined, 221; and number of regular integrals of an equation, 230, 233; of reciprocally adjoint equations, the same, 257. Chrystal, 7, 83. Circular cylinder, differential equation of, 164; (see Bessel's equation). Class, equations of Fuchsian (see Fuchsian type). Cockle, 49. Coefficients, form of, near a singularity if all integrals there are regular, 78. Collet, 20. Conformal representation, and automorphic functions, 491; and fundamental polygon, 493. Constant coefficients, equation having, 14-20. Construction, of regular integrals, by method of Frobenius, 78; of normal integrals, periodic integrals (see normal integrals, simply-periodic integrals, doubly-periodic integrals). Continuation process applied to synectic integral, 20. Continued fractions used to obtain a fundamental equation, 439. Covariants associated with algebraic integrals, 202; for equations of third order, 203, 209; for equations of fourth order, 204; for equations of second order, 206. Craig, vi, 411. Crawford, 474. Curve, integral, defined, 203, 205. Darboux, 20, 254, 475. Definite integrals (see Laplace's definite integral, double-loop integral). Determinant of a system of integrals, 25; its value, 27; not vanishing, the system is fundamental, 29; of a fundamental system does not vanish, 30; special form of, for one particular system, 34; form of, near a singularity, 77; when the coefficients are periodic, 406, 445; when the coefficients are algebraic, 481. Determinants, infinite (see infinite detenrinants). Determining factor, of normal integral, 262; obtained by Thome's method, 262 et seq.; conditions for, 265; for integrals of Hamburger's equations, 288-292. Diagonal of infinite determinant, 349. Difference-relations, 63, 417. Differential invariants (see invariants, differential). Differential resolvents, 49. Dini, 254, 256. Divisors, elementary (see elementary divisors). Double-loop integrals, 333; applied to integrate equations, 334 et seq. Doubly-periodic coefficients, equations having, 441 et seq.; substitutions for the periods, 443; fundamental equations for the periods, 444, 445. Doubly-periodic integrals of second kind, 447; Picard's theorem on, 447; number of, 448, 450; belonging to Lame's equation, 468; how constructed, 471, 475. Eisenstein, 525. Element of fundamental system, 30. Elementary divisors, of certain determinants, 41-43; of the fundamental equation, 55; determine groups and sub-groups of integrals, 62; effect of, upon number of periodic integrals when coefficients are periodic, 416, 450. Elliott, M., 424, 425, 474. Elliptic cylinder, differential equation of, 164, 399, 431-441. Expansion of converging infinite determinants, 353. Expansions, asymptotic (see asymptotic expansions). Exponent, to which regular integral belongs, 74; properties of, 75; to which the determinant of a fundamental system belongs, 77; to which normal integral belongs, 262; of irregular integral as zero of an infinite determinant, 368. Exponents, sum of, for equations of Fuchsian type, 126; for Eiemann's P-function, 139. Fabry, 94, 270. Factor, determining (see determining factor). Fano, 214, 218. Finite groups of lineo-linear substitutions, in one variable, 176; connected with polyhedral functions, 181; associated with equations of second order having algebraic integrals, 182; used INDEX TO PART III 529 for construction of algebraic integral, 185; in two variables, 192; their differential invariants, 195; the Laguerre invariant, 196; used to construct equations of third order having algebraic integrals, 197; in three variables, 200. First kind, periodic function of, 410. Floquet, 231, 234, 259, 411, 448. Fricke, 489, 515, 525. Frobenius, 78, 93, 109, 226, 231, 233, 238, 247, 254, 257, 259. Frobenius' method, for the construction of integrals (all being regular), 78 et seq.; variation of, suggested by Cayley for some cases, 114; applied to hypergeometric equation for special cases, 147; for the construction of integrals, when only some are regular, 235 et seq.; used for construction of irregular integrals, 379 et seq. Fuchs, L., 10, 11, 60, 65, 66, 78, 93, 94, 109, 110, 117, 123, 125, 126, 129, 156, 206, 208, 216, 399, 482. Fuchsian equations, 123, 495 et seq.; independent variable a uniform function of quotient of integrals of, 496-499; mode of determining coefficients in, 501; used as subsidiary to linear equations of any order, 515. Fuchsian functions, associated with linear equations of the second order, 500, 502, 515; associated with linear equations of general order, 515, 517; in the expression of integrals as uniform functions, 520. Fuchsian group, (see Fuchsian function, Zetafuchsian function). Fuchsian type, equations of, Chapter iv, pp. 123 et seq.; form of, 123; properties of exponents, 126; when fully determined by singularities and exponents, 128; of second order with any number of singularities, 150; forms of, when o is an ordinary point, 152; when o is a singularity, 155, 158; Klein's normal, 158; Lame's equation transformed so as to be of, 160; equations of, having five singularities, 161; Bocher's theorem on, 161; having polynomial integrals, 166; having rational integrals, 169. Fundamental equation, belonging to a singularity, is same for all fundamental systems, 38-40; invariants of, 40; Poincare's theorem on, 40; F. IV. properties of, connected with elementary divisors, 41-43; fundamental system of integrals associated with, 50; when roots are simple, 52; when a root is multiple, 53; roots of, how related to roots of indicial equation, 94. Fundamental equation when integrals are irregular, expressed as an infinite determinant, 389; expressed in finite terms, 392; various methods of obtaining, 399. Fundamental equations for double periods, 444, 445; their form, 447; roots of, determine doubly-periodic integrals of the second kind, 448; number of these integrals, 450; effect of multiple roots of, 451. Fundamental equation for simple period, 405; is invariantive, 406; form of, 407; integral associated with a simple root, 408; integrals associated with a multiple root, 408; analytical expression of, 419. Fundamental equation when coefficients are algebraic, 482; relation to indicial equation, 482. Fundamental polygon for automorphic functions, 490, 493, 500. Fundamental system of integrals, defined, 30; its determinant is not evanescent, 30; properties of, 30, 31; tests for, 31, 32; form of, near singularity, 50; if root of fundamental equation is simple, 52; if root is multiple, 53; affected by elementary divisors of fundamental equation, 57; aggregate of groups associated with roots of indicial equation make fundamental system, 95. Fundamental system, of irregular integrals, 387; of integrals when coefficients are simply-periodic, 408, 419; when coefficients are doubly-periodic, 449-457; when coefficients are algebraic, 480. Fundamental system, constituted by group of integrals belonging to a multiple root of fundamental equation (see group of integrals). Gordan, 182. Grade of normal integral, 269. Graf, 333. Greenhill, 466. Group of integrals, associated with multiple root of fundamental equation, 53; resolved into sub-groups, by elementary divisors, 57; Hamburger's sub-groups of, 62; general analytical form of, 65; can be fundamental system of equation of lower order, 72. 34 530 INDEX TO PART III Group of integrals, associated with multiple root of indicial equation in method of Frobenius, 80; general theorem on, 93; aggregate of, make a fundamental system, 96; compared with Hamburger's groups, 113. Group of integrals for hypergeometric equation, 144. Group of irregular integrals associated with multiple root of characteristic infinite determinant, 381 et seq.; resolved into sub-groups, 382. Group of integrals associated with multiple roots of fundamental equations for periods when coefficients are doublyperiodic, 451; analytical expression of, 452, 457; further development of, when uniform, 459. Group of integrals associated with multiple root of fundamental equation for period when coefficients are simplyperiodic, 415; arranged in sub-groups, according to elementary divisors, 416; analytical expression of, 419; they constitute a fundamental system for equation of lower order, 420; further expression of, when uniform, 421. Groups of substitutions, finite (see finite groups); infinite (see automorphic functions). Griinfeld, 259. Gubler, 333. Giinther, 11, 299, 399. Gylden, 462. Halphen, 254, 255, 281, 315, 316, 448, 464, 465, 473. Hamburger, 38, 60, 62, 63, 64, 113, 277, 280, 283, 286, 399, 482. Hamburger's equations, 276 et seq.; of second order with normal integrals, 279; the number of normal integrals, 280; of general order n with normal or subnormal integrals, 288 et seq.; of third order with normal or subnormal integrals, 301 et seq. Hamburger's sub-groups of integrals (see sub-groups of integrals). Hankel, 103, 333. Harley, 49. Heffter, 55, 156. Heine, 164, 166, 431, 441. Hermite, 15, 20, 448, 463, 465, 468, 473. Hermite, on equation with constant coefficients, 15-20; on equation with doubly-periodic coefficients, 465. Heun, 159. Heymann, 50. Hill, G. W., 348, 396, 398, 399, 402, 482. Hobson, 334, 337. Homogeneous forms (see covariants). Homogeneous linear equations, defined, 2; discussion limited to, 3. Homogeneous relations between integrals when they are algebraic, 203, 217; of second degree for equations of third order, 2 0; and of higher degree, 214. Horn, 333, 341, 342, 346, 347. Humbert, 167. Hypergeometric function, used to render integrals of differential equations uniform in special case, 509. Hypergeometric series, equation of, 1, 13, 34, 103, 126, 144-150, 173, 338, 501, 509. Identical relations, polynomial in powers of a logarithm, cannot exist, 69. Index, characteristic (see characteristic index); to which regular integral belongs, 74; properties of, 75. Indicial equation, when all integrals are regular, 85, 94; significance of, in the method of Frobenius, 85; integral associated with a simple root of, 86; group of integrals associated with a multiple root of, 86; roots of, how connected with roots of fundamental equation, 94; for equation with not all integrals regular, 222, 227; when coefficients are algebraic, 482; relation of, to the fundamental equation, 482. Indicial function, when all integrals are regular, 94; when not all integrals are regular, 227; degree of, as affecting the number of regular integrals, 230, 233; of adjoint equation, as affecting the number of regular integrals, 259. Infinite determinant, giving exponent of irregular integral, 368; modified to another determinant, 369; is a periodic function of its parameter, 375; effect of simple root of, 380, of a multiple root of, 381 et seq.; leads to the fundamental equation of the singularity, 389; expressed in finite terms, 392. Infinite determinants in general, 349; convergence of, 350; properties of converging, in general, 352 et seq.; uniform convergence of, when functions of a parameter, 358; may be capable of differentiation, 359; used to solve an unlimited number of linear equations, 360; applied to construct irregular integrals of differential equations, 363 et seq. INDEX TO PART III 531 Initial conditions defined, 4; values, 4; effect of, upon form of synectic integral, 9. Integral curve, 203, 205. Integrals, irregular (see irregular integrals). Integrals, doubly-periodic, irregular, normal, regular, simply-periodic, subnormal, synectic (see under these titles respectively). Integrals rendered uniform functions of a variable, when there is one singularity, 506; when there are two singularities, 508; when there are three, 509, 510; in general, 510 et seq.; by means of Zetafuchsian functions, 518, 520. Invariants, differential, Schwarzian derivatives as, for equations of the second order, 182; for equations of the third order, 195; for equations of the fourth order, 201, 213; Laguerre's, 196. Invariants of fundamental equation, connected with singularity, 38, 40; for irregular integrals, 398; connected with a period or periods, 405, 445; when coefficients are algebraic, 482. Irreconcileable paths, defined, 23. Irreducible equations, exist, 347; Frobenius' method of constructing, 248. Irregular integrals, in the form of Laurent series, 364; converge within an annulus, 366; formal expression for, obtained by infinite determinants, 376; groups and sub-groups of, obtained by generalisation of Frobenius' method, 379; these constitute a fundamental system, 387; made uniform functions of a new variable by means of automorphic functions (see automorphic functions). Jordan, 197, 200, 333, 334, 338, 341. Jiirgens, 65, 113. Klein, 150, 153, 155, 158, 161, 176, 185, 187, 190, 197, 206, 489, 515, 525. Klein's normal form of equation of second order and Fuchsian type, 158; method for equations of second order having algebraic integrals, 176. Kneser, 341. von Koch, 348, 359, 398, 399, 482. Kummer, 146. Kummer's group of integrals of the hypergeometric equation, 144. Lagrange, 251. Laguerre, 196. Lame's equation, 1, 126, 159, 160, 165, 168, 338, 448, 464-473. Lam6's generalised equation, 160. Laplace's definite integral, satisfying equation with rational coefficients, 318; contour of, 323; developed into normal integrals, where these exist, 324 et seq. Laurent series expressing an irregular integral, 364; proof of convergence within an annulus, 366. Legendre's equation, 1, 13, 34, 103, 126, 160, 163. Liapounoff, 319, 425-431. Liapounoff's theorem, applied to evaluate Laplace's definite integral, 324; method of discussing uniform periodic integrals, 425. Lindemann, 431, 434, 437. Lindstedt, 439. Linear algebraic equations, infinite system of, solved by means of infinite determinants, 360. Linear differential equation, definition of, 2. Lineo-linear substitutions (see finite groups). Logarithms, quantity affected by, can satisfy a uniform linear differential equation and determine its fundamental system, 66; identical relations, polynomial in powers of, cannot exist, 69; regular integrals free from, 106; condition that some regular integral shall be free from, 110. Lommel, 331. Macdonald, 333. Markoff, 169. Member of a fundamental system, 30. Minors of infinite determinants, 354. Mittag-Leffler, 399, 463. Modular function, used to render integrals of differential equations uniform in special case, 510; Eisenstein's function similar to, 525. Multiple root, group of integrals associated with (see multiple root). Multiplier of periodic integral of second kind, 410; is a root of the fundamental equation of the period, 406. Muth, 42. Normal form, (after Frobenius) of equation having some integrals regular, 227; of component factors of such an equation, and of a composite equation, 228; (after Klein) of equation of Fuchs 532 INDEX TO PART III ian type, 158; of infinite determinant, 350. Normal integrals, defined, 262; constructed by Thome's method, 262 et seq.; aggregate of, satisfy another differential equation, 271; conditions that Hamburger's equation of second order may have, 279, and the number of, 280; Cayley's method of obtaining, 281; of Hamburger's equation of order n, 288 et seq.; number of, belonging to equation of order n, 295, 298; belonging to equation of third order, 304, 308, 309; of equations with rational coefficients, 313; arising out of Laplace's definite integral, 329; are asymptotic representation of Laplace's integral, 340. Number of regular integrals of an equation and its characteristic index, 230; can be less than maximum value, 233; and the number for the adjoint equation, 257. Ordinary point, synectic integral in domain of, 4. Origin of infinite determinant, 349; can be moved in the diagonal without changing the value of the determinant, 350. Osgood, 85, 122. Painleve, 195, 198, 199. Papperitz, 142. Parabolic cylinder, equation of, 165. Paths, if reversed in continuation process, restore initial values, 21; deformation of, without crossing singularity, 22; reconcileable, and irreconcileable, 23; effect of, round a singularity, Chapter ii, passim. Pepin, 206. Period, fundamental equation for simple, 405; fundamental equations for double, 445; (see fundamental equation). Periodic coefficients, equations having uniform, Chapter ix, 403 et seq.; simply (see simply-periodic coefficients); doubly (see loubly-periodic coefficients). P-function, discussion of (see Riemann's P-function). Physics, equations of mathematical, and equations of Fuchsian type having five singularities, 161. Picard, vi, 317, 319, 341, 443, 447, 448, 460, 471. Pochhammer, 105, 159, 333, 338. Poincare, 40, 61, 105, 246, 270, 271, 315, 317 et seq. passim, 330, 338 et seq., 347, 348, 353, 399, 441, 482, 489 et seq. passim. Poincare's theorem on aggregate of normal integrals of a given equation, 271; development of Laplace's definite integral that satisfies equation with rational coefficients, 318 et seq.; asymptotic expansions, 338 et seq.; applications of automorphic functions to equations having algebraic coefficients, 488 et seq.; theorem on the integration of linear equations by means of zetafuchsian functions, 517,523. Polygon, fundamental (see fundamental polygon). Polyhedral functions, and finite groups, 181; associated with equations of second order having algebraic integrals, 182; used for construction of algebraic integral, 185. Polynomial integrals, equations having, 166; how far determinate, 167. Potential, equation for the, solved by means of Lame's equation, 465. Principal diagonal of infinite determinant, 349. Puiseux diagram used, 267, 269, 274, 285, 300, 310, 311. Quarter-period in elliptic functions, equation of, 1, 129, 337, 510. Rank, of differential equation, defined, 271; equations of, greater than unity replaced by equations of rank unity, 342 et seq. Rational coefficients, equations having, 313 et seq.; normal integrals of, 314; Laplace's definite integral solution of, 318. Rational integrals, equations having, 169. Real singularity, 117; conditions for, 119. Reconcileable paths, 23. Reducibility of equations, defined, 223; extent of, when some integrals are regular, 226, 248; if they possess normal or subnormal integrals, 273. Reducible, equations having regular integrals, are, 224, 226, 248; INDEX TO PART III 533 adjoint of a reducible equation is, 253; equation, having a reducible adjoint, is, 254; equations, having normal or subnormal integrals, are, 273. Regular, equations when only some integrals are, Chapter vi; form of coefficients, 221; equations having some integrals, are reducible, 224, 226; integrals possessed by an equation, number of, 230; equations having no integrals, 231, 233; integrals, when they exist, constructed by method of Frobenius, 235 et seq.; conditions that they exist, 237; how many integrals of adjoint equation are, 257. Regular integrals, defined, 4, 74; form of coefficients near a singularity if all integrals are, 78; construction of, by method of Frobenius, 78; conditions that all may be free from logarithms, 106; conditions that some may be free from logarithms, 110; equations having all integrals everywhere regular, Chapter iv (see Fuchsian type). Resolvents, differential, 49. Riemann, 137, 140. Riemann's P-function, definition of, 136; transformations of, 137; determines a differential equation, 141, 163, 165; forms of differential equation thus determined, 143; group of integrals deduced for hypergeometric equation, 144. Roots of fundamental equation and of indicial equation, how related, 94. Salmon, 43. Sauvage, 42. Schlesinger, vi, 113, 218. Schwarz, 492. Scott, R. F., 41. Second kind, periodic functions of, 410, 447; equation with periodic coefficients has integrals which are, 411, 447; number of such integrals, 411, 417, 448, 450; (see simply-periodic integrals, doubly-periodic integrals). Simply-periodic coefficients, equations having, 403 et seq.; possess integrals which are periodic of second kind, 411; analytical expression of these integrals, 415. Simply-periodic integrals of second kind, 411; their analytical expression, 412. Singularities of a differential equation, 3; real or apparent, 117, with conditions for discrimination, 119; how treated when coefficients are algebraic, 490 et seq. Singularity, effect of path round, 36; equation connected with, is invariantive, 38. Stieltjes, 169, 437. Sub-groups of irregular integrals (see group of irregular integrals, irregular integrals). Sub-groups, in a group of integrals associated with multiple root of fundamental equation, 57; Hamburger's, 62; number of, is equal to number of elementary divisors of fundamental equation, 62; general analytical form of, 65; can be fundamental system of an equation of lower order, 72. Sub-groups of periodic integrals, determined by elementary divisors of the fundamental equation of the period, 416; are analogous to Hamburger's sub-groups of regular integrals, 417; analytical expression of, 419. Subnormal integrals, defined, 270; how constructed, 270; aggregate of, satisfy another equation, 271; of Hamburger's equation of second order, 286; Cayley's method of obtaining, 284; of Hamburger's equation of order n, 299 et seq.; of Hamburger's equation of third order, 309, 313. Subsidiary equation for integration of any linear equation in terms of uniform functions, Fuchsian equations used as, 517. Substitutions, finite groups of lineolinear substitutions (see finite groups). Sylvester's eliminant used, 46. Synectic integral in domain of ordinary point, 4; is unique as determined by initial conditions, 8; vanishes if all initial values vanish, 9; is linear in initial values, 9; modes of establishment of, 10, 11; continuation of, 20. System of functions, when linearly independent, can satisfy a linear differential equation of which they are a fundamental system, 44; when the coefficients in the equation are rational, 45, 223; this property used to reduce an equation (see reducible equations). System of integrals, determinant of, 25; fundamental (see fundamental system). Tannery, 44, 94, 109, 129, 131, 135. 534 INDEX TO PART III Ternariants (see covariants). Thetafuchsian functions used, 520 et seq. Third kind, periodic functions of, 410; equation having integrals which are, 411. Thome, 74, 221, 231, 232, 233, 234, 257, 259, 262, 483. Thome's method of obtaining the determining factor of a normal integral, 262 et seq. Tisserand, 431, 441. Transformation of equations of rank greater than unity to equations of rank unity, 342 et seq. Type, equations of Fuchsian (see Fuchsian type); of equations, as associated with automorphic functions, 516. Uniform doubly-periodic integrals, 459; modes of constructing, 460, 468, 471, 475; illustrated by Lame's equation, 464 et seq. Uniform functions, integrals of equa tions expressible as, by means of automorphic functions (see automorphic functions); simple examples of, 506, 508, 509, 510; in general, 510 et seq.; Poincare's theorem on, 518. Uniform simply-periodic integrals, 421; Liapounoff's method of dealing with, 425. Vaientiner, 197. van Vleck, 169. Vogt, 399. Weber, 165, 333. Weierstrass, 42, 85, 117, 277. wesentlich, 117. Whittaker, 515. Williamson, 320. Zetafuchsian functions, 520; properties of, 521; used to express the integral of any linear equation, 522-524; most general expression of, 523. CAMBRIDGE: PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.