Production Note Cornell University Library pro- duced this volume to replace the irreparably deteriorated original. It was scanned using Xerox soft- ware and equipment at 600 dots per inch resolution and com- pressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Stand- ard Z39.48-1984. The production of this volume was supported in part by the Commission on Pres- ervation and Access and the Xerox Corporation· Digital file copy- right Cornell University Library 1991.A TREATISE ON Linear Differential Equations THOMAS CRAIG, Ph.D. ASSOCIATE PROFESSO!? IN THE JOHNS HOPKINS UNIVERSITY ASSOCIATE EDITOR OF THE AMERICAN JOURNAL OF MATHEMATICS Volume I EQUATIONS WITH UNIFORM COEFFICIENTS NEW YORK JOHN WILEY & SONS 15 Astor Pi,ace 1889 'A'A 'CORNELL" UNIVERSITY \ LIBRARY Copyright, 1889, BY THOMAS CRAIG. Drummond & Nett, Electrotypers, 1 to 7 Hague Street. New York. Ferris Bros., Printers, 326 Pearl Street, New York.PREFACE. The theory of linear differential equations may almost be said to find its origin in Fuchs’s two memoirs published in 1866 and 1868 in volumes 66 and 68 of Crelle’s Journal. Previous to this the only class of linear differential equations for which a general method of integration was known was the class of equations with constant coefficients, including of course Legendre’s well-known equation which is immediately transformable into one with constant coef- ficients. After the appearance of Fuchs’s second memoir many mathematicians, particularly in France and Germany, including Fuchs himself, took up the subject which, though still in its infancy, now possesses a very large literature. This literature, however, is so scattered among the different mathematical journals and publications of learned societies that it is extremely difficult for students to read up the subject properly. I have endeavored in the present treatise to give a by no means complete but, I trust, a sufficient account of the theory as it stands to-day, to meet the needs of students. Full references to original sources are given in every case. Most of the results in the first two chapters, which deal with the general properties of linear differential equations and with equations having constant coefficients, are of course old, but the presentation of these properties is comparatively new and is due to such mathe- maticians as Hermite, Jordan, Darboux, and others. All that fol- lows these two chapters is quite new and constitutes the essential part of the modern theory of linear differential equations. The present volume deals principally with Fuchs’s type of equa- tions, i.e. equations whose integrals are all regular; a sufficient account has been given, however, of the researches of Frobenius and Thom£ on equations whose integrals are not all regular. A pretty full account, due to Jordan, has been given of the application of the iiiIV PREFACE. theory of substitutions to linear differential equations. This subject will, however, be very much more fully dwelt upon in Volume II, where I intend to take up the question of equations having algebraic integrals and also to give an account of Poincaré’s splendid investi- gations of Fuchsian groups and Fuchsian functions. The theory of the invariants of linear differential equations has been several times touched upon in the present volume, and some of the simpler results of the theory have been employed ; but its extended develop- ment is necessarily reserved for Volume II, as is also the develop- ment of Forsyth's associate equations, about which extremely inter- esting subject very little is as yet known. The equation of the second order with the critical points o, I, oo has on account of its great importance been very fully treated. In connection with this subject it seemed to me that I could not possibly do better than to reproduce, which has been done in Chap- ter VII, Goursat’s Thesis on equations of the second order satisfied by the hypergeometric series. M. Goursat was kind enough to give me permission to make a translation of his Thesis, which is, I im- agine, not very well known among English and American students. In Chapter XIV I have given only a brief account of equations with doubly-periodic coefficients. I intend, however, to resume this subject in Volume II. I wish here to tender my thanks to M. Goursat for his kindness in permitting me to make a translation of his most valuable Thesis, and to Dr. Oskar Bolza, Mr. C. H. Chapman, and Dr. J. C. Fields for much valuable assistance. T. Craig. Johns Hopkins University, Baltimore, 1889.CONTENTS. CHAPTER I. GENERAL PROPERTIES OF LINEAR DIFFERENTIAL EQUATIONS. PAGE: Two General Theorems, ........... i Normal Form of a System, ........... 5 Independence of Solutions, ................4 The Adjoint System,........................9 Application to a Single Equation,.........11 Lagrange’s Adjoint Equation,..............18 Laguerre’s Invariants,....................19 Existence of an Integral,.................22 CHAPTER II. LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS Euler’s Method, ............. 23, Hermite’s Presentation of Cauchy’s Method,........24 Darboux’s Presentation of Cauchy’s Method,........33 Systems of Equations with Constant Coefficients,..36 Equations Reducible to Equations with Constant Coefficients, .... 41 Méray’s Method,...................................43 CHAPTER III. THE INTEGRALS OF THE DIFFERENTIAL EQUATION p_d«y. d«-'y dx« ^~^ldxn - 1 dxn - 2 ‘ • · · +pny = o. Linear Transformation of a System of Fundamental Integrals, Equations with Doubly-Periodic Coefficients, Values of the Cofficients p u p 2, . . · , pn , Formation of a System of Fundamental Integrals, The Characteristic Equation,.............. Independence of Choice of Fundamental System, . Hamburger’s Theorem, ....... Case of Equal Roots, ........ 51 52 54- 59 61 63 64 vvi CONTENTS. PAGE Investigation of the Forms of the Integrals,....................................65 Hamburger’s Determination of the Sub-groups of the Integrals, . . -76 Jordan’s Canonical Form of the Substitution corresponding to any Critical Point, ....................................................................84 Forms of the Substitutions S and S' in the case of Equations with Doubly-Periodic Coefficients,..............................................................84 Regular Integrals, 91 CHAPTER IV. FROBENIUS’S METHOD. Convergence of a Series and Proof of the Existence of an Integral, ... 94 Existence of Logarithms,.......................................................104 CHAPTER V. LINEAR DIFFERENTIAL EQUATIONS ALL OF WHOSE INTEGRALS ARE REGULAR. Definition of the Function E, .......... 108 Properties of Regular Functions, ......... 109 Fuchs’s Theorem,.................................................................m The Indicial Equation,.........................................................118 Forms of the Coefficients,.....................................................123 Fuchs’s Converse Theorem and Existence of an Integral,.........................123 Sum of the Roots of all the Indicial Equations,................................134 Applications, ............. 135 Reducibility and Irreducibility,...............................................148 CHAPTER VI. LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER, PARTICULARLY THOSE WITH THREE CRITICAL POINTS. Fuchs’s Transformation,........................................................154 The General Linear Transformation, . . . . . . . . .156 The Critical Points o, 1, 00, . . . . . . . . . . .157 Change of the Dependent Variable, . . . . . . . . .158 Differential Equation for the Hypergeometric Series,...........................159 The Twenty-four Particular Integrals, . . . . . . . . .164 Division of these into Six Groups, . . . . . . . . .166 Relations between the Integrals belonging to two different Critical Points, . .169 Markoff’s Two Problems,........................................................174 Heun’s Application of Abelian Integrals of the Third Kind, . . . . . 177 Riemann’s P-function,..........................................................185 Spherical Harmonics, Toroidal Functions, and Bessel’s Functions, . . . 193 Generalized Spherical Harmonics and Bessel’s Functions, ..... 200 Humbert’s Investigation,.......................................................202CONTENTS. Vil CHAPTER VII. ON THE LINEAR DIFFERENTIAL EQUATION WHICH ADMITS THE HYPERGEOMETRIC SERIES AS AN INTEGRAL ; BY M. EDOUARD GOURSAT. Part First. PAGE Summary of Contents, . . . . . . . . . . . .212 Jacobi’s Method,...........................................215 Table of Integrals, ............ 229 Application of Cauchy’s Theorem and Relations between the Integrals, . . 232 Application to the Complete Elliptic Integral of the First Kind, .... 252 Schwarz’s Results,.........................................258 Part Second. Transformations of the H y pergeô metric Series,.......................276 Tannery’s Theorem,.........................................277 Change of the Independent Variable: Linear Transformation, .... 279, General Theory of the Direct and Inverse Transformations,..281 CHAPTER VIII. IRREDUCIBLE LINEAR DIFFERENTIAL EQUATIONS. Frobenius’s Theorems, 362 Determination of the Reducibility or Irreducibility of a Linear Differential Equa- tion by the Study of its Group,.........................369 CHAPTER IX. LINEAR DIFFERENTIAL EQUATIONS SOME OF WHOSE INTEGRALS ARE REGULAR. Definition of the Order of a Coefficient of the Equation, ..... 373 The Characteristic Index,..................................374 The Indicial Equation,.....................................374 Thomê’s Theorems,..........................................375 The Characteristic Function,...............................377 The Indicial Function, ............ 379 Superior Limit to the Number of Linearly Independent Integrals, . . . 382 The Normal Form of a Linear Differential Equation, '.......382 Composite Differential Quantics, ..........................383 Irregular Integrals, ............ 386 Thomé’s Normal Integrals,................................ 388 CHAPTER X. DECOMPOSITION OF A LINEAR DIFFERENTIAL EQUATION INTO SYMBOLIC PRIME FACTORS. Definition of Symbolic Prime Factors, . . . . . . . . · 389 Method of Decomposition, ..................................39a Resulting Forms of the Coefficients of the given Equation,.392 A Transformation of the given Equation,....................393Vili CONTENTS. PAGE Conjugate Solutions,..............................................................398 Conditions for Commutative Prime Factors,.........................................400 Form of Linear Differential Equation possessing Commutative Prime Factors, . 402 Application to Regular Integrals,.................................................406 Composite Differential Quantics in general,.......................................409 Number of Linearly Independent Regular Integrals,.................................415 Theorems concerning the Adjoint Equation,.........................................421 CHAPTER XI. APPLICATION OF THE THEORY OF SUBSTITUTIONS TO LINEAR DIFFERENTIAL EQUATIONS. Products and Powers of Substitutions, . . ...............422 Canonical Form, ............. 423 Function-Groups, 424 Number of Distinct Functions in a Function-Group,......... 425 Prime Groups, 432 Characteristic Equation possessing a single Root,........433 Case of Incompatible Equations, .......... 444 Canonical Form of Substitutions in a Non-Prime Group,....446 Final Determination of Function Groups, ........ 448 CHAPTER XII. EQUATIONS WHOSE GENERAL INTEGRALS ARE RATIONAL.—HALPHEN’s EQUATIONS. Equations whose General Integrals are Rational, ....... 452 Form of the General Integral,.....................................................453 Halpken's Equations. (1) Equations whose Integrals are Regular and Uniform in every region of the plane that does not contain the point x1, ....... 454 (2) Equations whose General Integral is of the form y = cxe°-ixfx{x) + + . . . -\-cneanxf «(■*)» .... 455 Examples,.........................................................................461 CHAPTER XIII. TRANSFORMATION OF A LINEAR DIFFERENTIAL EQUATION.—FORSYTH’S CANONICAL FORM. —ASSOCIATE EQUATIONS. Change of the Dependent Variable and Removal of the Second Term, . . 463 Change of both Variables,................................................... . 464 Brioschi’s Invariant, ...........................................................467 Invariants of a Linear Differential Equation, ....... 469 Index of an Invariant............................................................469 Forsyth’s Canonical Form,........................................................470 Lagrange’s Adjoint Equation, . . . ................................. 471 Forsyth’s Associate Equations, ..................................................473CONTENTS. IX PAGE Properties of Lagrange’s Adjoint Equation,..........................479 Adjoint Differential Quantics,......................................482 The Adjoint of a Composite Differential Quantic,....................484 Relation between the Integrals of Adjoint Equations,................485 Frobenius’s Associate Bilinear Differential Quantic,................487 Halphen on Adjoint Quantics, . ................................488 Self-Adjoint Quantics and Equations, ......... 490 Examples,..........................................................490 Appell’s Theorem, ..................................................495 CHAPTER XIV. LINEAR DIFFERENTIAL EQUATIONS WITH UNIFORM DOUBLY-PERIODIC COEFFICIENTS. Elementary Properties of Doubly-Periodic Functions, ...... 496 Integrals of Equations with Doubly-Periodic Coefficients, ..... 503 Picard’s Theorem, ............ 503 Canonical Form of the Substitutions S and S', ....... 504 Formation of Functions submitting to these Substitutions and satisfying the given Differential Equation, ........... 504 Determination of the Constants entering into the Integrals, . . . . .510 Determination of Additional Integrals, .............................5x4LINEAR DIFFERENTIAL EQUATIONS. CHAPTER I. GENERAL PROPERTIES OF LINEAR DIFFERENTIAL EQUATIONS. According to general usage we will denote by x an independent complex variable; that is, x is the affix of any variable point in the plane. When fixed points, such as the critical points of the various functions we shall encounter, are to be spoken of, we will generally denote their affixes by a, by c, . . . Again, y is an unknown function of xy defined by a linear differential equation—that is, a differential equation whose left-hand member is a linear function of y and of the derivatives of y with respect to x; the coefficients in this linear function are arbitrary functions of x; the right-hand member of the linear differential equation is either zero or a function of x alone. The equation has thus the form (I) dny L A dn~ zy dxn ~^~i>'dxn-' A d”~*y dxn~2 + · · · + P»y — E(x). A first general property of linear differential equations is that they remain linear when we change the independent variable. Write, viz., x = (p(t)y and form the new differential equation connecting y and t: it will obviously be an equation of the same form as (i), having new coefficients containing t alone. The verification of this is so simple that it need not be given here. Another general proposition is (see Jordan’s Cours ct Analyse, vol. iii. p. 136) as follows: If y, z, . . ., w denote n functions of the same independent variable x, and satisfying the system of n linear differential equationSy (2) E1 = o, E^ — o, . . .) En — o ÿ2 LINEAR DIFFERENTIAL EQUA IVONS. and if V is a polynomial in y> z, . . w and their successive derivatives, the different terms of which have for coefficients arbitrary functions of x: then V will satisfy a linear differential equation whose coefficients are rationally expressible in terms of the coefficients of Ex, . . . Etn Vy and of their successive derivatives. Let fXy //, . . . denote the highest orders of the derivatives of y, #9 · · · , respectively, in equations (2), and let X denote the degree of the polynomial V. Form now the successive derivatives of V with respect to x \ each of these is a polynomial of degree X in y, s, . . . and their derivatives ; and, further, the derivatives dft + Iy d^z d^' + 'z u; dxt’ d^^y ‘ y dx*9 di^ny ···>··· are linearly expressible in terms of the derivatives of lower order by means of equations (2) and their derivatives. Substituting these values for the quantities (3), we have (4) V = xlpl +XS. +---- • df=Xt'Pl+X,'P,+----- where the functions X, Xf, . . . are functions of x of the kind men- tioned in the above proposition, and Px, Pa , . . . are products of the form the total number of factors (taking account of their orders of multiplicity) being at most = X. The number of these products P is of course limited ; let them be denoted by P\ 1 P* 9 * * ·> Pi · Eliminating these quantities between equations (4), we obviously arrive at a linear relation connecting V, , . . . From the dx dx* theory of ordinary differential equations we know that any simul- taneous system of such equations can, by the introduction of certainGENERAL PROPERTIES. 3 auxiliary functions and the performance of certain simple algebraical operations, be thrown into the normal form—that is, can be replaced by a system of equations in each of which there appears only the first derivative of one of the (old and new) dependent variables. In case the given system of simultaneous equations is linear, it is ob- vious that the property of linearity will be retained when the equa- tions are replaced by the normal system. In studying, then, a system of simultaneous linear differential equations, we may at once suppose the system to be replaced by the corresponding normal system. Suppose such a normal system to be ^ + a^yx + ax*y* + · ■ - . + axnyn = Tt , ^ + · ' • · ^2«J^« = 7^, 9 -^Jranxyl-\-0’«iyt-\- · · • · + &nnyn = Tn \ where 7\ , T9, ... 9 Tn and the coefficients aik are functions of x. As a particular case, suppose the given system of simultaneous linear differential equations to reduce#to the single one, (6) dn~'y I + d”~xy \ dx* dx*~'+A dx —2 + +Pny — T. Denoting by y', yr\ . . . , yn~x new auxiliary dependent variables, we can replace (6) by the equivalent normal system, y dx y = O, dl. - y" dx J = o, dy*-» „ , y»-* = o, dx ’dx - + ^—+·· - · -\-p»y = 71 <7)4 LINEAR DIFFERENTIAL EQUATIONS. As the reader is supposed to be familiar with the elementary theory of differential equations, and with the general properties of linear differential equations, it will be sufficient here to state some of these properties without proof. First : if y^, y9\ . . . , y„ ; yy . . . , yn* ; . . . are particular solutions of equations (5) when Tx, iTa, . . . , Tn are all supposed to be zero, then, denoting by C1 r , . . . , Cn arbitrary constants, (8) Ci yx + C^yJ + · · · + Cnyn\ Cxy? + Qy* + · · · + Cnyn, are also solutions. This is at once proved by substituting in equa- tions (5) when the right-hand members are all replaced by zero, that is, when equations (5) are of the form (9) jr: + 4- · dv + · dyn , , · , . · + ainyn = o, • . + a2nyn = o, . . —}— annyn — o. An obvious, but useless, solution of these last equations is yl — y2 = . . . — yn — o. Aside from this useless solution, suppose equations (9) to admit the following k particular solutions : y,, ƒ.*··■ , yn, R, y: , yn\ yf, y*, · · · , ynk- These solutions will be said to be independent if one at least of the determinants of order k formed from the different columns of (10) does not vanish. If k = n, then the solutions (10) will be independ- ent if the determinantGENERAL PROPERTIES. 5 y,\ y.'> · · y»\ y*> y', · · yn\ yy, yy, · yd*, does not vanish. The particular application of this to equations (7) and (6) is easily seen, but will be referred to later on. As an illustration of another well-known general theorem, stated below, consider the system (12) dy + ay + bz -j- ou + dv =0, dz ~fa + aiy + h* + + dxv = o, du , ^ + c*u + d^v = o, dv + a% y + + CZU + d%v = o, and suppose that we know two independent solutions of this sys- tem, viz.: y 1» ¿A «1» Vx\ y*> *f, v%. Suppose, for example, that the determinant ƒ 1 > *1> y z > , does not vanish. Let us write now y — cx yi + c*.y*’ /j,\ I ^ = £>, + u = Clul -)- C2u9 + - v = £>. + C'V, + v; where ¿r„ c„ S, tf, are new functions of x. Substituting the values (13) in (10), and remarking that the terms in cx and vanish (since6 LINEAR DIFFERENTIAL EQUATIONS. equations (io) would be satisfied if g and r, were zero and Ct and Ct constants), we have (H) (dCt dC, , * , . d^y' + ~dxy' + c% +dr1 dC; 1 + + c£ H~ diV = O, = O, dC, dC, dg ~dxu'^ ~dÌU· + Tx + ^ - °> dC, dC, . , _ dxv' + n + c%s + ^ - a „ , . . c dC, dC, dg dr, Solving these for ^^, we derive the system ('5) + dC, dx dS dx drj Vx, y*> u*> ^ > an independent system. (We of course exclude the obvious solu- tion, % = o, rj = o, of the last two of equations (15)). To fix the ideas, suppose g1 to be different from zero, and let C,1 and Ctl denoteGENERAL PROPERTIES. 7 the values of Cx and £7a as obtained by the quadratures; then we can show at once that the solution >3 = 0, + C.r2> /l6X I *3 = Qzx + cax, uz — C1lul -f- C,X -f- 4f\ <- v% = Qvx + C>a + is independent of the two known solutions, yx, zxy uiy vlf y*> in fact, we see that the determinant reduces to y,y *x , u *3 , u y>> ^3 , u 1 y, ’ *x y y,> **y a y s y and by hypothesis neither of these factors vanishes. The general theorem above referred to, and of which the preceding is a simple illustration, may be stated as follows : If we know k independent solutions (k < n) of a system of linear differential equations of the first order, the integration of this system can be conducted to that of an analogous system of n — k equations and to quadratures. Each solution of this nezv system will furnish a solution of the given system which will be independent of those already known. Another well-known general theorem which may be stated with- out proof is : Every system of linear differential equations of the form (9) admits n particular independent solutions, y', y,\ · · • , yny y'y y'y · · • , yny y”< y,*» · · . , ynn8 LINEAR DIFFERENTIAL EQUATIONS. and its most general solution is <·.A1 + + · · · + c„yf, cly,' + + · · · + c„yf, c,yf + c%yn* + . . . + c„y„” ; where c1, c79 . . ., cn are arbitrary constants. This is readily verified by substitution in (9), and by aid of the preceding theorem. Suppose Y' = 1 + ci yÌ + . . ■ + c.'y.\ y: = · · · , y«n, of equations (18); then denoting by ^1 > > · · • y ¿n y n arbitrary constants, we have r Ky,1 + + · · • + Y„yn = , (21) Y,y' + Y,yï + · · ■ . + = c,, . * · V „j,” + r;+ . . • + Yny„” = c„. Solving these last equations for Yx, Fa, . . . , Yn, we have the general solution of the system (19) involving, as it should, n arbitrary constants. If we knew only k solutions (k < n) of the given system (18), then we would have the k relations ' W + Y*y.'+ · · · + Y.yu' = cxf , + Y.y! + · · · + Y,yn' = . Y>y>*+ W+ · · · + Y.yl = c*. (22)GENERAL PROPERTIES. It These last relations would enable us to eliminate k of the fune- tions Y19 Fa, . . . , Yn from the adjunct system (19), and so the problem of the integration of this system would be reduced to that of the integration of a similar system containing only n — k equa- tions. This matter of the adjunct equation will be taken up again in connection with the study of equation (6) and the corresponding system (7), when we will assume for these equations that T, the right-hand member of (6), is equal to zero. As we shall have but little to do with linear differential equa- tions whose second members are different from zero, and as the general properties of such equations are sufficiently well known, we need only state here two fundamental propositions concerning them. Supposing the linear differential equation whose right-hand mem- ber is other than zero to be of the form (a) dny t dn~xy t dn~*y t ( _ dx* + +···+/*ƒ = «, where 3 is a function of x\ then if F be a particular solution of this, equation, and ^1^1 ~b · · · “l- £ny» the general solution of the equation when 3 is replaced by zero, we have as the complete solution of the given equation v = Wx + W* + · · · + cnyn + Y- Again, the integration of the equation (a) of order n can be made to depend upon the integration of a similar equation of order n — k, provided we know k particular integrals of (or). Proofs of these theorems will be found in Baltzer’s Theorie der Determinanteny Houél’s Cours d'Analyse, Forsyth’s Differential Equations, etc. We will now go on to the study of the linear differential equa- tion dny ) , ^dn~2y dxn ^dxn - 1 ' ^dxn ~ 2 + · · · + P«y = o. (23)12 LINEAR DIFFERENTIAL EQUATIONS. We have seen, equation (7), that by introducing new auxiliary variables n — 1 in number, say r7', vr\ . - · , we can replace this last equation by the system (24) Let (25) dV dx-* = O, % - * = O, dtin ~2 - dX = 0, drtn~x dx -JrPdf~1+pjr * + · · • +/«ƒ = O' » y", · · • » yx y*> yl, · · • . y. yn, y«, y»”, · · • , In denote n independent solutions of this system ; then the determinant (26) y xx y,, · · · 1/ « - i > 1 3 y;> y/. · · · y } y«, j>V, · · · y n — 1 must not vanish. It is perfectly obvious from the form of equations (24) that y*=ÂZ dyx dx' y" = ^L. dx2 y” dn - yn m dxn -1 ’ that is, all the integrals y* are derivatives of the corresponding inte- grals yy and also that yx, , . . . , yn are independent integrals of '(23) ; the condition then, by hypothesis, that a system of integrals ofGENERAL PROPERTIES. 1$ (23), viz. ylt yt, . • · , yn , shall be independent, minant * dn - y, dn - y1 y, dxn ~ 1 ’ dx*-* ’ · - · ’ dn ~ y, d” - y, (27) dxn ~ 1 ’ dx*~* ’ ' · · ’ y* dn - yn d" ~ y„ dx”~' ’ dx* ’ · ' · ’ yn , = A must not be = o. The nature of the independence of these integrals must now be shown. The non-vanishing of D does not mean that the functions yx, y2, . . . , yn are absolutely independent, but only that they are linearly independent; that is, if D is not zero, then there cannot exist any linear relation of the form (28) C,yt + C„y„ = o; where clf c9, . . . , cM are arbitrary constants. On the other hand, if D — o, then some relation of the form (28) exists in which the constants ciy ciy . . . , cn are not all zero. The following proof of these theorems is due to Frobenius, and is entirely independent of any considerations involving the linear differential equation : Suppose . . , yn to be defined as series going according to positive integral powers of x — a, and all convergent inside a circle of radius p, and having its centre at the point x — a. The variable x is now of course restricted to the interior of this circle. If among these functions there is a linear relation of the form (29) + cy, + . . . -f- c„y„ — o, where e1, cq, . . . , cn are constants, then between this equation and its n — 1 derivatives we can of course eliminate the constants c, and as a result haveLINEAR DIFFERENTIAL EQUATIONS. M (30) dn - yx dn ~ *yt dxn-’’ dxn-, ’ · * · » dn - yt dn - yt dxn ~1 ’ dxn-,’ · · · » y*9 * · 1 1 * dxn ~ 1 ’ dx*~* 9 · · · ’ y**> dky ■ For brevity we will write, as usual, yf instead of In order to prove the converse of the theorem, that is, to prove that if D = o there exists a linear relation of the form (29) between the functions J'i, y2, . . . , yfn we will first assume that the minor ZV* ~ S corre- sponding to the element ynn - 1 of Dy is not identically zero. Con- sider now the system of n — 1 linear equations r^yi + z,y, + · . · + z„yn = O, <31) z1yl' + + · • . + z„y„' = 0, .^y”~ 2 + *,y,a ~ 2 + . . . + z„y„n ~2 = O. Solving these for the unknown quantities z17 za, . . . , zn, we have for the ratios *n’ *n’ ’ ’ ’ * perfectly determinate finite functions; but, since by hypothesis D — o, these functions must satisfy the equation <32) zxyxn - 1 + z%y* ~1 + . . . + znynn “ 1 = o. Taking this equation into account, we have, as is easily seen, by differentiating equations (31), + · · · + *n'y« = °> " Z1'yif + + · · · + Zn'yn' = O, .z'yxn ~ 2 + - 2 + . . · + Zn’yn" “2 = o; (33)GENERAL PROPERTIES. 15 in which zf denotes ax Now since equations (31) determine defi- nitely the ratios *1 equations (33), which are identical in form with equations (31), must equally give perfectly determinate finite values for and these values are respectively equal to the values of that is, we have or *k **' d zk ,, i?- °r Tx~Z- = 0(*='. *.· Zk __ Ck . Zn~~ Cn' , n — 1), where c19 c9, . . . , cM are constants of which cn is arbitrary but different from zero. From the first of equations (31) we have now (34) + · · · + Cnyn — O. Let us now suppose Dnn~1 = o, but the minor of order n — 2, of D * ~ 1, which corresponds to the element to be dif- ferent from zero ; then by proceeding in the same manner we arrive at a relation of the form (35) + =°; in which cn _ x is not zero. Continuing in this way, we can show that if D = o there is always a relation of the form (34), in which the constants c19 c„ . . . , cn are not all zero; remembering, of course, that the equation yx — o is to be included in this form.i6 LINEAR DIFFERENTIAL EQUATIONS. We will speak in future of any number of functions being inde- pendent if there exists between them no linear homogeneous relation with constant coefficients, and the condition for this independence is expressed in the theorem: If n functions ylf y2, . . . , yn are independent, then their de- terminant {i.e.y the determinant D) does not vanish. Conversely, if the functions are not independent, their determinant vanishes. The theorem has only been proved for the region of the plane inside the circle of radius p and centre x — a, but by a well-known theorem in the theory of functions the truth of the theorem is estab- lished for all parts of the plane. Referring now to equations (24), let us write them in the form + oy — yf + oy" + oy'" + . . . + oyn~1 = oy + oy + oy' —y" + oy'" + . . . + oy* -1 = o, -f-oj-j- oy' -f oy" — ynr. . . -\-oyn~'L = or dvn ~ 2 —-----l· oy + 9/ + oy" + oy'" + · · . + oy’1 -2 — i/* -1 = or dx + P«y + Pn - , ƒ + Pn - +Pn - 3/" + . . +p,r~2 + piyn-1 = o- (36) dy dx dy dx df_ dx The adjunct system to this is: (3 7h - ~ + oF+'oF,+ or, + oF"/+. . . +pnY»~' =0, ax dVf Y+oY'+oY" + oY'" + . . . = 0, dx _^'+o7-F+oF" + or'+. . . 1 = o. iYizi dx ■oF+oF'+oF" + oF",+. . . + ƒjF*-1 = o-GENERAL PROPERTIES. 17 Multiplying equations (36) by F, F', . . . , Yn~xy respectively, and equations (37) by —y, —y't . . . , — yn~lf respectively, and adding, we have £ {Yy + yy + ... + r—ƒ·-*) = o, * or (38) Yy + ry + . . . + Y»-*y*-* = ¿7, where C is a constant. This result has already been obtained in the general case of a system of simultaneous linear differential equations of the first order. If we omit the terms in (36) and (37) whose co- efficients are zero, these two systems may be written in the forms : (39) +A y” ~1 +p%jr +p»y = O, = o, dy ^ dx - y = o ; and, for the adjunct system, (40) dY·· dx dY" ~ 2 dx —YPX“ + P,YM- -Y·- — y«- = O, = O, dY dx + PnY = o. Form now the (n — i)8t derivative of the first of these equations, the (n — 2)nd derivative of the second equation, etc., and subtract the sum of all the equations of odd order from the sum of those of even order; as a result we havei8 LINEAR DIFFERENTIAL EQUATIONS. (40 d*Y”-* d*~z , --------------( Vn — i\ dxn dxn - 1 yFl 1 d”~* dxn (//”-)- · · · + = o. Replacing Yn ~ 1 by M, this is . . dnM d"-1 , , d*-* , (42) 'IF {p'M) + 2,“.^) ~ This equation is the adjunct equation to • · +(—O'tMf =a 1 \ dny . dn~1y dK-2y (43) d* + A + A dx^ + * · +p«y = o. The meaning of (42) is easily found. Suppose we multiply (43) by the indeterminate function M and then integrate by parts; we have thus, indicating differential coefficients by accents, (44) Myn~x — M'yn~2-\- M"yn-*— + pxMyn -2 — {pxM)’yn"3 + · · - + pxMy ~1 + {p*M)n ~ 2 — . . . + (— i)npnM\ dx = const. If M is a solution of the differential equation (45) Mn - {pxM)* ~ 1 + (pjf)n -2 - . . . + (-1 )npn M= o, the integral in (44) will vanish, and we shall have a linear differential equation of order n— 1, containing an arbitrary constant, for the determination of y. If we know k solutions Mx, . . . , Mk of (44) we shall obtain, on writing M = Mx, M == , . . . , M = Mk, successively, k linear equations of order ft — 1 in y. Eliminating between these equations the derivatives yn ~ x, yn ~ 2, . . . , yn ~ we shall have for the determination of y an equation of order n — k containing k arbitrary constants. It is clear that if (42) is the ad- junct equation to (43), then (43) is the adjumct equation to (42); and,GENERAL PROPERTIES. 19 further, that the adjunct equation, say B — o, to a given equation, A = o, is simply one whose integrals are multipliers of A — o; that is, supposing B = o and A = o to be each of order n, then multiply- ing A (or B) by an integral of B (or A) and performing a quadrature, the equation obtained for determining the unknown function in A {or B) will be of order n — 1, and will involve one arbitrary constant. The subject of adjunct systems of equations will be resumed in another chapter. The question of the transformation of linear differential equations and the resulting theory of the invariants of such equations will not be dealt with in the present volume, but a few remarks may be made here on the subject. Suppose we have given a linear differential •equation, dn y 0 p. + p. dn~'y 1 D dn-2y , T" rî 7 ~1 „ I dxn ~ dxn ' + P»y = o. This equation can be transformed in two different ways, so that after each transformation it shall retain its original form. We may first change the independent variable by a relation of the form .x = f (/), and then after effecting this transformation we may change .the unknown function y by a relation of the form y — 0 (t)z. The different transforms of the given equation obtained by giving to f(t) and 0 (/) all possible forms may be considered as belonging to one and the same class. Thus all differential equations of the second order form but one class, and they are all reducible to a unique d*t? type, say —A Irf = o; but we do not know how, by means of simple quadratures, to actually make this reduction, nor, having given two equations of the second order, do we know how to find the transformations which will change one into the other. The cir- cumstances are entirely different, however, in the cases of equations of the third or higher orders. We will here only consider briefly the case of an equation of the third order, and show the existence of in- variants in this case. Suppose the equation to be20 LINEAR DIFFERENTIAL EQUATIONS. Transform first by making ¿c = f{t) ; writing dt d't . D(dt\ *dZd? + iP w (47) 3 P' = dt dx 3{t)z, and write (49) 3 P.= 30o = R. 3f+ 3^ 0 ^+3/>'$> + 3 we have fe=s-^·2Ó LINEAR DIFFERENTIAL EQUATIONSL Denote by S the sum of the residues of the function e** 11(a) F(a) which correspond to the roots of F(a) lying inside the contour of integration. The integral da has now the value 2mS. We will first suppose that the characteristic equation F(a) = F(a,+ A)~ h ^ ···"+-LINEAR DIFFERENTIAL EQUATIONS,. .28 also (12) ^*(*1 + *) Hx* hK'xK* “I The residue corresponding to or = aiy that is, the coefficient of 1 fi) W7+à) -hA)^ is found by multiplying together the corresponding terms in the right-hand members of (11) and (12). We find thus for this residue, and consequently for an integral of the differential equation, the expression 2rtiei V ~ , Cxrx C^x^ "I H_c'+T +"' + 77iT~ jJ· (Of course the residue alone does not contain the factor 27ti.) The general integral of the given equation is now of the form (13) y = + . . . + ; where Qkk is a polynomial in x of degree Since (K + l) + (K + l) + · · · + + l) = ny we see that the general solution contains n arbitrary constants. It is often desirable to determine the arbitrary constants in the general integral in such a way that y and its first n — 1 deriva- tives shall have, for a given value of x, certain specified values. In the case where the characteristic equation F[a) = o has all of its roots different, this determination is easily seen to depend upon the solution of a system of n linear algebraic equations. Suppose, in fact, that y — C,ea^ + Cj*** + . . . + Cne*nxCONSTANT COEFFICIENTS. 29. is, in the case of unequal roots of F[a) = o, the general integral of the given differential equation, and suppose x — o to be the particu- lar value of x for which y> yr, yn, . . . yyn ~x are to have the values y0 > y0'> W'> · · ·, y 7 and / will take the values _y0 and jj// for = c. A number of consequences arising from the general integral when x is real might be given ; but it is not necessary to give them, as the reader is of course supposed to be familiar with the ordinary elementary theory of differential equations. We will now briefly consider the case where the right-hand mem-CONSTANT COEFFICIENTS. 33 ber of the equation is not zero but a function of the independent variable ; say the equation is (21) d*y x Ad«~'y , Ad*~*y , + A,dx*~ 1 + — + - dxn .+A„y =f{x), fix) denoting an arbitrary function of x. The following method for integrating this equation is due to Cauchy and is given by Darboux in a short paper immediately following Hermite’s paper above referred to. We form first a solution of the equation, say y = #0, t\ which, with its first n — 2 derivatives, vanishes for x — t, and for the same value of x has its (n — i)st derivative equal to fit). Such a function being formed, we can easily verify that (22) y=f*. where xQ is an arbitrary constant, is a particular solution of the given differential equation. The formation of $ (x, /) is quite simple. From what precedes we know that the integral I pe^dot 2niJ Fia) (F{a) having the same meaning as before) taken round the circum- ference of a circle with very great radius is a solution of the differ- ential equation when its second member is zero, and that this solution, together with its first n — 2 derivatives, vanishes for x = o, and, finally, that its n — i derivative is, for this same value of x, equal to unity. If now in this integral we change x into x — t and multiply the integral by F(t), we obviously have the function sought given by the equation ${x, t) = lTi r V J 271iJr ea(x ~ *ida ’ F{d) ’ (23)34 LINEAR DIFFERENTIAL EQUATIONS. the integration extending round the circumference of a circle of in- definitely large radius R. The double integral (24) Y = eanda (29) =-=/«+*hO^r- F{a) Substituting the above values of Y and its derivatives in the differ- ential equation, we see that the equation is satisfied by since <30) F(a) = an + Ax an - 1 + . . . + An. Referring again to (24), let us write - t)rfa ~ 2rìJR~ F (a) 1 then R(t) is the sum of the residues relative to all the roots of F{a) = o. Suppose to be decomposed into simple fractions, and write i — — 2 i 4- ^ _L _J_ -Bp ~ 1 — I . a)~ ^\ac - ‘ ‘ ^ (a - a)p Ç ’ we have also e°-ix - 0 — - 0 [1 {at — a) (x — (a — a)p ) ’ +··■] and consequently the residue corresponding to the root a of F(a) = o is By (X — t) I ^-.(x-zy-n _ I . 2 I J"3^ LINEAR DIFFERENTIAL EQUATIONS. We have therefore for R(t) the value (3i) R(t) = I £. + -^-—9 -f . . . ' i . 2 . . .p — i ; * and, finally, for Y the value (32) Y = fj(t)R{i)dt. We will now give a brief account, taken from Jordan, of systems of linear differential equations with constant coefficients. We have seen that any such system can be reduced to an equivalent system of the first order. Supposing this reduction to have been made, we will assume (33) " dyx d~ + any, + a„yt + . . . + aisys = o, dy^ . ^fx + ai, y i + ai*yi + · · · + «2sys = o, dys . , , , dx aSS ys — o,. as our system of equations. Denote by A the characteristic deter- minant a11 -j— oty a12, . . ., ais a21, aM I ol , . . ., a2S &Si, ^2 > . . ., ass -j~ ol and let Axl, Ax1l, . . ., A ss denote the minors dA dA dA d^l d^: * 'CONSTANT COEFFICIENTS,. 37 of this determinant. Substitute now in equations (33) the following expressions: <34) yl = —. f A'A + A“A t ' ‘ ~+ AsAe~da, 2 HI'S A ’ 1 + AJ}* + · · · + AnPr .... T = ----------3----------^da’ J_ fA.A+A^+^.J^AJ - 27T**' A ’ where 0,, 02, . . ., 0, are functions of a, and where the integration extends around an arbitrary closed contour. From these equations we obtain by differentiation <35) dh _ I dx 271 dh _ I dx 2711' dys _ I A + A ^12^1 + ^32^2 4“ · · · H“ ¿/.T 27T^ eaxdocy eaxday ) -e*xda. Substituting in the first of equations (33), and observing that ^ {ail + a)^ll + #12^12 + · · * + aisAIS = Ay (3^) H a^21 4" * ‘ · + axsAzs = O, _ (^11 ~h* ^12“^i2 ~i" ... + €txs-^-SS we have as the result — f d e^da. 2 it id38 LINEAR DIFFERENTIAL EQUATIONS. Making the corresponding substitutions in the remaining equa- tions of the system (33), we have as the result of all the substitutions the expressions If we suppose 0,, 02, . . ., 6S to be arbitrary constants, the func- tions 61eaxJ . . ., 0seajc will be integral functions, and the integrals (37) will all vanish, and therefore the expressions in (34) will be solutions of equations (33), and these solutions will contain s arbi- trary constants. Let us assume that we have chosen a circle of infinite radius as the contour of integration. Now,*the initial value of yx for x = o will be • ♦ · 9 but we have (39) A — as -f- Bod ~ 1 -f- . . . AXA + Anet+. . . + An0, A and (42)CONSTANT COEFFICIENTS. 39 We can find in like manner the initial values of ya, y3, . . ., and so have finally for these values y° = 0., y* = e,, y> = >: = e. The solution which we have now found is the general integral, since by properly choosing the constants 0X, , . . ., Bs we can give yx, , . . ., ys arbitrary initial values. The values of the integrals (34) are easily found; each one is in fact equal to the sum of the residues corresponding to the roots of A = o of each of the func- tions which is to be integrated. Consider the integral yx, and let ax denote a root of A — o whose order of multiplicity is = ; then (43) + Atl0, + -\-Aslâs . + · (a — axY + (« - a,) + G· + * · ·5 where , . . ., Fx are linear functions of the constants 0X, , . . ., 6S. Again, we have (44) dy ^7 + ^21^1 “1“ ^22^2 ~}“ · · · + ^2sys — f£>%)'> dy + a**y% + · • “I ^ssys — Formulae (34) will give a particular solution of this system if we determine the functions 6X, 02, . . . , 6S and the contour of inte- gration in such a way that the equations (4 7) =/„ l-fd^da = ƒ„ 2 7tt shall be satisfied. This determination is easily arrived at if fl9 . . . , fs are of the form Qe^y where Q is a polynomial. Suppose, in fact, that (48) A = {F0 + Fjc + ... + Fmt-Y*; then in order to satisfy the first of equations (47) we have only to make 0, = + . . . + I . 2 .... m (a — A)« + > (49) a — ACONSTANT COEFFICIENTS, 41 and integrate around a small circle containing the point A. We see at once from this that if fx, /2, . . . , fs are polynomials of order m, then 0,, 0a, . . . , Bs will be sums of simple fractions containing powers of or — A in their denominators up to (a — X)m + *. We have consequently, pi being equal to zero, or, if X is a root of A- = o, pi being the order of multiplicity of this root, / v ^11^1+^21^2+· · -++A G A “* (a-A)* + * + i * " * ’ I 4· (M , a — X ' ' * and the corresponding value of yx, which is equal to the residue of +A++A+· · *-\-Asx&s A * for the point A, will be of the form Le^, where L is a polynomial in general of degree m + //, but will be of lower degree if the first coefficients G> Gx, . . . vanish. Similar results will of course be obtained ior y2, yz, . . . , ys. It is well known that every equation of the form {51) (ax + + Ai(ax + &)* " + · · · + Any — o can be thrown into the form of an equation with constant co-? efficients. Writing, in fact, ax -f- b = e*y we have42 LINEAR DIFFERENTIAL EQUATIONS. and, in general, (52) dky dxk ake ktPk ; dy where Pk is a linear function, with constant coefficients, of + d?y dky ’ · · > ~djk' Assuming(52) to be true for k (k = 1, 2, . . . , )„ it is readily seen to be true for k -f- 1 ; we have in fact dk + y dxk + Y a* + 1e~t de ktPk dt = a* + 'e~‘(- ke-ktPk + e~ki^) =dk + + ^Pk +1· Substituting these values in (51), we have for the determination of y as a function of x a linear differential equation with constant coefficients. If the corresponding characteristic equation has all of its roots unequal, the general integral will be of the form (53) y = O' + · · · + cj* = CSflx + b)a' + . . . + CJax + b)a" · If the characteristic equation has multiple roots, then to any one of them, say ax, there will correspond a solution of the form (54) ea^i\C + Cxt + . . . + Cy, _ Jp ~ *] = (*r + i)«t[C+ Cx log (ax + b) + . . . + - Jog*-'(ax¿)]~ We will find solutions similar to these in the more general class of linear differential equations which we shall presently study.CONSTANT COEFFICIENTS. 43. In the Bulletin des Sciences Mathématiques for August, 1888,. M. Ch. Méray gives an investigation of· the differential equation with constant coefficients, of which the following is an account : Consider the k series depending upon the same variable z; we will say that these series are co-recurrent if, for all values of the index m, we have the k rela- tions or recurrences of which the coefficients {a, by . . . , k) are arbitrarily given con- stants. The quantities (uy vy . . . , t) are now all known when we know (uQ, v0, . . . , tQ). The summation of these series is easily effected by aid of the theorem : These series are convergent for values of z with sufficiently small moduli, and writing U(z) = u0 + uxz + uj? + . . . + umzm + . . . , V{z) = v0 + vxz + v^ + . . . + vmzm + . . . , . T\z) = *0 + txz +/,** + ... + tmzm + . . . , (*) um x -f- axum —|— bxvm 4" · · · 4” hxtm — O, Vm + i 4“ 4“ 4- · · · 4“ ^a^*» Q tm _j_ x 4" akum 4“ bkVm 4" · · · 4“ — o ; I 4- axzy bxz, . ^ I b^zy . bkZ, . . . , I 4“ ^iZ (F(z) a polynomial in z of degree k)y also writing A}(z)y Bx(z), . . . , Hx{z)y (d) . Ak{z)y Bk{z)y . . . , Hk{z)y44 LINEAR DIFFERENTIAL EQUATIONS. to denote the minors of F corresponding to the like-placed elements in Fiz), we have TT( \ — uA&) + v,Alz) + . . • + tAkig) U(z) - m 9 V(z) = «.·#.(*) + + · · F(z) • + 9 n*) = v0H&) + . . • + tM*) F(z) The numerators in these are obviously polynomials of degree k— I. Admitting for the moment that the series {a) are convergent, we readily find that their sums are connected by the k simultaneous linear equations r (i +axs)U+ bxzV + . . . +hxzT = u„ a2zU +(i+^)r+. . .4- h^zT — v0, „ ak%U -f- V · · · —j— (i —hkZ) T — t0. For, adding the second members of equations (a) after multiplying them respectively, for example, by i + axz, byz, . . . , hxzy we see that the term which is independent of z reduces to u0 and, by the first of recurrences (b), all the other terms reduce to zero when we make m successively = o, i, 2, . . . , and the same holds for equa- tions (ƒ) other than the first one. The solution of these equations gives us (e). Now, since we have {g) F(z), = i + pxz + + . · · + PkZk, a polynomial which does not vanish with z, we know by elementary principles of the theory of functions that the rational fractions (e)CONSTANT COEFFICIENTS, 45 are developable by Maclaurin’s theorem for values of z for which mod. z is equal to or less than the least modulus of the roots of F(z) = o. The integral series obtained by these developments can- not differ from the proposed series (a) since they satisfy equations (/), which are equivalent to the recurrences (b). Each of these series considered separately is obviously recurrent in the ordinary sense of the word. If (k) a(z) , , —— = Wn -4- W Z -4- . F(z) 0 ^ 1 ^ + wmzm + is the development in a recurrent series of a rational fraction in zv whose numerator £l(z) is of degree < k, when k is the degree of the polynomial F(z), the integral series (0 w, wn wQ + —ig + —V + . . . + 0 1 I 1.2 ' 1 O'* I . 2 . , . m r + is convergent for all the values of z, and, representing its sum by t m y we have for its calculation U) F(z) f [O] I. When the rational fraction (A) reduces to the simple fraction (*) (i — ’4-6 LINEAR DIFFERENTIAL EQUATIONS. st being any constant other than zero, we have evidently ----1--- = 4~ 0 · · · (q + m i) m (i - sjf - - ... s. £i- I . 2 m = 1.2.....'(? - I) 2(**+ 0(**+2)· ··(** + ?— IK-J-. We deduce successively, and without difficulty, I T^f }= ¿T|^ + I)(M+2) · · · + ^ Smzn _I___3, ( hk further, let (^)j A(*)> · • · , rfx(s), o) - als\ 1 /»,(*), ■ • · » vJs)> L «*(·*)» Pk{s), . • · , n*(s), be the table of principal minors of this determinant. Calling now (/) UQ> Vo, · · ■ , ¿0CONSTANT COEFFICIENTS. 49 arbitrary constants (k in number), and making ' v(s) = U'tx^s) + v„at(s) + . . . + t0ai{s), (s)es^r ~ *o) ¡7^)1 t = % T(s)es(x. ~ [ƒ(*)“ which furnish thus the general integrals of the given equations. Making x = xQ in equations (m) and in all the equations deduced from them by repeated differentiations, we see immediately that the initial values of the integrals and their derivatives of all orders, viz., «0 7 «.» · · 1 * 7 , ■ · * 7 ^0 7 V,, . . ■ · > Vm , • 7 · · • 7 lm y . ! • 7 are connected by the recurrences (b). It follows then that the sums of the co-recurrent series in z which have these initial values as coefficients are given by formulae (e)y and consequently that the series in x — xQ are deduced from these last by replacing z by x — xQ and dividing the terms in (x — x0)m by 1.2.......m ; that is, the integrals of (m) having the initial values (/)SO LINEAR DIFFERENTIAL EQUATIONS. are obtained by applying formula (ƒ). These integrals are then the second members of equations (ç), for we have evidently /(«>=^(i), ■ and the elements of the table (o) are just what the corresponding elements of (d) become when, after changing z into j, we multiply each one by sk ~ x. The reader is referred to a memoir by J. Collet in the Annales de VEcole Normale Supérieure for 1888 for another treatment of linear differential equations with constant coefficients.CHAPTER III. THE INTEGRALS OF THE DIFFERENTIAL EQUATION P = d"y . dn~Ty . . dn d* + A d^ +A dx -\-p«y = o. The coefficients py, p.„ . . . , pn are uniform functions of x, hav- Ing only poles as critical points. Starting then at any neutral point, .say x0, moving along any path whatever (provided of course it does not pass through a critical point), and returning to x^y the functions p vary continuously and return at the end of the path to their original values. Let y19 . . . , yn denote a system of fundamental integrals: these integrals may or may not have critical points (in gen- oral of course they have), but whatever critical points they have, and of whatever description they are, they must be included among those of the coefficients p. When the variable starts from x9 and travels by any path back to x0, the equation resumes its original form, the coefficientsp having returned to their original values; the functions y,, . . . , yn vary continuously along this path, remaining always integrals of the equation; we must have then, obviously, at the end of the path that y1, . . . , yn have been changed into linear func- tions of themselves, i.e., a linear substitution has been imposed on yx, . . * , yn · Denote the final values of the integrals by Syx, . .., Syn; then we have Cuyx -f- . . . -J— cinyn, = cnPi + · · · + W*» . = cMIy, + . . . + Cnnyn; 5*52 LINEAR DIFFERENTIAL EQUATIONS. where cik is a constant. Or we may say that the integrals yi have submitted to the substitution (2) 5 = y« ; c«iy, + + clnyn 4" CnnPn Before proceeding farther with the study of the integrals it will be convenient to notice briefly the case of equations having uniform singly or doubly periodic coefficients. Suppose that in the equation P = dny , . dx* + P' dn~Ty ! dx n -1 ' + p„y = o the coefficients p19 . . . , pn represent singly periodic functions of x; let go denote the period ; then pz{x -f- go) = pz(x). If then we change .*■ into x -\-go, the equation retains its original form. In the investi- gation farther on of the integrals of this equation we will assume them to be uniform functions of x\ for the present purpose, how- ever, no such assumption is necessary. Suppose ffa), fj, the integrals fi will obviously submit to the substitution /. + · . . -|- (Xxnfn ƒ«; «*./. + · • · “f- annfn Again, if we travel from x to x + go', the integrals will submit to the substitution + · • · + a\nfn fn » «'». /, + · • · “f~ ** nttfn Finally, change x into x -f- go -f- go' ; the effect of this change is clearly the same as first making the substitution 5 and then making the substitution S', or first making the substitution S' and following it by S. We have therefore the relation (4) 55' = S'S. The order of the substitutions is the order in which they are written; e.g., SJSk means that first the substitution S* is made and then the substitution Sk. Returning now to our original equation, P = o, let yx, . . . , yn denote a system of fundamental integrals; we have then dnj\ , dn~ y, , dxn P'dxn - 1 ' ' ■ • + P«yx = o, dny, d"~y dxn ‘ P'dxn 1 ' • +P*y, = o, dnyn dn~yn dxn P'dxn * ■ +P«yn = O.54 LINEAR DIFFERENTIAL EQUATIONS. Regarding these as forming a system of linear equations in • · · , , we have d” ~ yx dn —1 + xyx dnyx d”-i~'y1 dxn ~ 1 ’' ' ' ’ dxn~i+l ’ dxn ’ dxn-i-i ’ · dn - yt d”~i + yi dy, d«-i- yx dxn - 1 ’ ' ' ’ dxn~ «‘+1 ’ dxn ’ dx«-i~ 1 ’ ' dn - y„ dx” ~ 1 ’ d”~i+y„ dXn-i+ i ’ dny„ dx” ’ d”-i~yn dx” -1 - 1 ’ ’ • · , yn where (6) dn ~ xyx dn ~ y, dxn~* ’ dx*~2 ’‘' • ·> Jl dn ~ yt d*-y, dx” - 1 ’ dx”- 2 ’ * ' dH ~ y„ d* - y„ • dxn -1 ’ dx” ~2 ’ ' ' • ? In particular, (7) A • · » 1 d”-yt <&· ’ dx”-' ’ dx”-2· ’ ' • · ,^x d”y n d*-y„ ··»ƒ« d”~y„ d”-y„ ·· ,ƒ« dxn * dx”~2 ’' dx”-1 ’ dx”-2 ’* Then we have (8) D= Ce-f***; where C is a constant necessarily different from zero, since the in- tegrals yx, . . . , yn are linearly independent. The formula for pi may be written for brevity as Di Pi—LINEAR SUBSTITUTIONS. 55 where D; denotes the determinant into which D is changed when its dny dnyn ith column is replaced by , . . . , The result of going round a critical point is to change the integral y{ into Syi, and, consequently, to change the determinants D and into AD and ADi, where A denotes the determinant of the substitution 5; viz., (9) A = Cn > Cn > · · * > ¿21 f ¿23 t · · · f ¿*2» Cn\y ¿«2 > · · · > Cnn Therefore, after going round the critical point, we have, as we should, AD; D; * = -2d=-d· It is of course to be noticed that the determinant A is not equal to zero ; for if it were, the system Syt-, and consequently the system yt·, could not form a fundamental system. If y1 , . . . ,yn denotea system of fundamental integrals, and z1, z2, . . . , Z\ linear functions with constant coefficients of yx, . . . , y\ , given by k — \ (io) *i — 2 Cik yk ; k — I then zx, . . . , zKy + · · · , yn form a fundamental system if the determinant c»> ca, ■ • · > (II) A' = cn, ^22 ’ * • · y C2 A C\i y C\2 , . • · 5 is different from zero; in fact, it is obvious that if A' be different from zero, there can exist no linear relation with constant coefficients between z 1 > · • · y y y^-\- I » ··· y yn ·56 LINEAR DIFFERENTIAL EQUATIONS. Suppose a particular integral yx (different from zero) to have been found for make now P dny dn ~ Jy dxn ' P'dx* ~1 • · - +AJ' = o; y = yjzdx ; then we have for z the equation dn ~Tz dn ~ 2z s = + · · ·+*»-!* = °; where _»dy _n{n— \)dy, np.dy, q* ~ I . 2 . ƒ, dx* ^ yx dx "r·A· (12) n(n — i) . . . (n — r + i) dryx 9r~ ~ if +A 1.2 . ... r.yx {n — \){n — 2) . . . (n — r + i) dr ~ xyx 1.2----(r — i) j/, dxr ~ + · · · +pk r.2....(r-k)yx d^*+■ --+A’ Suppose now that a solution zx, different from zero, has been found for Q = o; substitute as before * — zjtdx, and we have for t an equation of order n — 2. This process may be continued until we arrive at an equation of the first order in, say, w—an integral of which is wx. We have now as integrals of P= o the following: a > a=aAA> y^-yj^dx/tpix,... yn —y,f2idxftidxfuidx . . . fw^dx;LINEAR SUBSTITUTIONS, 57 and these constitute a fundamental system; i.e., we can have no such relation as Cj, + c,y% + . . . + CHyn = o unless all the constant coefficients C are equal to zero. For, suppose this relation does exist; then on substituting the above values of » · · · > yn and dividing through by yx it becomes O = Cj + C*ƒ zxdx + C%fzxdxf txdx + . . . + Cnf zxdx . . . f wxdx; differentiate and divide by zx, and we have Q + CJ fxdx + · · · + Cnftxdx . . ,f wxdx = o ; differentiate again and divide by tx, and so on; we come obviously to the condition Cn — o. Retracing our steps now, and we find Cn -1 —- Cn - 2 —— · · ·== c, = o. Denote by D' the determinant formed from a fundamental system of Q = o, in the same way that D was formed from the fundamental integrals of P — o. We found (13) D=Ce~f*d*; consequently D’ = C'e-ft^ = C'e*t?iJx~Itz? dx , or D' = C"Dy and {14) D = C"'D'y*. In like manner, {15) Dr — OyDnzn-\ and so on. Multiplying together these equations giving D, D', etc., and we have, finally, (i6) D — Cyxnzxn~ ztxn-2 . . . wx. All the results given here might have been incorporated in the first chapter, as they only require that the equation shall have58 LINEAR DIFFERENTIAL EQUATIONS. uniform coefficients. It seemed, however, better to have them here,, as some of them will be of immediate use. We will take up now the equation P= o, as defined at the beginning of this chapter, and see how its integrals behave when we travel round a critical point. For simplicity, let the point be the origin, x = o. As we know, the fun- damental system ylf . . . , yn has the linear substitution 5 imposed upon it when we travel round a critical point. Suppose 5 to be the substitution peculiar to the point x =o; then on going round this point we have ' = cltyt + . . . + ct„y„, (17) Sy, = ^y, + · · · + c*»y«, „ SyH = c„lyl + . . . + c„„y„ ; or y, ; cnyx + . . . + clKyn 11 00 y? ; v, + · · · + c^ny„ yn ; ctn.y-i “l- · · · -f· ^n.y» Let zx, . . . , zn denote another arbitrary system of independent integrals ; then Z\ — + · · · + ttinynj ( ^ ^ f. = a*yx + · · ■ + **nyn, ^ Zn — aniyx + . . . + annyn. The dij are arbitrary constants—such, however, that the determinant | a# | is not zero. Suppose that we travel round x — o ; then the y*s submit to the substitution S, and consequently the zs are changed into new linear functions of the ys. But by aid of the last equations we can express the ÿs as linear functions of the z*s9 and so, after going round x = o, the z’s will be changed into linear func- tions of themselves ; i.e., they will have submitted to a linear substi- tution. As this substitution depends in part upon the arbitrary quantities a#9 we can use this indétermination to affect a simplifica-LINEAR SUBSTITUTIONS. 59 tion of the substitution—viz., we may seek first to determine whether or not there exists an integral which, on travelling round the critical point, changes into itself multiplied by a constant. In the first place, if there is any such integral, it must obviously be of the form y = «iJV, + · · · + ecny„. Suppose the effect of going round the point is to change this into sy. Apply the substitution 5; t.e., travel round x = o; we must have, by hypothesis, aiiCu^i + · * · + C*n^n) + · · · + an{cnxyi + · · · + Cnny^) — + · · · + «*ƒ*)· As no linear relation with constant coefficients can exist among the integrals y19 . . . , yn, this must be an identity, and we have there- fore, on equating the coefficients of each y, the equations of condi- tion (20) ' 0» - ·*)«, + + ·-· + **.“« = O, f,,», + (c„ — s)a2 + . . . + c„2a„ = O, ^ C\n ^1 + ¿"2« ^2 · · · “f” (j'nn — O t giving for the determination of s the algebraic equation of nth order (21) Cm Cm = O. C m y Cnn $ This equation will be called the characteristic equation. Suppose A = o has all of its roots unequal; then if Si denote one of them, and if this value of s be substituted in the preceding equa- tion, these will obviously serve for the determination of the ratios of aY, , . . . , an, which is all that is necessary. The integral y will therefore be known, and as all the roots of A are unequal, we will obviously have n integrals each possessing the required property;,6o LINEAR DIFFERENTIAL EQUATIONS. or, in other words, for this set of integrals the substitution 5 becomes (22) ƒ» ; Wx y* ; s*y* yn\ sny>, It can be shown in a moment that these integrals constitute a fundamental system; i.e., are not connected by any such linear relation as Wx + · · ■ + Cnyn = o. Suppose such a relation does exist; then on applying the substitu- tion S, that is, turning round the critical point x = o, a similar relation must exist, and so for any number of turns round x = o. Suppose n — i turns to have been made; then we have c\yi + · · · + cnyn = o, cxsxyx + . . . + cnsnyn = o, cxsxn - yx + . . . + - yn = o. As the constants cx, . . . , cn are not zero, we must have This is the product of the differences — Sy , and cannot therefore vanish unless at least two of the roots s become equal; but by hypothesis the roots are all different; this determinant can therefore not vanish, and hence no such relation as Cxyx + . . . + cnyn = oLINEAR SUBSTITUTIONS\ 61 can exist; the integrals constitute therefore a fundamental system. If we had chosen any other set of fundamental integrals, say^ vx, . . . , vny the substitution 5 would have the form vi ; + · • · Yxn^n (23) Vn\ Y mV, + . • · 4" YnnVn The characteristic equation is now Yu — Sy Yn y • · · , Ym 'to' > II Yl* y Yn- Sy . . . , yn2 Yin, Y*ny • · · > Ynn The integrals yx, . . . , yn now become linear functions of viy z>2, . . . , vn and reproduce themselves, multiplied respectively by sx, . . ., sn, and it therefore follows that the equation Ax = o is identical with A — o, since it has the same roots. The coefficients of A being independent of the choice of fundamental integrals are therefore in- variants. A direct proof of this fact, viz., that the characteristic equation is the same whatever set of fundamental integrals is chosen* will now be given. We have (25) — ynVl 4“ · · * 4" YmVny SV2 — K2 ic\ 4~ · · · 4“ Y^n^ny „Svn = Ymvx 4” · ■ · 4" YnnVny with the characteristic equation Yi . — s, Yny · · ■ • y Ynx (26) A = Yu, Y22 $7 · · Yin y Y*ny • , Ynn — S*62 LINEAR DIFFERENTIAL EQUATIONS. Again: we must have V1 ~ ^*11 ƒ 1 + · · * + ^1« ƒ*> ^ vi — Kifi + · · · + Kn yn> Vn — A^J/, “h · · · -f" hnnjtn · The determinant | Aty | is of course not zero. Substitute these values of v1, . . . , vn in the right-hand members of equations (25), and we have (28) SVi = + VizKi + · · · + Yin^m)yi + · · · + (r*iA 1 n + Y22 ^2» « * « “l· · Again: in equations (27) apply the substitution S, and we have (29) Svi = (i„^n -f- ^21A/2 -j- · · · -f- cni\i^)yx . H“ fci» Azi ~l· ^2» Az-2 . . . -j- c„u\in)yn. From (28) and (29) we must then have k =■ n k — n ^ Ckj^-ik — ^ Yik^-kjy — &zj· k = 1 k = 1 Denote by the determinant | Azy | ; we see then that the two determinants formed from A by multiplying it by Ax and A2 respectively, have their elements equal each to each, and that each of these products is equal to djj Ajji, d12 Aia.T, ··· 9 ^21 ^21*^» ^22 A22^> · · · > ^2n A2W5 ______ ^ &ni & n2 .... S nn A nnS We have then that is, ¿A = 44 A = ALINEAR SUBSTITUTIONS. 63 This proof is due to Hamburger.* The following theorem is due to the same author, and is taken from the same paper: THEOREM.—If for a given value of s all the minors of order v in A vanish (and consequently all of higher orders), and those of order ( v— 1) do not all vanish, then all the minors of Ax of order v will vanish, and those of order (v — 1) will not all vanish. We have <3°) A = A, = *11— C21 y · · · » Cm v C* 2 · · · y Cm C\n y Cm y · · · y C-nti r*i— Sy Yu · · · · y Ym Yn— s, , Yn 2 Ymy Ymy · · · y Ynn — K, Ài2 , . . . , ~h\n ^21 » ^22 > · · · y ^-2 n 3-„i , A.W2 y · · · y A-nn - Àn^, d12 ^12^> . · . , K - d22 ^22^> . . . , L- liniS7 Ón2 \n2S, . . . , A Let Aiky Bik> Cik, Dik denote respectively the minors of order r of the determinants A, Alt z/2, As; where i and k denote any of the n(n — 1) . . . (n — v + 1) r. 2.3 . . . . V 1 = /*> combinations of v different numbers formed out of the series I, 2, We have now (31) Dik — Ci,Akl -j- . . . -f- Cip A — BtICkl -f- . . . + BIflCi kli· * Journal fiir reine und angewandte Mathematik, voi. lxxvi. p. 113.64 LINEAR DIFFERENTIAL EQUATIONS. Since by hypothesis all of the minors Ars are zero, it follows that all of the minors Drs are also zero; we have then the system of equa- tions 'BitCu + · · - · ~h BiflCIfJi — 0, (32) -< Biz Cn + · ■ • · ~{~ B{fJL B2fJ. — 0, ^ BVi 67x1 + · . . -j- BilxCw = 0. The determinant of this system, viz., r C r c r c is a power of A„ and therefore does not vanish ; consequently we have (33) Bit — B;2 = . . . = B,„ = o; and, as these equations hold for any value of i, it follows that all the minors Brs of order v of the determinant A1 are equal to zero. If now all of the minors of order (v — 1) of Ax vanish, it is clear from the foregoing that all the minors of A must also vanish, which is con- trary to the hypothesis. We will take up now the case where the characteristic equation has equal roots. By employing the results, not yet mentioned, of Hamburger’s paper the forms of the integrals can be obtained very readily; but it seems better to first obtain these forms in the same way that Fuchs obtained them, if for no other reason than that of the de- sirability of developing the subject in historical order. From what precedes it is clear that there is always at least one integral, say ux, which is changed into s1u1 when the independent variable travels round the critical point x = o. If yx, . · . , yn denote a system of fundamental integrals, we have (34) u, = + . . . + ot„y„,PROPERTIES OF THE INTEGRALS. 65 where the coefficients ax, . . . , aM are not all zero. It is clear now that we can replace one of the integrals yx, . . . , yn by ux and still have a fundamental system, provided that the coefficient a corresponding to the replaced y is not zero. Suppose ax different from zero ; then we can take for a new fundamental system the integrals «1, y*> y*> · · · > y»· The result of going round the critical point x — o is now to change these integrals into the following : (35) Sux = sxux, Ai^i ~f” A2^2 · ■ • · + A«7« ^y3 Ai^i —l- fi^y* “i- · * ' · + A*/* : „ fyn = /?«. 4- · ■ ■ · “h fin nyn The characteristic equation corresponding to this system is ob- viously A2 — ■*, A2 > ... , A (36) 0. - 4 A3 > A3 ... • A fi2.fi y A*» , A As sx is a multiple root of the characteristic equation A — o, and as the roots of the characteristic equation are quite independent of the choice of a fundamental system of integrals, it follows that the equation ^22 J fin2 (3 7) = o. $2 n y finn ~ * has at least one root s = sx. This being so, it follows further that there must exist a compatible system of equations, such as (38) (fit* — -0A + AsA ~l· · · * "f~ A« A — °> A2A + (As- -OA + · · · + A* A = O, A3 A · · · “f" {finn — Si)An — O.66 LINEAR DIFFERENTIAL EQUATIONS. If now we write ^2 2 “l- -^s^a + · · · “j- I^-nyn > and then travel round the critical point x = o, we shall have “ (fiuAi + An-^a + · · · + finidn)ux -f" SXU% , or (39) Su, = snux + sxu,, where s9X is a constant. The new function ua is obviously an inte- gral, and we can replace by it any one of the integrals y^, . . . , yn of which the coefficient a is different from zero. Suppose that u9 replaces y%; then we have a new fundamental system, yn· If sx is a triple root of the characteristic equation, we can find a new integral u% satisfying the equation SU^ --- *$3X^1 “f” *^32^2 + ^1^3 9 where s3X and ¿32 are constants. If then sx is root of the characteristic equation of multiplicity A, we can obviously form a group of X inte- grals ux, z/2, . . . , ux , such that 4i) Sux = sxux, Su2 = snux + sxu%, Su% = s%xux + snu9 + sxu%, w Suxx = + ^A2 + · · · s\, A - - I “h S · Finally, if the distinct roots of the characteristic equation A — o are sx, , . . . , , and if their orders of multiplicity are Xx , A,, . . . , A/, we can form in the manner above indicated l groups of integrals containing in all Xx 4- A2 4“ · · *4- — n integrals, and these integrals will, from what precedes, form a fundamental system. By aid of the above-found properties of a system of fundamental integrals it is easy to find the forms of the integrals.FORMS OF THE INTEGRALS. 67 Write (42) logj, 2 ni ’ log sx standing for any one of its values. Suppose ux an integral such that when the variable turns round the critical point x = O we have <43) -Sm, = s,u,; then it is clear that in the region of this point the product x~r> ux is a uniform function, say 0,(^) ; then (44) ux — xr' (px(x). It follows at once, if all the roots of the characteristic equation are •distinct, and if <45) _ log 5^ k 2 ni f that each integral can be written in the form <46) uk — xrk <(>k(x) ; where is a uniform function of and where the exponents r19 ra, . . . , rn do not differ from each other by integers. This last is clear since sk — e27trkt\ if n = rk -|- ffty where m is an integer, then we should have Sk — Si, which is contrary to hypothesis. The functions lx(x)f{x) is necessarily a uniform function, «- = 0.. log *j · Making > · · · y , having the properties set forth in equations (41), and they can be put into the following form, viz.,70 LINEAR DIFFERENTIAL EQUATIONS. (49) ' u, = Xr'ii , U, = xr'i* + 0« log x], " U, = xy1 [03I + 03, log X + 0„ log* x], „= ^ri[0x. + 0*, log * + . . . + 0M log*- ’x] 7 where r, = and 0,,, 0„, . . . , 0** are uniform in the neigh- 27tl borhood of the point .r = o. (I) The quantities 0 are such that any one of them, 0,y, where i is different from j\ can be expressed linearly in terms of those whose second subscript is i. (II) 0„, 022, . . . , 0aa differ from one another only by constant factors. Assuming the above statements to hold good for the first k — i of the integrals ux, , . . . , uK , they may be shown to hold good for the kth integral, Uk . For, by equation (41), (50) Suk — s^ux -f~ ^¿2^2 -j- · . · sk% k - 1 Uk - 1 -f- Sxtik· Also, we may always take (SO Uk — xrx\ki + kk \ogk-'x)', k3, · · · , kk being chosen at will, provided that 0^ is prop- erly determined. Assume then that 0*2, 0*3, . . . , (f>kk are uniform in the region of the point x = o, and let 0*x become

kx when the variable moves round the critical point. Equation (51) thus changes to Suk = sxxr 1 {'kx + (pk2 (log x + 2 ni) + . . . -f 0^(log jr + 2nif -1}. Expanding and dividing both members by sxxri, we haveFORMS OF THE INTEGRALS. 71 kk . . ■ / ..k—l-k—2 log* - **+(27ll)'-—y---4>kk k — I . k — 2 . k — \ + (2TTZ)8 kk -}-(27ti) —y k,k-i + k, k -1 log* - *x + · · · + (2*0 -f- (27Ct)k ~ 5 k — I . k — 2 . k —-3 i^-4---------:-------- /£ — 2 . k — Z .k — 4 -f- (2^r/)* ~ 6 _(.k — 3 . £ — 4 . £ — 5 + (2*0 k — 4. £ — 5 . £ — 6 !>* ~ 1 -----------^ 4>kk -6 — 4 2 ! 04·, * — 2 . / .. . £ — 4 . — 5 4“ (2W2) ^-| 0*, * _ 3 4" 4~ 4- (2*0 + 30^4 k, 3 log'* log*-3·* log’jf72 LINEAR DIFFERENTIAL EQUATIONS. + (27lt)k ~ \k- -I )kk log * + (27It)k ~ i&kk + (2nt)k ~ 3(4- -2)kik_2 + Sk, k — X k% + ·**, 2 + ^A-20/i-2,2 + J*, k-3<ï>*-3,2 + ^4 042 + S*3 03 2 log^r + j^ + ^ * - I kk + 0^, k - i] = ii0i,i-i+^, /6-10/fe-i, ¿-I» i # k—2 ) sA {27tty —y ^2 H- k - 1 0/4 - 1, 2 4” * · · > the terms in 0*2 are seen to vanish identically; hence the 2d, 3d, . . . , (>6— i)8t equations, in number k — 2, involve only the £ — 2 unknown quantities $kky 04, ,4 — xi · · · f 0-43» which can therefore be expressed in terms of 0’s of lower index supposed already known.7 4 LINEAR DIFFERENTIAL EQUATIONS. The determinant of the left-hand members is 27Cl{k—i), . k— l.k—2 {271 if {27Cif 2 ! k— l.k— 2.k—3 (27ti){k-2,) 0, O, , {27tif ———-, 27T*(/£ —3), O, o, o, , k—r.k—2.&—3 h k— 2. k— 3 · ^—4 (27r*)*"4---—------- , (27tl)k-S^-- ---- (27^)*-3 3 ! k—I .k—2 2 ! {27ti)k~2{k^l), {27ti)k~4 3 ! k-2.k — 3 2 ! (2?r/)*-3(£-2), , 3(2^·), o , 3(2 itif, 2(27ri) of which the value is s* ~ 2k - i . 2 ...3.2. (2*ri)* ~2 = (2«)* - *s* ~ 2 . {k— 1) ! Replacing the columns of this determinant successively by $k, k — I (pk — I, k — I J ^k, k — I 0<£ — I, k — 2 I k — 2 (pk — 2, k — 2 * ·£&, Æ — I 0Æ — 1,2 “j“ k — 2 ypk — 2, 2 “1“ · · · “f" S&' 2022 » and dividing the results by (2ni)k~2 . sxk~2 .{k — 1)!, the values of which establishes proposition (II), since Æ may have any value from 2 to À. Also,FORMS OF THE INTEGRALS. 75 (54) 27ttj “\~ *^81011 S 0 But it is already known that 0M = —~ ; hence 0a2 is expressed in terms oi 0n, 021. Proceeding in the same manner, it may be seen that proposition (I) holds good throughout. We have also the equation si{(27ttY *0** + (27tt)k ~ 2<(>ktk-14~ · · · 4~ (27tz)30¿4 4~ (2^)20>fe3 — *^i0*i 4“ Sk,k- i0* - i, i 4“ * * · 4" ^4041 4" Sk3>(t>31 4" «**2021 4“ ^*1011 - k2 is by hypothesis an arbitrary uniform function; it may therefore be so chosen as to satisfy the following equation : s^niy-'fak 4- ... 4~ (2?tO20*3 4“ {27tt) · ”f- Cm&n = 0, (58) ^12^1 + _ 0 **2 + · « . . —J— Cn2 Otn = 0, s. l + CinOt* + · · ' · {fnn — •0 = 0. If we had chosen v1, . . . , vn as our set of fundamental integrals, we should have (equations 23 and 24) ^ (7u — -0 ¿*1 + Y*i<** + · · · + YnCtn — O, Yn^i I (X22 Si) I · · · | “ Yn2^n — O» 1 ^Yrn JK v +1 ? · · · >yn\ and then <6o) 5 = u\ , tit , 3 Uv S\U\ , S\Ui, ... , S\Uv \ C y-j-i, i U\ -|- . · - -f~ ¿"V-f-i, v Uv v+i ···“}" nyn yn ; Cni U\ -f- . . . -f- c nv Uv -J- C *, v-f-iT" . . . -f- c'nnyn The characteristic equation becomes now (6i) A — (s — s,)v Af — o ; where (62) A’ - C *4-1, v + i Sy · · · 7 £ nt v-j-1 ^ v -|— i, j C nn $ As the characteristic equation is independent of the choice of the fundamental system of integrals, it follows that equation (6i) has the same roots as the original equation A — o ; and if s1 is a root of multiplicity À, we must have r^X. If v = A, then ulf . . , uy are all the integrals associated with the root sx and satisfying the rela- tion Su = sxu. We have in this case r, = X, sub-groups of integrals * As we have principally to do with linear homogeneous functions, it will be con- venient to drop the word ‘‘homogeneous,” so that a “ linear function” of any set of quantities will be understood to mean a “ linear homogeneous function” of those quantities.SUB-GROUPS OF INTEGRALS. 79 each containing one member. If, however, v , say V, which would satisfy the relation SV = sxV;. that is, there would be more than v linearly independent integrals satisfying this relation, which is contrary to hypothesis. It follows also that vf cannot be greater than r, since between any v -J- i of the functions U{ there necessarily exists a linear relation. Finally, since Ui is a linear function of uiy . . . , uv, and since each of these functions satisfies the relations Sui — sxuu we have SUi = s1Ui, and writing U=ZXUX+ . . . CyUs, wdiere C,, . . . , £„/ are constants, SU^s.U. If now v vr = A, then the A integrals corresponding to the root st of the characteristic equation are divided into vr sub-groups of two elements each, and v — v' sub-groups of one element each; viz., the sub-groups of two elements are (67) ' 0,, Ux) satisfying Sv1 = s1vl + Ux, SUl = sl Ul, {vvfiUv>) . . . Svv, = s1vV'-{-UV',SUv' = slUv'; and the sub-groups of one element each are the remaining v — vf linear functions of ux, , . . . , uv, which have no linear relation among themselves or with Ul, . . . , Uv>. There will obviously be no loss of generality if we replace simply by ux , . . . , uv>, and the remaining v — vf functions by uv> + x, . . . , uv; denoting then by II the sub-groups of two elements each, and by I the sub- groups of one element each, we have (68) where j II = (a,, v,), (a, ,vt), . . . , (uv,, vv), ( I zv + i t 2 > · ■ * } Mv ì Sv{ = sjji -f- Ui, Sili — sxUi. Suppose now that v -f- vr < X; then we can choose for our fun- damental system , · · · , , vx, . . . , vvt, yv · y y n »SUB-GROUPS OF INTEGRALS. 81 for which (69) 5 = U! , . . . , UV ; , . . . , Vl* * · . , Vv's 1V1 -j- Ui, . . . , S\Vv' + Uv' ’-(-!» ^ V-f-v'+I Wj-j- . . . —j—C */—{-*/'—f-1, v Uv~\~C . . 4”^ 1T ttVn ; c"nx «1 + ♦ · · -j- C nv Uv "4" C n, v -f-1 Z'i + ·. ■ ♦ 4“ <7 nn Vn The characteristic equation is now (70) A =(s — A" — o; where C v -}- v' -4- 1, v -f- v' -j- i $9 (7 0 A" = L v v’ 4- 1, n 5 L flj V —j— vf —(— I r" ____ C y nn Since v + v' < A, it follows that sx is again a root of A" — o, and this equation implies the existence of the system (72) OCx (c y _j_ v' -\- i, v -f- v' 4- 1 *^1) | * · · I & n — V — v’C n, v-\- v' 1 ------------------- O, ¿Tj ¿VI, » | · · · “f“ OC n _ y _ y' (c nn - O, serving to determine the ratios of or/7, . . . , ar n _ „ _ „/. Multiply- ing the n — v — yr equations corresponding to the last ;/ — v — vf rows of (69) by a, . . ., ann and adding, we have for the new function (73) ^ + 1 · · · “|” & « —v — v'J/n j which is linearly independent of the u's and the vs, and which satis- fies the relation (74) Szv — sxw + T; where T is a linear function of u1, . . . , uv, v1, . . . , vv>. Suppose now that there are only n — v — v* — vn independent equations in82 LINEAR DIFFERENTIAL EQUATIONS. the system (72). Then all of the n — v — vf constants, a/', . . . , a' n , are linearly expressible in terms of v” arbitrary constants, and consequently there exist v,r linearly independent functions wx, . . . , wv" which satisfy the relation (74), and every other function satisfying this relation is a linear function of (u, v9 w). For these functions w we have There can exist no linear relations between the functions of the form where , . . . , C*" are constants, and U is a linear function of uv; for if there could be such a relation, it would also be possible to find a linear relation connecting the functions wx , . . . , wvn9 say Wy alone and satisfying where U is a linear function of ux, . . . , uv. But by hypothesis the linearly independent functions vx, . . . , vv> are the only ones satisfying such a relation, and consequently we can have no such relation as (76). It follows also that v" cannot be greater than vf, and consequently cannot be greater than v. Suppose now that v -f- vf -|- v" = X ; we can obviously without loss of generality replace Tx, . . . , Tv" by vx, . . . , vvn, etc. We find then that the integrals corresponding to the root sx of the char- acteristic equation divide into v" sub-groups of three elements each, vr — v" sub-groups of two elements each, and v — vf sub-groups of one element each. Denote these by III, II, I, respectively ; then <76) (77) SW=slW+ U ; "III =1 (ui9 Vx , Wx) ... (Up", Vy»9 Wy»)y (78) I I --- (Uy/f _|_ I , Vyff -J- X ) · · ■ (fly* y Vyf ^ I ------ (Uyt j) . · . (Uy)ySUB-GROUPS OF INTEGRALS. 83 giving the relations ' from III, Suy = siul, Sv1 = s,vt -f- «,, = slwl -(- vx, (79) Suvft = SxUvn , SVvn Z=zSxVvn Uv" , Swvn = SxWy" -J“ Vvn J - from II, Suvt, + x = sxu^t + x, Svv" 4. x = sxvv" + uv», Suv>=sxuvty Svvf = sxvv> -(- uvf; from I, = sxuv>+l, . . . , Suv = Ey a continuation of this process we find a series of numbers, v, v\ ytr, . . . , r(*}, such that v -(- vr . . . + y(k) = A, and where no v is greater than the preceding one. For we shall have y[k) sub-groups of k -f- 1 elements each, y(k “ l) — y{k) “ “ k elements each, y v v yr a u a 2 il << “ i element each. The elements of a group containing m elements are, say, yl9 y7, ···,ƒ«; then & = + J, » · · · , Sfm = W*» +ƒ«-·· The substitution 5 corresponding to the point x = o^now takes the form J'.. , ... »y«; •y. . s,y,+y,> ··· > s,ya~r y«- ■ y*> . · · , X'! s,y,', ··· , si/«' +y'a' - 1 y®·. . ... ’ ·/«* * sty^·. . y2(fl + j.'0, · · · . s,y3+yl«-1 , ^ > · · · > #0; v. - ■Vs + · · · · , + 2$ - i84 LINEAR DIFFERENTIAL EQUATIONS. Jordan in his Cours cTAnalyse (vol. iii. p. 175) gives S a slightly different form and one which is a trifle more convenient. Jordan’s form is derived simply from (79), viz. : Write II II II y» F II M fl 1 v/ = yl *7 = Y,' = s, yt', . • · t y &11 ji “l- · · · n fn S = fn > &mfi I · · · I &nnfn fi, 11 fi + · ■ 1 * " 1 & i nfn fn 1 & nifi “h* · ■ ■ · Hr a nnfnDOUBLY PERIODIC COEFFICIENTS.. 85 and also (c) SS' = 5'5. We can by a method entirely similar to the preceding one reduce the substitutions *S and S' to their canonical forms. Let s denote a root of the characteristic equation corresponding to Sf and let yx, y2 , . . . denote those independent integrals of the differential equation which are multiplied by s when the substitution 5 is made, Le. when x is changed into x -f- go. The general form of the inte- grals possessing this property is then + oc2y2 +------- Suppose that on applying the substitution S' to yx we changeyx into Yx; now the substitution SS' changes yx into sYx, and S'S must produce the same result ; but S' changes yx into Yx, and so 5“ should change Yx into s Yx; it follows then that Y must have the form ai + a*y* “l· · · · · The substitution S' thus replacing each of the integrals , y2, . . . by linear functions of these same integrals, there must exist at least one linear function, say u, of these integrals which S' changes into s'u. We have thus shown that there exists at least one integral, u, which the substitutions 5 and S' change into su and s'u respec- tively. We proceed now to show that it is always possible to find a set of fundamental integrals, say j^u j · · · y y> y2i > · · · > y^h ? · · · > y><\ > · · · > y^K* Zxx , . . . , ZXMl , Z2I , . . . , Z2m^ , . . . , Z ^x , . . . , y such that the substitutions and S' shall take the forms {d)S = y*k y ··· y yik y ··· y S^ik y · · · y ~\~ ^ik)y · · · %ik y * * * y ^ik y ··· y ^2^ik y ··· y H- ^ik)y · · ·86 LINEAR DIFFERENTIAL EQUATIONS. where {sx, s/), (j2, s2')9 . . . are pairs of different constants, i.e., s9 is never equal to s2J s/ is never equal to s29 etc.; and where Y’ik are linear functions of the integrals y whose first suffix is less than i; Zik, Zik are linear functions of the integrals z9 whose first suffix is less than i; etc. Assuming this proposition true for substitutions containing less than n variables, we will prove it to be true of the substitutions 5 and S' containing n variables. We have seen that there always exists an integral it which is changed into su and s'u by the substitutions 5 and S' respectively. In the system of funda- mental integrals /,, f2, . . . , fn let us replace any one, say fn , by the integral u. The substitutions 5 and S' then take the forms the functions Ft and F/ denoting linear functions of x. Consider now the substitutions (j0 ^ — 1 f\ y · · · > fn - i y My -^i ~\~^i M, . . . , Fn _ T —{- Cln _ SU j {h) (*') of n — i variables. From the relation 6*5' = S'S follows at once (/) 22' = 2'2. Applying now the above-stated theorem to these substitutions, they may be written in the forms ( Stilus -\-Zik)-\-dtiu, .. u ; su y,k y . ’ * y yik y · •y si(yik+ Yii)+c!*U, · · · (/) 5' = Zik, ·· ■ - , *ik- - ■ ·; teik+dikU, ■ ■ ·, s'2{zik+Z'ik)+d'iku, ... U ; s'u Suppose now we replace the independent integrals yik by the follow- ing: 0») y'ik = yik + ociku ; the substitutions 5 and S' will retain their forms; but the constants cik and cik will be changed into [cik] and [c'ik\y where («) [>*] = (s — s^aik — s,Hik + cik, (p) \fik\ — (p ^i)^ik SiHjjg -f- » Hik and H'ik denoting what Yik and Y'k respectively become when in them we replace the functions y by the corresponding constants a. We will now first assume that s is not equal to sx ; in this case we can clearly assign such values to the constants a that all the new coefficients \cik\ shall vanish. It is easy to see, by aid of the relation SS' — S'S, that the vanishing of these constants will involve the vanishing of the constants (V^]. Equating the coefficients of u in the expressions SS'yik and SfSyik, and denoting by rik andrik what Yik and Y'ik become when the inte- grals y are replaced by the corresponding constants c and cwe have (P) si {f'ik 4" rik) -j- s'cik — Si{fik 4" P’tk) 4- sc'tk·88 LINEAR DIFFERENTIAL EQUATIONS. If now the constants cik are all zero these relations (ƒ) reduce to the form These equations (q) are linear and homogeneous in the quantities c'ik, and their determinant is a power of st—s ; but since by hypoth- esis sz is not equal to s, this determinant cannot vanish, and there- fore, in order that equations (q) may be satisfied, we must have all of the quantities c'ik equal zero. In the same way, if s' is not equal to sy we can make all the constants c' ik vanish, and their vanishing will also involve the vanishing of all the constants cik. Continuing this process, suppose that none of the relations are satisfied ; then we can make all of the constants cik, cfik ; dik, d[k ; . . . , disappear, and so the substitutions S and S' will be in the canonical form, and to the different classes of integrals y9 z, . . . we have added the class composed of one integral only, viz., u. Suppose now that s = sz, s' — s/; as before we can make all the coefficients dik , d'ik ; . . . , vanish. If now we write we will again have the substitutions 5 and S' in the normal form, the new integral u entering now into the category of integrals ylk belonging to the class of integrals y, which have unity for their first suffix, i.e.y the class which the substitution S' multiplies by sy and which the substitution S' multiplies by We resume now our original problem of determining the forms of the integrals of the linear differential equation with uniform coefficients. Starting from equation (8o'), Hamburger proceeds to determine the forms of the integrals; but Jordan gives a briefer and rather more elegdnt shape to Hamburger’s method, so we shall em- ploy it. Write as before is) 0, — s)c'ik + s.r't = o. Cik = s, yik, c a = s, y ik , log s 27tiFORMS OF THE INTEGRALS. 89 (dropping the subscripts for convenience). The substitution 5 being in the canonical form (8o'), let y0, y1, . . . , yk denote a group of integrals to which S applies, and let s denote the corresponding root of A = o. Write J„ = > Jl = 7 - ··, y*= ST**· Since after once turning round x — o, xr reproduces itself multi- plied by e™ir, = s, it follows that the functions z submit to the sub- stitution (81) 2 — | -S'o > · · · > 9 · · · y %k ~1~ %k — 1 | » since Sy. = sya, -$>, = .5 (ƒ,+ƒ„), ···. Syk = S {yk +yk _ ,)· It is only necessary now to find the forms of zQ, zY, . . . , zk. The first, zQ, is obviously a uniform function, since Sz0—z0. To get the forms of z1, . . . , zk we introduce a new function, 6X, de- fined by (82) _ log* 2 7tt The effect of turning once round x = o, that is, of applying the substitution S, is (83) Introduce now the series of functions^, . . . , dk, defined by the equations (84) 0, = ^..........„ x * 2m ’ * i. 2 . . . >6 for these we have, as the result of turning once round x = o, (85) ......^ =90 LINEAR DIFFERENTIAL EQUATIONS. Add and subtract 0k in S0k \ then for the right-hand member we have (0,+i)0,(0-i)... (0-k+2) 6,(6- x) ... (6A+ i> (86) 6, I . 2 . . . k — Ok + [(0, + l) — (#1 — k i)] l .2 ... k 6, (6,-i)...(6l-k + 2) 1.2 . . . k 1 I .2 ... k — 1 1 We have then finally for the functions 0, (87) 50, = 0,+ x, 50, = 0,+ 0l> ..., 50* = 0* + The result of A. turns round jr = o gives (88) 5*0, = 0,+A, 5*0,= 01 + A0, + ^y^ 5A0**= 6k j -|-—-0*_2-(- · · · -|-0î-a· If À = k, this last is (89) Skdk = ^ -|- k$k-14“ ”—"^-2+ . . . + 1. If X = k + /, the coefficients in this equation change into the binomial coefficients corresponding to the exponent (k -f- /) ; as there are no functions such as (i a positive integer), the last term is simply the binomial coefficient (/+*)(/+*-!) . . . (/+I) k\ If now we choose a system of uniform functions &0, u19 . . . , uk * we can write (90) ' Z. ; , = 0.». + » Zk — 6ku, -(- 0* _, u, -f- . . . And these values obviously satisfy the condition (81).FORMS OF THE INTEGRALS. 91 We have now for the integrals jyQ, . . . , yk the same forms as those in equations (49); retaining, however, Jordan's notation, we have (9I) „ yt = xr\.M* log x + Nil yk — xr \_Mr logk x Nk log k~1 x -f- . · .] . The uniform functions M0, . . . , Mk, Nx, . . . , Nk . . . are obviously linear functions of the k 1 independent functions u0, ux, . . . , Uk of equations (90), and in particular M0, Mx, . . . , Mk differ only by constant factors from each other and from uQ. It follows from this that the functions xrMx, xkM,, . . . , xrMk, which differ only from xrM0 by constant factors, are integrals of the differential equation. These functions M and N are so far perfectly arbitrary uniform functions; they will, however, in par- ticular cases be seen to divide themselves into two classes—one containing only a finite number of negative powers of x (or of x — if a be the critical point considered) and one containing an infinite number of such powers. When all of the functions M, N entering into any one of the integrals of the equation contain only a finite number of negative powers of the variable xy the integral is said to be regular in the region of the point x — o. A very important class of equations, first investigated by Fuchs, is that class in which all of the integrals in the region of a critical point are regular. The investigation of this class will be given in the following chapters. The substitution .S which we have been considering is of course only one of a number, finite or infinite, which belongs to the linear differential equation with uniform coefficients. If the variable be made to describe all possible paths enclosing one or more of the critical points a, b, c . . . of the equation, we shall have a certain substitution corresponding to each of the paths ; the aggregate of all these substitutions is called thz group of the equation. We will denote this group by the letter G. It is clear that G will assumeS2 LINEAR DIFFERENTIAL EQUATIONS. different forms according to the choice of the system of fundamental, i.e., linearly independent, integrals. This notion of the group of a linear differential equation is of the highest importance in the theory, and will be treated of more fully in another chapter of this volume, and still more fully in Volume II. From a knowledge of the group of an equation we can derive all the essential properties of the equa- tion. Suppose, for example, P — o is an equation having all of its integrals regular: it is obvious that among equations of this type are included all equations having only algebraic integrals. What, then, are the conditions which P — o must satisfy in order that all of its integrals may be algebraic ? It is obviously necessary that the different functions into which the substitutions of the group G change the chosen system, say yx1 . . . ,yn, of fundamental integrals must be limited in number, and consequently it is sufficient that the group G contain only a finite number of substitutions. The case of these equations will also be returned to later on. Among the substitutions which enter into a group there are, since we consider only the case of a finite number of critical points, only a finite number of independent ones. Suppose, for example, that S is a substitution corresponding to a given closed contour A", and that S1 is another belonging to a given closed contour Kx; then, if we describe successively the two contours K and Kx, we shall arrive at a substitution SS, which is the resultant of the two substitutions S' and vSj. All of the substitutions in a group therefore result from the combinations which can be made among the substitutions belonging to each of the critical points taken separately.CHAPTER IV. FROBENIUS’S METHOD. We will consider, as before, the region of the critical point x — o and let PQ{x)y Px{x), . . . , Pn{x) denote convergent series proceeding according to positive integral powers of x, and further assume that PQ(x) does not vanish for x — o. If all the integrals of the given differential equation, P(y) = o, are regular in the region of x = o, the equation, as will subsequently be seen, can be put in the form We will now seek to determine the form of the integrals of this equation by an extremely elegant and ingenious method due to Frobenius.* For simplicity we will assume P0(x) = i. In equation (i) make the substitution The limits v = o and v — oo will hereafter be omitted from the summation sign, but will be always understood. We see at once that this substitution gives rise to the equation (2) y = r) = 2gv (3) P(2g„xr +») = 2gpP(xr + -). * Ueber die Integration der linearen Differentialgleichungen durch Reihen. (Von Herrn G. Frobenius.) Crelle, vol. 76. 9394 LINEAR DIFFERENTIAL EQUATIONS. If now we write (4) f(x, r) = r(r — 1) . . . (r — n + 1 )P,(x) + r(r — 1) . . . (r — n + 2)P1{x) + . . . + Pn{x), we have (5) P(xr) = *rf(x, r), and consequently (6) P[g(x, r)] = 2grf(x, r + v)xr + ”. Since the functions Plx) are developable in convergent series going according to positive integral powers of x, it follows that the series (7) f(x, r) - 2fy(r)x* is also convergent, and that the coefficients of the different powers of x are integral functions of r of the degree n at most. The sub- stitution^ — g(xy r) being made in (1) gives us now (8) ^ [g,f{r+ v) + g,. tf Sr + r - 1) + ,(r + 1) + g«fSr)\xr +v — o. In order, then, that y — g(x, r) shall be an integral of (1), we must have 'g,f(r) = (9) gif(r + 0 + gfi (r) = °» gvf{r + v) +gv_JSr+v — 1)+ · . · +glfr-t(r+ 1) + gjv(r) = o. If we suppose now that gQxr is the first term in the series g{xy r) = ^xrJr% then g0 cannot be zero, and consequently r must be a root of the equation f(r) = o, which is of the nth degree in r. In whatFXOBEMUSTS METHOD. 95 follows we will consider rasa variable parameter, and g0 (or simply g for convenience) as an arbitrary function of r. Neglecting the first of equations (9), we can at once determine g,, g9, . . . as functions of r. Write (to) (- iykv{r) fSr+v— 1),/,(r+v—:2), ..ƒ„-.(*·+i),/,(r) f(r+v— I), /,(r+v—2),..ƒ„-,(?·+1), fy-Xr) O, f(r+v—2), -3(r-+i), ƒ„ _ 2(r) ; °> 0» ...,/(r+i), fjr) now from (9) and (10) we have , v , v_____________g{r)hy(r)___________ (11) /(r_}_ i)/(r+ 2) . . . f{r+v)· The variable parameter r will be so restricted that all of its values shall be found in the regions of the roots of the equation f(f) = o. Since the roots of this equation have their moduli all less than a cer- tain determinate finite quantity, it is easy to see that these regions can be chosen so small that the denominator of the rational func- tion gv(r) shall only vanish for the roots of the equation f(f) = o. This vanishing of the denominator of gv (r) would, however, make gv{f) infinite in general; this difficulty can nevertheless be avoided by a proper choice of the arbitrary function gif). We will suppose that the roots of f(f) = o are arranged in groups in the manner described in the last chapter, and suppose further that e (necessarily an integer) is the maximum difference of two roots in any of the groups. Writing now (12) g{r) = ƒ(r + 1) ƒ (r + 2) . . . ƒ (r + e)C(r), where C (r) is an arbitrary function of r, it follows that for all the values of r under consideration the functions gv{r) are finite, and consequently that if the series y — g(x, r) is convergent, then y — g(x, r) is an integral of the differential equation 03) P{y) = f{r)g{r)xr.96 LINEAR DIFFERENTIAL EQUATIONS. We have now first to investigate the conditions for convergence of the series g(*> r) = 2gvxr + v. If we assume v > e, then, recalling the definition of e, it is clear that f(r v -f- i) cannot vanish for any of the values of r to which we are restricted, and so, from equations (9), we have (u) ¿v+,= - f \ -+[¿v/.fr + y) + ¿v -»Ur + - I) + · · · + i7".+ .(r)]. Denoting now by Fv{f) and Gv{r) the moduli of fv(r) and^(r), we have at once the inequality* (! 5) Gr+I< F{r + \+l) [G„Fir +Y)+G._ ,F,(r + v - i) + · · · + GFv+i(r)\ We will now suppose a circle of radius K drawn with the point x — o as centre, and where K is as little less as we please than the radius of a circle inside of which the functions PY{x), P9(x), ·· - > Pn{x) are all convergent; also let f\x, r) denote the derivative of f (x, r) with respect to x. The series f(x, r), = 2fv(r)xv, and f\xy r\ — 2(v + i)fv + 1{r)xv, are both convergent so long as the inequality mod. x < K is satisfied; i.e., so long as the point x remains inside the circle of radius K, or, we may say for brevity, so long as x remains inside the circle K. Let now M{f) denote the maximum value which the modulus of f\x, r) takes on the circumference of the circle K; then by a well-known theorem we have (16) Fw+t (r)< < M(r)K~ *,FROBENIUS'S METHOD. 97 and consequently, from (15), (‘7) e, we will define certain quantities bv by the formula (20) f M(r+r) F{r + v) 1 L/r(r+y+ 1) ' KF (r + v+ i)J ’ also assume b€, as we obviously can, so that we shall have the inequalities (21) Gv < av < bv. We know that the integral function f(r) is of degree n in r, and therefore when v increases indefinitely the quotient f(r +v) f(r+r+ 0’ and of course its modulus, F(r -f v) F(r+ v + i)’ tends to the limit unity. Further: we have assumed PQ(x) = 1, and consequently f\x, r) is an integral function of r of degree at most equal to n — 1 ; also, M(r) denotes the maximum value which this function can have when mod. x = K. It is now easy to see (the98 LINEAR DIFFERENTIAL EQUATIONS. rigorous proof will be given immediately) that if v increases indefi- nitely we have (22) lim. M{r + r) F(r+ v+ i) ~ o. It follows now from (20) that (23) 1 ” Ky and therefore that the series and therefore also the series (24) g{x, r), — 2gv(r)xr + v, Is convergent inside the circle K. . It is necessary now to show, by aid of the results already ob- tained, that the series (24) is uniformly convergent for each of the values of r under consideration. If we denote by d a given arbi- trary small quantity, we must show that for all of the considered values of r it is possible to find a finite number, say j, such that the modulus of the sum V — 00 2gjr)x'+’ v =j shall be less than d. In establishing this we will first prove the truth of equation (22). Let s denote the modulus of r; i.e.9 s = mod. r — | r [ . (The symbol | X | , due to Weierstrass, will be used when con- venient to denote the modulus of the quantity X, whatever X may stand for.) Also, let Mj, M2, . . . , Mn denote the maximum values of the moduli of P9(x), P^(x), . . . , Pt!(x) on the circum- ference of the circle K; if then we write * (25) >[■(/) = s(s -I- 1) . . . (s + n — 2)M1 + s(s + 1). . . (s + n - 3)M, + . . . + Mn , * For the moment we will allow r to vary indefinitely until it is necessary to re-intro- duce r + v as the indefinitely-increasing argument.FROBENIUS*S METHOD. 99 we have <26) M(r) < The function f (r) is of degree n in r, and we may thus write f{r) = r» + [ ƒ (r) — r*\ and so obtain the inequality f{r) =r(r - 1) ... (r — n + 1) -f Pio)r(r - 1) . . . (r — n + 2) + . . . + PJp). Let JZ,, ... , IIn denote the moduli of Px(o), . . . , Pn(p) ; then, if we write (p(s) = s(s + i). . (s + n — 1) + nxs(s + i) . · .(j + n — 2) + ... + JT» provided only we choose j so large that the right-hand side of this inequality shall be positive; this can, of course, always be done, since (s) is only of degree n— 1. From the inequalities obtained above we derive at once F(r) ^ sn — | f{f) — rn | ; again, we have (27) and consequently <28) | ƒ (r) - r* | < 0(r), >s* — 0(s), M{r -\-v) ____________0 | y + y 1 but | r+v | < v + i and |r-j-i'+i|>y —IOO LINEAR DIFFERENTIAL EQUATIONS. and since the positive functions tp(s) and s" — (r + t- + 1) For the completion of the proof of the convergence of the series g{xy r), we notice that since all of the roots of f(r) = o lie in a finite region, and since the parameter r can only vary in the regions of these roots, then s must be always less than a certain determinate quantity, say t. If now we take v sufficiently large, we have ob- viously (32) and (33) M{r + v) jiv + t) F (r v 4- 1) (v — t)n — cf>{y — t) F{r + v) (r + t)n + (y+t)-\ K J; then if we choose , as we obviously may, we have generally cv > bv. Now if k be chosen as little smaller as we please than Kr then, since lim. the series 2evkv is convergent. Begin- cv A ning with the first term of this series, and counting forward, we canFROBENIUS'S METHOD. IOI cut off a finite number of terms such that the sum of the remaining v = 00 ones, say 2 cv kv, shall be less than an arbitrarily chosen quantity V=J dk ~s. Now, since we can of course choose j greater than it follows that we have I 2gr{f)xr+'' | < S y-J for all values of r inside the regions of the roots of f(f) = o, and for all values of x inside the circle of radius k and having the point x = o as its centre. The series g{x, r) = 2gvxr+* is therefore uniformly convergent and can be differentiated with respect to r, and its differential coefficient so formed will be equal to the sum of the differential coefficients of its successive terms. Having established now the convergence of the series g{x,r), we proceed to investigate the forms of the integrals of the equa- tion P (y) = o. We will consider the group of p. -|- i roots r0, rx, . . . , of the equation f(f) — o. These roots are so arranged (as already de- scribed) that for a < ft the difference ra — r$ is a positive integer. As certain of the roots of this group may be equal, we will further assume that r0, ra, rp, ry, . . . are the distinct ones; then r0 = rx = . . . = ra _ x will be an ¿t-tuple root of f{r) = o; ra = ra^.I — . . . — x will be a (¡3 — ¿*)-tuple root of f(r) = o; x = . . . = ry _ T will be a (y — y^)-tuple root of f(r) — o; etc. etc. We have assumed for^*(r) the form gif) = fir + I)fir + 2) . . · fir + e) C(r); where C(r) is an arbitrary function of r. Recalling now the definition of e, we have the inequality e > rQ — , and therefore g{r) cannot vanish for r = r0 = rx = . . . = ra _ x; but for r = ra = ra + I = . . . = r$ _ t, g(r) is zero of the order a, for r = Tp = rp + x = . . . = ry _ ,, g(r) is zero of the order y3, etc.;102 LINEAR DIFFERENTIAL EQUATIONS. and generally for r = rk, g{f) is zero at most of the order k. Again :: the expression f{f)g (f)xr is zero of the order a (or r = r0 = r1 ra _ x, of the order fi for r = ra = ra +1 = . . . = f, etc.; and generally f(r) g{r)xr is zero of at least the order k -f- i for r —rk, and consequently its kth derivative with respect to r must vanish for r = rk. Write now equation (13) in the identical form P[g{*, r)~\ = f(r)g{r)xr. and differentiate this k times with respect to r, and in the result write rk for r; we have as the result of this operation (35) *·*)] = °; dkg(x, r) where g*(x, r) = —--------· It follows at once that (36) y=zg*(x,rt) is an integral of the equation P{y) — o. Since the series g(x, r), = xr2gv{r)x'’, is a uniformly convergent series, the same is true of the series (37) g\x, rk), = x''k 2 | g*{rk ) + kg* - \rk ) log * + ' 2{rk) log2 * + . . . + gv(rt) log** j , which is an integral of P{y) = o. Since g{r) vanishes for r = rk at most of the order k, it follows that g{rk), g\rk\ . . . , g\rk) cannot all be zero, and so, by Fuchs’s definition, the integral (37) belongs to the exponent Tk · It *s obvious that (37) is of the same· form as (38) + 0» l°g* + · · · + 0*logM;FROBENIUS’S METHOD. 103 where the 0*s are uniform and continuous functions of x, and are not all zero for x = o. Suppose k < a ; then from (37) we find, as the coefficient of logkxy the series (39) xr*2g£rk)xv. Since k < a, we have g(r^) =g(rt), and so this series cannot vanish identically, and as a consequence we have that k is the exponent of the highest power of log x which can appear in the integral gk{xy rk)· It is easy to extend this result, and so observe that in general the integrals which belong to equal roots of f(r) = o have different exponents for the highest powers of log;r that enter, and consequently that these integrals are linearly independent. In par- ticular, if r is an «-tuple root of f(f) = o, then the n corresponding integrals of P(p) = o, viz., gk[xy r&)y (k = I, 2, . . . , «), are linearly independent- From what has been said it. is clear that the form of the integral gk(x, r£) is unaltered if we add to it a linear function (with arbitrary constant coefficients) of gk~ 1 (xy rk_x)y . . ., g(xy r0) ; we derive from this fact the conclusion that the most general value of the integral belonging to the exponent k contains k -f- 1 arbitrary constants. Let us suppose (which we may do without any loss of generality) that when the arbitrary function C {r)y which enters as a factor in all the coefficients of the series 2gv(r)xr+v9 becomes unity, the function g(xy r) becomes k(x, r) ; then g(x9 r) = C{r)k{xy r)y and consequently (40) gk{x, r) = Chk{x, r) + kC'hk~1 (x, r) + . . . + C ah(x, r), where ga, Ca, h* denote derivatives of g, C, h with respect to r. The functions hk{xy rk)y kk~'(xy rk)y . . . , h{xy rk) are linearly independent, and consequently the integral gk{xy rp) contains the ki arbitrary constants Cy C\ . . . , Ck. We have thus formed the general integral belonging to the exponent k of the differential equation without the aid of the integrals gk-'(xy rk_x)y ..., g(xyr0).104 LINEAR DIFFERENTIAL EQUATIONS. In the particular case of the root r0 the integral g(xy r0) contains only one arbitrary constant. It follows at once from what precedes that one integral of the differential equation P(y) = o is, to an arbi- rary constant près, completely determined by the following condi- tion : If the integral is divided by xr the quotient must be uniform, and must also be finite for x = o, r denoting a root of the algebraic equation f{f) — o which does not exceed any other root of this equation by a positive integer. We have above made an assumption concerning the form of the arbitrary function g(r)f in order to prevent the coefficients of the seriesg(xy r\ from becoming infinite: viz., we have assumed gir) = Ar + !)/(*■ + 2) ... . f{r + e)C(r) ; where C(r) is an arbitrary function of r, and where e denotes the greatest (integral) difference between two roots of any group of roots of f(r) = o. It is obvious that e may take any other value greater than this, and the foregoing results will still hold. The value of this liberty of choice of e is seen if we wish to actually compute the value of the series g(x, r) up to a given power of xy say xr+k. In this case assume e = k ; then all of the assumed coefficients gv(r) will be integral functions of ry and so the inconvenience of differentiating fractional forms will be evaded. Of course it is understood that the arbitrary function C(r) must be so chosen as not to become infinite for any of the considered values of r. For further remarks on this point the reader is referred to Frobenius’s memoir. We have seen both by Fuchs's method and that of Frobenius that logarithms generally appear in the integrals of a group. The groups of integrals arise, it is to be remembered, from the fact that certain roots of the indicial equation differ from each other by integers, in- cluding zero. The separation of these roots into groups, as already described, gives rise to the corresponding groups of integrals. It may be, however, that logarithms will not appear in a group of integrals. Fuchs investigated the conditions necessary to be satisfied in order that no logarithms should appear in a given group. The reader is referred to Fuchs’s memoir in Crelle (vol. 68) for his investigation of this subject, and also to one by Cayley in Crelle, vol. ioo. The test to be applied in order to ascertain whether or not logarithms exist isFROBENIUS'S METHOD. 105 obtained by Frobenius in a very simple manner in his memoir which we are now considering. Before taking up Frobenius’s remarks on this subject, we can show (Tannery, p. 167) that if in a group two roots are equal, say rx = r2, then logarithms necessarily appear in this group. The integrals belonging to rx and r2 are in general of the form (in the region of x — o) yx = an 0n , I* = ^*[0» + 022 1 ogx] ; where the functions 0 are uniform and continuous in the region of x — o. If now in P(y) — O we make the transformation y = yjzdx, we know that the integral zx of the linear differential equation of order n —- 1 in z belongs to the exponent r2 — rx — 1 ; but since ri — r1, we have in this case r2 — rx — 1 = — 1 as the exponent to which zx belongs, and therefore zx is of the form z>=-x+6{x)· where the constant A does not vanish, and where 6(x) is, in the region of x — o, a uniform and continuous function. From this we see that f z.dx is of the form A log * + ip(x), and consequently y% is of the form xr'[) 0 - 0 ------ f(r+ 0. Mr) then for r = rk the following equations must be satisfied: (47) hv (r) = 0 for v = rk_z — rk K(r) = 0 for v — rk — rk <“»= O y = rn for — rk .CHAPTER V. LINEAR DIFFERENTIAL EQUATIONS ALL OF WHOSE INTEGRALS ARE REGULAR. In what follows the critical point under consideration will be taken, as in the preceding chapters, as x = o. The results obtained for this point can be changed into the corresponding results for any other critical point xx by changing x into x — xx; the functions 0$>* or (My N) will of course be different for each critical point, but will always be uniform. It has been seen that functions of the form (1) F = Xr\_a + 0. log * + . . · + 0* log4 x\ r being the exponent above described, it follows that the uniform functions 0 contain only positive integer powers of. x in their devel- opment, and are not all zero for x — o. We have now dF dx + ^- r d4>r\ r + i+ x x + · · · -J log* jt. The coefficients of the different powers of log x cannot all vanish for x = O; for if they could we should have the system of equations rk — o for x = o, which is contrary to hypothesis. If, however, r — O, it is only necessary, in order that all the coefficients of log x vanish in dF —, i.e.y that equations (3) be satisfied, that we have dx k = k log4.*· ƒ/, · • · , yn r, y?, · • . , yn:EQUA TIONS ALL OF WHOSE INTEGRALS ARE REGULAR. 111 The coefficients of y, /, - ·· , yn are the principal minors of the determinant corresponding in order to these quantities, and they are obviously sums of regular functions, such as (6) xr{jPi log\* + . . . ] + *r‘[y>,·/ log'·.*· + ·.·] + ···· Now, when x turns round the critical point x = o, the integrals y yn, and their derivatives, y^a y * · · » y it y & — I» 2, submit to the same linear substitution ; the coefficients of y, y', . . . , yn therefore reproduce themselves each multiplied by the determinant, say â, of the substitution. [If we divide the equation through by the coefficient of yn , the coefficients, say . . , pn of ÿ, y", . . . , yn, will therefore reproduce themselves exactly after the substitution ; in other words, they are, as by the original hypothe- sis, uniform functions.] In order that an expression of the form (6) may reproduce itself multiplied by the determinant d after x turns round x =. o, it is clear that the logarithms must all disappear, and that the exponents ry rx, . . . can only differ by integers from the quantity log ^ 2ni 9 = , say, /3. The coefficients of the differential equation (5) are therefore of the form x?P, where P, like the functions 0, is a uniform function of x, having x = 0 only as an ordinary point or a pole. It follows also from the remark in brackets that px, p7, . . . , pn can have x = o only as an ordinary point or a pole. Fuchs's theorem regarding equations which have a system of independent integrals all of which are regular may be stated as follows: In order that the linear differential equation112 LINEAR DIFFERENTIAL EQUATIONS. zzz#y <2 system of linearly independent regular integrals in the region of the point x = o, z/ zj necessary and sufficient that each co- efficient, /,·, of the equation shall have the point x — o for an ordinary point or a pole ; in case this point is a pole, its order of mul- tiplicity must not be greater than z. The first part of this theorem has already been established. The groups of integrals which have been formed corresponding to the dis- tinct roots sx, s2, . . . of the characteristic equation have belonging to each of them a certain exponent r defined by log s{ -H:T=Ti5 the values r,, r2, . . . derived from sx, , . . . , and associated with each group of integrals, must then differ from each other by quanti- ties other than integers. As a change of r,· into rt -f- zzz, where m is an integer, would leave 5* unchanged, it is clear that the factor xr* in the group associated with might have different values of rt in the different integrals of the group, provided those values differed only by integers. In establishing the second part of Fuchs's theorem we shall, however, arrive at an algebraic equation (the equation f(f) — o of the last chapter) determining the exponent r for each group with- out ambiguity. Jordan proves this second part of the theorem in a very brief and elegant manner. Fuchs’s proof, which is rather longer, will be given later. If we transform the equation dny , d*-y , ¿¿n+Aj^rrr-l· · · · +P»y-o by the substitution y = Xv> where tj is the new dependent variable, and X is of the form l — 00 XK 2 CiX1 i= o and thereforeEQUATIONS ALL OF WHOSE INTEGRALS ARE REGULAR. 113 we have the equation (8) The values of q> , ... , qj are, from (12), Chap. Ill, , dX' ~n~d^JrP'X’ r_n{n— 1 )d'X dX ft = I . 2 dx2 + np' dx + AX’ , d"X d- - -x 9n ~ A ~d^-~ + · · · +t,«x; or, dividing by X, calling the new coefficients q2, . . . , qn, and f dlX writing Xw for —j—r, X’ , — n ~x +A* (9) »(»- i)X" , - , — 12 V 1 A> JT ~x qn = xix'*+p*x'*~*+ · ■ · +a^]. Xf X" Now -y. has obviously the point x = o as a pole of order i ; —- has X X« ^=oasa pole of order 2; and in general —^r has x = o as a pole of order i. If then /,, p^, . . . , pn have ;tr == o as a pole of at most the order 1, 2, . . . , respectively, it follows that the coefficients , q* > · · · > <2n in the equation (10) dnq t dn~'?? , , ¿ar” + · · * + ft.7 - °I H LINEAR DIFFERENTIAL EQUATIONS. have ^ = oasa pole of at most the orders 1,2respectively. (It is easy to see that the order of multiplicity of the pole x = o in may be less than i, but under the hypothesis as to the coefficients pi it can never be greater than i). If now the equation in y with coefficients p has all of its integrals regular in the region of the critical point x = o, it follows, from the relation tf — —yx~K^€i^, A i = o that all the integrals of the transformed equation (10) are regular; conversely, if all the integrals of (10) are regular and its coefficients possess the property in question, then all the integrals of (n) dny , dn~y , ^+A^v + · · · +Ar-o will be regular, and the coefficients p will possess the same property, since (11) is derived from (10) by the transformation V = 1 y ym x We have seen that the equation (u)with uniform coefficients always admits of at least one integral of the form y0 = *r3) % + Py = °· which admits a regular integral, The quotient, y = xr . i dy y dx ’ admits obviously the point ^r = oas a pole of order i at most, and therefore P must have x = o for a pole of order i at most; for if P had x — o for a pole of order, say, a > i, then there would be terms in P which could not cancel with any terms in - y dx Again, take the differential equation of the second order (i4) d*y d? + p% + & O, having all its integrals regular in the region of the point x — o. Sup- pose Yy = xr(p^, to be one of them ; then writing y= Vs, it is clear that z must be regular; substituting, we have dx* , f2dY , ~\dz , i Vd'Y , ndY , 1 + \_Ydx + P] dx + YLdxs' + P dx + QY\ ~ °’ dz or, making — = v (y is therefore regular), (i5) dv , \~2 dY , n d^+[_Ydx + P}v = 0·ii6 LINEAR DIFFERENTIAL EQUATIONS. As already seen, the coefficient of v here has x = o as a pole of order i dY I at most, and since — — possesses this property, it follows that P has x = o as a pole of order i at most. Write w = e~*Spd* y y — wv ; then substituting in (14), we have (16) (d'v d? + Iv = O, r ^ i dP /= Q~--r 2 dx If y1 and are two independent regular integrals of the given equa·*· tion, and corresponding to them are vx and v%, then (17) vx= y^Spd*y v2 = y^SPd*. From what has been found concerning P it follows that gk/Pdx is regular, and therefore that vx and v% are regular. The equation. d*v (18) ^ + ^ = has then its integrals regular, and consequently 1 d*v v dx2 admits x = o as a pole of order 2 at most, and consequently /, and therefore Qf admits jr=oasa pole of order 2 at most. (It is easy to see from (18) that P and Q have the required property.) A similar process might be employed for the differential equation of the third order, but it is rather long, and besides unnecessary. These two illustrations show the truth of the theorem for n = 1, 2.EQUATIONS ALL OF IVNOSE INTEGRALS ARE REGULAR. II7 d'n Returning now to equation (12), write -- = rf, and the equation dx becomes <'»> d* dn xrf dn ~ V . , ~ + q' + · · · + 9» - .7 - O. The integrals of this are the derivatives of the integrals of (12); but these latter are regular, therefore the integrals of (19) are regular. Suppose now the theorem to hold true for (19); then admits x — o as a pole of order / at most. The theorem is then true for equation (12) ; and since (12) is derived from (11), by the transformation <2°) v — It is also true for (11). We have then for the region of x = o the equations where Piy PQ9 . . . , PM are in the region of x = o uniform con- tinuous functions, such as (22) ' P* — a0 + axx + aj? + . . . ^2 = k + kx + b^x2, + . . . ^Pn— h + hx + hx* + ■ · · If we know the radius of the common circle of convergence of these series, say py then the values of the coefficients ay by . . . will be limited by the inequalities , =M /]ƒ mod·am <~j^’ mod-■ ' ' where M is a properly chosen constant. If we substitute in (11) for y the value <23) y =Il8 LINEAR DIFFERENTIAL EQUATIONS. we have as the result (24) F(r)xr + 4>(r)xr + l + ·2 + . . . , = xr$(x, r). where (25) F{r) = r{r - 1) . . . (r — »+ 1) + a,r{r— 1) . . . (r - n + 2) + · · · + b,r{r — 1) . . . (r — n + 3) + . . . , (26) m{r) - amr(r — 1) . . . (r — n + 2) + - 1) . . . (r —«+ 3) + . . . The equation F (r) = o is called by Fuchs the “ determinirende Fundamentalgleichung/” following Cayley (Quart. Journ. of Math.y 1886) we will call it the Indicial Equation. We will now give Fuchs’s investigation concerning the forms of the coefficients in the case where the differential equation has all of its integrals regular. Let o, xx, , . . . , xp denote the critical points of the integrals, and write f(x) = x(x — Xl) . . . (x — xp) ; then the differential equation under consideration has the form (27) dMy 1 Pp(x) dn - y P2p(x) dn - y dxn tp(x) dx1 dxn ”2 + · · · + where Pkp denotes a polynomial in x of the degree kp> or of a lower degree. We have seen in Chapter III that a system of fundamental inte- grals. of thq equation (28) dy , d—'y dx“ ' P' dxH - 1 + · · · + p«y = o can be obtained of the form (29) y 1 * y, = y^fz.dx, y% = yjzylx ft ¿lx, . . .EQUATIONS ALL OF WHOSE INTEGRALS ARE REGULAR. 119 where zl9 tl9 . . . are integrals of linear differential equations of orders n — 1, n — 2, . . . respectively. It is obvious that these auxiliary functions can be so chosen that the fundamental integrals yl9 · · · > yn shall submit to the substitution (80) (Chapter III) when the variable turns round the point x — o. The integrals of the equation of the nth order are yif y*> · · · > y»; those of the equation of order n — 1 are y *^2 y · · · y - i » those of the equation of order n — 2 are t\y ^2 y · · ♦ y - 2 » etc. It is easy to see that the integralsyi, z19 t19 . . . are all of the form x* x — 4 dy x — 3 _ did 2x(x — 2) dx ' 2x‘(x — 2)^ °’ this admits as solutions the two independent functions and (x2 — 2x)* , which both belong, in the region of x = o, to the exponent ; we can, however, replace them by the independent functions X*, (x* — 2 xf — V — 2.x* ,120 LINEAR DIFFERENTIAL EQUATIONS. belonging to the exponents J and f respectively. This “ case of ex- ception ” being borne in mind, we need not refer to it again unless it is absolutely necessary in some particular case. Suppose now that yx is known and is of the form (30) y, — xrHp{x), and satisfying from (80') the relation Sy, = w ; suppose further that we write, as we may, (31) y, =yjzldx·, where Sy, = sx + sxyx, and where'belongs to the exponent r, r we have (32) from which follows z, — d y, to) s*~ = isj, = * dxy,’ d s,y, + s,y, d yt dx yx ' From this it is clear that zx is a uniform function belonging to the exponent — rx — 1. It is easy to show that the remaining in- tegrals z3, . . . , zn„iy which with zx form a fundamental system for the equation of the (n — i)st order belong to the exponents rz — rx — 1, tx 1, . . . ; and also that in y* -yjz^dxft.dx the function tx is uniform and belongs to the exponent r3 — r3 — 1. By continuing in the manner indicated, we find finally that the functions ?xf i> ·· · , y^xt 1 . . . wx belong respectively to the exponents r 1 > r, — i, r, — 2, rn — (« — i).EQUATIONS ALL OF WHOSE INTEGRALS ARE REGULAR. 121 From equation (16), Chapter III, we have, (34) D — Cy"z" ~1 . . . w19 where C is a non-vanishing constant. It follows then on substitutin for yA its value, and remembering that zx, tx, . . . , wl are all uni- form functions, that ' (35) D = xr*+r«+ · · ·+rn - ~r~ 4 (x); where tp(x) is uniform and continuous in the region x = o, and does not vanish when x — o. A similar form is at once obtained for the point infinity: it is only necessary to change the variable by the relation i J and, as before, investigate the point t — o. point in a moment. We have (36) We shall return to this denote, as above, by S the determinant of the substitution arising from travelling round x = o, and write (37) log _ 2ni It follows now at once that (38) D — xm+£ tp(x), Di — xm,+f*ip\x) ; m and m! being integers, and tp(x), $>'(*) uniform and continuous functions of x in the region of x — o. In developing these deter- minants no logarithms will appear. These results have already been obtained. Suppose now we multiply each element in D{ and D by x raised to the power denoted by the negative of the exponent to which the corresponding element belongs. The determinant D{ will then be multiplied by A, _\sn-- = x 1 1 (n — 1) f, bjo.122 LINEAR DIFFERENTIAL EQUATIONS. and D by The quotient, B, = < i 2 ) . ^ ’ differs from , z>., — , by the factor —*. It follows now at once that piX* is a uniform and continuous function of x which may be- come zero for x — o, but may not become infinite for this value of x. We have then, finally, (39) where P{ is a uniform and continuous function in the region of x = o ; and if the integrals are regular in the regions of all the critical points o, x1, . . . , xp, then (4o) A = £'; where ip = x(x — x^y . . . (x — xp), and A now denotes a uniform and continuous function in all the plane. The equation has now the form , , dny P,dH-'y P, d”-*y 1 dx” Ip dx" ' 1 " if·1 dx" ~ 2 . . -1----- y = o. ~ tpn^ To study the point 00, write x we find without difficulty dn ~ y j. Transforming the equation, (42) . 1 dtn n’n-'t dr- i ^ «, « - 2 ^ 3 ^rrr+-----o; i’-i lj> f - 1, n - 2 rp t3 £ tEQUATIONS ALL OF WHOSE INTEGRALS ARE REGULAR. 12$ where x — j has to be substituted in ip, P1, . . . , Pn. The quan- tities aktk-i > #k,k-2j · · · are integers, the first of which = k(k — i). The point t = o is now a pole (or ordinary point) of order at most = i,2j . . ., ny for each of the coefficients of dn~'y dn~2y IF^7’ dr-*’ · · · ’ y~' p p it is therefore necessary that —1, . . . be developable in positive ascending powers of t, and that the first term in each of these de- velopments shall be of the degree i, 2, . . . , ti, respectively. Now,, i 0 we must therefore have (43) p, = ^+j^t+ · · · = 4*+4*'- + . . . e e P — —L -\------i 2 ¿2P t pp- ... — ej2? + ef? - 1 + . . . That is, Px, , . . , Pn are polynomials of the degrees P, 2p, 3p, . . . , np; where p —|— I is the number of finite critical points. We shall now give Fuchs’s proof (which has only so far been faintly indicated) of the converse of the theorem just proved ; viz., we shall prove that every linear differential equation of the form (44) dny M1 dn Jy M2 dn2y Mn _ dxn ' x dxn ~1 x* dxn ~2 ' ’ * * xn y ~ °3 where Mx, , . . . , Mn are uniform and continuous functions of x in the region of the critical point x = o, admits in the region of this- point a system of fundamental integrals all of which are regular.124 LINEAR DIFFERENTIAL EQUATIONS. The method employed in Fuchs's proof is the same as that used by Weierstrass in his general proof of the existence of an integral of an algebraic differential equation. Instead, however, of referring directly to Fuchs’s memoir, we shall follow Tannery’s exposition of the same. From what has been shown we know that (44) admits at least one integral of the form (45) y = xr(x), where (x) is a uniform and continuous function of x, and is not zero for x = o. If now we substitute (46) y = xrr) in (44), we shall have (4 7) dnrj dxn where Mx dn~xr] dn " 2rj MJ _ ~x~ ~dxn - 1 “I" ~dx* - 2 + · · · ~ ° (48) n(n — 1) k-\- 1) --------------zTI----------:---T\T 1) ... (r — k 1) (7t—i)(n—2) . . . (n—k-{-i) r(r— 1) . . . {r—k^2)Mx{x) + . . . + 0 - k + i)rMk _ X(x) -j-M^(x); r(r — 1) . . . 1) -)- r(r — 1) . . . (r—n+2)Mx(x) + ... + rMn_l(x) + Mu(x). Now if (44) admits as solution the value of y in (45), then (47) must admit a solution of the form (49) V= C0+Cxx + CX + . . . ; where C0 is necessarily different from zero. Before goi-ng farther take, for example, the case n — 2, viz.:EQUATIONS ALL OF WHOSE INTEGRALS ARE REGULAR. I2J Substitute here y = xrrj, and we have (»■) S + ^ ■) Mx{x) dtf dx + r{r i) + rMi(x) +M&) __ ----------------------v = o. Now this must have an integral of the form (S2) V — C* ~h Cxx -f- C^pd where C0 cannot vanish. This value of 7, substituted in (51), must give a result which is identically zero ; Le., the coefficient of every power of x must vanish. Bearing in mind now that Mx and M2 are uniform, and, in the region of x — o, continuous functions, it is clear that x~ 2 is the highest negative power of x that will appear after making the substitution ; the coefficient of x~2 is obviously the quantity CMr- 1) + rMt(o) + ÆT2(o)}; that this may vanish we must have, since C0 is not = ot (53) r{r — 1) + rMt(o) + MJo) = o. In the same way we see that if (49) is a solution of (47) we must have that the coefficient of x~ M vanishes ; that is, Mnf{ o) = o, or (54) r(r - 1) . . · {r — n + i) + r(r - i) . . . (r - n + 2)^(0) + .·■ + r(r— 1) ... (r—n-\- 3)^(0) + . . . + rMn_ t(o) + Mn{o) = o. This is the same as equation (25), and will be called the indicial equation. If r,· is a root of the indicial equation, and the corre.- sponding root of the characteristic equation, we have a relation which has already been shown.I2Ó LINEAR DIFFERENTIAL EQUATIONS. Notice that if Mn(x) = o, and of course then MJ d'~2rl 1 w dxn -1 ^ * w dxn -3 dr} + . . .+N'n_I{x)x^+Nn{x)r?. We have to show that this equation admits in the region of x — o an integral of the form (60) r} — CQ 4- Cxx + C^x1 -f- . . . ; substituting this value of rj in (56), and equating the coefficients of xk where k is an integer, we have (61) l(k+i)k(k-i) . . . (k-n+2)+Nl(o)(k+i)k(k-i) . . . (k-n+3> + ^(o)(k+i)k(k — 1) . . . (k — n+ 4)+ . . . +A^.I(o)(i-+i)]G + I = A& + A& + AtCr. where the coefficients A0, A1, . . . , Ak are made up from mere numerical quantities and from the coefficients of the different powers of x in the development of the functions n;{x\ n;{x\ N’n_ix\ Nn{x). The coefficient of Ck-f x equated to zero gives (62) (k+i)k{k-1) . . . (k-n+2)+N1(o)(k+ i)k(k— 1) . . . (k-n+3) + ... + Ar._I(oXi+i) = o. This is simply the indicial equation corresponding to equation (57), divided through by r, and then r changed into k1. Now the roots of this equation have been shown to be r2 — r1 — 1, ... f rn — rx — 1, none of which are either zero or positive integers; there- fore, since k is a positive integer, no such equation as (62) can exist, and consequently the coefficient of Ck + I can never vanish for any positive integer value of k, nor for k — O. From (61) we can now obviously determine each coefficient C in the form (63) Ca = 2U·EQUATIONS ALL OF WHOSE INTEGRALS ARE REGULAR. 129 We have now to establish the convergence of this series V — Co + Cy* + + · · · in the region of x = o; for this purpose we compare it with another series, the convergence of which is readily ascertained. Let ill9 il99 . . . , iln-if denote the maximum moduli of * Nx'(x)9 . . ., N’n _ Nn(x) ; then, by a known theorem in the theory of functions, (64) mod. mod. < a < Of A P*’ n. mod. daNn (x) - dxa I < « —1 p* where p denotes the distance from the origin to the next nearest critical point. We form now the auxiliary differential equation (65) dn ~ lv dn ~ 2v 1 — Y'X dx” dn- dxn 3 + · · · “l· Yn - I dv dx n. - , dn~'V , A , d" ' ^ I , A-. „ ^ - I “T x ^ - 2 "T · · · + x I — - P I — , nH H-------zv'> where the quantities yx, . . . , yn _ , are arbitrary positive quanti- ties subject merely to the condition that no root of the equation (66) yx w(w — i) . . . {w — n -f- 3) -f- yjv(w — 1) . . . (w — n 4- 4) + · · · + Yn -1 = 0 shall be either zero or a positive integer. We seek now to satisfy this equation by a series 00 (67) v = 2gaxa, o \130 LINEAR DIFFERENTIAL EQUATIONS. where g0 is not zero. As above, substitute this value in (65), and equate coefficients of xk : we have without difficulty (68) . . .{k — » + 3) + y,(k-\- i)k . . . (k — n +4) . . . + Yn - + l)]gi + I = B*gk + Bxgk - I + · · · + Bkgt ; where the coefficients B are formed in the same way as the coeffi- cients^ in (61); viz., the B’s are formed out of the coefficients in the developments of nx nn j · · · * _—— x x9 I---- I--- P P according to ascending powers of x, and out of certain numerical quantities, just as the A's are formed from the coefficients in the developments of n;{x\ n;{x\ . . ., Nn{x), and the same numerical quantities. Since the B’s are obviously all positive it follows from (64) that (69) Ba > mod. Aay a = I, 2, . . . , k. Equation (68) gives now (70) g® = &g., where 3Ba is a positive quantity ; if then we take g0 positive, as we obviously may, all the coefficients gx, g9 , . . . will be positive. It is obvious that there must exist a finite limit for k, say k — t> such that for k ^ t we have always (71) mod. \k{k— 1) . . . {k-n-\-2) -f- Nx(o)k(k—i) . . . {k-n-\-3) + · · · + _ ,(o)] > yjz{k—l) · · · {k~n+2) + yjzik— 1) . . . (k—n+3) + · · · Yn - x - Now from (61), (68), (69), and (71) it is clear that we shall have gk > mod. Ck, if this can be shown to hold for t.EQUATIONS ALL OF WHOSE INTEGRALS ARE REGULAR. 131 Let A denote the largest modulus in the series 1, B>, Ha, . . ., H/, and let B denote the least of the quantities 1, Ba, · . S*, and suppose we choose C0 so that (72) A mod. C0 < Eg,. Since A, B, g, are different from zero, this inequality is of course always possible, and gives for C0 a value different from zero. It is clear now from (63), (70), and (72), remembering the hypothesis as to A and B, that for k ^ /, and consequently for all values ky that we have always gk > mod. Ck · If then the series <73) v = go + gx* + g^1 + · · · is convergent, it follows a fortiori that (74) V — C<> + Cxx + C^c1 + . . . is convergent. To establish the convergence of (73) in the region of x x — o, we proceed as follows: Multiply out (65) by 1 — —, and then substitute the above value of v; this gives, on equating the coeffi- cients of xky {75) [(^+I)^(^“I) ···(&— n + 3)y, + · . · + {kAr\)y„_ l]gk + 1 = [/&(£ - — n + 2){fj + n^j + k(6- i)...(k- n + i)(^+n) + · · · ^\ n f A. - 1} +132 LINEAR DIFFERENTIAL EQUATIONS. We have from this, for the limit of the ratio of two consecutive coef- ficients ¿*+1, ghy lim ih±l = y'+Bf1' t * = “ Sk PYl and for the limit of the ratio of the two corresponding terms in the series (73) If lim k = » .g·*+■■**+ gkX* y i + Pn, PVi mod. py, y*mod. x, < i, PYl then the series is convergent; that this may be, we need only limit x by the inequality mod. x < PYi Yi + pH, ' Since yx and p are positive non-vanishing quantities it is obvious, that (73) has a circle of convergence having x — o as its centre. It follows at once that (74) is convergent inside this same circle. As yt can be taken as large as we please, we have lim Yi= 00 PYi Yx + pA = Py so that it is clear that the series (74) is convergent in the entire region of x — o. This can also be shown by a well-known process in the theory of functions. We have shown now that (59) admits as a solution the uniform and continuous function V — ^0 + Cxx + + · · · > and it therefore follows that (44) admits the solution yx — xr^y or, say, yx = xr*(p(x) ; where ··· 9 Put then, since rx, , . . . , rn are so arranged that for ft > a the differ- ence r$ — ra is never a positive integer, it follows that the same property’ exists for the roots pt, . . . , pn; viz., pp — pa is for ft > a never a positive integer. The differential equation in z there- fore admits a regular integral of the form *1= ip(x) being a uniform and continuous function of x in the region of x — o. Corresponding to zx we have y, = yjzfa, an integral of (44) belonging to the exponent r, - rx - 1 + 1 + rx, = rt. By the same process we can find a third regular integral of (44), y 3 = yj^dxf txdx, where tx is a regular integral of a differential equation of order n — 2, and belongs to the exponent r% — — 1. The integral y% therefore belongs to the exponent r% — r, — 1 + 1 + r2 - rx - 1 + 1 + rx, =134 LINEAR DIFFERENTIAL EQUATIONS. The integrals ziy tiy . . . , wx of the auxiliary differential equations; are always of the form where 0(x) is a uniform and continuous function of x. The loga- rithms which appear in the integrals of (44) can then only enter through the different integrations which have to be performed. We have thus established the existence of a fundamental system of regular integrals of equation (44), the elements of which system belong respectively to the exponents rxy , . . . , rny which are the roots of the indicial equation. It is also obvious that these expo- nents are respectively equal to log log log sn 2ni ’ 2ni ’ * 2711 where syy siy . . . , sH are the roots of the characteristic equation— a result arrived at in what precedes. Resume for a moment equation (41), viz., dy_ + Eidinz+...+p-:r = 0, dxn ‘ ip dxn " 1 ' ipn where ip = x(x — — xj . . . (x — xP)y and consider the critical point x — xt. We can show that the sum of the roots of the indicial ft! fi__ j \ equations for the critical points is equal to p —-. Denote by 2 ib'(xt) the derivative [^\ ; the indicial equation relative to this \dx)x = xì point is then (76) r(r — 1) . . . (r - n + ï) + r(r - i) . . . (r - n + 2) + Pi*h The sum of the roots of this equation is r(r — 1) . . . (r — « + 3) + . . . = a n(n — 1) I\(x) tvy 2EQUATIONS ALL OF WHOSE INTEGRALS ARE REGULAR. 135 Again: the indicial equation corresponding to the critical point X = 00 is r(r— i) . . . (r-n-\-i)-\-(aHiX.I—d,)r(r— i) . . . (r—n+2) + . .. = o. The sum of the roots of this is _ n(n — 1) -1 + di — — n(n since an>n_I = n(n — i). The sum total of the roots of all the indicial equations is P + « - 2 2 ip (xt) Now by a well-known formula for the decomposition of rational fractions we have ■2 Pfad____I___= Pjx) . $'(**) X — iix) ’ multiplying this by x and then making x = oo, we have = d therefore, finally, we have for the sum of all the roots the value P In concluding this chapter we will give a few illustrations of the development of the integrals in series, and the method of obtaining the coefficients. We shall use for convenience only the point x — o, and begin with an equation of the third order, viz., <£y ,_________________OW_________________ dy dx3 x(x — at)(x — a^) ... (x — adx3 + + ___ Qlx)_________________ x*l(x — a,) ... (x — a*)}2 dx __________flfr) y = x’\(x — a,)... (x — y o; (Or36 LINEAR DIFFERENTIAL EQUATIONS,. where the critical points of the coefficients are x = O, x — ax, . . ., ^ = ay, in number jx -}- i ; and Qx{x\ Qt{x), . . . are polynomials in x of degrees jxy 2jxf . . . respectively. Equation (i) may be put into the following form: (2) cPy Pjx) d'y Pjx) dy Pjx) _ dx% x dx* x* dx xz o; Px(x\ Pt{x) being rational functions of x for which x = o is not a pole. The indicial equation is r(r — i )(r - 2) + P^rir - 1) + P,(o)r + P,(o) = o, of which the roots {a, b, c) will first be supposed all different, and the difference between no pair of them an integer. Equation (2) will then have, in the region of the point x = o, three integrals of the form xa2c{x\ xb'2ci'xty xf2c//xiy (i = O, I, . . . 00.) Equation (2) may be conveniently written (4) d.V~ 1 dx‘ * dx + P*y = o. Substituting in this equation the series y 2 Ci X{+ay and equating to zero the coefficient of xfi + af we find (5) i) + Cp _ 202(^+# — 2) + · · · = O. The symbols F and , 0,, . . . have here the following meanings: The point x = o being an ordinary point for Px, P2y P9J we may write Pi = ao + *1* + + · · · , Pi = ^0 + bxX + + · · · » P% = ^0 + + dyX* + . . . ;EQUATIONS ALL OF WHOSE INTEGRALS ARE REGULAR. 137 if now in (4) we substitute ƒ = x*, then F(r) = r{r — i)(r — 2) + aar(r — 1) + b,r + d„, •PSr) = aAr — 1) + bS + di. = a,r(r — 1) + b%r + d,, o can, as seen above, be expressed as a product of c„, which cannot = o, into a function of the root a, and so 1 = 00 the series y — ^cixfl + a is completely known when any value arbi- i = o trarily chosen has been assigned to c0. In a manner precisely similar the series corresponding to the roots b and c may be calcu- lated. Suppose one of the roots, as ay to be zero: the corresponding138 LINEAR DIFFERENTIAL EQUATIONS. I — CO integral will be y = 2 CiX% which on being substituted in equation i = o (4) gives for the coefficient of x* - i0,O — i) + C» - ^0.0 — 2) + ···=: O. Making /i = o, we find c9F(o) = o, but F(o) = o; hence c0 is arbitrary, since it cannot vanish. It may be noted that P3 must be divisible by x in this case, or d0 = o. For jj. = 1, ^I) + i.0,(o) = O. For fi = 2, C*F{2) Ci0i(O + ^O02(°) = Therefore the coefficients c are to be obtained just as before. The case of all three roots of the indicial equation distinct is thus com- paratively simple; but a numerical example may not be found entirely useless in this connection. Suppose the coefficients P1, P3, P3, when developed by Taylor's series in ascending powers of x, to have the following values : P, = i + ajc + + tf3;ir3 + . . . , P*= | + b,x + bj? + b^3 + . . . , p, = -* + 4* + 4** + 4**+------------ Thus aa ■— da = — £, and the indicial equation for .*■ = o becomes P(r) = r(r — JXr ~ 2) + — 0 + i*" — i = °. or (6) r»_J^r’+r-i = o. The roots of this cubic are I, and Since no two of them differ by an integer, there are no logarithms in the integrals, and we mayEQUATIONS ALL OF WHOSE INTEGRALS ARE REGULAR. I391 obtain them by putting iy h i successively for a in equation (5). We: have then, first, (7) ^F(2) + *.0i(O = o. But F(2) = 2s-VX2l+2-i = 2i, 0i(O — 4" i equation (7) therefore becomes 2£ "I" (Pi 4~ ^1) ^0 = °y or c\—-----~ C° * Again, F{a + 2) = F(3) = 3s - V" X 32 + 3 ~ i = I3i> 0i4~ 0 — 0i(2) — 2^i 4~ 2bx -[- dx, 0a(I) — ^2 4" ^2 i whence equation (7) becomes 13icft — (2ai 4" 2^\ + ^1) ~ co 4“ co (K 4~ d^) = o; that is, ^ 4“ di)(2ai 4“ 2^i 4“ ¿i) — 5(^2 4- Ol · In the same way, as many of the coefficients may be calculated as are desirable. It will often happen that cQ, which is a common factor to all the terms, may be so chosen as to simplify the series more or less. The roots and J may now be treated in the same way, and each will give rise to a convergent infinite series all of whose coefficients except the first are determinate. Returning to the supposition that a, b, c are the roots of F{f) = o, let two of them, as a and b, be equal, and corresponding to the double root a we shall have two integrals; the first being the convergent infinite series y 1 — ^ bjXa~^t, (2 — 1, 2, · . . , 00,)*4° linear differential equations. and the second of the form 7a = 2 C{Xl -f- ^ dxî . log X\. Substituting ya in equation (4), the following results are obtained for the coefficients of xa+*L and log x> which must separately vanish. Coefficient of xa+*t· : (8) F(a-\-}d) Cp. +0i (*+/*“ l)c n - i+02 (0+/*— 2)c ^ _ 2+ . . . +^/(/2+/+/+0/(a+/*— iyV-i+02/(^+^—2)^V-2+ . . . = 0. Here F\a + /¿) denotes the result obtained by differentiating F(r) with respect to r, and substituting a + ja for r in the result ; and similarly for the other functions, 0/, 0/, ... It may also be noted that if ++!· For xr logA;r = xr, and it is easy to see that it makes no difference whether we perform the differentiation X times upon xr and substitute the result, or substitute xr and differentiate the result X times with respect to r. In fact, the result of sub- stituting xr is A dnxr dxn + B + · · · = r)xr > where the coefficients A, B, . . . do not contain r, and since dK dn dn dK drk dxn ^ ' dx11 drK Hence, knowing the result obtained by substituting xr, that for xr logA x is given by the following identity: 0> rK] = r)x" log^r + 1 logA ■ 1 ■* dr1 , , dk \\a ~\~ l) + c0/(p2(a) — ° i From equation (9), in like manner, c0'F(a) = o ; c\F(a + 1) + c*,(PJf) — o ; From the latter set of equations the coefficients ¿r/, „ are all obtained in terms of c0\ which is still undetermined. We know by what has been previously shown that the series 1 = ca 2 c{xa + i, the coefficient of log x, can differ only by a constant factor i = o from^ = '2bixa^i {i — o, 1, . . . , 00). The root b of the equation F{f) = o corresponds to a series entirely similar to ylf which may be found in the manner already amply illustrated. Let us now suppose that the three roots of the indicial equation are all equal, say the common value is a. In this case the three integrals in the region of the point x — o are of the following forms: yx — xa'2clfx\ (z = O, 1, . . . , 00 ;) y,3 = xa{^c-"xi + 2c/'x{. log x}, (/ = o, 1, . . . , 00 ;) yi — xa\^ c{x*' -)- 2 c/x*. log x 2 c"x. log2 x], (z = o, 1,..., 00 .)142 LINEAR DIFFERENTIAL EQUATIONS. The manner of finding the integrals yx and yz will now present no difficulty ; as to y3, remembering the useful formula given above for substituting xr log*x in the differential equation, which is mentioned by Jordan (Cours d'Analyse, iii. p. 82), we find for the coefficients of xa+>*· log2 x, xa+fL log xy xa+*i the following expressions, all equal to zero : (10) c' yJLF{a +/0 + c”il - M—i) + cr'** (a + M~~2)~\~ · · · = o; (11) cf^F{a -f- p)-\-d^ _ z xx + bp? + . . . pmxm — m0 + mxx -f m^x* + . . . Form the indicial equation, F{f) — r{r — 1) . . . (r — n 1) -f- aar(r — 1) . . . (r.— » + 2) ~h · · · H~ mo — °y and obtain its roots. If no two roots are equal, or differ by an in- teger, the equation has m integrals of the form 0' = o, 1, . . . , 00,) where a is one of the roots. If, however, two or more roots are equal, or differ by integers, some -of the integrals will in general con- tain logarithms which occur in a manner already explained. Let yx, y3, . . . , y\ be the integrals corresponding to the root a of mul- tiplicity A. Substitute y2 in the equation, and equate to o the co- efficients of every term in x and every term in log x separately. From the system of equations thus formed the constants of the series may be successively determined, with the exception of two which will remain arbitrary; and the series multiplying log x will be the integral yx. In the same way the constant coefficient in the integral y may be determined, and the series multiplying log\r will again be the integral yx. Proceeding in a similar manner with each root of the indicial equation, the integrals corresponding to the region of the point x = 0 may all be found.144 LINEAR DIFFERENTIAL EQUATIONS. As a further illustration, let us obtain the integrals of the equa- tion dky 4x 2x% —|- x% d*y dx4 ' 8x(x + i)(x — i)(x — 2) dx3 ( 64 4“ i6;r2 -f" 4*a + x* d2y 32^ (x + i)2 (x — i)2 (4· — 2)2 dx2 512 —|— 64^ 4~ 8.r6 + ^ 64x3{(x + i)(x — i)(x — 2)[3 dx 51200 -f- 3200X4 4~ 25^·18 _ 204%xa{{x 4" *)(x — l)(x — 2)Vy ° in the region of the point x~o. This equation is seen upon exami- nation to satisfy the conditions that all of its integrals shall be regular, and each numerator is of the maximum degree in x. We have AW = 4X 4“ 2-T2 4“ x* 8{C* 4- i)(* - i){x - 2)\ = Ì\X + X* + K4-4K + IK + IK 4- ¥K + · . <1_ 64 4- ~f~ 4^4 4~ x f% 32\ix 4- l)(x — 00* “ 2M2 1{ I 4. ^4. 3^4- J^*3 + f£r4 + + \3K + flK4- . ) ft __ 512 + 64*’ + 8*6 + 4T* A 64{(*+iX*- i)(^-2)r ) it 4 _ 5 1200 + 3200T* + *> w-^yv +0,(*+^- iy v -. +0a+/*-2 y v +3[^/(f+-/‘y,v+0./(H-/‘- ly'v - I+0/a+^-2y,v -. + ···] — °; (®) 0c'n - i + 2)iV - a + · · · f 2[/-'(i+/iyv+01,(i+^- iy v _ ,+ 0,,(ff)«-2y/(. +3[^*,'(i+/‘y,v+-, + ^"(H^-ay", -. + ...] = o;(Q (4 + ~— i “I- “f" 2)^ __ 2 ~f" · · · -\-F- .+0/(4+/^—2)c'(1 _ 2-f- · · · +^"(4+^V+0O^'V - I + 0/,(i+^-2)c'V - a + ... +^"(^M^Vh0 IKV - .+0a'"(i+/i-2)^"V - , + · · · = o. (D) The meaning of the symbols F, F', . . 01( i--4 + ¥ = 'i4; -FOi) = 11|- Therefore nK" + iK" = °. which will not alone suffice to determine c"' and c”'. However, making pi - o in equation (C), the result is 3^"(iK"' = o. SinceEQUATIONS ALL OF WHOSE INTEGRALS ARE REGULAR. 147 ^ is a root of multiplicity 2 only, F”($) is not = o ; hence c"' = o, and consequently c"' = o. By making fx = 2 in equation (A), we find F(2%) c"f = o, whence it may be concluded that c"' is arbitrary. In equation (B), making // = 1, we obtain 1i·^/' + i&o” — °» an<^ making //= o in equation (D), F"{%)c9ff = o ; hence c0" = o and c" = o. Assigning the value 2 to F in equation (B), the result is, since F'(2£) = o, F(2i)c" = o, showing that ¿r," is also arbitrary. The only conditions by which to determine c0 and c9' are the follow- ing, derived from (C) and (B) : F(iW = o and F(i)c0 + F'Q)c.' = o, both of which are identities, so that cQ and cj are also arbitrary. Returning to equation (A), let jx = 3. Then whence F(3iW"+ = O 0i(2i) ^ /// ^(3*) ’ ‘ In like manner, by making jx = 4, 5, . . . , the coefficients ¿r/", . . . may all be found from equation (A). Aided by these values, \ c"> · · · are t° be obtained in like manner from equation (B), . . . from (C), and cx, c%, . . . from equation (D). Thus we have an integral of the form announced containing four arbitrary con- stants ; it is therefore the general integral of the given equation in the region of the point x = o.* Among functions of the kind considered is obviously the func- tion y, defined by the irreducible algebraic equation f(x, y) = o of, say, the nih degree in y. If the n branches of the function y (ue the n roots of f — o) so defined are linearly independent, then y satisfies a linear differential equation of the above form of the nth order; if, however, there are only m{m < n) linearly independent branches, it is clear that y will satisfy an equation of the mth order. If a differential equation is satisfied by a particular root of the irreducible algebraic equation f (x, y) = o, it must be satisfied by all the roots. Since the remaining roots are branches of the one func- * The preceding illustrations of the general theory are due to Mr. C. H. Chapman.14^ LINEAR DIFFERENTIAL EQUATIONS. tion y obtained by travelling round certain critical points, and if y% were the chosen integral, it will remain an integral during all the motion of the variable, though the branch yx changes into, say, _y2, etc. Suppose the given differential equation to be of order n, and suppose that among the n branches of the function y there are only m(m < n) linearly independent; then the functions , . . . , yn satisfy a differential equation of order my and so this equation has only algebraic integrals, and the given equation has these same in- tegrals and some additional ones. The first equation, having all the integrals of the second for integrals, is said to be a reducible equa- tion. We have here the first notion of reducibility and irreducibility in differential equations, the notion being entirely analogous to that of reducibility and irreducibility in algebraic equations. This sub- ject will be taken up later on for a fuller discussion, but it is con- venient to give here a few theorems in connection with the notion of reducible equations. Suppose the linear differential equation P = o has among its integrals all of the integrals of Q = o, where Q is of a lower order than P. Since among the integrals of P = o there are functions which do not satisfy Q — o, it is clear that the equation P = o may have critical points which do not belong to (2 = o. Conversely, in spite of the fact that the integrals of Q = o are all integrals of P = o, it may happen that Q = o has critical points which do not belong to P — o. It is easy to see what the character of the indicial equation is for these points, and conse- quently the character of the integrals in the region of*the points* Suppose a one of the critical points of Q = o which does not be- long to P = o; now, remembering that the only critical points the integrals of a differential equation can have are those of the equa- tion itself, it follows that in the region of x — a the integrals of (2 = 0 must be uniform and continuous functions, since otherwise, as these integrals are also integrals of P = o, this last equation must have x — a as a critical point, which is contrary to hypothesis. In the region of a non-critical (or neutral) point, say a, of Q = o (or of any other linear differential equation), the m integrals will be developed in series of powers of x — ar whose first terms are re- spectively (x — d)\ (x — a)', (x — a)\ * · * r EQUATIONS ALL OF WHOSE INTEGRALS ARE REGULAR. 149 if a is a critical point of Q = o of the kind just mentioned, the developments of the integrals in positive integral powers of x — a will begin respectively with (x — d)ri, (x — of*, . . . , (x — ; where the positive integers rx, ra, . . . , are all different, and do not coincide with the numbers o, i, 2, . . . , m — i, and can consequently not all be less than m. It is also clear that if any one of these roots r,, ra, . . . , rm is greater than n — i, the point ;r = a is a critical point of P = o, and that the integrals in the region of this point will be developed in positive ascending powers of x — a, the first terms of the developments being as above, (x — a)ri, (x — a)r*, . . . Suppose for a moment we call these points quasi-critical points; we have then for m < n the theorem : If a linear differential equation of order n has among its integrals all the integrals of a linear differential equation of order my the?i all the critical points of the latter equation ivhich are not critical points of the first are quasi-critical points, and the roots of the indicial equations belonging to such points are all integers each of which is less than ny the order of the given equation. Among the different branches of an integral of a given differen- tial equation (uniform coefficients always understood) of order n there can, of course, exist at most only n which are linearly inde- pendent ; if, however, there are only m(m < n) (it will be assumed hereafter, unless something is said to the contrary, that m is always < n) linearly independent branches of the function, it will satisfy an equation of order m, and consequently the given equation is reducible. It follows conversely from this that the number of linearly independent branches of a function which is an integral of an irreducible linear differential equation is exactly equal to the order of the equation. It is also evident that among the integrals of a reducible equation there are always some the number of whose linearly independent branches is less than the order of the equation. Again, if a function y is an integral of a given equation, then all of its branches are integrals of the same equation. Suppose the given equation to be an irreducible one of order m; then among the150 LINEAR DIFFERENTIAL EQUATIONS. branches of y there are just m linearly independent ones, viz., j'uj'j) · · · >ƒ*· If now y satisfies another equation of the same kind but of order #, then yx, y9, . . . , ym satisfy it, and conse- quently, denoting by cx, , . . . , cm arbitrary constants, y - W + *.ƒ.+ ··· + cm?m also satisfies it. But Y is the general integral of the irreducible equation; and so it follows that if a given equation has for an in- tegral one of the integrals of an irreducible equation, it has among its integrals all of the integrals of the irreducible equation. Suppose now that we have a reducible linear differential equation of order n; it must have an integral y in common with an equation of lower order m: and suppose that among the branches of the function y there are / which are linearly independent; then / can be at most equal m. From what has been said, we see that y satisfies a differ- ential equation of order /, of which the general integral is Y- y, + + · · · + ctyt. Now every differential equation which is satisfied by y must be satis- fied by yx, jk2 , . . · , yi, and consequently Y satisfies the reducible differential equation of order n. If, therefore, a given linear differ- ential equation is reducible, there exists a linear differential equation of lower order all of whose integrals are integrals of the given equa- tion. As this last equation may be again reducible, we have that if a given linear differential equation is reducible there must, as is easily seen, be one or more irreducible equations all of whose in- tegrals are integrals of the given equation. We have seen that if an integral of a linear differential equation changes into itself, multiplied by a constant s when the independent variable turns round a critical point, s is a root of a certain alge- braic equation, viz., the characteristic equation corresponding to the critical point. If s is a simple root of this equation, there is only one integral satisfying the condition Sy = sy; but if s is a multiple root of the characteristic equation, say of order k -j- i, then there are corresponding to it k + I independent integrals.EQUATIONS ALL OF WHOSE INTEGRALS ARE REGULAR. 151 If these are regular integrals, we know from what precedes that their general form is, in the region of x = a> = 0. + 0i log {x — a) + 0a log* (x — a) + . . . + 0a log- (x — a); where 0#, 0,, . . . are functions which may vanish for x = a, and which when x turns round the point x = a change into themselves each multiplied by s. The highest exponent of log (x — a) which can enter into this group of integrals cannot be greater than k; if this highest exponent is equal to k, then there can exist but a single integral corresponding to s which contains no logarithm. This has already been seen, but it is convenient here to prove it in another way.* Suppose ya to be an integral corresponding to the root s of the characteristic equation for the point x = a; then ya is of the form ya = 4>a, o+ 0*, I log (* — a) + . . . + 0., « log® (x — a); where h 9 · · ♦ > yu · Now since to a k -|- i-fold root of the characteristic equation there can exist only k -f- i linearly independent integrals, it follows that if x turns round the point x — a, any one of these integrals, say y0, will become Sy. = y* + cxyx 4- . . . + ckyk; equating the coefficients of the different powers of log (x — a), we have Ck — Ck -1 — · · · — c\ — o, and consequently Sy„ - c0y0. We will say that two integrals whose ratio is a constant do not differ from each other; e.g., yQ and Sy0, = , do not differ from each other. Suppose now, conversely, that the given equation has only one integral in the region of x = a which, when the substitution is applied, changes into itself multiplied by the {k-\- i)-fold root s of the corresponding characteristic equation: we must have then that k is the exponent of the highest power of log (x — a) which can enter into any of the integrals belonging to the root s. Suppose y0 to be the integral considered ; then Sy0 = sy0. If k > o, the equation has other integrals than y0 which correspond to the root s; and since y0 is the only integral satisfying the relation ■Sy. = sy,, these other integrals must involve logarithms, and one of them, say yx, will be of the form y, = 0i, o + 0i, 1 log (x — a). Now we know that (f>h t can only differ from yQ by a constant factor, and of course by choosing yx properly this factor maybe made unity, and so we have lt x=y0, and consequently y 1 = 0,, 0 + y. log (x — a).EQUATIONS ALL OF WHOSE INTEGRALS ARE REGULAR. 153 Suppose now there exists a second integral in which log (x — a) enters to the first power, say ; it must have the form yt' — \, o + y. log {x — a). We have, therefore, — 01,-0; but the difference, yl — yx, is an integral of the equation, and, as it contains no logarithms, we must have yi—yx = or yl, = ctyt+yl; where c0 is a constant. It follows then that every integral which only contains the first power of log (x — a) is a linear function of y0 and yx. If k > i, then corresponding to the (k -(- i)-fold root s there must be more than two linearly independent integrals, and among them there must be one, say j/2, of the form y» = $2,0+ $2,. log {x — a) + 02> 2 log5 (x — a). It can be shown just as before that all integrals which contain log (x — a) only to the second power are linearly expressible in terms ■of y0, yx, and . By continuing this process we see that if to the root s of order of multiplicity k-\- i there corresponds only one in- tegral free from logarithms, then k is the exponent of the highest power of log (x — a) which can appear in the group of integrals be- longing to the root s. We arrive thus at the theorem : In order that a linear differential equation shall possess only a finite number of in- tegrals which satisfy the condition Sy = sy (where S refers to any critical pointy and s is any root of the corresponding characteristic equation), it is necessary and sufficient that the roots of the characteristic equation corresponding to the critical point in question shall be all dis- tinct , or, in the case of a root of order of multiplicity k -f- i, that k shall be the exponent of the highest power of the logarithm in the integrals of the group belonging to this root.CHAPTER VI. LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER,. PARTICULARLY THOSE WITH THREE CRITICAL POINTS. The linear differential equation of the second order with three critical points is the one which has been studied more than any other, and is of the highest importance, as its integrals include many of the most important functions with which mathematicians are, as yet, thoroughly familiar. Before taking up the case of three critical points it will be convenient to prove a theorem due to Fuchs* con- cerning a transformation of differential equations of the second order—a transformation which does not apply to equations of a higher order. As shown in the last chapter, the most general form of a differential equation of the second order with p finite critical points and all of whose integrals are regular, is (I) I Pp) dl _L AM . = o· de ^ at) ~dt-r 4'\t) where W) = . . . {t — tp), and where Px(t)y P£t) are polynomials of the degrees p — I and 2p — 2 respectively. Fuchs proves that such an equation can by a change of the dependent variable z be thrown into the form (2) rn + <2,(0 % + my = °; where the degrees of tp, Qx, and Q,2 are p, p — i, and p — 2 respec- tively. * Heffter : Inaugural-dissertation. Berlin, 1886. 154EQUATIONS OF SECOND ORDER—FUCHS'S TRANSFORMATION IS5 Suppose Ti a root of the indicial equation corresponding to the critical point ; also suppose that r,· is not zero. The indicial equa- tion is (3) r(r — i) + r 5(A) + m) no = o. Make the substitution (4) * = n(t — t,)sÿ ; (* = 1, 2,. . . , p;) equation (1) becomes now (5) I, 2, P i] ë - («+!)(/?+1) $ = o ; dy or, xi Tx = n, (I3) x(l-x>æ+b+l-ia+i+fi+i+i>i% (“+!)(/*+ 1)7 = Pr which is of the same form as (12), and differs from it only in having a 1, /? -f~ 1, and y -f- 1 in place of a7 /3, and y. It is obvious that dmy when = p we shall have (14) x(\ - x)*£+[y-\-iH+(a + M + fi + m+iy\^L — (« + m){P -\-m)v — Or differing from (12) by having a + § + m, and y + m in place of a, §, and y.\ In order to get the indicial equation corresponding to the point x = 00 in (12), transform by writings = j ; we have then (ic\ fl-L. 0+ fi - 0 — (r — 2V dy_ V T /(i - I) dt a 6 W=i)y~ o. The three indicial equations and their roots are now, from (12) and (15): (16) For x — o, For x = i, Forx= 00, { I I r(r -1) + yr = o, r, = O, = i - y; r(r — 1) — — a — /Î — i) = O, r, = O, r, = y — a - /3 ; r{r — 1) — r(«+ fi — i) + aft = o, r, = a, r, — /?.EQUATIONS OF SECOND ORDER—THREE CRITICAL POINTS. l6l In the region of x = o we have corresponding to the root rx = o an integral of the form y* = C. + C,x + + . . . ; substituting this in (12) and equating to zero the coefficient of we have \k(k - 1) + k(a + fi + I) + afi\Ck = [>(£ + i) + ^ + I)]G+I > giving (i7) - __ (<* + ^)(/? + i+I (I + ¿)(r + A) * and so, making the arbitrary constant CQ equal to unity, we have for yx the value (18) I + « . p a{a -|- W -f~ 0 * ! I.2.7(y+I) ^ ‘ ” or, employing the usual notation for hypergeometric series, yx = p, y, x). We have seen above that there are six ways of transforming the differential equation in question into an analogous one when the independent variable is changed by the relation (19) au 4- p x — ----, yu ô and corresponding to each of these there are four transformations of the dependent variable of the form (20) y — x\x — 1)^; where X and ju are respectively roots of the indicial equations corre- sponding to x = o and x = 1.IÓ2 LINEAR DIFFERENTIAL EQUATIONS. The critical points of (12) are o, 1, 00, and the six different forms of (19) are given by the following table: (21) X — 0, I, 00 X = u u = 0, I, 00 X u u 0 00, I li — I X = I — u u I, 0, 00 X u — I u T 00, n u u X I u 0 I I — u X == I u u = 00, I, 0 111 IV VI As an illustration of these transformations take the case of u — 1 x — —~—. The critical points corresponding to the original ones x = o, x — 1, x — 00 are respectively u — 1, u = 00, u = o, and the roots of the indicial equations are: For u — I, r, - 0, i — y ; (22) -< For u = 00, r, = 0, rt — y — a — fi'■> For u — 0, rx = a, r.> = A These values are written at once by aid of what precedes. The transformed equation—for which we have, however, no use—is (23) d*y \ —1 — —Y— I)u dy di? uiu — 1) du a/3 u*(u — l)^ — o. — tiy-z, Make now the transformation <24) y — u\ 1EQUATIONS OF SECOND ORDER—THREE CRITICAL POINTS. 163 X and pi being roots respectively of the indicial equations corre- sponding to u = o and 8=1. Referring to the formulae (or), we have \ — a or fi, pi = O or I — y. Suppose now we choose X = o[y ¡1 = 1 — y; the formula of transformation is then {25) y — u\ 1 — u)x ~ iz. From (a') we have at once, for the roots of the indicial equations belonging to the (z, u) equation, U — O ; r, = O, r* = fi~ a ; (26) X u = I ; r, = O, r* = Y - i ; 2/ = £ 1 + « II c 8 r* = 1 - A The C®’» «) equation is / 2 7^ 1 ^ - a—1)——ot — $)u dz —1)(«- -r+ \z7) du* u(u — 1) du u{li — •I) giving at once the roots in equations (26). Write a' = a — y + I, ft = I — A y = a — + I ; then this equation has in the region of u — o the integral 2 = A, y\ u). Transforming back to the variables y and x, we have, on neglecting a constant factor which is merely a power of — 1, (28) y — x~a[l — p[a — y + I, I — A « — /?+!, Proceeding in this way, it is clear that we can form 24 particular integrals of (12).LINEAR DIFFERENTIAL EQUATIONS. 164 A full discussion of these functions is contained in the follow- ing chapter, which is a translation of a Thesis by M. E. Goursat; the reader is also referred to another memoir by the same author in the Annales de VEcole Normale for 1883. In the following table of the 24 functions we must make some supposition as to the values of the different powers of jt, i — x, 1 — which appear as factors: suppose we choose the values such that the different powers x, 1 — x, 1------shall reduce to unity when we make respec- tively x = I, H II p H II 8 I. n a, fi, y, x). II. 0 - *)■>-*-f>F(y -a, y - fi, y, x\ III. (I - x)—f(oc, y fi, y, ^ _ ,) . IV. (I -x)-^fi, y-*,y, V. X1 ■ ~yF(a — y-\- 1, fi — y + i, 2- - r> x). VI. Xx -v(i — — a, I — fi , 2- - y, XY VII. Xx _T(i — x)y ~ a~ *F^a — y i> i - P — vjirò- Vili. Xx -*(i — x)y ~ p ~1F^fi — y-\- I, I — ot, 0 IX. H a, fi, a -+- fi — y -|- i, i — x). X. X1 ~yF{a — y+l, a-\- fi- y + i. , I - -X). XI. x~ aF{<*> or-y+i, a + fi — y + I, —)· XII. x~ PF[fi, fi—y + i, ff+ fi— y-{- i, ~L> XIII. (I — x)y « PF(y — a, y — fi, y - — a - fi+ I, l—x)- (1 - a, i-fir y — a— fi-\- i, i-x). XIV.EQUA TIONS OF SECOND ORDER—THREE CRITICAL POINTS. l6$ XV. (i — xp—— a, y — a, y — a— /? —I, ~~~)· XVI. (i — x)y~ a~P xP-y_F(^i — 0, y — 0, y — a — /S + I, ~~~)· XVII. x-*F(a, a — y-\-i, a — 0I, ^). XVIII. x-*(i -l)V'a_V(i -0, y-0, a-0+1, i). XIX. x-*{\ — .^(a, y — 0, a - 0 + I, xx. i,(a-r+i.i-A“-/»+I.7r;)· XXI. .r - /? — j'-j— 1, /? — a-}-1» ~)· XXII. ^-^(1 -^)Y_a'V(i -a, y-a, /J-«+i,^)· XXIII. *-f(i r-«, /?-*+!, XXIV. ^-^(1 — V(/?— ^4-1,1 The general condition for the convergence of the series F(a, ft, y, g) is mod. g < 1 ; from this we can see at once what are the regions of convergence for the above series of functions, viz.—(1) : for the argu- ment x the region of convergence is the interior of the circle of radius unity, whose centre is at x — o; (2): for the argument — the region of convergence is the entire plane outside of this circle; (3): for the argument 1 — x the region of convergence is the interior of the circle whose centre is at the point x = 1 and whose radius is unity; (4): for the argument --- the region of convergence is all166 LINEAR DIFFERENTIAL EQUATIONS. £ of the plane outside of this circle; (5): for the argument op- pose x = £ -f- irjy and make £ = J, then mod. = mod. £ + iy — £ + *v = I, and so the region of convergence is all of the plane to the left of X — I the line £ = £; (6): for —the required region is obviously all of the plane to the right of the same line. The dissection of the plane which is necessary in order that the above functions may remain uniform while the variable travels all over the plane without crossing any of the cuts is obviously affected by drawing a cut from o to — 00, and another from -f-1 to -f- 00. It is now easy to see that the above 24 functions divide into six groups of four each. In the region of the point x — o the indicial equation has for roots o and 1 — y ; now the functions I to IV inclusive are uniform and convergent in the region of the point x = o, and they reduce to unity for x = o; there can, however, be but one integral in this region possessing this property, viz. the integral yi = -F(«> A y, x) belonging to the exponent zero; therefore each of the integrals I, II, III, IV represents the same function. Again, the functions V to VIII inclusive are the products oi x1 into expressions which are uniform and convergent for the same region x = o, and which reduce to unity for x = o ; they represent therefore the one regular integral, say y9, corresponding to the root I — y of the indicial equation for x = o. The expressions IX to XII, and XIII to XVI, represent in the same way the two regular integrals, say y% and yk, corresponding to the point x = 1; and XVII to XX, and XXI to XXIV, represent the two regular integrals, say ys and y&, belonging to the point x = 00. It will be convenient to insert here a direct determination of the integrals of (12) which belong to the critical points o, 1, and 00 respectively, and to show what conditions must in each case beEQUATIONS OF SECOND ORDER—THREE CRITICAL POINTS. 167 satisfied by the constants a, fi, y9 in order that the integrals may be of the forms sought. Writing equation (12) in the form (29) O2 - x) - [r - («+ P +OO - aPy = we have as above, for the roots of the indicial equations: For x = 0, rx = O, I II For x — 1, rx = 0, II 1 ft 1 For x — 00, rx = o', ^ =P- When y is not a negative integer we have in the region of x — o the regular integral yx = F{a9 y9 x), where F{ay /3f y, x) is a convergent series inside the circle whose centre is x — o and whose radius is unity. Inside this same circle there must exist another integral belonging to the exponent 1 — y; to find it, write y = x1 - y Yj and we have for Y the equation (3°) (*’ -*)^-[2-y-{a+fi-2y + 3)*] £ - (« - Y + l)(P — Y + 0 y = O; which differs from (29) only by having af ¡3, y replaced by a — y + 1, /3 — y 1, 2 — y. When 2 — y is not a negative integer, equation (30) admits the solution (30 Y = F(oc — y + I, P — y + J, 2 — y, x),i68 LINEAR DIFFERENTIAL EQUATIONS. and consequently (29) has the solution (32) y, = X1'yF{a - y + I, fi - y + I, 2 — y, x). It follows then that if y is not an integer, the general integral of (29) is, in the region of x = o, (33) y = Cyl + C'yt, where C and Cr are arbitrary constants. For the region of x = 1 make x = 1 — £ ; equation (29) becomes now (34) ¿3 J3> cKyKy c6y&, c6y6; then it is clear that we must have the relations , . j + Bc,yt; ( chyh = Ec,y, + Fctyt; '37) 1 ctyt = Cc1yi + Dcjt; U M c,yt = Gc^+Hcj,; where the constants A, ... , H have now to be determined. Suppose x to describe a circle of radius unity around the origin; the integral yx will not change, but y2, yb, y6 will be multiplied respectively by ~ ?>, e~2,r/a, e~2nip , and equations (38) become (39) ( *~™''acbyt = Ec,y, + ~y)c,yt; 1 Gc.j\ + He™iuLINEAR DIFFERENTIAL EQUATIONS. 170 Now in equations (37) make x = o; also in (37), (38), (39) make* x = 1, and we have the following eight equations for the determina- tion of the constants A, . . . , H: (40) ' clyX = AclyX + Bc,(y,)„ ; ct(yX = Cct(y,\ + Bc,(y,\ ; f.(= Ac>(yX 4- BclyX ; **00. = CcXyX + DclyX ; • Xj'X = £ct(yt\ + Fclyù ; *.(ƒ.), = -y)^^ ; [ “*<:.( A). = ƒ,). + ~ Y) 4a)> · (4i) The values of (y,)0, , (yi)1 are known to be (yX — 1; (yX = o; ( v\ — r (a+/3-y+i)r(i-r). /\ _ r(r-<*-P+i)r(»- VJV# r(ytf-^+i)r(it-x+i) ’ '-.TV· — r(i _ a)r(i - /5) (a). (A). = 1 ; (a), = O; r (1 - /jjrjy - ft) ’ (yX— r(i - a)r(r - a) / \ r(y)r(y-a — fi) V J iA p _ a^p (y _ jjy r(2 — x)r(x — a - fi) T(i ~ a)r(i -/?) ( — r(o--/?+i)r(r-nr~/g) # _ r(/3-a+i)r(y-a-/3) These values introduced in (40) give us the means of determining the constants A, H in terms of the constants cx, . . . , ctr exponentials and T1-functions. A choice of values must now be made for the constants cx, . . . , c6; if we make them each equal to unity, then A, . . . , H are determined as functions ‘of the expo- nentials and the T-functions. A still simpler result is obtained by writing (42) -{ _ r(a)r(y -a) _ T(l - fìr {fi - y + l). />) ■’ i '(2-r) _ r(a)r(0 - r + 1) _ r(i - fir{y-_fi. C% r(a + /? - y + I); e* - r(r - a - § + 1) ’ r(a)r(i-fi r(p-y+T)r(y-a) f‘“r(«-/3+'i)’ c*~ r(/8 —a+i)EQUATIONS OF SECOND ORDER—THREE CRITICAL POINTS. J/I Equations (40) are now (43) f 7»r(i -y) r(a)T(r ~ a) r(a-y+ 1) r(y) ' ’ r(l — y)r{y — a) _ cr{a)r{y — a) A r{ 1 - «) r(a)r{0 — y+i) r(a+/3-y+i) r{y — a — fi)r (a) r{y ~ P) + B o=C = E r{y-a- P)r{a) r(y - P) r(y — a — P)r(p^y + 1) r{I - «) - * - /?)/» r(r - /?) — a — p)r (P — r+ 0 r( I — «) r(K — a — P)r(a) r(r - P) r(v -a- P)r(P - • r+ 0 r(l - a) If we divide each of these equations by the product of the /^-func- tions which appear in each numerator and then employ the relation r{p)r{i-p) = -Tt sin p7t' we have finally sin (y — oi)n = A sin yn ; sin an = C sin yn ; sin (y — a — 0)n — A sin (y — fi)n -f- B sin an; O — C sin (y — fi)n -f- D sin an ; (44) ^ sin (y — /3)n = E sin (y — f3)n + F sin an ; sin an = G sin (y — fi)n -f- H sin an; g — nr ¡a sjn (y= E sin (y — /?)# -f- Fe2iri^ ~ ^ sin ^-27T^sJn _ Q sjn ^ _J_ //^2TTZ (1 - 7) gjn172 LINEAR DIFFERENTIAL EQUATIONS. npi , ese give A, H in terms of sines and exponentials only, viz.: A = sin (r ~ «)*. (45) £ = C = sin yn sin(y-a—/3)n- sin an L sin an sm{y—a)ns\n(y—0)7r~^ sin yn sin yn £ __ sin an sin (y — /3)n t sin an sin yn E = E=z G = — y) ____ 2ir/a ¿2iri{i -y) ____ J £- 2nia _ j gjn ^ ¿,27r/(i - y) _______ J I sin an sin an e- 2iri'P — j sin (y — fi)n sin (y— fi)n e21ti(l ~y) — i ’ e - 2ÌT//3 ___ J e2iri{i - y) — j ’ These forms have been taken from Jordan; they are to be found in a number of places, e.g., Forsyths Differential Equations (here in a somewhat modified form) and Goursat’s Thesis, above referred to. If y (and therefore also i — y) is an integer, then one of the integrals in the region of x = o may contain a logarithm; so if y — a — /3 is an integer, or if a — ft is an integer, a logarithm may appear in one of the integrals belonging to x = i and to x = oo. It has been shown in general that logarithms enter in such cases as these; they may, however, fall out even when these conditions are satisfied. This question will be entered into later. A single illustration of the . entrance of logarithms will only be given. We have already seen that if the two roots of an indicial equation are equal, then a logarithm must necessarily appear. Sup- pose now that we have y— i, and consider the region of x=o. The roots of the indicial equation are now each zero, and the dif- ferential equation itself isEQUATIONS OF SECOND ORDER—THREE CRITICAL POINTS. 173 (46) x(x - I) ^+ [(« + /?+ l)x~ i] afy = Q one integral of which is F(a, /?, 1, x), — say, A0 + Axx + A^x1 r when for brevity we have written 4 __+ !) · · · (a + & — (fi + 0 · · · (fi + & —0 k ~ F\ For convenience write 2> = F{at, /?, i, x) ; then a second integral of (46) is of the form y = 0 + 0 log x, where 0 is a uniform and continuous function of x in the region of x = o. Substituting this value of y in (46), and equating to zero· the aggregate of terms which do not contain the logarithm as factor, we have for 0 the equation (47) x(x — x) + [(^+/^+I)jr“ 0 + afi . d = - 2y(2a + 2)8 + I) — 4«/?, -? = 2^’ — / = + A g = — 2afi{2y — I). It remains now to find all the cases where this last differential equation admits the solution If n denote the degree of the function, the conditions are found to be as follows: (A) n even : 2/ X2d*z . , v/ , ,^d*z x (I ~ x) ~d? + ^ dz where a — — (« + p + 0» z = integral function of x. 2 1 3 2 ’ • » 2n — 1 2LINEAR DIFFERENTIAL EQUATIONS. i y 6 (B) n odd : n ii (1) a — —, Y =-, , N „ n i i (2) /î=--, r = 2’ ~2’ (3) a-\- ft = - n, Y = 3 n — 2 2’ ■ _ 2 j ft, fi~ I, · · · j it 1 n « 1 3 n — 2 2’ ’ — 2 1 a — I, . . . n — i 2 r I 3 n — 2 2’ “ 2’ • · · > 2 ’ n 2n — i In his second note * (closely allied to certain investigations by Schwarz) Markoff seeks to find all the cases where the same differ- ential equation of the second order admits an integral of the form Xÿ'+Yÿy + Zf, [y' = %) where X, Y, and Z are rational integral functions of x. The con- ditions found in this case are as follows (n an integer): (0 a + ft = — w> Y — h — h — 1» · · · , — » + i, (2) a + fi = n, y = h h · i, (3) « + ft — 2 y = - n, y = h h · · • , n — i, (4) oc + ft- $f II M 11 -i, - h · · · > " - n + if 2 a — y ) (5) - ■ = — n, a + fi- II 1 gH ~ f » · · · > — n + Ì r 2 ft — y ) 2a — y ) (6) or \ = n, a + 0 — y = $, f, . . . , n — 2ft — y ) * Math. Annalen, voi. xxix. p. 247.EQUATIONS OF SECOND ORDER—ABELIAN INTEGRALS. 177 a — ft + Y ) (7) or > = — n, y = h — h — h · · · . —»+ £; P — <* + r ) a — 0 + y ) (8) or > = n, y = f, /* — « + r) An interesting case of the differential equation of the second order has been studied by Dr. Heun in th Mathematische Annalen* If ,· - · > £P are the finite critical points of the equation, and if ip(x) = (x — 4;,), (x — (x) Heine has shownf that we have (55) f{x) y=p— 1 ƒ lv + i ( =*c'l dz z) ip(z) -Z; where Z is a rational integral function of x, and Cl9 C3, . . . , Cp _ x are constants determined by aid of a certain system of algebraic equations. This last equation can be written in the form (56) _r2 - vyx + E, where V and E are functions of x satisfying the relation * Heine : Handbuch der Kugelfunctionen, Bd. I, § 136. f Crelle, vol. 61.EQUATIONS OF SECOND ORDER—ABELIAN INTEGRALS. 179 viz., from (56) we have (53) and from (54) <59) d (y. ,\ dV d fE\ dx\y, J - dx + dx\yj ’ '\ _ e dx\y J- - ' 2 * equating the right-hand members of (58) and (59), we have (57). This result of Heine’s is, of course, restricted by the condition imposed upon Qt(x), viz., Starting from equation (57) we can, however, determine Vand E in the general case. Write (60) yr - r = r> 2> · · · . P- Since the polynomials Qipc) and *p{x) are respectively of the degrees p — 1 and p, we have (61) QS.x) _ r, 1 y, . . y, >p{x) x — Zx x — S, * ' ' x — ' Write, again, (62) (x — S,)Mx — ■ ■ ■ {x — Stf* = 6(x); now in (57) we can make (63) V =ƒ ; where Z(n) — N{n) is the nth convergent of the development of W in continued fractions, and R{n) is the corresponding remainder. We can now write (67) W = fl(-r) P 4>{x) *p(x) J &(x) dx, ZW=Z, N^—yx, R* = f{x) if we can establish the following conditions : I. That W is developable in a continued fraction of the form W = C. «0 + b.x + C, + K* + C, at+b^ + . . . . which, for a given value of n, is regular up to, and including, the nth partial denominator. II. If there exist as many values of rf>{x) as there are rational solutions yx of the differential equation, viz.,. (n -f i)(« + 2) . ■ ■ (n -f- p — 2) 1 . 2 ... (p — 2) III. If the function R{n) ii*) 6{x) is a second solution of the differ- ential equation.EQUATIONS OF SECOND ORDER—ABELIAN INTEGRALS. l8l To show that W is developable in a continued fraction of the form in (I), it is necessary first to show that W is developable in a convergent series containing only negative powers of x. Write i\x) 6{x) — il(x) ; then, by a known theorem of Abel's,* <68) (z — x) 0(x) = m n ^ S \ Wl-\-l ^(w-)_„+I)_______________________________ m —o n — o n-|-i } Jm,m where T ___ p xax p Jm>n ~ J « J - U(z) 9 6(x) H(x) d£l (x) dx dd(x) dx ¿(o) _|_ a . . + à(p - *)x* - *, d°> -f ¿l)x + - . . + c(p “ ~1. The roots of the equation 6{x) = o are 5,, -1) nSp), and consequently the function W is a sum of entire Abelian inte- grals of the third kind.EQUATIONS OF SECOND ORDER—ABELIAN INTEGRALS. 183 We proceed now to the complete determination of the function (x). (One coefficient only in ' Dropping the x when it is not necessary to write it, we have from these last equations or giving or (76) -R(K) 0 = y> ftdx ’t 7(«) 0* y 0 R{n) Z<"> e · y, = J 0 d 6 ' y, ' d P R—

must drop out. Writing now = h0xn + X ' X x • + ~+ 1 xr 1 we have at once the following equations of condition : r #1 hn “I” aJln - i + · . . -f- an -f i ^0 — 0 ; (78) H ajin ~"b ^3fon -- i + · • · “j“ +2^0 — O » ^ &n + p- 2 fon — i fon -1+ · • · “j~ -j-p — 2 foo —— O· The quantities a have the form ar — a{l)Cx + a{2)Cz + . . . + *(p~i>Cp-x; where the coefficients a are again linearly expressible in terms of the primitive moduli of periodicity of the Abelian integrals which enter into the problem. From the first n of equations (78) we can now ex- press the coefficients hoy foxy . . . , hn as functions of these moduli, and of the ratios cx cx cp_,’ cp_,’ c„- 2 ca and consequently also as functions of the ratios *0 3 Introducing the found values of h0, k19 . . . , fon in the remaining p — 2 equations of (78), we have a system of equations giving (n + i ){n + 2) ...(» + p — 2) 1.2.3 . . . (p — 2)EQUATIONS OF SECOND ORDER—RIEM ANN S P-F UNCTION 185 groups of values for the ratios and, from equations (72), the same number of groups of values for the ratios of the constants C. The results of the investigation may be summed up as follows: If the regular differential equation ^)g+e,w£+aw/=o has a rational integral function of degree n as its first solution, then this function is the denominator of the nth convergent in the de- velopment of the function w=2c, f*nm. /=I Jii (* — *)* z)f(z) dz in a continued fraction, where »/’(■?,) = O, 6(z) — {z ■ £?i(£i) (?i(£a) £,) - £,) and where the constants C are so determined that the first n partial denominators of the continued fraction are linear, while the(n -(- i)st tb(x) is of degree p — 1. Further, the remainder multiplied by is a second solution of the same differential equation. In particular, if the quantities om MS*) are proper fractions, then the second solution, ya, contains no higher transcendents than entire Abelian integrals of the third kind. It is not the intention here to go into any extended account of the differential equations which give rise to Spherical Harmonics and the other allied functions of Analysis, but it is interesting to show how these equations are connected with the differential equa- tion satisfied by Riemann’s P-function. (We will use this form of the186 LINEAR DIFFERENTIAL EQUATIONS. letter P to denote Riemann’s function, in order to avoid confusions when the P of ordinary zonal harmonics has also to be employed.) Riemann’s definition of the P-function is as follows: Denote by r a b c P a fi y x . ol' fi' Y' - a function of x satisfying the following conditions : (I) For all values of x other than a, b, c, the function is uniform- and continuous. (II) Between any three branches of the function, say P', P", P"'r there exists a linear relation with constant coefficients, viz.: c? P' + c" P" + c?" P" = o. (III) The function can be put in the form cTPT + ¿VPT' (r = a, a'f fi, fi'y y, y') where the products T*T{x — (t)~t , ~PT'(x — o-)-T' (o* = a, b, c) are uniform for x = a\ 6y fi', y, y' must be real. The first columns of the function " a b c P - a fi y x - fi' Y* - can be interchanged among themselves, also a can be interchanged with a, fi with fi\ and y with y, y'. Further, let x, = ex+f. gx + h’EQUATIONS OF SECOND ORDER—RIEMANN S P-F UNCTION. 187 where the constants are such that for x = a, b, c we have x' = a\, b', cr; then " a b c 1 " a' b’ c' + a /3 r *\- a fi Y * . a! fi' ÿ J _ a fi' Ÿ It is shown by Riemann that all the P-functions having the same values of a, aJ3, ft, y, y', can be led back to the function O 00 I a fi y af fi' y' which is denoted more briefly by the symbol P fi fi' y yr In such a function the quantities in the exponent-pairs (a, a), (ft, ft), (y, yf) can be interchanged, and the pairs themselves can be interchanged if we replace x by ex +f gx-\-h' which for the first, second, and third exponent-pairs gives for the variable the respective values o, oo, i. In this way the functions P can be expressed by P-functions of the arguments188 LINEAR DIFFERENTIAL EQUATIONS. and having the same exponents, but having them arranged in dif- ferent orders. From the definition, it is easy to see that we have " a b c - a ft y x lx — a\« P 1 II ►d . «' P y' . a b z3 + we have (84) A = 3 — 2, = I — a — a'4-i_ /? — ^'4- 1 — y — yf. Multiplying the first of equations (82) by ay the second by b, the third by cy adding, and observing that a\b — c) 4- b\c — a) c\a — b) {a — b)(b — c)(c — a) we have — (a b c), (85) B=-[(i-cc-a')(b+c)+(i-p-p,)(c+a)+(i-y-y,){a+b)]. In like manner we find for C the form (86) C — {1 — a — ar)bc 4~ (1 — ft — ft')ca (l — y — y')ab. Multiply now equations (83) in order by (c — a){a — b)f (a — b)(b — c), (b — c)(c — a), and adding, we get the value of F; multiplying again by (c—d){a—b)(b-\-c), {a-b){b—c)(c-\-a)y {b—c)(c—a)(a-\-b)yEQUATIONS OF SECOND ORDER—F IE MA N N S P-FUNCTION. I9I and adding, we get G ; finally, multiplying by (c — a)(a — b)bey {a — b)(b —c )cay (b — c)(c — a)ab, and adding, we get H. The values of Fy Gy H thus found are ' F — — {aa\c — d){a — b) -f- — b){b — e) +yy\b — c)(c — a)}, G = 1 aa'(c — a)(* - b){p + c) 4- Pfi'ia — b)(b — c)(c + a) i?) i +yYXb-c)(c-a)(a + b)\, H — — {aa\c — a){a — b)bc -{- — b){b —c )ca + yy\b-c)(c-a)ab\. The values of Ay By . . . , H from (84), (85), (86), and (87), substi- tuted in equation (79), give us for this equation the form (88) or d2y dx2 + (1 — a — a')[x2 — (b -f- c)x -[- be] + (1 ~ /? — — if + *)* + ca] + (i - y - y ){x2 - (a + b)x + ab] (x — a)(x — b)(x — c) a a (a — b)(a — e)[x2— (b c)x -f- be] + — e)(b — a)[x2 — (e -j- a)x -f- ea] + yy\c — d){c — b)[x2 — (a -f- b)x -f ab] (x—a )\x — b)\x — e)2 dy dx y — o; {x—a)(x—b){x—c) j aa'(a-b)(a-c) /3/3'(b—c)(b—a) x—a ' x—b YY\c—a){c-b) X — c y — o. This form of the differential equation satisfied by the P-function was first given by E. Papperitz ; * Riemann has given indications as * Ueber verwandte s-Functionen, von Erwin Papperitz. Math. Annalen, vol. xxv. p. 212.192 LINEAR DIFFERENTIAL EQUATIONS. to the formation of the equation, but does not seem to have worked out the form. In (79) make x = j ; the equation then takes the form / x d2y , (2 _ A -[- Bt -f- Ct2______________) dy dt2 ' ( t t( 1 — at)( 1 — bt)(i — ct) ) dt F+Gt + H? , ' (1 — at)\ 1 — btf(\ — ay y The coefficient of y does not contain t in the denominator as a dy factor; the coefficient of -7- is at 1 2(1 — at)(i — bt){i — ct) — A — Bt — Ct2 t (1 — at)( 1 — bt){ 1 — ct) substituting for A, B, C their above-found values, we have for the numerator of this coefficient the value 2(1 — at)( I — bt){ I — a) — (3 — a — a'— /Î — /?'— y — y') + [(I ~a-a'){b+c)+{i -p—p'){c-\-a)+{i —y—y)(a+by]t — [(1 — a — a')bc -f- (1 — fi — /3')ca -)- (1 — y — y')ab]t\ Now remembering that a a' -f- -)-/3' y y' = 1, we see that this last expression is divisible by t, and consequently that the dy coefficient of -jj in (90) does not contain t as a factor in its denom- inator. It follows then that t — o is not a critical point of (90), and consequently that x = 00 is not a critical point of (88) (or (89)). The points a, bf c are then the only critical points of our differential equation. If in (88) we write *=_I, b = ~, + a = a' = y — yf = o, ^ — fly fir = n + I, lim e = o,EQUATIONS OF SECOND ORDER—RI EM A NN* S P-FUNCTION. 193 we find, after simple reductions, the equation for zonal spherical harmonics, viz., (90 d*P 7.x dP , n(n+i)„ dx* 1 — x1 dx + i — x* P — °‘ Again, writing __ T I CL — 6, u — —, C -— e 7 a = v, a' — — v, 0 — too, 0' = — too, Y = iO + — 40^), ?/ = £(1 — V1 — 4&32)r lim 6 = 0, lim a? = 00, we have the differential equation for Bessel’s functions, viz., (92) dx* · X dx^\ — o. If, in space of three dimensions, we consider two points, one at a distance unity and the other at a distance p from the origin, and if co denote the angle between the lines drawn from these points to the origin, we have for the reciprocal of the distance between the points (93) ^ = (1 — 2p COS CO -f- p2)-*. K The ordinary zonal harmonics are the coefficients of the different powers of p in the development of (93) in ascending powers of p, viz., (94) I = p0 + piP + Pj? + . Writing cos go = xy these functions Pn satisfy the differential equa- tion d2y dx2 2x dy , 1 — x2 dx )dmJ is the associated function of the first kind of degree n and order m. This function, as is well known, satisfies the differential equation (96) d2y ~dx2 2X I — X2 dl_ _1_ ”(«+ — *2)— nil dx ( i — x2y y The second integral of this equation is the associated function of the second kind, and is denoted by Q„tm. If instead of n being an inte- ger it is of the form then n — — i, n + 1 = l·1 + h and (97) becomes , ON d*y 2X dy I (M2 - j)(i — X2) - rn2____ te8) dx* i — x2 dx^~ (1 — xl'f y which is, Basset’s Hydrodynamics, Vol. II. page 22, the differential equation for Hicks’s Toroidal Functions. The spherical harmonics for the cases above considered present themselves in the theory of attraction in space of three dimensions ; we will speak of them as being of rank 2. In considering the theory of attraction in space of k 1 dimensions, we get spherical harmonics of the rank k; these harmonics appear as the coefficients of the powers of o in the de- velopment of * _ k — I (1 — 2xp + p2) 2 ·EQUATIONS OF SECOND ORDER—TOROIDAL FUNCTIONS. 195 The differential equation of these functions is known to be d'y kx dy n{n-\-k— i)(i — x') — — 2) <99) d?~T=*di +--------------------------------y=°- * Por k — 2, let x = cos —, and then for an indefinitely increasing value of n we get the Bessel’s functions, r 2* ( «\ /- = lim —=P«.m \cos -1, * = 00 v rnt x n 100) ·{ K, - lim 2-“-’Vff V n Q, (cos3; the differential equation for which is P„ + x are connected by the sequence relation (106) (2n + 1 )Pn+* — AnCP« + (2« — i)Pn -1 = 0. By aid of this equation Hicks finds the value of Pn as follows : In (106) write n (2n — 2){2fl — 4) ... 2 “ — (2n — i)(2« — 3) . . . 3 . 1 “* ’ with = P, = «1; then „ , (2tt + iV Ux + 2 - 2Cu„ + l + 2n^2n _|_ 2-j «. - ° , or, writing — (2n "t" I)2 _ (2n *)a _ T 1 L( 1___________L_____\ n ~ 2ti(2n + 2y ~ (2n + i)a — 19 2 \2n 2n + 2) 9 £0 — \ j Un + 2 = 2Cun + i — cnun. andEQUATIONS OF SECOND ORDER—TOROIDAL FUNCTIONS. IÇ/ It is clear from this that un is of the form where an, fin are rational integral functions, algebraic functions, of 2C; an of degree n — i, and fin of degree n — 2. Writing 2C = ty we have For, supposing an of this form, i.e., wanting every other power of /, it follows at once that aM+1 is of the same form; it is seen above that ¿r3 is of this form, and so the statement is generally true. The same remarks apply to the function /3n. Now an satisfies the equation (109) an = tan _ f — cn _ 2an _ 2; with a0 = o, ax = 1. Hence, substituting the above value for any we must have (I I O) &nr — - i, r - 2 - 2, r - 1 » {10;) = tut — ; ^3 = (¿* — C^UX — . We can now show that an, /3n are of the form (l08) an — anotn 1 + CLnx t* 3 + t* 5 + · · · ** ” ^ “ 1 + . . . also, (III) Hence (I I 2) anr — {fn - 2 &n - 2, r - 1 —{” From this we have i + · · · + ^2r - i ^2T - I, r - 1)· — “ (^* - 2 + Cn - 3 + · · · + ^i) Î 0*2 = - 2 (fn - 4 “l· * · * + ¿0 + Cn - 3 (ƒ* - 5 + * * * + Cl) + * * * + C*€\ *I98 LINEAR DIFFERENTIAL EQUATIONS. From this last we observe that an2 is equal to the sum of the products cfy taken two and two, with the exception of all products where the subscripts are successive. Assume that (—)ranr = sum of products of the c s taken r at a time up to cn_2, excepting such products in which successive sub- scripts occur. Then an,r+i = (—)r+I|^„-2(pr°d. up to cn_4, r at time, etc., . . . ) + '-3( “ “ 4.-5 “ “ ··· ) + · · * · · · ] = (— )r+Ijprod. (r + 1) at a time up to cH_2 without succes- sive subscripts. \ Whence by induction the assumption is seen to be universally true.* It may be thus stated : anr is the sum rat a time of the terms 3* 5a 72 {2n ~ if 2.4’ 4.6' 6.8’ ’ {2n — 2)(2n — 4)9 all products being thrown out in which, regarding the numbers in the denominators as undecomposable, a square occurs in the de- nominator. We have ^«o — I 5 __ (4 n — 3 X« ~ 2) 4(» - 1) ' This result is of very little use for application. If the coefficients an are needed for particular values of t, they can be very rapidly calculated by means of equation (109), while if their general values are to be tabulated, equation (no) will serve to calculate them in succession. Further, 2fin is the same kind of function as an in every way,, except that it does not contain cx; in fact, 2/3n is the same function of c2i c2, . . . , cn _ 2 that an _ x is of cy, c2, . . . , cn _ 3. Calling this afn -1 > we can then write Un — n - 1^0 ·EQUATIONS OF SECOND ORDER—TOROIDAL FUNCTIONS. I99 The functions n0 and nx are expressible as elliptic integrals, viz.: Writing 2 S i _ b'* — ____ ____ f · A3 I 7/2 _ X * - C + S’ ~~ (C-f S)a’ ^ ^ or P — 1 — e~2*, k/2 = e~ 2119 and t — 2C — k’ -j— -jT j we have ”0“7o i/c^s = 2 V~kr de S cos 6 J Vi — Æ2 sin2 d «y o = 2 VI'F ; (i 13) n, = J VC- Scos0.d0 = -^=J Vi - J? sin1 0 . dd Vk' E. Hence (II4) Pn = 2 (2fl — 2){2U — 4) ... 2 (2n — i)(2n — 3) ... 1 J^E-kVy.aV,.^. For n — o, Pn — n; for n — oo, Pn = oo. By very simple operations the second integral Qtt can be shown to have the form (I I 5) Qn = 2 (2n — 2) (:2n — 1) Fr and Ef being the complete elliptic integrals of the first and second kinds with modulus kf. The functions Pn and Qn thus found can obviously be put in the forms of the hypergeometric series. For further information on the Toroidal Functions the reader is referred to Hicks’s memoir, and also to Basset’s Hydrodynamics, Vol. II.200 LINEAR DIFFERENTIAL EQUATIONS. Chap. XII. It is interesting to derive the above results directly from equation (103), but limits of space will not allow us to enter into the investigation. Returning now to equation (99), we will write it in the form (116) d'y dx1 fx dy . gkg+f—'il· — x')-h{k+f-2) i — x1 dx ' (1 — #y y where the parameters ƒ, gy h are rto longer restricted to integral values.* We will define now as spherical harmonics of the first and second kind of rank fy degree g, and order hy two definite linearly independent particular integrals of (116). These generalized functions therefore depend upon the three arbitrary parameters fy g, hy and have the points + 1, — 1, 00, as critical points. The P-function above defined depends essentially on the differ- ences of the pairs of exponents (a, a'), (y3, /?'), (y, y'), and upon the critical points a, b, c. At first sight it might appear that these new functions, i.e.y the integrals of (116), were identical with the general P-functions. That this is not the case can, however, be easily seen. The differential equation of the P-function is, equation (89), (”7) I j I — a — a' , i fi , I Y y' ) dy_ dx* '( x — a x — b ' x — c j dx I ( a a’(a — b)(a — c) ' (x — a)(x — b){x — c) ( x — a) PP'(b-c)(b-a) yY'{c - a){c — b)\ +------1---------+-----------------K = °· This is changed into (116) by the following substitutions: ' a — — I, b = QQ, C — I ", h P=-g, h (118) - a — 2 ’ y = i; a! = h -)- ƒ — 2 2 Y’ = -k+f-2 2 * Studieft uber die Kugel- und Cylinderfunctionen, von R. Olbricht. Halle, 1887.EQUATIONS OF SECOND ORDER—GENERALIZED HARMONICS. 201 We have now the P-function ' - I, <*> + 1 h — gf h (i 19) P - 2’ 2 h+f- 2 0 ’ g+f- I, h + ƒ — 2 0 In this the differences of the pairs of exponents are two of which are equal. We arrive thus at the important theorem : The theory of the generalized spherical harmonic is identical with the theory of the J*-function, in which the differences of two pairs of corresponding exponents are equal. We will merely state, without proving, two other theorems found by Olbricht, referring the reader to his memoir for the proofs and also for other interesting results concerning the generalized spherical harmonics and Bessel’s functions. (a) The generalized spherical harmonic of rank f degree g, and order h are derived from the P-function " — I 00 + 1 h* ti P* 2 g' 2 X h' 2 g' + 1 _ h' 2 which represents the spherical harmonics of rank z, degree g', and order h' by replacing g', k' by g' = g + f- h! = k + f- 2 2 2202 LINEAR DIFFERENTIAL EQUATIONS. and multiplying the result by g —f (I - X1) 4 . (b) The generalized BesseVs functions are obtained from those of the second order by multiplying the P-function In the next chapter, M. Goursat’s thesis, will be found an investi- gation of the differential equation of the second order satisfied by the complete elliptic integral K. At the time of writing the preceding pages the author had not seen an interesting paper by Humbert,* of which the following is an account, and did not know that some of Heun’s results (see above) had been previously given. Humbert investigates the most general polynomials satisfying the equation where, as before, ip, Q1, Q2 are polynomials of the degrees p, p — I, p — 2 respectively. If we wish (120) to have as a solution a polynomial in x of degree ny the three functions tp, Qx, Q,a cannot be arbitrarily taken. Heine * “ Sur l’équation différentielle linéaire du second ordre,” par M. G. Humbert. Journal de VEcole Polytechnique, cahier 48, 1880. limP À g7t 00 O ig ig h X - ■ ig — k — h g -/ by x 2 , and replacing h by iV(2-fy + k(k+/-2). (120)EQUATIONS OF SECOND ORDER—POLYNOMIAL INTEGRALS. 203 has shown* (see above) that if tf>{x) and Qx{x) are arbitrary poly- nomials of degrees p and p — i respectively, there exist only (n + I ){n +2) . . ■ (n + p — 2) 1 . 2 . . . (p — 2) integral polynomials Qipc) of degree p — 2, such that equation (120) admits as a solution a polynomial in x of degree n. Denoting by x1, xp the finite singular points of the differential equation, we write, as before, *K*) = (* — X* -x,) ... (x — X9); also write Qix) — iix) a polynomial of degree p — 1. Defining a new function K(x) by the equation (121) 2_dK _Q± K dx ~ f ’ we have K(x) = (x — xjvi (x — . . . (x — XpfP . This function K (x) has an important property; viz., if we make in (120) the substitution (122) y = u-K{^y this equation retains the same form ; it becomes in fact (123) m ~+\_2f\x)- Qixj\ * Handbuch der Kugelfunctionen, zweite Auflage, p. 473.204 LINEAR DIFFERENTIAL EQUATIONS,. For simplicity we will first assume p — 3; we have then f{x) = (x — xt)(x — x#x — *,) ; <2,0) _ ,_________________, >«» x — xx ^ — jr4 ‘ x — x%' We will also first assume that the constants px, pi9, /¿3 are all posi- tive. Consider a path of integration between the points xx and x%, and another going from x2 to ;rs; the integrals / iis)(x~z) f ? c/ JTx «7 -T, *0) '¿0)0 - *) will then have a sense and be perfectly determinate so long as these paths of integration do not pass through the point x. Suppose now Pn(x) to be a polynomial of degree n satisfying equation (120); then equation (123) will admit as a solution the function K{x) tp(x) P«{*\ Substituting in the left-hand member of this last equation the func- tion 024) V, = 'K{z)PH{z) rp{z)(x—z) dz, we readily find, by aid of a known formula due to Heine, (I25) + l2f(x) - <2,0)] + [Qi*)+ Ì'\x) - &'0)>. = O0)]^ + X y{z,x)ds\EQUATIONS OF SECOND ORDER—POLYNOMIAL INTEGRALS. 20$ where xif — K{z) p /\ i '{x) - S(z) 9 ~ {x — zf + >p(z) Ki } x-s #0 r® X — Z L tp{z) S=2lf>' - &, t= &+r-&'· Developing we find « = K{z)Pn(z) _ K{z)Pn'{z) (x — zf X — z This is to be taken between the limits xx and x^ and since K{z) vanishes at these limits, it follows that # disappears from (ilg). As to t£w(z,x)dz, we see that it is a constant in x. We conclude then that vx is a solution of the equation (!26) + +iQlx) + f(x)- <2/O0> = const., the constant being determinate. So also the function f- J*. t K{z) Pn(z) tp(z) x — z dz (12 7)200 LINEAR DIFFERENTIAL EQUATIONS. is a solution of this same equation with a different value of the con- stant in the right-hand member. It follows now that v1 and are solutions of the equation solution of (123), is therefore also a solution of (128). Between these three solutions of this last equation we can show now that there exists no linear relation with constant coefficients. The functions v1 and developed according to descending powers of x commence, at the highest, by a term in —; on the other which is greater than — 1, since n > 1, and rlx , , /¿s are all posi- tive. If then there exists any linear relation, it must be between the functions v1 and % alone,, and so be of the form (I28) iix) + [vP'(x) - <2.(*)] ^ + [>"(*) - 2QX*) + &(*)] g + [<2,'w + rv) - 0.»] « = o, X hand, the function is of the degree A + M* + Mm + n — 3> where \ is a constant, orEQUATIONS OF SECOND ORDER—POLYNOMIAL INTEGRALS. 20? We can write fr(*) PJ& ip(z) x — z dz = P„(x) K(z) ip{z) dz X — z /»'r2 f X. iK*) W - ^,/2) dz x — z The second term of the second member of this last equation is an integral polynomial in x of degree n — i at most. We have then 'K(z) dz tp(z) x — z dz x — z IIn _ X{x) being a polynomial in x of degree n — i at most. It is easy to see that the expression in { } satisfies the differential equa- tion {129) iix) ddx = [Q,(x) - f'(x)~\ y-\-ax + f3, where a and f3 are constants. This equation admits then as a solu- tion the rational function n„ _ lx) n*) ’ which after suppressing common factors will be written JM P(x)· We shall have then iiH'P -P’ll) = (Px - ip')PII+ (ax + /3)P7, and consequently ip . Pf. TJ is divisible by P. Now P(x) is prime to II(.x), and therefore it must admit at least one of the factors of tp, say the factor x — xx. Again, the factors of P(x) are also factors208 LINEAR DIFFERENTIAL EQUATIONS, of Pn{x) ; PS*) therefore vanishes for x = xx. The differential equation fPn"(x) + QJ>J(x) + &/>„(*) = o shows that Pn\x0) = Taking the successive derivatives of the first member of this equation, we find in like manner that P«'{x„) = O, PJ"(xa) = o, . . . , Pnm(xa) — o, . . . , provided only that none of the polynomials Qkx) +P’P'(x) vanish for x = x0 · In order that they should vanish we must have M, + P=o, which is impossible, since /q > o. We arrive thus finally at the con- clusion that the constant Pnn(x0) is zero, and consequently, as is easily seen, that the relation ? vx — \va is impossible. Denote by Q the coefficient of the first term in QJ^x) ; then, as is easily seen, the indicial equation of (120) for the region x = 00 is 0 3°) r(r J) + Oh + Mi + M*)r + Q — o. Since Pn(x) is to be a solution of (120), n must be a root of (130); the two roots are then n, Mi /h ft -\- 1. For (128) the roots of the corresponding indicial equation are readily found to be - 1, n + Mi + M* + M* — 3» — n — 2.EQUATIONS OF SECOND ORDER—POLYNOMIAL INTEGRALS. 20g The root # + /*, + /¿a + /¿s — 3 is obviously greater than — n — 2. Equation (128) has therefore a solution developable in a series, ' of the form I P /3 y “1 (‘3‘) ^*+2 = [_“ + + · · · J ’ consequently (132) yn+2 = vx + eotv, + 6 Pu (x). In this it is clear that 6 — o, since the degree of greater than — 1. There remains, then, , K{x) tp{x) Pn(x) is (133) v, + œ.v, = denoting by (I j a series going according to descending powers of xy and commencing with a term in . Now, (‘34) v, = K(z)PAz) ip(z) x — z dz = PM(x) ƒ K/ X i K (z) dz f{z) X — + rK(z)Pn(z)-Pn(x)^ -r, n*) x- z The second term of the third member of this equation is an integral· polynomial of degree n — 1 at most. Write \{z) dz tp{z) x — z’210 LINEAR DIFFERENTIAL EQUA FIONS. /,= K (z) dz ip(z) x — z 9 the relation (133) now becomes (135) ^ (*)[/, + = na.t(x) + From this we see that Ix -f- ooj^ is represented, to terms of the order près, by the quotient Pn(x) * In other words, having given the functions Ix and /„, if we form the product P{*V, + where P{x) is a polynomial of degree n, we can by a proper choice of the coefficients of this polynomial, and of the constant gox, cause the terms in — , , . . . , —of the product to disappear. Equa- tion (135) shows that Pn{x) is such a polynomial. As we have assumed Q£x) to have any one whatever of its de- terminations, such that (120) admits as a solution •an integral poly- nomial Pn(x), the theorem just proved holds true for all polynomials of degree n which satisfy the differential equation $(*) + Qkx) -fa + QXx)y = °> whatever be the choice of the polynomial QJ^x)· Heine has given an analogous theorem for polynomials satisfying Lamé’s equation. The previous results are readily extended to the case where the differential equation has p finite critical points instead of only 3 ; for these generalized results the reader is, however, referred toEQUATIONS OF SECOND ORDER—POLYNOMIAL INTEGRALS. 211 Humbert’s paper. In concluding his paper Humbert gives, without going into details, a theorem for the general case in which the constants Pif Mt > · · · y Mp are not all positive; for this also the reader is referred to the paper.CHAPTER VII. ON THE LINEAR DIFFERENTIAL EQUATION WHICH ADMITS THE. HYPERGEOMETRIC SERIES AS AN INTEGRAL. By M. Edouard Goursat. I. The hypergeometric series F{a, /?, yy x)y considered as a func- tion of the fourth element x, is only defined for values of this variable which have moduli less than unity. In order to define it for any value of x, it should be regarded as a particular integral of sl linear differential equation of the second order; the problem then comes to finding what this integral becomes when the path described by the variable ends at a point situated outside the circle of radius unity and having the origin as centre. By a generalization which immediately presents itself, one is then led to propose the same problem for any integral whatever of the equation, the path followed by the variable being simply subjected to the condition of not passing through any critical point of the equation. This celebrated equation, studied by Gauss* and Kummer,f ad- mits, as the latter has shown, twenty-four integrals of the form ~ *)qF{a\ ft', /, *), where z is one of the variables I I X X — I X, I — x. —, ----- , ----- ,-------, ' X I — X X — I X provided none of the numbers y, y — a — /?, ft — a is an integer. In the same memoir Kummer gives the linear relations with con- stant coefficients which connect any three of the integrals; but, as he only considers the case of a real variable, the formulae which he obtains present some difficulties when we wish to pass to their applications, the more so as he does not always sufficiently define * Collected Works, vol. iii. p. 207. f Crelle, vol. xv. p. 39. 212GO UR SA T: H YPERGEOMETRIC SERIES. 213 the sense of his integrals. Suppose, in fact, that we wish to pass from a real value of x, positive and less than unity, to a real value of x, positive and greater than unity ; we cannot do this by giving x a series of values all of which are real, because we may not pass through the point x = 1, which is a critical point of the differential equation. It is necessary then to give to x a series of imaginary values, which can be done in an infinite number of ways, and, the final value of the function depending on the law of succession of the different values of the variable, we see at once that, for the proposed object, it is necessary to introduce the consideration of imaginary values in the differential equation. The general theory of this equation, when no restriction is im- posed on the value of the variable, has not, so far as the author knows, up to this time been treated completely. Tannery,* how- ever, has, by employing Fuchs’s method for linear differential equa- tions, shown that we can find all of Rummer’s integrals ; in another memoir f he has determined the linear relations between the inte- grals for a numerical example previously studied by Fuchs. The results obtained are in accord with those deduced from the general case. 2. In Part First the author develops first a method, due to Jacobi, for finding the twenty-four integrals of the differential equa- tion when no one of the numbers y, y — a — ft, /3 — a is an integer; then, by applying Cauchy’s theorem to the definite integrals which represent the hypergeometric series, the relations between the inte- grals themselves are obtained. If among the numbers y, y — - * -1 + <*gfi( i — g)y~%i — xgYa-x*216 LINEAR DIFFERENTIAL EQUATIONS. If we take e = i and give^* one of the values o, I, + oo, this result will be zero provided i — a has its real part positive, and also pro- vided that the product u\i — u)v-P(i — xu)~ a~ 1 vanishes for the limit u — g\ i.e., provided that the integral J*~x Vdu has a meaning. We have now, if all of the above conditions are satisfied, six functions, defined by definite integrals, which satisfy the given dif- ferential equation. The following is a table of these six integrals together with the conditions which must be satisfied by the quanti- ties a> y in order that the integrals may have a meaning. The inequality A > o indicates simply that the real part of A is positive. (0 y = J Vdu . . > 0, y — p > 0; (2) y = J Vdu . • . P > 0, a-\-i—y>0; (3) y = j Vdu . • · r — P > °> a +1—y>o; (4) y = 1 f* Vdu . • . . P> 0, I —a > 0 ; (5) y = 1 Jyvdu . . . . y — p> 0, i — a > 0; (6) y = °° Vdu , . . . a + ! — y > °, I — a > O. X 4. Before going farther it is desirable to define exactly those values of the function V which we can take in the above integrals, and also to examine more closely the properties of the functions represented by these definite integrals. 5. The integral y — f ~ x(i — u)y ~ p ~ J(i — xu)~ adu oGOURSAT: HYPERGEOMETRIC SERIES. 217 has a meaning provided that the real parts of fi and y — /3 are posi- tive, and provided in addition to this that x has no real value greater than unity. We will take this integral along the straight line o — i (Fig. i) ; we will also take o for the value of the argu- ments of u and i — u, and for the argu- ment of i — xu we will take that value which reduces to zero when u = o. The function y thus defined is a uni- form function of x in the entire plane if only the path described by the variable does not cut the line i-------------1- oo . Suppose now the point x describes a closed curve which does not cut this line; then the point xu, corre- sponding to a value of u lying between o and i, will describe a closed curve homothetic to the preceding one, and leaving the point i out- side. The argument of i — xu will then resume its initial value; the same thing will be true of the element of the integral which corresponds to this value of &, and consequently for the integral itself. It follows from what precedes that, inside a circle of radius unity and having the origin as centre, the function y can be developed in a series going according to ascending powers of the variable x. The coefficient of xm in this development will be / / Fig I. [dmy\ \dxm Jo that is, 1.2.3. · · m y a(a + O · * « (« + m — 1) Cl a . / \v « yj — —--------^---!-------1 / +** - *(1 — u)v -£-xau, 1.2.3 · · · m do v y or a(a -[- 1) . . . (a -f- m — 1) r(/3 -\-m)r(y — 0) 1.2.3 · · · nt * r(y^\^în) ’ or, finally, a(a -f 1) . . . (a -f m — 1 )/?(/? -f 1) ... (¡3 -f- tn — 1) ^ r{0)F{y — 0) 1.2.3 . . · m . y(y 1) . . . (y -j- m — 1) l\y)2l8 LINEAR DIFFERENTIAL EQUATIONS. We have then, for every value of x for which \ x \ < I, . F (a, fi, y, x). The preceding integral can take three other forms, which can be at once obtained by the following transformations due to Jacobi: By the first of these substitutions the integral Vdu is changed into (i — x)~a'tfo vy ~ p “ x(i — vf ~ — ———dv. This integral is taken along the same path as the one already con- sidered. If we make the same suppositions as above concerning the arguments of v, i — v, i — ^ - - - v, it will be necessary, in order that the two integrals may be identical, to take for argument of i — x that one which is zero when x — o. We have then J*o ~ *(i — îif - y - *(i — xu) " °-du U — l — v\ V u — -------. I — vx u Make now the second transformation. v u —------------7----; I — X -j- vx the integral becomes (i — x)~^So ~ X(x v)y ~ ^ ~'\l ~ ~~— d"V· ---v x— i >GO UH SA T: HYPER GEOMETRIC SERIES. 219 This integral is taken along an arc of circle 0M1 ; it is easy to see that the area comprised between this circle and the line o — 1 does x — 1 not contain the point ———, which is a critical point for the new X — I function under the sign of integration. In fact, the value v = ——— corresponds to u = 00; the point, say A, which represents * must thus belong to the circumference of which oM\ is an arc. We can consequently take the integral along the straight line o — 1, and, making the same conventions concerning the arguments as before, we can write fo Vdu =(1-*)-*£ -»)»-*-« (1 - iy ) dv~ The transformation 1 — v I — vx gives in like manner Jo Vdu = (1 — *(i — vx)~ - a)dv. Each of the new integrals can be developed in a convergent series for values of the variable lying between certain limits. Thus, for values of x such that 1 * 1 < 1» we have (1 — x)y - a - P j* vy - ? ~ j(i — vy ~ J(i — vx)- " F\a’ r~fi>r> (i — x)~ vj* z>p ~ *(i — v)y-P _i^i X • v dv X — I F{fi)r{y-P) / * \ F{y) ( Ì FV P' V’ x - I/' For brevity we will denote by C0, Cx, the circles of radius unity having the points O and I respectively as centres. Further, denote by E0 and Ex the portions of the plane limited by the common chord (produced indefinitely) of the circles C0 and Cx, E0 containing the point o and Ex the point i. It results from the previous considera- tions that the proposed differential equation admits an integral which is uniform throughout the plane, provided only the variable is subjected to the condition of not crossing the line i-----------(- oo. De- note this particular integral by Y’ = (i — x)-* F{y - a, p, j). the argument of i — x being supposed to lie between — n and -f- **· The previous results have only been established on the hypothesis that the real parts of /? and y — /3 are positive, but it is clear that these results will still subsist, provided only that y is not zero or a nega- tive integer. Suppose we wish to verify that one of the precedingGOURSAT: HYPERGEOMETRIC SERIES. 221 functions satisfies équation (i); the sign of the quantities /?, y — /? will not enter into the calculation, and consequently the verification will be the same in all four cases. The same is true if we wished to verify that two of these functions are equal for values of x which make the two series convergent. This remark is made once for all in order not to have to refer to it in analogous cases. If the reaL parts of /3 and y — § are positive, we have has a sense if only the real parts of fi and a -f- 1 — y are positive,, and if x has no real negative value. It is readily demonstrable that the function y is uniform in the entire plane, provided only that the the same is true if the path described by the variable is further define the sense of this integral it remains to choose the arguments of u, I — u, and i — xu; for the argument of I — u we will take o, for the argument of I — xu that which reduces to zero for u — o; but for u we will take the argument ± n. Suppose first we take arg. u = + ^ and make the change of variable defined by 6. The integral y — ~ T(i — u)y - * - x(i — xu) - adu path described by the variable x does not cut the line— co-------o; restricted to not cut the line i -f- oo . In order to exactly u v varies from o to I, and u from o to — oo : i I — u I — V I — XU — I — v(i — x) I — V222 LINEAR DIFFERENTIAL EQDATIONS. If we take o for the arg. of v and of I — v, we ought to take n for the arg. of (—- i), and the arg. of i — v{\ — x) will be zero for v = o. By this change of the variable the integral becomes f v& - ’(I — v)a ~ y[i — v{\ — xJJ - adv. In this last integral the arguments of v, i — v, i — v{\ — x) have the same sense as in the integral already studied. We can therefore apply the preceding transformations to this new integral, and so conclude that, subject to the conditions indicated above, while a + § + 1 — y is neither zero nor a negative integer, the differ- ential equation (i) admits a new integral, 0a, which is holomorphic in the entire plane. In the circle Cx we have 0i = + 1 — r> 1 — x) = x1 _ yF(a + i — —+ — y>1 ~x\ and in the space Ex Ìa> 0a = x-aF[a, ex -j- I y, a-f /? + i — x — i\ = x-?F — y y * + fi + i — y y arg. x lying between — n and -f- n· When ¡3 and a + i — y have their real parts positive the func- tion 0a can be represented by a definite integral, viz., fut ~1 (i — u)y ~ ? ~ *(i — xu)~ adu = +„j—0 J* V ' V * r(a+fi+i — yY" supposing in the integral arg. u = + tt. If we take arg. u = — ny we will have fut - x(i — u)y ~ ~ J(i — xu)~adu = e~ do ' ' V J r(«-L ft 4- T -y) r(a+fi+l-y) 02GO URSA T: HYPERGEOMETR1C SERIES. 223 7. The integral /+°® UP - _ uy/-0 - 1(1 — xu)~adu has a sense provided the real parts of y — ft and a -f- 1 —- y are positive and x has no real value lying between zero and unity. It is easy to see that if the variable describes a closed path enclosing the line o------- 1, the function y takes its original value multi- plied by e± 2nai; the function can therefore be rendered uniform if we make the convention that the path of the variable must not cut the line o---------\- 00 . As to the path followed by the variable, then, this new integral presents an essential difference from the two integrals already examined, which arises from the fact that we can- not pass from the upper half of the plane to the lower half, or con- versely, by crossing the line o------ 1. The integral ceasing to have a meaning for a point of this line, there is nothing to indicate that the analytical continuation of the function would be represented by the same symbol after crossing the line; in fact, we shall see that the same symbol will not answer after the crossing. In order to definitely arrive at the sense of the integral we will take o for the arg. of u, and for arg. (1 — xu) that which is zero for u — o, and which varies continuously when the variable describes the positive part of the axis of x. As to 1 — uy we will take arg. (i — u) — ± 7t. Suppose arg. (1 —//)= — zr; make 1 — dv u — —, du — —^—, V V If we take arg. v = o and arg. (1 — v) = o, and for arg. ^1 which is zero when z/ = o, we ought to take arg. (— 1) = for arg. x a value lying between — n and + n. - -j that XI — n, and224 LINEAR DIFFERENTIAL EQUATIONS. /+o° Vdu now becomes — -^dv. This new integral is of the same form as the one first studied, and on applying to it the same transformations we arrive at the follow- ing results: Whenever a-\- I — /3 is neither zero nor a negative integer, the differential equation (i) admits as an integral a function 03 which is uniform throughout the plane, provided the path described by the variable does not cut the line o----------l· oo. This function can be developed in a series as follows: Outside the circle C0 we have 03 = (— x)-aF [a, a + I — y, ot + i — /?, = (- - »(i - · ·- w t1 - a y—a «+1 ■- a ¿)· Outside the circle Cx we have 0, = (I — x)-aF[ot, y — fi, «+ 1 - fi, 7“) - (—xy ~ v(x — x)y~a~ ^(«+1— y, i —/?> a+i—p, 73"“)· We suppose arg. (— x)y as also arg· 0 x)y to lie between — n and + 7r. If the real parts of y — /? and of or*-f- i — y are positive, we shall have /+«> #0_I(i —u)y~P~z(i —xu)~ adu ■ r(a + I — /?) on assuming arg. (1 - a) = - rt- If arg. (1 — *)=+*, we shall have J^ — *«)- 0-1)»* r(a+i-y)r(r-/g) T(« + I - /*) 03GOURSA T: HYPERGEOMETRIC SERIES. 225 8. We can study in the same way the remaining three integrals: J^^Vdu. As this study involves no difficulties it will be sufficient merely to give a summary of the results. The integral y — J up ~ *(i — u)y ~ & _I(i — xu)~adu has a sense if /3 and I — a have their real parts positive and if x has no real value between zero and unity. If we subject the path described by the variable to the condition of not cutting the line o--------f- 00 , the function y will be uniform throughout the plane. v This integral can be thrown into the first form by making ^ ; we have then J* u^-z(i — h)y-0-i(i — xu)~adu = e±nPt'(— x)-t Jl vP~J(i — v)-a[i — ^ ^dv. In the first integral we take for arguments of 1 — u and 1 — xu those which vanish for u = o. As to the argument of uy denoting by go the argument of (— x)> we will have for the coefficient of the second integral e± according as we take for this argument the values (— go ± n). We have now for our differential equation a new integral, 04, which, like the preceding ones, can be developed in a series. Outside the circle CQ we have 0, = (- x)-fiF[/3 + I — y, P, P + i — a, = (— x)a -»(i — x)y-a~pF^i — a, y — a, /? + I — a,22 6 LINEAR DIFFERENTIAL EQUATIONS. and outside the circle Cx we have = (i — y /? + I — a, = I—«, ¡3+1-a, the arguments of (— x) and (i — x) lying between — n and + it. For properly chosen values of /3 and i — a we have f uP-1(i — u)y-P-x(i--xu)-adu = e±____?2 Jo 9 V } r{p+l-aVA 9. The integral /+°° uP-x{i — u)y-t-x(i — xu)~adu has a sense if the real parts of 1 — a and a -)- 1 — y are positive, and if x has no real value greater than unity; further, the function y will be holomorphic if the path described by the variable does not cut either of the infinite lines — 00-----------o, 1-------------f- 00. This integral can be put into the first form by the transformation u = —, and we then have xv /M-°° Jr uP ~ x(i — u)y -1 ~ x(i — xu)~ °-du X == e± irtXy-P-i)±m'a X1 -y J* _ V)~ a(l — XV)i “ 0 “ ldv. The arguments of u and i — u are fixed by the continuity by sup- posing that we start from the origin with the argument o and de- scribe the infinite radius oL passing through the point -· As to I — xu, we can take arg. (i — xu) — ± n. In the preceding for- mula we will take the sign -j- or the sign — before ni{y — /3 — i) according as the point represented by x is in the upper or in the lower half of the plane, and the sign + or the sign — before nai according as we take arg. (i — xu) = -|- it or — n.GOURSA T: HYPERGEOMETRIC SERIES. 227 While 2 — y is neither zero nor a negative integer, the differ- ential equation(1) admits a new integral, 05, which is uniform under the condition enunciated above as to the path of the variable. In the circle C0 we have + i — y, a + 1 — y, 2 — y, x) = jrI-i'(i — x)y-a~^F(i — a, 1 — 2 — y, x), and in the space E0 0* = - Y(I - x)y ~ “ " + 1 -y,i—p,2-y, — = - Y(i — p -f- i — i — «» 2 — 7, j~), the arguments of x and i — x lying between — n and n. For properly chosen values of a, /?, y we will have /-(- 00 £¿0-1(1 _ u)y-0-t(i — xu)~adu - ± wKy_ „ _± r(« + 1 - y)r(l - «) T(2-r) 10. The integral y = U&- 1 (1 — u)y -P ~1 (1 — xu)~~ adu, which has a sense provided the real parts of i — a and y — /? are positive, and x has no real negative value, is holomorphic under the same conditions as the preceding one, and can be thrown into the first form by the transformation u — ■V+ I, ■giving then x228 LINEAR DIFFERENTIAL EQUATIONS. The arguments of u and i — xu are defined by the continuity; viz.,, we start from the origin with the initial value o, and describe the straight line o------1 ; then starting from the point x = i, we will describe the straight line joining this point to the point ^ with per- fectly determinate values of these arguments. In the case of i — u there is, however, some ambiguity. In the preceding formula we must take the sign -j- or the sign — before ni(y — — i) accord- ing as we take arg. (i — it) — arg. (i — x) — arg. x ± n. While y-\-i — a — is neither zero nor a negative integer, we will have a new integral, 06, of the differential equation (i), which will be, uniform under the same conditions as the preceding one. In the circle C1 we have 0, = (i — x)y-a-^F(y — a, y — p, — a — p, l — x) = x'-y (I — x)y-a-PF(I — a, I — p, y-j- I — a — p, i — x)„ and in the space El 06 = x*-y (i — x)y~a-pF = xP-v(i — x)y-a~?F (y-<*> (r —A , „ X — 1\. i — a,y+l—a — p, ——J i - a- p,U^y. the arguments of x and i — x lying between — n and -f- n. Foir proper values of a, /?, y, we have u&-1 (i — (i—xtt)~°-du = riy—0)ril— a) r(y+1 —a—p) 0,. ii. In summing up all that precedes we see that, so long as no- one of the quantities y, y — a — /?, — or is an integer, the proposed differential equation admits six particular integrals each of which, in different parts of the plane, can be expressed in four different ways by~ hypergeometric series: these are Rummers twenty-four integrals. We can render these integrals uniform by imposing certain con- ditions upon the paths described by the variable; thus, for the in*-GO U R SA T: HYPERGEOMETRIC SERIES. 229 tegrals l9 if 8 * '(1) F(a,0,y,x), (2) (i — x)y ~ «· ~ PF{y — a, y — 0, y, x), (3) (ï — x)~°-F{a, y - 0, y, (4) (I - x)~>f(p, y — a, y, — '(1) F (a, 0, a + /3+ I -y, 1 — x), (2) 4T1 - vF(a + i — y, 0 + I — y, a + 0 + I — y, i — x), (3) x-'fLx, « + I — y, a + 01 - y, ~f), (4) x-*f[p,P+1 -y,a + 0+l -r,^)· (1) (— xy*F{oc, a + I — y, a -f I - 0, (2) (- xf~y{I - - 0, y - 0, a + 1 - 0, (3) '(! - x)-aF(et, y — 0, «+ I - 0, -ff)' (4) (-xY-yil-xy-'-'F^a+l-y, 1-0, oc+l-0,230 LINEAR DIFFERENTIAL EQUATIONS, (0 4>t (2> (3) (4) '(0 (2) 0* (3) (4) (1) (2) 0. (3) (4) (~ x) {p + 1 ~ Y< fi + I — OC, (— xy-i{l — x)y-*-tF[\ — a, y— a, /S + I— a, (_ Jr),_y(I —xy-*-'F(p+i—y, i—a, fi+i—a, y^)- ·*’ yF(a + I — y, /S+ I — y, 2 — y, x), x1~y{\ — x)y~a-PF(i — a, \ — fi,2 — y, x), x'-v{i — x)y-*~'F(a+ I — y, 1 —/?, 2 - y, *I-7(l x T _a>2 _y> (1 — x)y~a-pF(y — a, y — /?, y-j- I — a — /?, I — x), x'-y(i — x)y-a~l3F(I — a, 1 — /?, y-\- 1 — a — fi, 1 — Xs), xa-y(i — x)y~a~PF(y — 0!, 1 — a, y -j- 1 — a — /?, 4T^”Y(l — x)y-a-?F (y — fi, \ — fi, y — a — fi, 12. Between three of these integrals there is a linear relation with constant coefficients in all that part of the plane in which the inte- grals are holomorphic. If we consider three of the integrals denoted by 0a 1 03, and let M and Mr denote two points, one situated in the upper half of the plane, and the other in the lower half. There exists no path joining M and Mr which will not cut at least one of the two lines — oo---------o, o----------1- oo ; one at least, then, of the three functions represented in the region of M by 0t, 02 , 03 willGOURSAT: HYPERGEOMETRIC SERIES. 231 not, after travelling such a path, be represented in the region of M' by the same symbol. The preceding remark is essential, and it will be useful to de- velop it. Suppose Ey Ef (Fig. 2) two areas with simple contours T, T', neither of which encloses in its interior a critical point of a differential equation of the sec- ond order; suppose that be- tween these two areas there is a critical point, A, of the dif- ferential equation, and suppose further that the areas have in common two separate areas, C and C'—that is, such that we cannot pass from a point of the area C to a point of C* without cutting at least one of the con- tours Ty T\ Let P and Pr be two linearly independent par- ticular integrals which are uni- form in the area Ey and let Q and Qf denote two such integrals in the area Ef. In the common part C we have the relations (I) and in Cf we have (II) i Q = \P + MP', t Q’ = VP + //>'; ( Q == XxP -f- ( q: = \;p + Ml'p>. It is easy to show that the relations (I) and (II) must be distinct; that is, that we cannot have simultaneously A.J A, fly - fly Aj - A , fly - fl . Suppose, in fact, that these last equations were satisfied ; let us start from a point m in C with the particular integral Q and de-232 LINEAR DIFFERENTIAL EQUATIONS. scribe a path situated inside the area E> and so arrive at a point mf of C\ All along this path our integral will be given by \P -j- fxP'y and under our hypothesis will, at the point m\ coincide with the particular integral Q. If now we return to the point m by a path lying in E\ we will evidently arrive at this point with the original integral Q. We would therefore have described a closed curve con- taining the point A and have returned the integral to its original value. The same would be true if we had started out with the par- ticular integral Qr. The point A could not, therefore, be a critical point for the differential equation. It is clear that in the case under consideration the areas C and C' coincide respectively with the upper and lower halves of the plane, and the contours T and Tr with the lines — oo-------o, i---------oo , and o----------1- oo. It is thus established that, as they have been defined, there should exist between the integrals 0,, 0S two different linear relations according as the point x is in the upper or lower half of the plane. 13. Let us assume the points in the upper half of the plane, and V= u&-J( 1 — n)y ~ (1 — xu)~ a will be holomorphic inside the area bounded by these semicircles and the portions of the straight line L'a', ab, brL. By Cauchy’s theorem we have suppose further that the real parts of /?, y — /3, a -f- I — y are positive. X Describe around the points x — O and x = 1 two semi- circles ama! and bnb' with very small radii r and r' respectively, and around x = o describe also a semicircle LML with a very large radius R; the function where the integration extends around the entire contour just de- scribed.GOURSA T: HYPERGEOMETRIC SERIES. 233 This can be written in the form f Vdu 4- f Vdu 4- /*+ Vdu = — f Vdu — f Vdu — f Vdu. «'-tf V -|-r V LML· Va'ma °bnb· If now the radius ^ increase indefinitely, and if the radii r and r* tend towards zero at the same time, it is easy to see that each of the three integrals in the second member of this last equation tends towards zero. Take, for example, the integral ƒ Vdu = f - J(l — u)y -0 - x(l - xuY adu ; ” LML> ” ML· for values of u for which mod. u is very great this integral can be written {-Xs)-0·] uy-a-*(i + €)du, °LML· where e is an infinitely small quantity. Write now ' u — Reie ; then / Vdu becomes dLML· i{-x)-° f R(y-°--'V\i + e)dO. t/0 Let now y — a — 1 = p -\- iv; then fLJ[du = *(— x)~ +«^+•’0(1 -(_ e)M = i{— x'y ‘ - »vd*·+ -£(*)!(! -f €)dd. Since by hypothesis fi is a real negative number, we can take R so great that the maximum modulus of the function under the sign of integration shall be less than an assigned number 7?; the modulus of the integral will then be less than ^[mod. (— x)~ a]; that is to say, it can be as small as we please. We can show in like manner that •each of the two integrals / Vduf / Vdu has zero for its limit. 'Jama· °bnb· We have therefore the equality f° Vdu +jT Vdu +f“Vdu = o. If we take o for the argument of u and 1 — u along the path aby it234 LINEAR DIFFERENTIAL EQUATIONS. is clear that we must take -f- n for the argument of u along the path L'a!, and — n for the argument of i — u along the path b*L. By referring then to the definite-integral expressions for the functions 0i, 03, 03, we see that the preceding equation gives the following relation connecting these functions: W F(a + ft + I - y) V* _ r(ji)r(y-p) _ yWr(a + i - r)r{y - P) r{y) + " r(a+l-fì 08 This relation has been established by supposing the real parts of fi, y — fi, a-\- i — y to be positive. In order to demonstrate that, the relation is general it is only necessary to employ the well-known method of procedure which consists in showing that if the relation holds for values of /?, y — ft, a + i — y comprised between certain limits, it is still true when we diminish one of these values by unity. We will show first that the relation holds whatever be the value of If (I) is demonstrated for certain values of a, yff, y, it will hold for a-{- i, /?, y; we can therefore write, replacing +i-r) />+/? + i -r) F{a, ft, a + ft+ I - Y, I *) + ‘?(g+1~Y)’r' · (-x)'aF{a’ a+l~V’ «+I-A j)» a «+* + * - * i - *> = A * *) 4- ¿e+i-yi«ir(a + 2 - r)r(r — P), r(a-\- 2 — ft) K X)GOURSAT: HYRERGEOMETRIC SERIES. 23Î Multiply the first of these relations by (a — ff)x -f- y — 2a, and the second by a (i — x), and add the results. The coefficient of nr) [(a — §)x + Y — 2a~\F(a, /?, y, x) + a(l — x)F(a + I, /?, y, x), that is, (y — a)F(a — I, /?, y, x), by a well-known formula in the theory of hypergeometric series^ Further, in the second member of our equation we shall have λ+.-y»·r^+ 1 ~ r)r(r_zJ),_xY a r(«+i-/î) <■ x> x | [(<* — 0)x + Y — 2a]F^a, a + I — y, a + I — /?, a(a + I — y)(l — x) (“ + I» « + 2 — y, a + 2 — /?, i) I. (« + i - P)x The quantity in -j j- reduces to {a — 0)xF{at — I, a — y, a — 0, and the coefficient of F [a — I, a — y, a — fi, becomes -<<#+' (» - «- *)-<~> =- “X- We find in the same way for the first member of our new equation the value ^ir i^)r {« — y) r „ , r, X T(a + /3 — y) (r-a)F(a~ l> A «+/î-y, I —x),236 LINEAR DIFFERENTIAL EQUATIONS. and so have finally -l, fi, a + fi-y, l-x) r(a+ P — y) = r{p)r{y-p) r{y) ,,0 ^n«-r)T(r-fì + r{a-fì F(a — I, p, y, x) (—jr)_(“_l)^(ar— l, oc y, a — p, -) which is simply equation (I) with a changed into a — 1. This rela- tion is therefore exact whatever be a. In the same way we can show that if (I) holds for two values of y differing by unity it will also hold for a value of y differing by unity from the least of the preceding values, and consequently that it holds for all values of a. and y. It remains then only to show that ft can also be arbitrary, which is done by a process entirely similar to the above and which need not be reproduced here. 14. Formula (I) is then perfectly general, and we can readily de- duce from it the following formulae : ^ r (ac + pi — y) r{y) + +1 ~ r)r(r - P) é + r{a+i-p) .r(a)r(P ~f~ I — y) jl r(a)r(y - a) .. tn+1-yïH r(a + 1 ~ r)r(r- 0) r(a+l ~0) 3 , r(a+p+i-r) r(y) , ,(«+*-*>/ F(P+ I - y)r(y-a) + r(fi + i — a) V" (3) e^rf~ ~ j 0, = r^)r± ~ 0, r(y +1 -«-/?) r(y) + rtf+i-a)*'»LINEAR DIFFERENTIAL EQUATIONS. - r)H«) ^ = r^+^i-^rc-ff r{tx + P+i — y) r{2-y) + r(a+I-/S)0” - «K(i - fl 0e = r{a)r(y —a) ^ r(y+l-a-fi) r{y) ^ r(« +1 - /?) 03> ^+I-r,^^L±lZLZm 01 = r(«+i-,)r(i-«) r{a + fi + i — ^ T(2 - r) + r(/s + i-*)04’ ~ ~ «) 0s = r(r - ftryi + i -y) ^ r(y + i — a —A) T(2 - r) + cu-n»ir(r - +1 ~ r) . + ros+i-a) 04 ’ ^-aW,r(I ~ «)r(r - /?) 09 _ -T(I — a)r(a+ I — r(y + i -a-0) r{2 - y) i i-.w-r(r-^(«+ i - y) + r(a+ I-/?) 0” ^ _ r(y)r(y -a-0) ^ , /»F(« + >8 - r) ^ 01 - r(r _ a)r(y- 0) *' ■+■ F(«)r(/S) 04 ’ . _ r(2 — y)r(y— a— 0) r{2—y)r{a+0—y) 0‘ ~ r(i - a)r(i - 0) 9'^-I\a+i-r)I\0+i-yf' _r(«+/?+I-r)r(l-r) JXa+/J+I-y)r(y-i) 0’-r(«+l-y)r(fi+i-y)*'^ r(a)r (0) 04 _r(r+i-ff-/3)r(i-r) r(r+i-«-)g)F(r-i) 04 r(l-«)r(l -yS) r(y - a)r{y - 0) *'GOURSAT: HYPERGEOMETRIG SERIES. 239 .. _ r{Y)r{0 ~ a) ^ , r(y)r(a -0) < 3) 0. r{y_ a)r{fi) & + r(y _ fir (a) & · / I4\ é = r (2 ~ rKO? ~ g> -»)- 0 1 4) 0* /’(j _ a)r{fi 4- I - y) 03 r(2-y)r(a-^1_y> ^ T(l - /3)r(a + l-y) . r(l -y)T(a+I -fl < 5) ^ r(I-/5)r(a+ ! -r)01 _ r (y)r (l — y)r (a + I — /Î) r(2-y)r(y- p)r(a) /lfiv M _ r(i-y)ro?+i-«)^ _ r(r)r(i - y)r(/? + I - a) r(2 - y)r(y - a)r{fi) (17) r(>8 + i - r)r(jS) + r(a + /? + I - y)r(a - 0) (l8) 0, r + /?-r)T(«+ i-/?) (9) 0» r{l_^r(y-p)r(a+p+l-y) 03 J> + /î-r)r(« + 1 - ft) (v_p)i (20) 0, = r(«f+ I — y)r(a) r(y + I - a - /?)*> +0 - r)r(/î + 1 - a) 0. T(l - a)r(y — a)r{a + p+ 1 — y) r{a + /3- y)r(fi + 1 -a) rtf+i -y)r(p) ^*‘03 e*y—'>i 0,.LINEAR DIFFERENTIAL EQUATIONS. Lower Half of the Plane. + i -r) _ r(fj)r(y - ¡3) t 2 - r{a + (S + i - y) nr) 0, I + 1 - r)r(^ ~ A) 0 + rca + i-/?) 03 (/? + i - r) , r(«)r(r - «) * — :——---------r 0, =-----TFT\--- 01 r{a-\-p + I — r) r{r) ,-.w^+' -r)r(y-a) r(fi+i-a) 04 > ^nr^mLzA 0 0 r(y + i-«-/») 06 01 + I - «) + 1 - rK» . _ r(/?+ I - y)nI-/S), ^ F7—I-/»'' f t--\ 0« ----F?r--c----0. ^ (« + ^ + 1 - r) r (2 — r) + -J-) A, + r(a+ I-/?)0’* s(a-^nr-«)n l-ft) 06 = na)r(y-a) ^ r(y+l-a-p) r(r) + r(a+ I - /?)0S* y^-.^n«+l-y)r(/Q r(g+i-y)r(i-g) r(« + /3+1-r)03 r(2-r) 03 , /T_,_.w,-r(/?)r(i - g) ^ r(/j + I - g) 04' r&- + .-«-AT· r(2-r) ft + r^+.-«) *·■GO URSA T: HYPERGEOMETRIC SERIES. 24I (8)' ^,wr(» ~ = r{i-a)r(a+i- y) . - 0)r(°L±0 + r(«+i-^) 05· (14)' = r(2 - ^)r(l 0 V 4) 0» r(i — «)/"(/? + 1 — y) r(2 - y)r(a -„A_Vt_1>b . ^r(i -/*)/> + i -r) ,ieV ^ _ r(l - y)r(ar + I - >3) ^ ( 5) 03 r(i - ff)r{a + I - r) 01 r(K)r(i - y)r(g+ I -fl fl r(2 - y)r{y - fir (a) y)iti 06 ,lfiV ,* - r(T ~ r*rV +1 ~ «).. ( } ^ r(l - a)r(/3+l- y)*' r(y)r{l-y)r(fi+l-a) r{2-y)r(y- a)r{fi) , w r(a + /S + 1 — v)r(/S — a) . < 7) 03 r+£ + i - r)r(« - fi .. r(a + I — y)r(a) r(y+I-g-/>)rO>-flr) ^ ^ 04 . /tqv * _ V/ , . (^l8^ 0g T-F/ \T~1f \ ^ 1 {i — ¿r)/ (}/ — a) ~f~ 1 r( I - /î)r(r - /3) + rfr+ ; ~ “ ~ fj)r(l ~ fj) e*-* 0, - ('9) ^ ~ rU-flr, This differs from (1) in that n is replaced by — tt. All the other formulae for the lower part of the plane are deduced from this by permutations of the letters just in the same way as in the upper half of the plane. 16. Consider a path of arbitrary form joining any two points M and Mr of the plane, but not passing through either of the points o or 1. If we start from the point M with a particular solution of the differential equation, this solution will be defined all along the pathGO U R SA T; HYPERGEOMETRIC SERIES. 243 described by the variable, and we will arrive at the point Mr with a determinate integral. The preceding relations permit us to find this integral when the path from M to M' is given, and when the particular solution with which we start from M is given. Suppose we start from a point A corresponding to a real value of x and lying between x — o and x = i, and go to a point Mr of the plane by a direct path which cuts neither of the lines — 00------o, -|- 1-------1- 00. In the region of A the integral can be represented by Cx + C'cp2, and describe a lopp in the direct sense round the critical point x — o. To see how this integral behaves, we have only to replace 03 by r(« + /j+i-r)r(i-r) r(« + /?+i-r)r(K-i) r{a + I - y)r(fi +l-yV1^ r(a)r{/3) ‘ When the variable describes the loop, 0, does not change, but 06 changes into ~y)*iri6, so that we come back to the point of de- parture with the integral244 LINEAR DIFFERENTIAL EQUATIONS. or where n« + '- r)r[fi+i-y) r r(a + f}+l-y)r{l -r) l9' r(«+i-r)ro?+i -Y) c& -f- C/0, > + C'/'-vW/l r_r,r r(« + /?+I-r)ni-r) ‘ c + c r(«+i-r)ros+i-r)( So also for a loop round ;r = i the integral Cx -f- C*0„ changes- into Cx(px -f- ¿7/02 ? where Ci = Ce± ^ r> — r'A-r r(r)r(r ~ — P) 1 · C r{y-a)r(r-p) (i — ¿»±2ir/(y-a-0)).. The sign -f- or — goes before 27rz( y— a — 0) according as the loop is described in the direct positive) sense or in the opposite (i.e.v negative) sense. If the variable describes several loops in succession, the formulae will have to be applied a corresponding number of times, and we will finally be conducted to results similar to the preceding. Let us take now the general case where the path of the variable joins any two points of the plane. This path can be replaced by a direct path going from M to A> followed by a perfectly determinate path going from A to MIn order to be conducted to the preced- ing case, it will be sufficient to determine with what integral we arrive at the point A in following the direct path from M to A~ The above method enables us to do this without difficulty. Let us consider as an example the differential equation Jx\ dy 3 ' dx o; this admits as an integral the hypergeometric series ^(I, i> h x)·GOURSAT: HYPERGEOMETRIC SERIES. 245 Suppose we start from the point A with this solution and describe the closed curve ABCDA (Fig. 5), surrounding the two points o and 1. Fig. 5. This contour reduces to two loops described in the negative ¡sense around the points x = i and x — o. After describing the first loop around ^ = i we return to the starting-point A with the integral i, x) + r(y)r(r- a-ft r(r — <*)r(r — P) (l ___e-2*tXy-t *); after describing the second loop we will have an integral which, in the region of A> can be represented by CF{\, -J-, £, x) + CxF(i, i, V» 1 — x\ where — e—27Ti(Y-a-0) , sin (y — oi)7t sin (y — , Y ‘ sin y?t sin (y — a—J3)7T v ^ ^ c = r(^rSr ~__a~ & 1 r(r-a)r(r- f>)e ^ >· Making now a = i, § — y — we have Sm C — e3 + 2(1 — e3) = 2 5« ^3 . 5™246 LINEAR DIFFERENTIAL EQUATIONS. 17. When y is an integer, or indeed when y — a — /3, or a — /5T is an integer, we have only one integral in one of the groups; in order to find a new integral we employ the following well-known process* Let y - F(x, r), y, - F,{x, r) be two distinct integrals of the differential equation which become equal for a particular value, r — ri, of the constant r. We will ob- tain another integral by seeking the limit for r = rx of the expres- sion F(x, r) — Fix, r) r-rt which is also an integral of the differential equation whatever be the value of r. Let us suppose y to be an integer; we may also assume y posi- tive; for, if it were negative, we could make the transformation y — x* 1~yyl. There are two cases to be distinguished according as yr is equal to unity or greater than unity. First Case : y = 1.—The two integrals F(a, /?, y, x) and x'-yF{a -f I — y, /?+ I— y, 2 — y, x) become identical for y ~ 1. From what has been said we must seek now the limit for y = 1 ol the expression xJ~yF(a -|- 1 — y, I y, 2 — y, x) — F(a, /?, y, x) l - y This limit is obviously equal 0i ^ + dcf) 1 da + d/i ^ dy 9 in which we make y = 1. As the function 0, is susceptible of four different forms, there must equally be four different forms for the new integral. Let 01 = e(a, p, y, x),GO URSA T: HYPERGEOMETRIC SERIES. 247 and write » /■ \ ¿ft . ¿ft _i_ -,^ft and denote by the coefficient of in the development of F (a, P, y, x)· We find then readily ft(*) = AmBmx”, m—i where “ + £* -f- 7+··· + or 4“ ^ 1 ^ + ^ + 1 P^ P + I + · · · + /3 m -----------2 (1 -f I + - + . . . 4- — m — 1 \ ' 2~ ~ ml If we take x = (i — — a, y x)y and denote as above by Am the coefficient of the general term in the series F{y — a> y — /?, y, ;r), we shall have for if>x(x) the new form i*=4-oo ft(*) = ^ AmBnxm{\ — x'y-«-$: where + 1 I — a: 1 2 — a + · · · + 1 +T-4 + ··· m-a ' 1 — yd ‘ 2 — p _|_____Ï______2(- . 1 ■ 1 ■ -i ‘ m — 6 \ 2 1 3 1 1 m Each of the two expressions for 0, will giVe a different expression for tpx. Thus we find ft(*) = - log(i - *)(i ~ x)~aF(a, i — j _£\ X — 1/ I f m — 00 / \ + (I ~ x)-«2 AmBm(-?—) , A" _ a(a + I) . . . (a + m - l)(l - fife-. /3) . .. (m - p) (1.2... mf~ >248 LINEAR DIFFERENTIAL EQUATIONS. ¿ + ··· + 2hÌ^ + I37 + ^ + ··* + ~r^-2(I+I+i+* · · +i)· Also, *,(*) = - log (i - i - 1, J+-) ** = °0 W + (I +Ì , m = 1 \x — l} _ p{fl + 1) ...(/? + m — i)(i - «)(2 — a) . . . (ot — g) “ (1 . 2 . . . ntf ^« = 4+*-+ + ··· + 1 ' 1 1 1 + + · · · fi /?+ I ' ’’’'/? + W — I 1 I — a 1 2 — a + —1— 2(i + i+i+ .. . + -V m — a \ 2 3 W Whichever be the expression adopted for ^,(;r), we will denote by Q the new integral, viz., Q = 0i log * + ^0)· This new integral, like the preceding ones, will be uniform through- out the plane provided the path described by the variable does not cut either of the lines — 00-o, 1--------f- 00. The method employed for determining the new integral enables us also to find the linear relations connecting it and the integrals already known. To fix the ideas, suppose that the sum a + fi is neither zero nor a negative integer, and give to y a value differing but little from unity. We know the three integrals F{ot, A r, x), xx~yF{a + i — y, /? + I — y, 2 — y, x), F(a, A a + ^ + I — y, I —x), between which exists the relation r(a+fi+l-y)r(l-y) ^ , r(a+fi+l-y)r(y-l) ^ V* r(a+l-y)r(Ji+l-y)01 + J»r(/J) ^GO U R SA T: HYPERGEOMETRIC SERIES. 249 Replace in this relation 0B by x and Qx become under this hypothesis. We have r(a + /J+i-y) r(«)ro*)r(«+1 - y)r(/i + 1 - r) £r(i — y)r{a)r(/3) + F(y — i)r(a + 1 — y)r(p + 1 - y)]0, r(g + /?+ I -r) r{a)r(Ji) r(y)Q>· Let y tend to unity ; the coefficient of Q reduces to — 'ctjrffi)'· m · e , 1 -T “l·* $) r . the first factor of the coefficient of x has r / .:-,a for limit ; the second factor can be written in the form [r(a)r(P)J r(2 - y)r(a)r(ft) - r(y)r(a + I - y)r(fi + l - y) “I - y · The limit of this will be found by taking the derivative of the nu~ merator for y = 1, and is 2r'{i)r{a)r(fi) - r{a)r'(ji) - r{0)P {a). We have, therefore, (21) F {a, /?, a + /?, I — x) r{«+P) -r(a)r(fi) _ r(fi ryf) n«+0)n r (a)r (p) ^ Second Case.—Let us suppose y greater than unity, and write y — 2m where m is an integer and may be zero. Let us examine the series F(a + 1 — yy fi + 1 — y, 2 — yy x) when y, supposed at first to differ a little from 2 + m, tends towards this value.250 LINEAR DIFFERENTIAL EQUATIONS. The first terms of the series are i , (« + 1 — r)iP + * - r) „ + I · (2 - r) , O + 1 — y)( ^ (/?+!— x)(/? + 2 — rX/^ + m + 1 — r) ^ 1 I . 2 . . .(m-\-i)(2-y)(3-y) . . . y) | #*+... -xm+1 + . . „ As y tends to the value 2 4- my the terms in x, ;ra, . . . , ;trm preserve finite values, while the coefficient of ;rw+I becomes infinitely great unless some factor in the numerator vanishes. In order that this may happen it is necessary and sufficient that a or fi take one of the values i, 2, . . . , m + i. In this case all of the other terms will retain finite values for y = 2 + *0, and we shall still have an integral. The integral is uniform in the region of the origin, and admits this point as a pole; the integral is thus seen to be different from F(a, ft, y, x). Starting from the term in xm, the aggregate of the remaining terms may manifestly be written CF (a, ft, *« + 2, ; as to the preceding terms, they may be replaced by C,F{a, a + I — y, a + I — /S, ^jxm+1—, so that after multiplying by ;tr(w+l) the integral becomes Cix~aF(a, a + I — y, a + I — + CF(a, fi, y, x). Take for example the differential equation *(I-^id+2%+2*=0’ which corresponds to « = i, p = — 2, y = 2.coa/iSA T: HYPERGEOMETRIC SERIES. 251 This equation admits 1 . 3? *=I-*+y as particular integrals; the general integral is then It is to be remarked that when the circumstance above signalized presents itself, the origin is not a critical point for the general in- tegral, but may be a pole. Removing this special case, we see that the series F(a + i — y, P + 1 — y, 2 — y, x) presents terms which increase indefinitely when y tends to the value 2 —j- m. We will now seek the limit of this integral when y tends to 2-\-m\ (m-f-2 — y)F{ot + i — [m-\- 2 — P + i — y> 2 — r> *) (a + I - y)(/3 +i — ÿ)_ t + 1.(2-y) ^ + The first term in the [ ] to become infinite is («+1 — r)l(tf+^+I — y)(P+ 1 — r)\(P+m + l —y) xm+, Q+ i)| (2 — y)\{m + 2 — y) Letting now y tend to the value 2 -j- my the terms in I, x, x2, . . . xm vanish, and the remaining terms are finite; for example, the term in xmJrl becomes (a — m — i)|(flf - i)(P - m — i)|(/? - i) + m -|- i | . — m | 9 where — m| = (— m){— . . . (— i). The series tends there- fore towards the value 1 (a — m — i)| (a — i)(j3 — m -j- i)l(j3— i) (— \)m m |. m -f- 11 F {a y fiy y y x\.LINEAR DIFFERENTIAL EQUATIONS. '252 If, therefore, we consider the integral ^ = (m -j- 2 — y) {a—m—1 )!(<*— 1 )(fi—m—i)|(/?—1) x'~vF(a+i—y, /3+i—y, 2-y, x), we see that for y = 2 m this becomes F(a, /?, y, x). As before, a new integral can be found by seeking the limit for y = 2 -f- m of the expression this is I — P, y, x) 2 m — y ’ i ma \ i dF . dF . dF iog^(«,AJ-,*) + s + ^ + 2j?, in which y = 2 -J- We can then express this new integral in terms of the already known integrals 0, and 02 by the same process as that above employed. We operate in exactly the same manner if y — a — /?, or a — /3, is an integer. 18. We will now apply the general theory to the case of the equa- tion (I)' which is obtained by making <* = /?=£, y — This equation presents itself in the theory of the elliptic functions when we wish to define the complete integral of the first kind, K= fl-r- —= , Jo - x‘){x-k'S) as a function of the modulus for imaginary values of the latter. The equation has been studied by Fuchs* from this point of view, and studied directly by Tannery.f The results which these writers have obtained are derived with- out difficulty from the general case. For this example we will em- ploy Tannery’s notation. * Creile, voi. 71, p. 91. \ Annales de TEcole normale supérieure, 2® sèrie, t. viii.COURSAT: HYPERGEOMETRIC SERIES. 253’. Equation (i)' admits an integral, P say, which is uniform throughout the plane, provided the path of the variable never crosses^ the line i------1- oo. Inside the circle C0 we have /-= Fa, i,= i +2"[- 23 4s;6 ; ’>] Let (P(x)=zF(i, i, i, x). In the space £0 we shall have, in like manner, P = Since y = i, we shall have a new integral containing a logarithm: Denote this integral by Q. This integral will be uniform throughout the plane if the path of the variable does not cut either of the lines. — oo-------o, I---------|-oo. Write ip{x) = 5 1.3... (2 m — 1 )' = i L 2.4. 2 m Bmxm, I+“+^‘ + · * * + I 2m — 1 1 I 2 ~4 then inside the circle CQ we have Q = 4tp(x) + 4>(x) log 1 2m ’ and in the space E0, Q = j0 tin) [l0s * - i°g (1 - *)]+4* (^y) I · Equation (1)' does not change form when we replace j by 1 - x,, and it therefore admits two other integrals, Pr and Q\ which, subject to the same conditions as the preceding ones, are uniform through^ out the plane. In the circle Cx we shall have Pf = 0(1 —x), Q' = y — 1 ; we find then F = - 7t - r(j)j P—u& also, We have now p,=ihglp_l Tt 71 p = 4jog2p,_L 7t 71 (22) p' = ^p-\q. 16 log2 2 — 7T2 4 log 2 _ G' =------------P-l—r-Q, 7t />= 7T 7T ?r 0 = 16 log* 2 - «* ^ _ 4 log 2 These are the relations found directly by Tannery. These formulae suffice to integrate the equation (i)', a fact which has been already observed in the general case. As an application, we will take the example treated in Tannery’s memoir. Suppose we start from a point a (Fig. 6) very near the point x = 2, and in the upperGO UR SAT: HYPERGEOMETRIC SERIES. 25$ half of the plane, and describe a closed path including in its interior the points o and i. This path can be reduced to a path aMA, where A is a point of the line o-----i, say the point x = followed by two loops de- scribed successively in the direct sense round the points x — o and x = i, and then the path AMa. Let us start from the point a with the integral Pf; we will of course arrive at the point A with the same integral. Now to find the change in P' when we go round the loop x — o, we replace P' by its value, 4 log 2 7t After describing the loop o, P returns to its original value, but Q changes into Q -f- 2niP, and therefore Pr becomes F - 2 iP. Now to see how this integral behaves on going round the loop i, we replace P by i> = ilog2p,_1 7t It F will not change, but Q! becomes Qf + 2niP, so that F — 2iP changes into — 3 P — 2 iP. We return to a, therefore, with the integral256 LINEAR DIFFERENTIAL EQUATIONS. If we start from a with Qf, we arrive at A with 16 log2 2 — 7T3 4 log 2 Q- After the loop o the integral will be represented by 16 log2 2 — 7T2 „ 4 log 2 ^ -----------P-^—^Q-Silog2.P-r TC 7t or, what comes to the same thing, by Q’ — Si log 2 . P; or, again, by 1X + 8i log 2 _ 32* log” 2 p, ^ n it After the loop 1 we shall have Q, it + 8* log 2 2in {n -f Si log 2) — 32i log2 2 p*\ that is, 2i{?t 4- 4i log if n, 7i 8? log 2 7t ' 7t ^ * We will of course return to a with this same integral. The equation (i)r admits also two other integrals, susceptible of development in series, analogous to the preceding ones. First we have the integral = 0 x which is uniform throughout the plane provided the path of the variable does not cut the line o---------f- 00. Denote this integral by P". To find another integral, suppose first y = 1, a = and let /3 differ a little from the value The differential equation will admit the two integrals V GO U R SA T: HYPERGEOMETRIC SERIES. 257 The expression will also be an integral. The limit of this expression when fi = J will be a new integral Qf/, viz., also, e" = - + 7=4)· C" = ~=j[-^(r^)‘»gc-·>)+#■(—)]· Q" is uniform throughout the plane under the same conditions as P". For all values of .r in the upper half of the plane we have the relations (23) „ 4 log 2 i P = —F — - Qn, 7t 7t 1 p> _ ^ 4Z 2 jr)//_|_ ^ 7T for values of in the lower half of the plane we have (24) 4 log 2 I P =—~P"~ -Q", 7t 7t p, _ Tt -f- 4? log 2p, _ 1_ 0„' n 7t These formulae are established, like (21), by starting with the rela- tions (13) and (17) which exist in the general case. Remark.—Formulae (23) and (24) do not appear to be in accord with the formulae given by Tannery (loc. cit. p. 188). This arises from the fact that the functions Fr and Q" here used are not pre- cisely the same as the corresponding ones employed by Tannery.258 LINEAR DIFFERENTIAL EQUATIONS. Instead of the system of integrals F' and Q" consider the following system: e,"= ^[-(»(i)iog» + 4i(.(i)], where the argument of x is supposed to lie between — n and n. We find easily for tlie upper half of the plane P" = %p;\ Q” = iQ" - np;'\ and substituting these values in the second of (23), we have · 4 log 2 ^ „ 1 _ _ 7t 1 7t^ 1 which is identical with Tannery’s relation. 19. The formulae established in what precedes enable us to deter- mine whether or not the differential equation admits an algebraic integral and whether its general integral is or is not algebraic. This question has been treated by Schwarz (Crelle, t. 73, p. 292), who, by employing the differential equation of the third order satisfied by the ratio of two particular integrals of equation (1) and by the aid of Riemann’s surfaces, was led to a question in spherical geometry where regular polyhedra presented themselves. Later, Klein con- sidered the more general case of a linear differential equation of the second order with rational coefficients. In the following, Schwarz’s results are arrived at by quite elementary considerations. If equation (1) possesses a single algebraic integral, this integral must reproduce itself to a constant factor près, when we turn round a critical point. In the domain of the point x = o it will be represented by one of the integrals 0,, , x + Çx y,2ÓO LINEAR DIFFERENTIAL EQUATIONS. The ratio admitting only a limited number of values, the same is true of this logarithmic derivative, and as besides it admits of no other singular points than critical points and poles, it follows that it is an algebraic function of x. This being true for any integral, let zx and z2 denote two inte- grals of (i); we have then the relation The product of any two integrals, zx, z9, is then an algebraic func- tion of x. If zx and z2 are two integrals, then zx and z1-j- z9 will also be two integrals. The product zx -f- zxz2, and consequently zx9 will be an algebraic function of the variable. Every closed path starting from and returning to a point A can be reduced to a series of loops described round the points x — o and x — I ; we are therefore led to examine the behavior of the preceding ratio when we describe one of these loops. We will take forjj/j and the integrals x -f- C'02 and describing the loop x — o in the direct sense, we shall, as we have seen above, return to A with the integral Cx2 , where z2zx — zxz/ = Hx~y(i — x)y-a-P-'j which can also be written *1*1 = Hx~y(i — z, z< c, — c + c,r^a + ^ ~t~1 ■r(a + /? + i-y)r(i - r) (l _ r(«+i - y)r{0+ i - yf h Cx —COURSAT ; HYPERGEOMETRIC SERIES. 2ÔI C Let p be the initial value of , and p' its final value; then y oe-~iC-y) _|_ r{a + fi+l- r)r(l - y)fl - H ~ r(a + I — r)r{/3 + I - r)\ I ’ Making r(«+/?-(- I — y)r{I — y) _ r{a+l-y)r(fi+l-y) ~ this last formula becomes {A) p' — a = K{p — a). If the loop were described in the negative sense we should have (A)' p' - a = -^(p - «). - Now let the variable describe the positive loop around x = 1; the integral ¿70!-f- £7'0a changes into ¿7^ + ¿7/0,, where c:= c+c ~ go - r{y — ct)r(y — py Denoting again by p and p' the initial and final values of the ratio ^7-, we have p' Making + r(r)r{y~a-P)h-^y-^\ p ' r(y — a)r(y — P)\ /’ r(y)r(y — a — P) _ r(y-a)r(y-p) ~2Ó2 LINEAR DIFFERENTIAL EQUATIONS. we have, from the last equation, (B) if the loop is described in the negative sense, we have (BY The problem which we seek to solve can be stated as follows: Having given a series of quantities such that each is deduced from the preceding by one of the formulce (A), (A)', (B), (B)\ in what cases can we arrive at a limited number of different quantities, the order in which these formula are successively applied being perfectly arbi- trary ? The geometrical method seems to conduct most readily to the sought result. (20) Draw in the plane two rectangular axes OB and Op. Represent as usual the quantity p = B + ip by the point whose co-ordinates are B and p; let A and B denote the points (on the axis OB) which represent the quantities a and b respectively. Let M be the segment is turned round the point A through an angle go (go = 27t(y — 1)), we obtain the segment AM', of which the ex- tremity M' represents the quantity p. If the loop were described in the negative sense, we should come to the point MJ obtained, as before, by making the segment AM turn round the point A through an angle go, the turning, however, being in the negative sense* point representing the initial value of p, and M' be the point represent- ing the value of p' obtained by letting the variable travel round the loop x — o in the direct sense: p' — a — K (p — a). Fig. 7. The quantity p — a will be repre- sented by the extremity of a segment, drawn through the origin, equal and parallel to the segment AM; if thisGOURSAT: HYPERGEOMETRIC SERIES. 263 Fig. 8. When the variable describes a series of loops round 0, the different values of p will be represented by the vertices of a regular polygon inscribed in the circle of radius AM, one of these vertices being the point M. Suppose now (Fig. 8) that the variable describes a loop round the critical point x — 1. Let M be the point denoting the initial value of p; ^ will be represented by the point Mlf ~ — b by the extremity of a segment drawn through the origin equal and parallel to BM\, and K' — b^j by the extremity of a segment equal and parallel to the segment BMxy which is simply the segment BMX turned round B through an angle go' = 2n(a -f- — y). Consequently the point Mx represents the quantity —ty and the point Mf represents the quantity p'. Let the variable describe several successive loops round the point x — \ \ the point ^ will then coincide successively with vertices of a regular polygon in- scribed in the circle whose centre is at B ; the point p itself will be on the circumference of a circle transformed from the preceding by reciprocal radii vectores having the origin for the pole of the trans- formation, and unity for the modulus. If we apply this transforma- tion to all of the circles having B as the centre, we evidently obtain a system of circles passing through two fixed imaginary points. It is easy to see what are the point-circles of this system : first is the origin, which corresponds to the circle of infinite radius with centre at B, and then the point C, = corresponding to the circle of radius zero. All the circles of this system are conjugate with respect to these two points, which are two double points of the homographic transformation2©4 LINEAR DIFFERENTIAL EQUATIONS. Before going farther we shall demonstrate the following geometrical theorem : Having given a circumference and n points, A, B, C, . . . , L, upon this circumf erence which are so disposed that on making an inversion, taking for the pole a point O of the plane, the corresponding points A', B\ . . . , L! are the vertices of a regular polygon, if O' is the con- jugate point to 0 with respect to the circumfere nee, all the points of the circumference described upon 00' as a diameter and perpendicular to the plane of the figure will possess the same property as the point O. Let 5 be a point of this circumference; then, from an elementary property of the inversion, consequently A'B' = AB OA . OB p, aa = ab SA . SB™1 ’ fiA - ffi\ x OA v OB A'B' ~ \P) A SA X SB' _ . . , . OA OB . Each of the ratios is constant, and so the same is true of the ratio ^7^7. If the polygon A'B’C'D'E'F’ has its sides equal, the same will be true of the polygon . Q. E. D.GOURSA T: HYPERGEOMETRIC SERIES. 265 Fig. 10. Returning now to the proposed question, consider the circum- ference described upon OC (Fig. 10) as diameter in a plane perpen- dicular to the plane of the figure. Let 5 and S' be the points of intersection of this circumference with the perpendicular to the plane of the figure through the point A. Conceive a sphere •described upon SS' as diameter, and make a projection upon this sphere, taking 5 as the point of sight. All circles having their centres at A are projected upon circles having for poles the points S and SAs to circles con- jugate with respect to two points O and Cf they are projected upon circles having PP' for axis. It follows from what precedes that having given upon the surface of the sphere a point m representing the value of p, we shall find the point m' representing a new value •of p by turning the point m through an angle go round SS' or through an angle go' round PP'. This is evident for the axis SS'; as to the axis PP' it suffices to remark that, having given two points M, M' in the plane representing two values of p, of which one is deduced from the other by one of the formulae (B)y (B)', if we make an inversion with the point O for pole, the angle MXBMX is equal to go'f and, from the preceding theorem, the angle mpm' must have the same value, p denoting the foot of the perpendicular let fall from m upon the axis PP'. We can now replace the above enunciation of our problem by the following: Having given two diameters SS' and PP' in a sphere and a series of points upon the surface which succeed one another hy a law such that we pass from any one to the following one by turning through an angle go round SS' or an angle go' round PP', in what cases can we arrive at a limited number of points, the order in which the construc- tions are applied being perfectly arbitrary ? All depends, evidently, upon the order of symmetry of the axes SS' and PP' and on the angle, V, between them. Let Y = y — a — f = Pf266 LINEAR DIFFERENTIAL EQUATIONS. P P The fractions - and being irreducible, SS' will be an axis of sym- metry of order q, and PPf one of order q’As to the angle V, we have from the preceding figure V = 2 AOS, cos A OS = cos V — OA _ QA os ~ VOA X OC OA 2^ — 1 = 2 ab — OA OC’ 1. Replacing a and b by their values, we get v_ . n« + fi +1 - y)n1 - r) , r(y- <*— fir(y) r(pc —(-1 r)r(fi+ I -r)x r(y- a)r(r - ft) lr or Tr sin (y — ol)tz sin (y — /?W COS V — 2 ----—------------—---—— T sin yn sin (y - a - /?)* Remark.—When we diminish or increase the value of one of the quantities o', f3, y by any number of units, q and q' do not change, neither does cos V. We may then suppose 1 — y and y — a — ft to lie between o and 1, and can thus conduct a — fi to lie between — 1 and -j- 1 ; but as there is symmetry between the elements a and /?, we may suppose a — fi to lie between o and 1, a supposi- tion which will be adopted in what follows. (21) The geometrical representation upon the sphere is not possible unless the point A lies between the points 0 and Cy or, what comes to the same thing, unless the product sin (y — a)7t sin (y — /3)7t sin yn sin (y — a — fi)n is comprised between o and 1 In the contrary case we can demon- strate directly that the ratio p can take an infinite number of values in each point of the plane. The establishment of this last state- ment is based upon the following remarks : I. There exists a circle having its centre at A, and conjugate with respect to the segment OC, If the point M which representsGO U R SA T: HYPERGEOMETRIC SERIES. 267 the initial value of p is on this circle, all the other points derived from it will likewise be on the circle. If the point M is outside or inside the circle, the derived points will themselves be all outside or all inside. II. Supposing that we have taken the point A as origin, let us denote by z and zf the quantities which are represented by the points M and M* in the new system. The first transformation is zf — kz\ in the second transformation z' is connected with z by a relation . clz —I— b z = where a = a ifi, c — a —f- b — y -(- id, d—y-f- id'. Let us seek the locus of the points z such that mod z — mod z\ we have . . mod (az 4- b) m0d * = moTfc+H) ’ az + b = (ax — /3f + y) + i(&% + ay + #), mod (az -f- b) — V(a* + + y*) + + ny + A mod (cz -f- d) = V(an -f- fi/9)(x* + y9) + m>x + n'y ~l·"P * The equation of the sought locus is («’ + /i3)(V +/) + mx+ny + p = + /T)(x3 + yy + {m'x + n'y +p')(x' +/)..268 LINEAR DIFFERENTIAL EQUATIONS. This is a quartic curve having the circular points at infinity for double points. The circle with centre A (Fig. u), conjugate to the segment OC, is evidently part of the locus, and as the points 0 and C also belong to it the quartic breaks up into two circles, viz., the circle (A) and another circle through the two points 0 and C. If the point M, which represents the value of zt is on the ex- terior or the interior of the two circles, we have mod z' < mod z; but if zr is outside one circle and inside the other, we have mod zr > mod z. This granted, let M be the point representing the inital value of p, and suppose M outside the two circles (A), (A)'. Applying to the point M the second transformation as often as it gives us different points, we shall arrive at a certain number of points upon a circle conjugate to the segment OC. Suppose M, that one of these last points which is nearest A ; M1 must therefore lie inside the circle A\ and we have AMx < AM. Apply now the first transformation to the point Mx so as to obtain a point M\ outside the circle Af; we shall have AM2 = AMX. Apply the first trans- formation to the point M\; we shall find as before a point M% nearer A than the point , etc. We shall find by continuing this process a series of points M, Mx, Ma, M%, . . . following one another according to such a law that we shall never have a new point coinciding with one of the preceding ones, and this will go on indefinitely. The ratio p takes therefore an infinite number of values. (22) The question in spherical geometry to which we have been led, which, as we shall presently see, is identical with the question to m2 mGO URSA T: HYPERGEOMETRIC SERIES. 269 which Schwarz was led, has been solved by Steiner.* We can also de- duce the solution from a memoir by Jordanf upon Eulerian polyhedra. There are two particular cases where the solution is immediately perceived. 1. If we have two axes of binary symmetry, the angle between them must be commensurable with 2n. 2. If we have one axis of binary symmetry, and one axis of order of symmetry = K perpendicular to the first, we shall always end with a limited number of points. Discarding these particular cases, let us suppose that we have two axes of symmetry PP\ SS', one of which, PP', is of order K{K^_ 3). The repetitions by symmetry of the axis PP' will be axes of symmetry of order K of the figure formed by the symmetric repetitions of a point M of the sphere. These repetitions ought to be limited in number. This comes to studying the figure formed by taking P itself as the point of departure. We can demonstrate without difficulty that, if we arrive at a limited number of points, these points are the vertices of a regular polyhedron. The axes PP', SS' ought then to be the axes of symmetry of a regular poly- hedron having one of its vertices at P. It is evident that the plane of the two axes will be a plane of symmetry for the polyhedron. This necessary condition is also sufficient. Conceive in fact this regular polyhedron, and consider the spherical polygons on the cir- cumscribed sphere which correspond to its different faces. The repetitions by symmetry of one of these polygons will always lead to an analogous polygon; as, further, one of these polygons can only coincide with itself in a limited number of ways, it follows that the symmetrical representations of a point will be limited in number. Example /.—Let y = a = o. We have two axes of binary symmetry, It is sufficient that a be commensurable. This can be verified directly. The differential equation is cos V — 2 cos 7ta cos nfi — I = cos 2an. * Crelle, vol. xviii. p. 295. flbid. vol. lxvi. p. 22..27 o LINEAR DIFFERENTIAL EQUATIONS,. or 2.X (i From this we have d x(i — x dy adx Vx(i — x) sin 1 If a is commensurable, we deduce from this an algebraic relation between x and y. Example II.—Schwarz’s example : There is one axis of binary and one of ternary symmetry; their angle is given by This is exactly the angle which in a regular' tetrahedron is in- cluded between the altitude and the line joining the middle points of two opposite edges: the integral is then algebraic. (23) The preceding results can be placed in a little different form and so bring better into evidence their identity with Schwarz’s results. Suppose, as explained above, that we have conducted the three numbers 1— y, y — a — /3, a — fito values lying between o and I, r = 1 > « = — tV> £ = i> y — a — §, x —/? = y — « — P=i- I . 2 7t sin — 3 4'GO URSA T: HYPERGEOMETRIC SERIES. 27I and denote by A, yu, v the three positive numbers, less than unity, so obtained: A — 1 — y9 p = y — a — r=a — ft. We will now find out for what sys- tems of values of these numbers the integrals are algebraic. As before, let SS' and PF (Fig. 13) be the two axes of symmetry. Construct a triangle having for base SP, and for angles PSQ— (1 — y)7r, SPQ. = (y — <* — P)n· For the angle Q we have cos Q — sin P sin 5 cos (SP) — cos P cos 5 2 sin (y — a)7t sin (y — 0)n “1 )7t ~ J _ sin yn sin (y — a — + cos yn cos (y — a — /3)n, = sin yn sin (y — a—J3)n or, finally, cos Q = cos (a — 0)n. The angle Q is comprised between o and n, and further we have supposed a — fi comprised between o and 1 ; .·. Q — {a — fi)n. The three angles of SPQ are then 5 = A 7t, P = fxn, Q = vn. What conditions should this triangle satisfy? In the particular case where y=^,a-\-j3 = oy S and P are right angles; the point Q is then one of the poles of the great circle SPSfPr. If we con- struct a double pyramid having for vertices the poles Q and Qf, and for base the regular polygon of which 5 and P are two vertices, the three planes of the trihedron SPQ will be three planes of symmetry for this double pyramid. If the axis SS' is a binary axis, and if the arc SP is a quadrant, the three planes of the trihedron will still be three planes of symmetry for a double pyramid whose vertices are at the points P and P', and whose base is in the plane SQS Q'.272 LINEAR DIFFERENTIAL EQUATIONS. These singular cases being examined, let us see what takes place in the general case, supposing there exists a regular polyhedron hav- ing 55' and PP' as axes of symmetry and having one vertex at P. The three planes OSP, OPQ, OQS will be three planes of symmetry for this polyhedron; it is evident from what precedes that OSP is a plane of symmetry. The point P", symmetrical to P with respect to the plane OSQ, is one of the vertices of the polyhedron; as we can start from P' instead of P to find the remaining vertices, it is clear that the plane OSQ is a plane of symmetry. So for OQP; viz., the point S" is an extremity of an axis of symmetry of the same order as the axis SS'; if we replace the axis OS by the axis OS", we shall obviously form the same figure, and consequently OQP is a plane of symmetry. Thus the three faces of the trihedron OPQS are three planes of symmetry of a regular polyhedron. The converse is easily demon- strated ; in fact, every body admitting these three planes of symme- try OPS, OSQ, OQP will also admit OSP" as a plane of symmetry. If we take the body which with respect to the plane OSQ is sym- metric to the first body, then take the body symmetric with respect to the plane OSP' of this new body, this last will coincide with the first; but these two operations are equivalent to turning the body through an angle PSP" = 2(1 — y)n round OS; .'. OS is an axis of symmetry for the body. Finally, then, in order that the general integral shall be algebraic, it is necessary and sufficient that the three planes of the trihedron OSPQ shall be the three planes of symmetry of a double pyramid or of a regular polyhedron. It is necessary first that the three angles Xn, jin, vn shall be the angles of a spherical triangle; this requires X + M + r> I, l > fl + V, fi+l> v + \, V + 1 > * + ¿V. These conditions are equivalent to the above-found condition that sin (y — a)n sin (7/ — (S)n sin yn sin (y — a — 0)n shall lie between o and 1.GOURSA T: HYPERGEOMETRIC SERIES. 2/3 It evidently comes to the same thing whether we consider the triangle PQS or one of the triangles PQSQPfS'y QSP' ; the angles of these triangles have the following values : PQS..........................X 7t }l7t V7t, PQS'.........................Xn (i — ju)7r (i — v)ir, P'QS..................(i — X)7t ¡X7C (i — r)7i, PQS'..................(i — X)7t (i — P)7t V7t. We will choose the triangle for which the sum of the angles is the least. Let Xr7Cy fxfny v'n be the angles of this triangle, and let X", p", v" denote the numbers A', //, vf arranged in descending order of magnitude. The following table gives the systems of values of X", p", v" in order that the general integral shall be algebraic: A" M" v” I, . · • i i ii Double pyramids. 11, · . ■ i * *1 >- Tetrahedron. Ill, . . • 1 * *J IV, . . • i i i 1 V, . . ■ 1 i r Cube and octahedron. VI, . . • i i il VII, . . 2 • ■§■ 1 3 i VIII, . . • 1 i 1 IX, . . • i 2 z X, . . XI, . . • 1 2 • 7T i t * f - Dodecahedron and icosahedron. XII, . . • f i i XIII, . . 4 • ^ i i XIV, . . . i 1 XV, . . • Î 2 T 1 3 J (24) We have neglected the intermediate case where the circle274 LINEAR DIFFERENTIAL EQUATIONS. described upon OC as diameter is tangent to the perpendicular SS'; that is to say, the case where the two homographic transformations, (A) p'^-a = K(p-a), <*> have a common double point. The double points of the first are £> = a and % = oo; those of the second are % — o, £> — If we b refer to the values of a and by which may become zero, but not in- finite, we can see that there are three different ways in which the two homographic transformations may have a common double point. (1) a — o; this will happen when one of the numbers a -l· i — y, P I — y is zero or a negative integer. (2) b — o; this will happen if y — a or y — § is zero or a nega- tive integer. (3) ab — 1 ; this condition gives sin (y — a)7t sin (y — p)n = sin yn sin (y — a — /8)nr, or cos {a — p)7t — cos {a + P)tt9 from which (a — P) ± (a + p) = 2m; one of the numbers a, p must then be an integer. In order to see what takes place in each of these cases, suppose the common double point at infinity; the homographic transformations will be defined by z' — z0 — K(z — zQ), z' — zx = Kr {z — £,), the points z0 and zx being the other two double points. These give rise to a simple geometrical construction. Having given in the plane a point M representing z, we shall find the point Mf repre- senting zr by turning the radius z^M through an angle go round zof or the radius zrM through an angle go' round zx. Starting from any point of the plane, it is clear that we can apply the constructionsGOURSAT: HYPER GEOMETRIC SERIES. 275 successively in such an order that we shall never arrive at a point already found ; for example, we can arrange so that in following the process the radius z0M never decreases. The common double point is a case of exception, as it will always coincide with itself. There will then be a particular integral whose logarithmic derivative has only a single value in each point of the plane ; this derivative is therefore a rational fraction. Under the adopted hypothesis, i.e., where the numbers a, /?, y are real and rational, the corresponding particular integral will be an algebraic function. We see, further, that there is no other such integral. We must, however, remark that this will cease to be true if the other two double points are the same ; in this case all the integrals will be algebraic. This case arises when we have simultaneously a — o, b — o. For example, consider the differential equation where d^y dy x(y- ^+(* -2x^Tx-^y = °’ 1 2> r = 3 n · The equation admits a particular algebraic integral^ = —-, but the \ x general integral, y = C C' sin 1 Vx V x V x is transcendental. On the contrary, take the equation —+( dx^X I 2x\ dy , 20 3 3 ) 9/_°’ 1 II = r = b where We have simultaneously a — o, b — o; the equation admits the two particular integrals, and so the general integral is algebraic.276 LINEAR DIFFERENTIAL EQUATIONS. (25) In what precedes we have supposed the two integral's (f>x and to be distinct ; Le., neither of the numbers a, /3 is zero or a negative integer. If this circumstance does, however, arise, we can replace. 0a by another integral, for example 05, and operating as above, we shall be led to consider two homographic transformations. These two transformations will have a common double point, since equa- tion (1) admits as an integral an entire function whose logarithmic derivative is a rational fraction. If the other two double points are different (which will generally be the case), there will not be any other algebraic integral. But if these two double points coincide,, the general integral will be algebraic. In order that this may be so, one of the elements a,, ft must be zero or a negative integer, and the other a positive integer. Thus the differential equation admits two particular algebraic integrals, and therefore the general integral will be algebraic, viz.: (I) The memoirs of Gauss and Kummer on the hypergeometric series contain a great many formulae not found among those given above, and which only exist when the constants a, fi, y satisfy cer- tain conditions. The general type of these formulae is where oc — — 2, /? = i, y = Ì, Part Second. X~f(\ — x)-tF(a, 0, y, x) — t? {1 — ty F (a', 0', y', i), where t is an algebraic function of x. The function xt{\ — xyt?{i — ty F (a!, 0', y', t)GOURSAT: HYPERGEOMETRIC SERIES. 277 is then an integral of the differential equation (1) x(i ~ + -(« + /?+ 1)*] = °· We are thus led to seek the cases in which equation (i) admits integrals of the above form. Such is, very nearly at least, the path followed by Kummer. Kummer, however, indicates no means of finding all the cases where such integrals exist. This question we propose to treat in what follows. (2) We will adopt a slightly different point of view from that of Kummer. Let us denote, as Riemann does, by P(x) a non-uniform function of x possessing the following properties: (1) It admits in the entire extent of the plane, or of the sphere, only three critical points, viz., x — o, x — I, x = oo ; it is holomor- phic in any region of the plane having a simple contour which does not contain either of the points x — o, x = i. (2) Between any three branches, P9 P\ P", of the function there exists a linear homogeneous relation CP + C'P" + cmPm = o with constant coefficients. (3) Each branch of the function is finite for x = o, x = i, and also for x = oo when we multiply it by a proper power of x or i — x, Riemann has shown that certain branches of the function P can Be expressed by products such asx~p{\ — x)~qF(a> /?, yy x), In the light of the more recent analysis, this can be demonstrated by a simpler method than that adopted by Riemann. It results, in fact, from a theorem given by Tannery,* viz.: The different branches of the function P are integrals of a linearf differential equation of the second order having uniform coefficients, and having no other critical points than the points o, I, oo ; further, all these integrals are regular in the region of a critical point. This differential equation, as Fuchs has shown, is of the form <2) ^(i-*)’^+[/-(/+*»(l-*)^ + (Ax· + Bx+ C)P = o. * Annales de lÉcole normale supérieure, 2e série, t. iv. p. 130.27% LINEAR DIFFERENTIAL EQUATIONS. We pass from (i) to (2) by writing y — xp{\^xfP\ A, B, C, l, m are given by the formulae (3) \ m — 2?+<* + /? + i — y, A ={p+q+oc){pArq-\-0)^ \ C —p{p — i+y), A + B + C = q(q + a + /J — y). Conversely, we pass from (2) to (1) by making P — x~\i — x)~ qy ; ot, ft, y, p, q will now be determined by aid of equations (3) in terms, of A, By C, ly m. Riemann’s theorem can be established in the same way. [In order to completely define the function P it is necessary to- add the following condition, viz.: if P\ P" are two linearly distinct branches, the determinant P, P" dP dP" dx ’ dx must be different from zero for every point of the plane other than the points x =. o9 x = 1. If, in fact, this determinant vanished for x — ay the point a would be an apparent singular point for the differ- ential equation. Consider, for example, the integrals of equation (1); it is clear that they satisfy the conditions which serve to define the function P. At first sight, the same is true of the products ob- tained by multiplying each of these integrals by the factor (x — 2) nevertheless, these new functions satisfy the differential equation -v* - -a- + I [r — (« + + 0*]C* — 2) — 2*(i — x)\(x — 2)~x + \2x{1 — x) + [(« + P + I)Jr — J'] (x — 2) — afi (x — 2)s} Z — O,. which is not comprised in the form (2); but the determinant D is· evidently zero for the point x = 2.]GO URSA T: HYPER GEOMETRIC SERIES, 279 Remark that having given a system of values for Ay B, C, /, my there result four systems of values for a, fiy yypyq\ equation (2), and consequently equation (1), admits then four integrals of the form x~p{\ — x)~qF(ay /?, y, x)—a well-known result. Let t be a new variable given by the equation x = 0(/). If, when in equation (2) we change the variable by the relation x = the function P satisfies the same relations as relatively to x, we ought to find a new differential equation (4) analogous to (2), viz.: d'P dP (4) f{i - v - o-*)^ + (A'i2 + B't + C')P= o. It is easy to see that the problem treated by Kummer is herein contained; if, in fact, equation (1) admits the integral x*{l — /3', y', t), equation (2) will admit the integral -tyF{a\ /3', /,/). If then in equation (2) we change the variable by the relation x = 0(/), we must obtain an equation of the form (4). The problem to be studied may now be stated as follows: Required to determine for what values of the constants A, B, C, /, m9 there exist transformations such as x — {t)y and , dx „ d*x writing x = —, x = —, we have dt df dP_ _i dP dx ~~ x! dt ’ æp _ 1 r , d*p __ „ dP~\ dx3 ~ df ~X dt A'280 LINEAR DIFFERENTIAL EQUATIONS. and the new equation is ** (i - xf d'P [7 - (/+ m)x x,q di' + L , . x"xq(i - xfl dP <■-*)--------p—in -f- (Ax9 -(- Bx —j— C)P = o, where x, xf, x,r are to be replaced by their values as functions of t. In order that the new equation shall have the desired form, we must have (5) (6) X9(l — xf fii-ty x,q(Axq -f Bx + C)~ A'? + B't + C l - (/ + ni)x , _ _ 1' - {1' + my x(l — x) X xr ~~ t(l — t) This last equation can be integrated once, since it can be written in the form J, log [V(i - 4-] -~ log P = i log [/”(. - /)-']. We can then replace equations (5) and (6) by a system of two differ- ential equations of the first order, viz.: (5) (7) VAxq + Bx +~C dx _ V'A'f-f B't + C x{i — x) dx t{\ -1) Kdt dty x1 (1 — x)m ti' (1 — ty ■ The constants A', B\ Cr, /', K are arbitrary, and we have now to see in what cases we can determine them so that equations (5) and (7) shall have a common integral. (4) The following five transformations are readily seen to exist in all cases, whatever be the values of A, B, C, /, m; viz.: x — 1 — t, ^r=—, x = i — t' X = t — I / — I ’ X = This result is also evident if we refer to the definition of the func- tion P. If a multiple-valued function satisfies the required condi-GOURSAT: HYPERGEOMETRIC SERIES. 28l tions when we take x as the variable, it is clear that it will also satisfy these conditions when we take as variable one of the quanti- 1 1 x x ~ 1 r 1 1 ties 1 — x* — , ----, ------, -----; for, the values of any one of these quantities for x — o, 1, 00 will also be o, 1, 00 taken in a cer- tain order. From these considerations arises a very simple method •for finding Kummer’s twenty-four integrals, but we will not take up that point here.* We can show now that there can exist no transformation of the first order between the two variables other than the above-mentioned ones. Consider, in fact, the transformation __ at -f- b X “ ct + d' If the values of t corresponding to the values x = 0, x =1, x= 00 :are also o, 1, 00, taken in a certain order, the transformation will ob- viously be one of the preceding ones. Suppose on the contrary that for x = o, for example, t takes a finite value tx different from o and from 1. Let x (Fig. 14) describe a small loop round the origin ; the point t will describe a small loop surrounding tx. After describing such a path, any integral whatever of (4) will return to its initial value: the point x = o itself can then not be a critical point for equation (2), which is contrary to hypothesis. We see further that, if for proper values of A, B, C, /, m, we can make the transformation x = (/), we can also, for the same values of the constants, make the five other transformations * This method has been given in Chapter VI.—Author.282 LINEAR DIFFERENTIAL EQUATIONS. From the form of equations (5) and (7) we can deduce still further consequences. If for the values of A, By Cy /, m we can make two different transformations x = (x) = Thus, from a transformation x—(i) we would be able to deduce all those which would be obtained either in re- placing x by 1 — x, ^, x— 1 y X x 9 or *n rePlac*nff t by 11 t t — 1 . , . 1 — t, —, ——, ------, —-—, or in making the two transforma- t I f i 1 t tions simultaneously, making thus in all thirty-six transformations,, not all of them, however, being different. We will determine, at the same time, all of these transformations, together with the inverse transformations. (5) If equations (5) and (7) admit a common integral, this integral will also satisfy the equation obtained by dividing the equations (5) and (7) member by member, viz., VAx*-\-Bx+C. x?-' (1 —x)9"-' — K VA'f+B't + Cf. tl'~\ 1 -ty~K Taking the logarithmic derivative, we have 2Ax -f- B _j_ l — 1 m — 1J dx . / — 1 m 2(Ax* -|- Bx C)' x ' x - -jtY_-2A'A+]y__ , ” L2{A}f\B't-\-cry V- I + £t]· Replacing, from equation (5), dx and dt by the quantities to which they are proportional, and squaring, we have, finally, (8) |(2 Ax + B)x(x - 1) + 2 {Ax' + Bx + C) [(/-!)(*- I)+(M-lK)P (Ax*+Bx + cy {(2A't + B’)t{t - 1) + 2 (A’f + B’t + C') [(/’- ip- 1) +(m' (A'f + B't + cyGO UR SA T: H YPE R GEOMETRIC SERIES. 28j If this relation is not an identity, we see that x and t are connected by an equation of at most the sixth degree in each of the variables.. Equation (8) will be an identity if the two members reduce to zero or to equal constants. 1. In order that each member of (8) shall be identically zero it is necessary and sufficient that VAx'1 + Bx -f- C . xl~ x(i — x)m~z and VA'f + £'t+ C'. 1 - ty-* reduce to constants; Ax*Bx C = o should then admit of no other roots than o and 1, and the same should be true of A'f+ B't + C'= o. All of the possible combinations are given in the following table: 1st case. 2d case. 3d case. 4th case. 5th case. 6th case. A+B= 0, A = 0, A = 0, A = C, A — O, B — O, C — 0, C = 0, B+C= 0, B—2 A =0, B — O, C — O, II 3 II 5*- l=%, m=1, l— 1, l=i, m=0, l—m— 1, 1=0, m—i, X = ± +1 II HH II =Sk M — ± 1, v=± 1, X — dz I» l·1 — ± i> II H- y = ± h A = ± v, X = ± y, M — ± r- In the last two horizontal lines we have given the corresponding conditions to be satisfied by the elements a, /?, y, where X = l — y, ju = y — a — /3, v — fi — a. The conditions for A\ B', C/', tri will be identical in form with the above. In each of these cases equations (5) and (7) reduce to the same, and so an infinite number of changes of the variable can be made which will leave (2) unaltered in form. For example, take A + B — o, C — o, / = m — £; it is sufficient now to take for ;r284 LINEAR DIFFERENTIAL EQUATIONS. an integral of one of the following differential equations, in which K is arbitrary: dx Kdt dx _ Kdt 1 it 1 Vt{ I — t) ’ Vx(i — x) ~ (1 — t) Vt dx Kdt dx Kdt 1 1 tVi-t’ V x{\ — x) t ’ dx Kdt dx Kdt Vx( i — x) I — ty Vx{i — x) /(i - t)' It is easy to assure one’s self that, in all these cases, the general integral of (4), and consequently of (2), is expressed by means of elementary functions. Remark first that the second and third cases can be conducted to the first by changing respectively x into X I x___j or int° ~ ‘j so also the fifth and sixth cases can be led to the fourth by changing x into or into i - Xm It remains then to •consider the two equations / .d*P .dp <'-x)dS+{*-x)-£: + AP = o, ,d*P dx1 dx dP X' ' * +Xdi + AP = o. The general integral of the first of these is sm-'^ = VAsin->^+c„ and of the second is P = Cxxr ^ where r = VA. In particular, if A = 0> the ^ integral P=C* + C%]ogx. ISGO UH SA T: HYPERGEOMETRIC SERIES. 285, 2. It may happen that the two members of (8) reduce to con- stants ; for this it is necessary that Ax2 -f- Bx -f- C admits x or x — 1 as double factors or that it reduces to a constant. The following are all the possible combinations: 1st case. 2d case. 3d case. B — 0, II A = 0, C = 0, B — 2A =0 B = 0, II p m = 0, l m — We find for a, y the same conditions as in the last three- examined cases. Suppose, for example, A = C, B — 2A = o, m = o; we can make: any one of the transformations x = Kf, x = K1{i-ty, * = where Kx and n are arbitrary. It is easy to see now that the general· integral of (2) is of the form P = Cxxr + Cxxr'. Summing up: When (8) reduces to an identity we can effect an infinite number of changes of the variable in (2) so as to leave the form of this equation unaltered, and can so deduce an infinity of relations included in the type of Kummer’s formulae ; but it is to be noticed that all of these relations exist between functions which are expressed by means of the elementary functions, exponential, circu- lar, or logarithmic. (6) Discarding these particular cases, suppose that (8) does not re- duce to an identity. If now there exists an integral common to (5) and (7), x will be an algebraic function of t defined by an equation of, at most, the sixth degree in x and in t. It is therefore only among algebraic functions that we must seek the functions which will per- mit us to transform (2) in the desired manner. We shall determine first the rational transformations, and after will show that all the other transformations can be conducted to these. p Let x = ~ be a rational transformation ; R and S are two poly-.286 LINEAR DIFFERENTIAL EQUATIONS. normals of degrees at most = 6, and at least one of them is of a degree higher than unity. Among the values of t which correspond to the values o, I, oo of x there will be at least one which is differ- ent from o, i, oo. Suppose, for example, that for x — o we have t — o, and for x = oo, t — i ; then R will be equal to Ktr, and 5 to K\ i — i)s. The values of t which correspond to x — i will then be the roots of the equation Ktr — K'(i — t)s = o. One, at least, of the numbers r, s being greater than unity, the left-hand member of this equation cannot reduce to a constant; further, the equation does not admit either o or I as a root. There are therefore finite values of t which are neither o nor I and which correspond to == i. Sup- pose then that for x = o, for example, t takes a value t1 which is neither zero nor unity. The point x — o will be a critical point for that value of t which becomes t1. In fact, let the variable x describe a, small loop round the origin (Fig. 14): if t were a holomorphic function of x in the region of the point x — o, it would return to its original value after having described a small loop round tx, and we could then conclude, as above, that the origin would not be a criti- cal point for equation (2), which is not the case. It is therefore necessary that several values of t become equal at tx when x = o. Suppose n to be the number of these values; then x — {t OVOO, f(i) being a rational function of t which is neither zero nor infinite for t = tx. In the same way we shall have dx . . _.. w = v-v-m fx(t) possessing, relatively to the point tx, the same properties as f{t). From equation (5) we have, further, dx __ x(i - x) VA'f + B't+ C1 dt “ /(1 — t) VA x* + Bx + C * The quotient i — x tiy -1) is different from zero for t = tx; if neither Ax2 -f- Bx + C nor Arf -f- Brt -f- Cf were zero for x = o and t = tx>GOURSAT; HYPERGEOMETRIC SERIES. 287 ~we should have jjjr = <« - '■>·*«· tp(t) being a uniform function of t in the region of the point tx and different from zero for t — tx, which is impossible. If Ar f -\-B' t-\-Cf were zero for t — tx and C not zero, we should have /7v //r or ¿f = (t-ty+'W), *p{t) having the same meaning as above. This is also impossible. We therefore conclude that: If for x — o, t takes a finite value tx which is neither zero nor unity, it is necessary that the constant C be zero. (7) It remains to find the values which the integer n may take. Several cases are to be examined, as follows : First hypothesis : Art;+Bftx + C'< o, i? whence n = 1. This hypothesis is therefore to be rejected.288 LINEAR DIFFERENTIAL EQUATIONS. Third hypothesis: A't*+B'ti+C'= o, 2A'tl+B/<:o, B< o; it follows that dx % | j — =(/ — /,) 2 ip(t), whence—= n — i, n — 3. Fourth hypothesis : A't* + Bftx + C' = o, 2A% + B' $ o, ^ = o; which gives — = (i — ty ip{t), whence n = f. The transformation being supposed rational, this hypothesis is to be rejected. Fifth hypothesis : Ad; + B% + C = o, 2^47, + Bf = o, £ x = Rntr{\ — t)‘, (*) x = Rrr, (0 X = Rn(i — t)% {d) x = R", (') R"r (ƒ) Rn( i — ty x= r ' (g) Ra X = T' (A) R" (0 R· X ~ tr{ I — //’ R denoting an entire function which has no double factor, and which is not zero for either t ■ = 0 or t — 1 ; n is one of the numbers 2, 3, 4, and r and s are positive integers. We can reduce the number of these transformations. Thus, we can suppress the forms (c), (f),290 LINEAR DIFFERENTIAL EQUATIONS. (//), which by changing t into 1 — t are conducted to the forms (b\ {e)f (g). Take now the transformation (g) and let r' be the degree 1 RH of the numerator. If r' < r, then, by changing t into —, — changes R? into ^-17 or into R?f r', and we are thus conducted to the form (b) t__ j R* or the form (d). If r' > r, then on changing t into —-—, — changes Rn into and so we are conducted to the form (i). Take now the form (e) and let s' be the degree of the numerator. If t Rntr s' < s, changing t into --changes ------r-s into t — 1 (1 — t) R”tr (i -1) —s = R-ni -1y and we are conducted to the form (a) or to the form (b). If s' > s> 1 , Rntr . R,n changing t into — changes ^—-y into , and so the trans- ts'~s( 1 — t)s’ ' formation is conducted to the form (i). We are therefore confined to considering the four following forms : (a) = Rnr{ i — ty, (b) X = RHtr, (0 x — Rn, (0 R" x - t\i-ty (9) We now proceed to calculate the unknown coefficients which enter into these transformations. Suppose first that for x — 1 there is no value of t different from o, 1, 00. If the transformation has the form (a), the values of /, for x = 1, are given by the equation Rnt\ i — ty — i = o.GOURSAT: HYPERGEOMETRIC SERIES. 291 It is clear that this equation does not admit either t — o or t — 1 as roots, and that the first member does not reduce to a constant. Take in the same way the form (b); the equation K1? —1=0 must only admit of the root t = 1; consequently Rntr = 1 +H{ 1 — t)\ Now the first member of this admits multiple factors, while the second does not; the equality is therefore impossible. If we take the form (d), the equation R* —· 1 =0 ought to admit no other root than o and 1 ; this requires that n be equal to 2, and that R be of the first degree. It is easy to see that we must take R—2t— I ; from this results the transformation x — {2t — i)\ For the form (i) we must have that Rn — tr( 1 — t)5 reduces to a constant H. The equation Rn — H = o cannot admit t — o or t — 1 as roots. We are thus led back to the preceding case, which gives now the transformation x = (2/ - ^ 4— 1) * In order that these transformations may be effected we must have C=o and / = £. Following is the table of transformations, deduced by the above process, together with the inverse transformations. On the left-hand side of the table are given the conditions to be satis- fied by the constants A, B, C, /, m and the elements a, /?, y them- selves. The quantities À, /*, v have the same meaning as before. C — o, l=h 1- X = {2t- I)’, * = (—)’’ ii T~ (2-v 4/(/-0’ 4(! -'/) _(I +0' 41 ’ À — db292 LINEAR DIFFERENTIAL EQUATIONS. 4f+£+C=o, III... x-Ate-ty V 1 1 V 1 - ¥ ’ yl = O, V... I Y x-( * )' X- /I - A* / -f- w = |, - (2Ì-I)- [2 -t) ’ 4(1 — i) -V- ' y Y Vi + t) *' 1 VI... T •ts» >-s» 1 4i i' = ± i- (2i-l)a’ (2 - tf ’ (i+ir r \^='+'r‘. 1+ VV=i -T = , ^ _ 1+ Vi — T-~ 1 A -j- B = O, / = m, = ± ƒ<· VII. 2 Vi _i -f- Vi — t _ Vt — [ + Vi 2 VT^T’ * ~ 2 V7^T~’ _ Vt -}- Vi — I 2 V? VIII. B + C = o, / -j- 2;« = 2, /<= ± v. IX. ^=(>i+vrEirj ~_(i + Vi -/y 4Vt(t—i) ’ 4 V1 — t ,r_(i +^)3 4 V/ 2 2 2 V? »·*· = i+Vi’ i—|— V1 —tJ 14- Vt 2 Vi — t 2 Vt — 1 X -------^^=1 . X — i + Vi — t Vt — 1Vt 2 Vt Vt + Vt — 1' X. 4 Vi(i - 1) r _ 4 Vi — * ( Vi + Vi — i)j ’ (r. + Vi — tf _ 4 Vi1 *’“(!■ + VifGOURSA T: HYPERGEOMETRIC SERIES. 293 A — C — o, 2,1 —|— fH — 2, " 1 = i v. XI... _ V? — i _ V1 —t — 1 1 — Vt x 9 x 1’ 1+1/y i — VT^~t _ V~t — Vt — i * 1+V1 —V/+V/—i’ Vt — 1 — Vt X —..........-Z , Vt — i + V t XII... ( ~ 1 + ^ _ A+ Vi — Aa \ V7 — i — V// ’ * ~ V — Vi —7/ The transformations VII, IX, XI are the inverses of the rational transformations. Transformations VIII, X, XII are obtained by- combining transformations I and II in the manner already ex- plained. They can also be obtained by combining III and IV or V and VI. The preceding table contains all the transformations given by Rummer in the case where two of the three elements a, /3, y are arbitrary. (10) Suppose now that, for x — 1, several values of t are different tfrom o, 1, 00 ; we shall have simultaneously C — o, A + B =■ o. Ex- amine now the corresponding forms of transformation. Let x = Rnt\ 1 — ty; the values of t for x = 1 are given by the equation Rnt\ 1 — £)s — 1 = o, admitting neither o nor 1 as roots. We must then have RHtr{ 1 - t)s — 1 + Sn\ where 5 is an integral function of the same nature as R; n and nr are each one of the numbers 2, 3, 4, and r and s are positive integers. Remark further that each of the members of this equality294 LINEAR DIFFERENTIAL EQUATIONS. must be of a degree less than or at most equal to the sixth, and that we cannot suppose n — nf — 2; for we would have at the same time / = m = J, and should thus be led back to a particular case already examined. These remarks made, we shall now demonstrate the impossibility of the above equality. We see at first that 5 can- not be of the firfet degree, for then the second member could have no multiple factor, whereas the first member has such factors. Let us assume »S af —j— bt —|— C, 71 = 2 J n will be equal to 3 or 4, and the first member will be of a degree higher than the fourth. Let 5 = af -j- bt -f- c, nf -= 3 ; both members will be of the sixth degree; the second member ad- mits a multiple factor of order of multiplicity at most = 2. The same cannot be true of the first member, for, if R is of the first degree and n = 2, one at least of the integers r, s will be greater than unity. Suppose, finally, S — af -f- bf - ct d, n' = 2 ; n will be equal to 3 or to 4, and the first member will admit either a quadruple factor or a triple and a double factor. The second mem- ber cannot admit a quadruple factor, neither can it admit a double and a triple factor; in fact, the double factor would have to be either / or 1 — t. Suppose it to be t; we must then have c = o> d= 1, but in such a case the equation af -J- bf -f- 2 = o could have no triple root. Examine in the same way form (d). In this case we must have Rn — 1 = Sn’tr{ 1 — t)s. This equality is identical with the preceding one, save that here r and s may be zero. If R is of the second degree and n = 2, we should have n' = 3 or ri = 4, and then the second member wouldGOURSA T: HYPERGEOMETRIC SERIES. 295 admit a triple or a quadruple factor, while the first member has no such factor. The remainder of the discussion is the same as above, and we can show in the same way that the preceding equality is impossible. In order that the form (b) may hold we must have Rntr — I = Sn’(i — e)% where s may be zero. If s is zero, we are led to the preceding case. Suppose then r and s not zero. The functions R and 5 will be at most of the second degree; and as one of the numbers n, n! must be greater than 2, it is necessary that at least one of the functions be of the first degree. Let R = at -f- b ; Rntr — 1 can only have one multiple factor, and that must be a double factor. In fact, every multiple factor must be a divisor of the derivative, that is, of Rn - ltr " x\jiat -f- r(at -j- b)~\ ; the only suitable root of the derivative is given by the equation of the first degree nat -|- T{at -j- b) = o ; further, this root does not annul the second derivative n(n — 1 )a*f + 2nrat(at -\-b) + r{r — i){at -f- Sf = o. From the first we get at at + b r ~ — n 9 and substituting in the second we find — nr(r -f- n), which is always different from zero, since r and n are positive integers. 5w/(i — t)s can then only admit one double factor ; now, if nf 2, n is greater than 2, and the first member is at least of the296 LINEAR DIFFERENTIAL EQUATIONS. fourth degree. Then 5 should be of the second degree or s > 1 ; in both cases the second member admits more than one multiple factor. The only form which can hold, therefore, is the form (/), R" X-f{\ -if' The values of t for x = 1 can be neither o nor I, and they must all be roots of Rn — r(i — t)s — o of the same degree of multiplicity. We are thus conducted to the following problem: Required to find two entire functions R and S, and two integers n and ri, so that the equation Rn _ Snf = o admits only the two roots o and 1; R, S, n, ri being subjected to the restrictions already indicated. As shown above, we cannot suppose n — ri — 2; neither can we suppose n = nf = 4, because the equation R* — S* = o always ad- mits more than two distinct roots. The only admissible hypotheses are, supposing n ^ nf, n — 3, n = 2, n = 2, n = 3, ri ==3, ri — 4, ri =3, = 4. (11) Let n — ri — 3; the equation i?3 — 58 = O is equivalent to the three equations R = 5, R=jS, R=z j*S. The first member of one of these must reduce to a constant, and the other two ought each to admit a distinct root: this requires that they shall be of the first degree. Let R = t —j— u, 5 — t —J— v j we must have u=j*v, 1 +*=/+>,GO U R SA T: HYPERGEOMETRIC SERIES. 2 97 from which v=f, u=j. In fact, we have the identity (*+/)· - (f+jy = 3ju - imi - *), and we deduce the transformation vu-'W-ty The following are the transformations which conduct to this form j is of course one of the imaginary cube roots of unity: A -f- B — o, £7 = 0, XIII... I — m — 3’ XIV...- 1 — ± b M — ±b , A = o, C = o, 1= m =\, X = ± v = ± ^ 'A = o, 5 + C = o, ' XV... I = m — i, lM = ±h y = ±b J _ 3/0 - iKi - t) ’ „_(i±yy (¿4-/7’ „ _ 3/(/ - iXi ~ t) (t+jy Inverse Transformations. A = C - - B, I = m = §, f fyt—j Vt— i fyt—i—fyt * - * =r Yt_j VCT /* V/-/ i-V/ x“ I-V/ ’ x-j'Vt-j’ fWi-t-j i —Vi — t ’ X I -fT~t f Vi —t—f XVI... 4 x = ± M> \ = ± V.298 LINEAR DIFFERENTIAL EQUA IVONS. (12) Let n — 2y nf = 4. It is required to determine two poly- nomials R and 5 such that R*-S4 = Htr{ 1 - t)% where H is a constant introduced for convenience. From this we deduce R - S*= H't% R + S* = H"{ 1 -1)\ In fact, the equations R — S* = o, iv! + ’S* = 0 can have no other roots than o and 1 ; besides, these equations must admit no com- mon roots, because a common root would annul both R and 5. One of the equations admits then only the root t — o, and the other only the root t— 1, at least unless the first member of one of them re- duces to a constant, a case which will be examined later. 5 is compelled to be of the first degree, but R may be of the first, second, or third degree. Suppose R to be of the first degree, R — at -\- b, S — mt -[- n, R — S* = — nf? -(- {a — 2mn)t -f- b — #*, R + S2 = -f- (a -f- 2mn)t -f- b -f- n\ As we can always suppose n = 1, we ought to have simultaneously b — 1, a — 2m, m* + a -f- 2m -f- 2 =0, 2n? -j- a -j- 2m = o, which is impossible. Take now R of the second degree, R = af+ bt+ c. If neither R — S* nor R -f- 52 reduces to the first degree, we shall have c = 1, b = 2m, a -f- m* -f- + 2=0, 2a-f- 2m1 -f- 4m = o, giving m = — 1, and consequently 5 = 1 — t. Suppose R — S* reduces to the first degree ; we must then have c — 1, a = m\ 2M* -f- 2m + 2 + b = o, 4m* b 2m = o>GOÜRSA T: HYPERGEOMETRIC SERIES. 299 giving m — 1, ¿=—6, a = I, and consequently the identity {? _ 6t + i)a + (t + i)a = - 16/(1 - t)\ If R is of the third degree, we must have R — (mt + tif — /3, R (mt -f- nf = (1 — /)3, from which we derive [(1 — /)3 — /3] = 2(mt + nf, which is impossible since the two roots of the equation (1 — t)z — t* = o are distinct. If one of the functions R — Sa, 7? -f- S* reduces to a constant, R will necessarily be of the second degree. Let R = m*f -f- 2mt 4-- c, S = mt -[- 1, R S* = 2n?t -f- 4mt -(- c -f- 1. We must have c — — 1, m — — 2, and consequently the identity (4*a — 4/ — i)a — (2/ — i)4 = 16/(1 — /); this can be deduced from the preceding identity by changing t into / t — 1 ' The following is a table of the corresponding transformations: XVII... A. -\~R — o, C — o, l=h w = h ^ = ± i, M = ± i*, (¿2—6/-f-i)a __ (/a+4/-4)2 — (r ~f~ 4^ — x — 16/(1 -1) ’ 'A-\-B = o, C = 0, XVIII l=\, m = \, (2—ty ~ 16/(1 - ty 16/3( i - ty _____2/ - i)4 x ~ — 16/(1 — /) ’ „ ^ — ± h M=±bLINEAR DIFFERENTIAL EQUATIONS. 300 XIX... - XX... - XXI... \ XXII... < XXIII. .. - XXIV. .. - A = o, C — o, l = \, m = I, ^ = ± i, v= ±\, ^ x = x — (f— 6t + i)a (i+O4 ’ * (1 4? — 4**)* 21 - i)4 ’ 4)* (2-/)‘ A = o, C = o, ^ l=m = A. = ± 3-, v = di i- ^ _ (* + 0* _ (2 — 0* - (/■ - 6t + if ’ * - {?+4t- AT x- (2t~iy (1 + 4/ - 40* ’ A — 0, B-\-C — of / = 1, m = i6/(i - /)* 16/*( 1 - ¿) (i + O4 ’ (2-i)4 ’ — l6/(l — /) M = ± i> v — =1= h a A = 0, JB—\~C = 0, l — m — (2/- I)4 ’ — 16(1 — ty — i6f(i — /) (/*-<* + I)” *”(<■+4* - 4)* ’ l6/(l — /) M=±b v = ±£·j *“(1 +4/-40*’ Inverse Transformations. A = C, B — 2A, l=i, ^ X = ±^= ± v. C—4A, B=—4A, l=i, m=i, X ~ = ± M = ± v. (S-6Z+1)' (i +x)· _ , — i6jr(i — *)’ ’ i6x(i — xy ’ (x‘-6x+iy _ (I +xy _ (I + XY -r’(x‘-6x+i y-*’ i6x(i — x'y —i6x(i—xy (1 + x)' = *' (**—6*+iy = *’ {x'+ 4^ — 4)' _ (2 - x)' _ — i6x*(i — x) * i6x*(i — x)~ ’ (X‘+4X — 4)1 _ (2 — xy _ (2 — xy ’ (x,+4x-4)* ~ ’ i6x* (1 — x) _ — i6jt2(i —jr) (2 - xy ~ (**+4*—4y = tyGO U R SAT: HYPERGEOMETRIC SERIES. 301 XXV... ' A=4.C, B= — 4C, l — h = V u = ±M = ±~. (i-f4*-4*y _ (2X - I)* _ i6x(i -x) ’ -i6x(i-x) - ’ (1+45-4^7 _ O* - i)‘______ (2* - i)4 ’ ( 1 +4x- — i6.r(i — x) \6x{i — (2X-1 y ~ f’ (i+4*-4*7 - ** (13) Let n — 2, n' = R and 5 are to be determined in such a. way that we shall have JP - S3 = R can be of the first, second, or third degree, and S of the first or second degree, giving in all six cases to examine. First Case.—Suppose R and 5 each of the first degree, R = at by S — mt -f- n. The polynomial R* — S* will be of the third degree, and will admit one double factor and one simple factor. Let t be the double factor; we must then have b1 = nZy 2ab — 3 mr?y (a -f- by rif. "Ifyi We can always take b = 1, n = 1. On doing this we have a = — and consequently (? +.)■=<»+.)■; this gives m — — f-, a — — f, and therefore the identity (91 - 8)’ - (4 - 37 = - 2jf(\ — i). Second Case.—Suppose R of the second and 5 of the first degree,, viz., R ” at1 -j— bt —{- Cy S — mt —j- ti· If we had R* -Sz= Ht\i - if,302 LINEAR DIFFERENTIAL EQUATIONS. we should have S' = R' - Ht\i - t)\ Now the equation — Ht\i — ff = o breaks up into two equa- tions each of which is at most of the second degree, viz., R = ± VTT. t( i - /), and consequently cannot admit of a triple root; £3 — R* will then have a triple factor and a simple factor. If t be the triple factor, we shall have the conditions d = n\ 2 be — 3 mtf, R + 2ac = 3 n?n, {a + b + cf == (m-\- n)\ Taking n = 1, c = 1, we get and so , 3 m yri1 qm4 , qm3 b= T’ a = ~8-> &r + -V = < m—— -§, i f, a = ft, from which results the identity (8/a — 36/ -f- 27)® — (9 — 8^)s = — 64/3(i — f). Third Case.—Let R be of the third and 5 of the first degree, R — at3 —|— bt% —ct —fcS = rut —|— ti. / R*— S3 will admit of a quadruple factor and a double factor, or two triple factors, or a quintuple factor and a simple factor. We cannot have R2 — S3 = /4(i — ¿)2, for then S3 = R* — t\ 1 - /)\ In order that i?2 - /4(i - t)* = o should admit only one triple root, we ought to have R = t\ 1 - t) + H;COURSAT: HYPERGEOMETRIC SERIES. 303 but then the equation R + — t) = o becomes 2t\ 1 — t) + H — o and has no triple root. Neither can we have R* — S3 = Ht\ 1 — /)3, for then R*= S* + Ht\ 1 -t)\ and in no case can the second member of this admit three double factors. It remains to be seen whether we can have R* - S3 = Ht\ 1 - t). Taking n — 1, d — 1, we must have 2c = 3m, 2d zbc -j- 2a — mz, b2 -f- ac = o, (a b + c -f- i)a = (w -f- i)3> giving yn yn2 m3 c = £ = -g-, a = - and substituting in the fourth relation this becomes gm* 3 m* . . - 75- = o, giving « = O. This hypothesis must then be rejected. Fourth Case.—Suppose R of the first and S of the second degree, R == at —b, tS = wild -j- 4” P% R2 — S* cannot admit two triple factors ; if we had R2 - Sz= t\ 1 -t)\ then R2 = 534-/3(i - t)\304 LINEAR DIFFERENTIAL EQUATIONS. and the second member evidently admits more than one distinct factor. If we had S3 = R* — t\i — ff, each of the two equations R = ± f(i — /) should admit a triple root: R — f{ i — t) — {ut -{- vy, R -f- t*(i —■ /) = (uxt-\- vf. Then 2R = {ut -f- v)3 -f- {up -|- vx)3; R being of the first degree, this is impossible. If we had S2 - R3 = Ht\ 1 - /), we should find as condition n = o, ab — o. This hypothesis there- fore gives nothing. Fifth Case.—and S of the second degree. We can demonstrate as in the preceding cases that this hypothesis is to be rejected. Sixth Case.—R is of the third and 5 of the second degree, R — at3 -f- bf ct -\- d, S — mf -f- nt p. R3 — S3 may be of a degree lower than the sixth. By simple trans- formations the second member of the identity R3 - S3 = Htr{ 1 - t)5 is conducted to one of the following forms: Hf% Ht\ Ht, Hf{i - f)\ Hf{\ -1\ Ht{\-ty If, for example, we had R3 - 5s = Ht\\ - f), changing t into — and multiplying by fy we deduce R3 - S* = - Ht{ 1 - t). If we had R3 - S3 =Ht3,GO U R SA T: HYPERGEOMETRIC SERIES. 305 we should conclude that the equation R% - Hf = S3 had three double roots, which is impossible. Supposing we have then R* - Sa = Ht\ R* _ Hf = S\ and we should have R - VJf. t = {ut + v)\ R + V7T. t = (uxt + vx)\ and consequently 2 VH. t = {ut + v)z — (uxt + vy, which is inadmissible. If we had R* - 5s = Hf{ 1 - f)\ then we should have R + VH. t{ 1 - t) = {ut + v)\ R — VH. ¿(1 — t) = («,/ + and consequently (&/ -f- VY —· (#/ ~l· ^)3 — 2 VH . t{\ — t). That this may be so we must have ut + v = / +y, ¿¿y + = * +72; from which r = k* +/)3 + k* +>2)3 = *3 - r - f +1, S — {t +7 X* +78) = ? — t + 1, and consequently the identity (2/3 - 3f -31 + 2)2 — 4(/2 - / + i)3 = - 27^(1 - t)\306 LINEAR DIFFERENTIAL EQUATIONS. If we wished to have R* - S3 = //¿3(i - /), we should, in supposing d = / = I, get the conditions n? = #2, these give 3m2# = 2ab, 3m2 + 3m#2 = ¿2 -f· 2mn -f- n% = 2a -j- 2bc, 3m -f- 3#2 -f- 3» = ¿:2 -f- 2^ -(- 2^; these give ,— . xn Vm 6mn + n* —* 2m 4/m a ~ m Vm, b —-----------, c ~--------------—-------; 2 3 n Vm substituting these in the third equation gives n* — 12 mn -j- 16m Vm = o, or, making n — u Vm, u% — \2u -V 16 = (u -(- 4)^ — 2)3 = o.GO UR SA T : HYPERGEOMETRIC SERIES. 307 "We have therefore either n = zVm, or n = — 4 Vm. If we take n = 2Vm, we come to the same identity as above. Take 11 ~ — 4 Vm; now a — m Vm, b = — 6m, c — Vm. Substituting these in the last relation, we find Vm = 4; therefore ni — 16, « = — 16, # = 64, b = — 96, = 30, giving the identity (64/* — 96/* + 30/ + i)* — (16/2 — 16/ -j- i)3 = — 108/(1 — /). If R2 — 5 s = Ht, the conditions are a2 — ms, $m*n = 2ab, 3;/za + 3^«* = ¿a -f- 2ac, 6mn n3 = 2a -f- 2be, c2 2b = 3m 4- 3«2, giving n = — 4 Vm, a = m Vm, b = — 6m, c = J 5 Vm. 2 these substituted in the last relation give m — o. The following is a table of the transformations deduced from the preceding identities: A B = o, ¿7 = 0, l = h m — f, A. = ± -J-, _ (9/ — 8)a (1-9*y (9 - — 2fjt2{\ — ty — 27/(1 —/)2’ 27(1 — /)’ __ (1 (* ~ *) _ (/ — 9)2/ 27/ ’ ~ — 27(1 — /)8> _ (* ~l· 8)2(J — *) L - 27/2 XXVI...308 LINEAR DIFFERENTIAL EQUATIONS. XXVII.. XXVIII.. XXIX. A + B XXX. XXXI., XXXII. XXXIII. (8/2 — 36* + 2 7)' (8/* + 2ot — 1)’ -64/*(I-/) ’ -64/(1 -/)3 ’ (8 — 3 6t + 2 ’jf'f (2 yf — i8t — if x ~ 6i(T^7) ’ x — Kft ’ (f 4- 18/ — 27V — 2ot — 8V * = 64? ~ ’ * = 64(1 -if ’ _ (2f — 1? — 31 + 2f X~ - 2 7f (I - if ’ _ (64/'—96^+30Ì+ i f _ (/·■-f 30/" — 96/4-64)’ ‘ X~ — io8t(i — t) 'x— \o8t'{\ — i) ’’ (<* - 33^ ~ 33* + 0* x io8t(i — tf o, C — o, / = -f, = è, A = ± // = ± i· (3* ~ 4)3 (3*+0* (3 ~ 40s * —2yf(i—t)’ 2-jt{i—tf' 27(1—t)’’ —^L~-iT. r—(4~*y „_ (*4·3)3 27# ’ 2jf ’ 27(1-if’ r (8/ - 9)3 (8/ + i)s (8 - 9/)V x ~ - 64?(f - /)' — 64/(1 — jf)3’ x — 64(1 - t)!' (91 - O’O - 0 _ _ (* - 9)3(i ~ i) X~ 64/ ’ * _ 64/* l -64(1-¿r __4(^* — i ~t~ 0* 27/5(i - tf ’ r (i6*3 - 16t + I)· _ {f - i6t + 16)* * — io8*(i — t) ’ X ~ -* 108/* (1 — if) _ (* ~l~ T4* ~l~ i‘f ~ — io8*(i — tf ' A = o, C=o, l = i, m = £, ^ = ii ** = ±GOURSA T: HYPERGEOMETRIC SERIES. 309 XXXIV. XXXV. XXXVI., XXXVII.. A =0, XXXVIII XXXIX. (9* - 8)’ r (1 - 9tf (9 - 8 t)'t h -(3^-4)3’ (3* + 1)” {At- 3)3’ I _ (t - 9)V _ (¿ + 8/(1 -/) _ (i+8/)*(i-/) l (¿+3)3’ (4-03 ’ (1-4 ty 3 (8f — 36t + 2JŸ (8f + 201 — i)" *~ (9 - 8+ ’ *- (81 + i)s ’ (8 - 36t + 27f y (27? -181 — I)· {gt — 8)7 ’ *“ (1 -9/)*(i -/)’ (/’ + 18/ — 2 7)* (f — 20/ - 8)* (9-ty{i-t)'x~ /(8 + /)3 ’ (2/3 — 3/* — 3* + 2)* 4('3-'+i)3 ’ f (64^-96^+30/+1)2 _ (/»+3o/»_96/+64)* *~ (i6/J — 16/ + i)3 ’*“ {f- 16/+16)3 ’ ~ 33** ~ 33* + i)" r- (I + 1++03 ' C — O, l =\, *n — f, >1 = ± -J-, v = ± r ~(3<~ 4)3 (3* + O3 (4* ~ 3)3 (9^ — 8)a ’*-(1-9/)" *-(9-8/)·/’ _ (* + 3)3 _ (4 — *)3 _ (1 — 4ty { (t- 9P’ ~ (t + 8)3(I - /)’ * - (i+8/)\i-/)’ f (9 — 8/)3 (8/+I)3 “ (8/’ - 36/ + 27'Ÿ ’ ~ (8/s + 20/ - 1)· ’ {gt — 8)7 _ (1 — gi f ( 1 — t) ' X ~ (8 - 36/ + 27ty x ~ (27/*- 18/-I)·’ (9-/)3(i-/) (8 + tyt (f + 18/ — 27)a ’ {f — 201 — 8)’ ’ 4(/3-/ + i)3 (2/* — 3/* — It + 2)’ ’ XL..* =3io LINEAR DIFFERENTIAL EQUATIONS. XLI.. A = O, XLIL. XLIII.. XLIV., XLV. A = o, XLVI. _ (16/2 - 16/ + i)2 ' _ (/2 — 16/ + 16)’ x ~ ^64/”—96/2-|-30/-}-i)a ’ X ~ (is+30/s-96^+64)2 ^ (1 + 14/ + 0s f ~ 33*a - 33* + i)a‘ £ + <7 = 0, / = m = i, M = ±h v = ±\. — 27^(I - ^ - (3/ - 4)’ /) 27/ (1 — /) ’ (3/ + i)s ’ * 27(1 -/) (3-4^)s ’ 27/ 2 7/2 27(1 - ty l (4/ i)s ' ^ — (4 — /)s ’ (i + 3)* 64/’ (1 — t) 64/(1 — /)* 64(1 - /) (9 — 8/)s ’ x ~ (8/ + i)2 ’ x ~ a ** ON 1 00 e4/ 64f r X — (9*- i)3(i ~t) {*- 9)3(i - /)’ -64(1 - /)2 x - (8 + /)3/ ’ 27/'(1 - ty 4(/3-/+i)” ' ^ _ 108/(1 — t) (16/2 — i6t -f- i)3 — 108/(1 — /)* — io8/4(i — /) (?- 16/ —|— 16)2’ x = (i + B + C = o, / = f, tn = f, = ± r = ± i·. — 27/* (1 —/) — 27/(1 — ty 27(1 —/) * = ~l9* - 8)3 ’ * = (i-90s ’ Jr = fe^)2/’· 27/ 27(1 — /)2 " x : (1 + 8/)2(i - /) ’ ^ = (/ - 9)2/ ’ — 27/“ ƒ =(/ + 8)2(i -/)’GOURSA T: HYPERGEOMETRIC SERIES. 311 XLVII XLVIII.. XLIX. LI... 4 '____— 641’(i— t) _ —641(1 — ty X = (8/2 - 36t + 27)a ’ x - (8f + 201 - if ’ 64(1—/) _ 64/ x = (8 — 36/ + 27?)' ’ X = (27? - 18/- 1)” Ó4/3 64(1 — ty x~ (? + 18/- 27)’ ’ x~(f- 20t - 8)’ ’ _ —27^(1—/)’ ‘ * _ (2f — if - 3/ +2)” — 108/(1 — /) _ io8/4(i —/) x ~ (64/3-96/2+30/+i)2’ * ~ (/8 + 30/3 - 96/ + 64)” 108/(1—/)4 ƒ-(/»_ 33#· — 33* 4- 1)3· Inverse Transformations. yi—o, 4B y — ± h (9X 8)3 _ (3* ~ 4)3 _ — 27^(1—.r) ’ —27^-2(i — ;r) * (9-y - 8)· _ (4 - 3xY _ (4 - 3*y ~ ’ (9x - 8 y ~ X 2 — ± M· ~27x\I—·*■)_. -27^(1— jr) ^ (3X - 4Ï ’ (9^ — 8)3 -A A = o, B = 3C, I = m = ^ = ± i, M X = ±-. 2 C=o, 4&+3A = 0, I ^ X = ± i, (I- - 9XY _ . (3^+03 — 27^(1 — ;r)2 ’ 2Jx(l—x)% (I - 9XY . (3^+i )3 , (3* ■+ 0* 1 (r - 9xf ~ *> 27;r(l ~XY , -27x(i-xY (3* + !)3 ’ i1 ~ 9XY ~ (9- Sx)'x , (3 - 4*)3 27(1 -x) ~ ’ 27(1 -x)~ ’ (9 - 1 H Ç» 00 (4X-3Y _ (4*·- -3)s “ ’ (9 - Sxfx ~ ’ 27(1 , 27(1 - x) ^ (3 - 4XY ~ ’ (9-&r)V _ · LII... ^312 LINEAR DIFFERENTIAL EQUATIONS. LIII..J L1V. LV... \ LVI... C = o, B = iA, l ^ 3- — dz . ^ v = ± 2 {x - gfx (x + 3)a _ —27(1—Jr)3 ’ 27(1 — xy ’ (x - gfx _ (.x + 3)a _ -- A ^\2 ^ - ¿4 (*+3)s 27(1 - *)* (* + 3)’ (x - gfx - * -2rtl~xY - f ’ (■*■ — 9fx ~ C=4~A, B-fsA—o,' l — m — \l M — ± h X — = ± v. 2 (x+8f(i-x) (4 -xy , — 2"Jx‘ ’ 2JX* ’ (;r+8)*(i-*)_, (4 ~ XY . (4 -*y -'’(x+mi-xr ’ 27X’ — 27X* -a —-x = t. (4 — *f ’ (·*· + 8)3(i — x) A—4C, B-\-$C = o,' l = b m = i> M — ± i> V A- = ± . 2 (i+8*)’(i-*) _ , (+r - i)* _ — ly — ty 27 X 2JX (i+%xf(i-x)^^ (I— 4Jr)3 -I O „A 2 / _ \ (i-4*)s ’{i+8xf(i-xy 27 x ; = *, 27 X A. —o, 9^-|— 8£7—o, / = £, m = £, A -=±„. (4^ — i)s ’ (i + 8jt)3(i — *) (8-r3 — 36-f + 27)’ = A — 64^(1 — x) {8x - 9) = A = A — Ó4^3(i — ;r) (8jt3 — 36* -f- 27)* (9 - 8^)3 (9 - 8^)3 = A (8-r1 — 36J? + 27)' — 64^(1 — x) 7 = A = A (8* ~ 9)3 — 64x\\ —x) {8x* — 36^ 4- 27)1 =LVII invili.. J LIX... 4 GOURSAT: HYPERGEOMETRIC SERIES. (Sx* + 20X — ï)’ 3*3 A = o, B = 8 C, I — m = i, λ=±* C— o, 9 B-\-SA—o, I — b m = b Λ. - -C y, y -=±μ. C = ο, B = 8^4, — 64^(1 — λ·) (8*4-1)* =, 64x(i - xf (&X1 -f~ 20X — lf (8J+ïÿ ____(8*+i)3 (Sx2 + 20X — if Ô4x(i — xf __ ~W+i)r = A — 6φ*·(ι — xf (8-T1 + 20^· — ï)* (ï — $6x + 27 xy 64(1 — X) (8 - gxfx _ 64(1 -x)^ ’ (8 — 36.*· + 2yxy (gx — 8 fx (gx — 8fx (8 — 36X + 2JX*f 64(1 - *) _ (8 - gxfx - 64(1 — x) (8 — 36X + 27x*f = t, = t, = t. = t, = t, - t, (8 -f- 20X — x?f 64(1 — xf — (8 4" xfx = t, 64(1 — xf = t, = f, m = i, (8 + 20X — xy (8 + xfx - *' * = ± b (8 + xfx • f* (8 + 20* - XJ — *’ y — zt —· 3 64(1 — xf -(8 + xfx — ’ 64(1 - xf (8 +2οχ — = t.314 LINEAR DIFFERENTIAL EQUATIONS. LX... 1X1... \ LXII..J A=gC, B-{-ìoC=o, l = m = = ± i> V \ = ±~. 3 C=gA, B-\-ioA = o, l = m = f> ' (27^·’— 18^ — 1)’ 64X {gx - - i)3d - 64X (2 ?x' — iSx - I)3 ii- ■9*)\ i -X) ~ ii - 9*)\i -X) (2 7** — l8^r - 17 ~ 64^· {gx - - I)3(I ■ -x)~ 64X (2 7*a — iSx - I )* “ (*3 + iSx — 27)a _ 64X2 (* — 97(1 - 64X3 — TJ (*3 + 18x — 2 77 (9- ^)3(i - -X) - = ± & \ - = ± v. 3 (9 ~ ^)3(i ~ x) _ {x1 -{- iSx — 27)* * 64^® (·*■ — 9)\i — x) ~ *' 64*’ (*’ + It* - *77 ~ A = C= - B, ' (2X* — 3X* — 3X + 27 — 27Jr2(l — x7 4(x3 - * + 0* . 27^(1 — .*y ’ (2^5 — 3x* — 3X + 2 y 4(^ - * + iy 4(X* -x+1 y X = ± fi — ± V. (2X3 — 3X1 — 3X -f 27 2jx\\ - x7 4L** -x+i y ’ — 27^’(l — x7 {2X% — 3x‘ — 3 x+ 2 7GOURSAT: HYPERGEOMETRIC SERIES. LXIÏÏ... LXIV. f i-^B 3= o, A = 16C, ' (64^* — g6x9 + 3ox + !)* — 108^(1 — x) (i6;tra — i6x -f- i)3 ioSx(i — x) tf (64X* — 96^ 4- 30^ + O8 i6;ra — lôx + i)3 (i6x* — 16* + i)3 (64a:3 — gôx* -|- 30^ + i)a 108^(1 — x) _ (i6x3 — 164; + i)3 — — 108^(1 — x) (64X3 — çôx* -f- 30X -|- i)a ' (x* -j- 30^ra — 96x -f- 64)* io8^r4(i — x) {pc’1 — i6x —(— 16)3 — io8^‘(i — x) ~ (x* + 30^’ — 96x + 64)2 rB+C=^o, C^ 16A, l— 1, m ^ -< {x* — \6x 4- i6)s (** - 16x 4- 16)3 X (x3 4- 3°x — 96* + 64.4 - — ± ¡X— ± v, l 4 — Io8jr4(l — .*-) (x1 - i6x 4- 16)3 “ *’ io8^-4(i — x) _ (*s + 30^2 — g6x 4- 644 f À = C, B — 14 C, ' (*3 - 33x* - 33-^ + i)a 108^(1 — ;r)4 (1 + 14X + xy — 108^(1 — x)4 ’ /= i m = b u \ = ± — = dt y 4 (xa — 33^ — 33,y 4- i)3 (1 + iz^ + ^4 i + 14*· + XJ {x1 - 33X2 - 33X 4- 1)* (x* - 33** - 33X + i)2 LXV... -{3i6 LINEAR DIFFERENTIAL EQUATIONS. (14) We still have the hypothesis n = 3, n' = 4 to examine. It is easily shown that, under the imposed restrictions, it is impossible to have an identity R' - S* = Ht\ 1 - t)s; there exist therefore no other rational transformations than those already determined. (15) In combining among themselves the transformations which we can effect for the same values of A,B, C, /, m, we obtain new alge- braic, but not rational, transformations. It remains to demonstrate that all such transformations are obtained in this way. Resume the two differential equations (5) and (7) which are to be satisfied by the function x of t. <5) VAx* + Bx C , } " dx X{1 - X) VA’t'+B't + C ^ /(I - t) (?) dx Kdt ^{1 — x)m ~ ?{i — t)™' ‘ The relation deduced from these is (8) where [0MT imi_______ (Ax* + Bx+ Cf ~ {A'f + B't + CJ’ t(t) = /(/_ iy2A,t+B,)+2{A,f+B't+C,W'-0(*~ i)+K-1)/]. Differentiating (8) and taking account of (5)>we get the new relation ,, - i)[2 ,'(t)(A'B + B't + C’) - + B')-] ( A 'f B't + C'yGO UR SAT: HYPERGEOMETRIC SERIES. 317 Every integral common to (5) and (7) satisfies (8) and (9). Con- versely, every algebraic function which satisfies simultaneously (8) and (9) is a common integral to (5) and (7). In fact, every function which satisfies (8) satisfies also 2(f{x){Ax*-\- Bx-\-C) — 3d>(x)(2Ax-\-B) \Ax* + Bx + cy VAx*-\- Bx+ C . dx 2l'{t)(A't'+B't+ £7')- 3frW?A't + &) (A'f + B't+CJ VA’f+ B't+ C'.dtr or, taking (9) into account, equation (5). Taking account of (5), (8) can be written d. log [ VAx* + Bx -f- C . x*~r(i — x)m ~x] = d. log [ VA'f+£'t+C' . tv - *(i - ty - ’]. We have then VAx*+Bx+ C.x?-\\—xY~'= VA'f+Bft + Cf. and consequently, still taking account of (5), dx Kdt xl{\ — x)m “ tl\ 1 — t)m" We are thus reduced to the problem of finding the conditions to be satisfied in order that equations (8) and (9) shall have one or more common roots. (16) To this end we shall first demonstrate the following theo- rem : Whenever (8) and (9) have a common factor of degree higher than the second with respect to one of the variables, they are identical. Suppose, for example, that they have a common factor of degree n in x (n ^ 3). Let x and xx be two values of x corresponding to the same value of t. Between x and xx there will be a symmetric relation of degree n, at least, which should be contained in each o£ the following: [(*)]’ {Ax' + Bx + C)' ~ {Ax{2+ Bx, + cy (10)3i8 LINEAR DIFFERENTIAL EQUATIONS. (T x(] + 2x(x — \)(x) = 2A(x — d)\x(x— 1) -f (x — a)[(/ — i)(x — i) + (m— l)x] Relation (10) reduces to the fourth degree: /iox 4A{x(x- 1 ) + {x- a)\{l- i)(x — i) + (m - i)-r]p ^ ; A\x-ay _ 4A {Xt{xx - i) + (*, - «)[(/ - i)Q, - I) + (m - iKl \* A'ix.-a)* In like manner (12) can be written /.„x AAx% (x- i)*+ {*—a)ƒ,(*) 4Ax? (*, — l)" + {x — «)ƒ, (*,) X ’ A\x-ay ~ A*(x,-ay Relations (10) and (12) should be the same. Subtracting them, member from member, we get a new relation, <13) ƒ,(■*) /,(*,) A'(x-ay- A* fa —a)” which should be an identity, as it is at most of the third degree.320 LINEAR DIFFERENTIAL EQUATIONS. Suppose n = 3 ; if (12) is not an identity it should be, at leastr of the third degree, since two of the quantities A, A B -f- C, C cannot vanish at the same time. Let There are several cases to be examined. First Case.—Ax2 -f- Bx + C = o admits two simple roots x = a, x = ¡3 different from zero. 0(ar)0(/J) $0. We cannot have simul- taneously ip {a) = o and tp(/3) = o, for then (12) would be of a degree lower than the third. Suppose ip(a)ip(/3) ^o; for xx — a, two values of x become equal to a and two to /?, by (12) ; let us assume that the two values of x which become equal to a belong to (10). From (12) we should have and from (10) ,im ='· ,im (j^)’=■- two incompatible conditions. We reason in the same way if tp(/3) = o. This hypothesis must therefore be rejected. Second Case.—Ax* + Bx A- C — o admits the simple root x = o and another root x = a different from unity. Equations (10) and (12) become [#*)]' _ OK)]’ x(x — a)3 xx (xx — a)3’ (12) ${*) __ t (xò . x(x — af x1 {xx — Oif 9 ip(x) is of the third degree, and we should have tp(a)^ o, 0 Reasoning as before, we find that this hypothesis must also be rejected. Third Case.—Ax2 -f- Bx -|- C — A{x — a)*. Replace the system (10) and (12) by (10) and (13). This last equation must be anGO URSA T: HYPERGEOMETRIC SERIES. 321 identity, otherwise it would be of the third degree, and we should derive from it while (12) gives Summing up : When n is equal to or greater than 3, one of the equa- tions (12) or (13) reduces to an identity which requires that one of the rational fractions $(*) /,(*) {Ax* + Bx-{- cy ’ A*{x- af reduces to zero or a constant. It is easy to see directly that the numerators tp{x), /¿x) cannot be zero while (8) is not an identity. One of the two expressions ought then to have a constant value different from zero. (17) It follows now that, if j'{x) {Ax*+Bx+C) a reduces to a constant, we can make all the transformations !>(*)]* _ 1>.(*)T {Ax* + Bx + cy {A't* + B't + cy ’ where A \ B\ C\ fri have values such that the quotient {A'f + B't + cy reduces to the same constant. Among these transformations we can find one which is of the first degree in t; it is sufficient to take A' -f- Bf — o, Cf — o, V — m! — ^ ;322 LINEAR DIFFERENTIAL EQUATIONS. then 0.(0 = y ?{t— i). AY(\-tf *■« = -~i 0.(0 I (AY + B't + CJ — 3 A" If we take a proper value for A\ we shall be able to make the trans- formation [0(*)]a _ * [Ax' + Bx + CJ gA'(t-i)’ which is of the first degree with respect to t. It is clear that every transformation which we can make in the same case will conduct to the preceding followed by a new transformation, equally of the first degree with respect to t. If the ratio ; -- has a constant value, we will take in like A*(x—a)9 manner A' + B' = o, C'= o, lf = h tn’ = h Now 0(0 0/(0= -4’/(3jf-2)· r 0.(o]a _ * (AY + B't + CJ 4A' (t - i) 0.(0 31-2 (AY + B't + CJ ~4A'(t-iy The difference of these is ^7 ; if we take a proper value for A', we can effect the transformation !>(*)]’ _ * (Ax'+Bx+CJ 4A'(t-i)’ from which we derive the same conclusion as above.GOURSA T: HYPERGEOMETRIC SERIES. 323 In order then to obtain the new transformations, it suffices to combine the rational transformations which can be effected for the two systems of values of the constants A> B, C, /, m, C— o, A+ B = o, / = i, m = §; C— o, A -f- B — °, / = i, m — f. (18) It remains to examine the case where there exists a transforma- tion of the second degree with respect to the two variables. Let xa be a value of xy and tQ, tx the corresponding values of t. To the value /0 correspond for x the values x0 and xx; to the value tx correspond for x the values xQ and x2. If the relation F(x0, xx) = o between two values of x which answer to the same value of t will be of the third degree, and we thus come back to the preceding hy- pothesis. If xx — x^y the relation F(x0, xx) = o is of the second degree and breaks up into two relations of the first degree, viz., *1 ax0 + b cx* +d' From what we have seen above this last relation should have one of the forms X\ — ^ Xo f *1 = Xo - I Suppose, for example, xx = 1 —x0. The relation between x and t will now be (2X— i)* =/,(/), J{\ (t) being a rational function of t of the second degree. The two differential equations VAS + Bx+C^ = VAx,- + Bx±£ x(l—x) Xt(l—X,)324 LINEAR DIFFERENTIAL EQUATIONS, dx______Kdxx Xi (i — x)m ~ xf (i — X$* should then have for integral x = i — xA; for this we must have l — m, A -]-£ = o; but then we could make the transformation {2X — i)2 = U, and the proposed transformation would be a combination o£ the* two following: (2X— l)a =U, U=f1(t), which is one of the transformations indicated above (VIII, X, XIL):. (19) For brevity we will denote by U, V, W, respectively, anjr one of the transformations contained in the following three tables: U. (21 — (2 - tv /i+/\* (2t-iy (2 — ty (i+o? >’ \ t ) ’ \i -// ’ 4*(*-i)’ 4(1 - ty 41 * V ~ 6/ + i)a {t* -\- 4t — 4)* (1 + 41 — Afy — 16/(1 — ty’ — i6/a(i — ty 16/(1 — t) (9( — 8)* (1 - 9ty t(9— 8/)a (1 - /)(i + 8/)* — 2jf{\ — /)’ — 27/(1 — ty' 27(1 — /) ’ 27/ ’■ / (/ - 9)’ (/ + sy (i - /) (8/’ - 36/ + 27)’ — 27(1—/)” — 27/’ ’ — 64/’ (1 — /) ’ w... (8f + 20/ — i y — 64/(1 — ty ’ (/* “I- 18/ — 27)" 64? ’ (8 — 36/ + 27/7 (27/* - 18/ — i)a 64(1 — /) ’ 64/ ’ (/* — 20/ — 8)a (2/s - 3/* -3/4- 2)a 64(1 -tf ’ -27/a(l-/)a " (64/" — 96/” -f 30/ i)a (/a + 3Q/a — 96/ -|- 64)” — 108/(1 — /) ’ 108/'(1 — /) ’ if - 33** ~ 33* + O’ 108/(1 — ty Employing these conventions, all the new transformations will be found in the following table :GO URSA T: HYPERGEOMETRIC SERIES. 325 7LXVI 7LXVII LXVIII. LXIX LXX. 7LXXI. LXXII 3L.XXIII 'A -|-B = o, C = o,' / ■=. m = £, b (2x — i)’ . A = ± M = ± i· A —o, ¿7=o, / = I, w = £, _ A = ± y = ± I (i+*)' (1 - x)· ' A = o, ^ = °> l — tn = J, _ ju = ± ^ = ± i- A —|— B — o, ¿7 — o, -1 * l =m = f, A = ± // = ± ·£. ► (2X — i)’ A = o, ¿7=o, _ A — ± v = ± -J·. M = o, ^ + ¿7 = 0,'! ·< / = m = f, ^ = ± r = ± f A — o, B = o, / = /* = «, ^ = ± 1, A = ± /i. B = o, ¿7 = o, - I — l, m = f, (2* — i)J 4* (* — 1) (2 - xy 4(1 - x) = V, V, V, = w, = ÎF = W, = fF, ÎF .A = ± // = ± v. J326 LINEAR DIFFERENTIAL EQUATIONS,: LXXIV... A LXXV... A LXXVI LXXVII... -| LXXVIII... A LXXIX... A 4 — ¿7", .Z?-|—2£7 — o, / = fi m = /« = ± f, A = ± v. A = C, B = 2,4, / = f, = i, o+*r 4-* = W7, 1 = * = ± v. 2 - = +4, 5 = — 4*4, / = I, m = f, - 6*· + i)5 _ — i6x(i — xf ~ (x1 - 6x + i )3 _ -=±M=±r. A — 4 C, B — — 4 6"; l = m = b A — ±7* — ± —· <4 = 0, 4^+3 C = o, /=f,;« = £, v = ± — = ± /<· A — o, B = 3C, t=h m—b , , ^ v = ± A = ± — l6jr(l -xy (*’ + 4* -4)3 - i6^s(i - ■*) (x1 -f- 4^ -4)! — i6^2(i - x) (T + 4^ - - 4*7 76^(1 - - X) (I + 4* - - 4X1 l6x(l ■ -X) (gx- sy — 2 7.^(1 -x) (gx — 8)’ — 27^(1 -*) (1 - gxf — 2Jx(l -xY (i -9*Y — 2?x( I -xy = i7, = u, = f7- = = F, = <7, = W, = u, = r,.GO U R SA T: HYPERGEOMETRIC SERIES. 3 27 LXXX... 4 LXXXI... LXXXII... A LXXXIII... -I LXXXIV... -I LXXXV... 1 C — o, — o, / = *· — ± è> n—±—· C = o, B = $A, ] l—h m = b l = ± i, ~= ± r- U — ¿^A, B—j— §A — o, l — b m = b (9 - Sxfx 27(1 - x) (9 — 8jr)2jtr 27(1 - ·*·) (x - 9)ax - 27(1 - x)' (x - 9)2a- = U, = ^ M = ± i, ~ = ± »/· ¿4 — Æ+5C — o, l — wz — I» - 27(1 - x)' (x -j- 8)a(i — x) — 27X1 (x ~l~ 8)Xi ~ x) — 2JX1 = u, = W, = 0; = n — ± i, A — ± — · (1 + foQ'O - x) 27 X (I + 8^(1 - X) 27X = 0; = ÏT, ■ o, <)B-\-%C — o, ' l—\l m — \l \ v = ± i» - = ± /<· (8-y* - 36* + 27)' _ — 64^3(i — x) 9 (Sx1 - 36^ + 27)a _ — 64jf3(i — x) ’ A = o, B = SC, ' l = 4’ m = b M v= ±l,\=±~. (Sx1 4" 20^· - i)’ _ - Ô4X(1 - x)3 ~ ' (8**+ 20-y - i)a _ w - 64x(i - x)3 ~ ’328 LINEAR DIFFERENTIAL EQUATIONS. LXXXVI LXXXVII LXXXVIII LXXXIX XC. C — o, qJB—j— SjI — o, i — \i m — ^ = ± M — ± ~· C = o, B = 8A, l—\, m — \, \= ±%, - = ±v. A=gC, B-\-ioC=o, * = b m — b (8 — $6x + 2jx‘)' 64(1 — x) (8 — i6x -f 27 x1)' 64(1 - x) (8 + 20* — xj = u, = W, 64(1 - xy = u, (8 + 20x - xj _ 64(1 - xy - w' fX — ± \ — ± —. (27^ — iSx — i)’ 64X (27 x* — i8x — i)3 64X = u, = w, C—gA, B-\-ioA=o, ' l—\, m — = ± b J = ± v- A = C, B C=o,' l=m = „ \ — ± H— ± v. («', fi', y', t"), which is reproduced, to sign près, when x describes a loop round the origin. This integral will be of the form x*{i — x) ç'F{otl, fi,, y„ x); and so, denoting by B a constant, we shall have Bxi{i — x)-tiF(a„ fi,, y„ x) ¡3’, /, f) _ t"\x-t’Ì F {a', fi', y>, t"\ If the values of t which become = tx for x = o are in number greater than 2, it is easy to form the integrals of (2) which, when the vari- able describes a loop round x = o, reproduce themselves multiplied by a constant. Suppose, for example, that three values of t become equal to tx when x = o, and let tr\ t,n be these three rootsGOURSA T: HYPERGEOMETRIC SERIES. 331 arranged in the order in which they present themselves when the variable describes, in the direct sense, successive loops round the origin. The integral I - ft', y', + - t"fF(a', ft', y', t") + ?"'{i ~ f'f F W, ft’, y’, t'") is evidently uniform in the region of the origin while the integral t'P\l - t’)“ F(a’, ft', /, t’)+ft”r (I - t")' F (a’, ft', y', t") + -t"iF{*', ft’, y’, t'") reproduces itself multiplied by j when x describes a loop in the direct sense round the origin. We deduce now relations analogous to the preceding. One or several values of t may be zero at the same time as x. If the value of t which vanishes for x ~ o is unique, the integral t* (1 — if F(a', /3\ y\ t) reproduces itself, to a constant factor près, when x describes a closed path surrounding the origin. It ought then to be equal to an integral of the form Cx~* (i — x)~q F {a, y, x). The case when several values of t are zero at the same time as x requires a fuller explanation. Suppose first that x = 0(/), where 0 denotes a rational fraction. Equation (2) admits an integral of the: form x~\\ - x) -?F(a, ft, y, x), and the same is true of equation (4). If we consider t as the inde- pendent variable, then when t describes a loop round t = o, x wifi return to its initial value after having described several loops round x — o, and the preceding integral will reproduce itself multiplied by a constant factor. We have then *-* (I - x)~q F{«, A Y, *) = C't'i I - trf'F{a’, ft', y', f), tf denoting one of the roots of the equation x = 0<7) which are zero when x = o. Since every transformation can be led back ta332 LINEAR DIFFERENTIAL EQUA IVONS, rational transformations, the preceding conclusion holds when x is no longer a rational function of t. The calculation of the constants which enter into our formulae is effected without difficulty, wThen for x = o we have at the same time t'*' t — o; it is sufficient to seek the limit of the ratio--- for x = o. x Now take the case when for x = oy t takes a finite value, t19 differ- ent from zero and unity. In the formulae written above make x — o, t = ; we get A = 2tf\l - t/F(a', (V, y', /.), B = lim. ^'(i -t’ fFW, P, /, p, y\ t") X* for x = o, t = tx, 1 AtP’f +n\q’ i B = [¿''(I - t'YF{a\ p, y>, *'],=,.· , dt The following examples show the method to be followed in each «case. (21) Consider the differential equation d*y dy x(i — x) -j-f + [i — (<*+/? + i)x] — apy — O. dx Writing x — {2t — i)a, this becomes *{t- i)^ + [« + yS + i- (2« + 2/? + -4«/?y = o. For ;r = o the two values of t are equal to -J·. We shall then have aF(a, f3, x) — F^2a, 2fi, a + /? + i, —F^2a, 20, a + /? + £, -— i VxF(a 4- -i, -i') = ^2«, 2/J, « + /? + *, F ^2«, 2 ft, a + /? + *, 1^?^.GOURSA T: HYPERGEOMETRIC SERIES. 33X For x = i one of the values of / is = o; then F{a, ft, a + ft + i> I — x) ~ F^2a, 2ft, a -f- ft J,- This last formula enables us to calculate a and b. We have in fact a — 2F(2a, 2ft, a -(- ft -f- i, i), or, from the preceding formula, a-2F(a,(3, a + /J + i, i), - + + ^ · In like manner, b = a-\- ft + Ij + l> a + 0 + i> £)· In the value of a change a into a + i and ft into ft + f; we find, then 4a/3 +/? + |) _ 4^r(« + /? + J) a + ii+jr(«+W+i)" r(«)r(fi) (22) Consider again the differential equation Writing ; and at the same time making (/* - 6t + i)1 - i6r(f-r)” y = ta (l — r)2% we have the new differential equation * (i — *) ^ + [2« + f ~ (6a + *)*] % ~ 4« (2« + = 0. For x = o two values of t become equal to 3 — 2 V2.334 LINEAR DIFFERENTIAL EQUATIONS. Let tT and t,f denote these values; in the region of x = o they are developable in a convergent series going according to ascending powers of jr*. Let t' = 3 — 2^2-\-V — i (3 hfi — 4)x*+ . . . , t" = 3 — 24/2 — 4/ — I (3 4/2 — 4).** + . . . . The integral /a(l — t'YaF(4.a, 2a -f- 2a + f, + tff\I - /")2a^(4«, 2a + i, 2a + I, t") is uniform in the region of the origin. We have then, A being a constant, AF{a, 4 — a, £, x) = t’\I — t'Y^F^a, 2a + 4, 2a + /') + 2« + b 2a + t"). In like manner, denoting by B a new constant, Bxl‘ F(a 4“ £>£—«, f, *) = t,a (i — t'YaF(4a, 2a + 4> 2a +1, O - t"a{I - 02a^(4«, 2a + 4, 2a + |, t"). Making x = o, we get A— 2^1 —2 Vly(2V2 — 2)mF{4a, 2a4-4, 2a +1, 3 — 2 4/2), .5=2 4^^i (3 4/2 — 4) ^ [/“ (I — ¿)“.F(4a, 2a 4- 4, 2a 4-1, /)] for t — 3 — 2 Vz Observe now that, for x — oo, one of the values of t becomes zero. The proposed differential equation admits the two integrals t°-{I — tY*F(4a, 2a 4- i, 2a + f, t\ x-“F^a, a 4- 4, 2a 4- f, ^) which belong to the same exponent in the region of the point x — co.GOURSAT: HYPERGEOMETRIC SERIES. 335 They ought then to be identical, to a constant près. We deduce from them the relation F(4af 2a + 2a + f, t) = & ~ 6t+ « + i, 2a + f, , which can also be written -F{4oc, 2 a + i, 2 « -f f, ¿) = (i + t)~*F(a, a + J, 2a + f, In this last formula make £ = 3 — 2 4/2 ; then ^(4«, + 2a+|, 3-2 V2) = (4—2 V2)~**F{pt, a+{, 2a+|, i), and so _ r(3 — 2 V2)(2 — 2 VifY 2 Vhr(2a-\-Z) L (4— 2V2)* J F (a $)r(a -)- f) __ / 1 2 \/7tT(2a -)- |) = \l6j jr(« + i)r(«+ Î)· In the same way we have ¿“(i — t)*aF(^a, 2a -f- b 2a f, t) = [(!+/)* J F [“’ a + 2a + T’ (I +tj~ J _ pXl^ÆTj *SrF(2a + i) r , t x (¿a-6/+i)p - L(i+i/ J 1 ¿'(«+i)r(« + /L"’a+T’a’ (i + P J , r(-*)r(2a+})/*-6# + i r{a)r{a +1-) (I + ty Taking the derivative of the second member and neglecting all terms which are zero for t — 3 — 2 V2, we find j------ f(3 ~ 2 ^2)(2 ife - 2)a~|a v L (4-2 v2y J (3 4/J - 4)(- 4 IZ) n- i)r(2flT + j) (4—2 v'i)’ · r(a)r(a + i) ’ £ — 2336 LINEAR DIFFERENTIAL EQUATIONS. or, on reducing, n _ V---7 llS 4 i^r(2« + D V16 j F(a)l\a-i-i) ' 23. The following formulae have been obtained in an analogous, way: (25) aF(a, /?, i, x) =F^2a, 2/?, a-\- /?+£, —---^+/r^2ir, 2fi, a-|-/S4-£, - a — - VnF (a + P + i) r(a + £)F(/3 + i)’ (26) cF(ot, p, J, *) _ (>t/ -y· ____ r _L 4/T\ 2a, 1 - 2/?, a + 1 - /?, ƒ (4/jk _Z t l/ v\ 2a, I - 2p, a + I - p, —J, c — 2 F (« + I — 0) F(« + i)F(i -/S)’ (27) £ VxF(a, P, f, *) = F^2a - 1, 2/? - I, a + p - £, — F^2a —1,2/3— I, f + P — I — (28) (32) F{a, p, a p — i, x) — (1 — xY'F^ia — 1, 2^—1, « + /? — i, 1 — Vi — -)· (33) fi, et + P — b, x) = =(,-^-,(1+.^· *y ^ /^2ar — 1, « — /? + £> a 4~ ~ ^ > \ Vi — x-j- 1/ (34) F (a, P, « + 0 — i, *) = (1 — *)"*( i — x-j-V—x) 2 V — x '^2a — I, a + p — 1, 2a + 2P — 2, VI — x-j- jz —2 a Vl —x-\-V (35) F (a, a + h y, x) '1 -j- Vi — _ /I_l_'^1 — x \ r? ( ~ „ 1» I — VI — jr\ -( i ) F(2*· *« + '-r.r. , + v~)· (36) F(a, <*+$, y, x)=(i—x)-‘f(2a, 2y—2a—l, y. \ 2 r 1 — x J (37) F (a, a+i, j', *)=(l+ Vx)~™F (2a, y—\, 2y—i, ~x \ t \ I + V*J338 LINEAR DIFFERENTIAL EQUATIONS. (38) F{«, fi, i+4 + i, ,) = f, i±|+1, *r(, - *) £* + 1 + I fl' + ytf+I 2 ’ 2 ’ 2 (39) A = (i — 2x)F « + /?+! >4^(1 -*)J, , Jr) = (i — 2Jr)- a + + 1 4*{x — 1) .2 ’ 2 2 ’ {2X — l)a J ’ (40) — ( I71 —X + V— x) oc, --—- , a -ft, X-- « -- 1 4 — 1) f] _ 2 9 2 r + a y -¡- I — a (Vi —x -j- V — xfA J , yy 4x(i—x) 2 » r, 4^(1 — x) (41) F(a, I — oc, y, x) = (1— x)y~'F = (1 — jr)i'-I(i — 2x)F (42) F(a, 1 — ar, x) , \ , v _ |“v — or v + 1 — O’ 4^(^r — On = K)- v F \J—, ----, r, il—y*] , (43) F(a, I — a, y, x) = (1 — x)y~T( V1 — x V— x)*-**-** F \y + « - r. Y ~ h 2y - . ----- /_=^- L (4/1 — x-j- V— x) (44) F(a, ft, 2ft, x) = (1 - x)· F [|, ft - ~, ft + ~, ~A~V- 1J = M<- „--4=, [,+i^, 1±£,, + Ì, , (45) fi, * ,) = (,- r V[f, =±i, A+i, (j^)·] = (—*)’-(' - ;)‘ X/>- f. />+Lr5. />+|. (rr^)’]. 2GOURSAT: H Y P ER GE OME TR IC SERIES. 339 (46) F (a, fi, 2 fi, x) = {l — x) ïfL 2fi ~ a, fi+ iî— i- — 4 V1 — x -1 <47) A 2A *) <48) F(a, fi, a — fi + 1, x) = (l- r)-*Fra “ + 1 ~ 2P „ a 1 r -4*1 (49) ƒ■(«, fi, a — fi -f 1, x) f+ l~A oc-fi+i, (i 2 2 (50) ^(a. fi, a — /î+ 1, x) « " + 4* 2 ' ' (i+*)a]' (51) F(a, fi, a— fil, x) = (l — xy-^(i -|- ~- /" denote those two roots of the equation (f — 6t + i)2 + 16/(1 — ifx = o which are equal to 3 — 2 V2 for x = o: t' = 3 — 2 V'i’ + V^l(3 V2 — 4)j4 + . . ., t" — 3 — 2 4/2 — V— 1(3^2 - 4)j4 + . . . , — CiW 2 V+T^a + j·) r(« + i)r(« + f)> B — ^ — i(A)« 4 4/jr F (2« + f)] r(«)r(« + i) (55) AF(a, J — «, i, x) = A( 1 - - «, « + b h x) = (- t’)\i - t')aF{4«, i, 2« + f, O + (- t")\ 1 - 0^(4*, i, 2« +1, /'% (56) Bx*F(a + i, f - «, I, + - A+I — — a, a + |, #, ■*) = (- t')\i - /')^(4«, i, 2«+ I, O - (- t")\i - t")*F{4*, i, 2a + f, Ov tn are those two roots of the equation (1 -f- 4/ — 4 Z2)2 — 16/ (1 — /).# = o 1 — V2 which are equal to---------- for x = o* 2GOURSAT: HYPERGEOMETRIC SERIES. 341 • t = 1-4/2 t" = __ V7 — V— i -x4 2 4 i — V'i , ,--- V2 -J- — 1 .*4 -(- .... ‘(57) CF(a, i~ a, h x) = C(I — — a, a + x) = <'”(1 - 0^(4«, 2« + b 4« + i, o + - naF(4«, 2a + i, 4« + i, 0» •(58) DxiF(a + i, I — a, f, x) — Dx*( 1 — x)iF(i — a, a + £, f, *) = i/2“(i - ¿0a^(4«, 2« + 1, 4« + i, O - ¿"“(I - 0^(4*, 2« + b 4« + i, O; /, t" are those two roots of the equation (r“ -j- 4^ — 4)1 "4* i6/*(i — =r o «which are equal to 2 4^2 — 2 for x = o. ¿/ = 24/2 — 2+4/— l(3 4/2 — 4)44 t" = 24/2 — 2 — V— I(3 4/2 — 4)44 , _ 2 4/ nr{2a -f I) _ ^—· 4 \Tnr{2a -j- |) “ r(« + i)r(« + i) ’ r(«)r(« + i) ’ <59) « + i> i» ·*) = (i + 0^(4*, 2or + b 2a + f, O + (! + ¿T-F(4*, 2« + i, 2« + f /"),342 LINEAR DIFFERENTIAL EQUATIONS. (60) Gx^F (a i, a -j- f, ·§·, x) = (I + tTF(4a, 2a + 2a + f, f) ~ (I + 2« + i, 2« + t") f, t" are those two roots of the equation (f - 6t-\- i)* — (i +t)'x= o which are equal to 3 — 2 V2 for x = o. t' — 3 — 2 V2 + (3 t/2 — 4)^ + . . ., t" — 3 — 2 V2 — (3 V2 - 4)x* -f . . . , p Vnr(2a + I) _4 4/^r(2« + f) il-2r(«+i)/’(«+ f)’ tz- r(«)r(« + i)· (61) £ƒ■(«, « + i, h x) = (1 - 2/')4a^(4«, i, 2« +1, f) + (I - 2t’TF{4*, h 2« + 1, *")> (62) GxiF{a + i, a + f, f, ·*) = (1 - 2trTF(Aa, i, 2a + f, t') -(i-2^(4«, i, 2a + f, O; are those two roots of the equation (1 -f- /\.t — 4.f)* — (2^ — i)‘x - o 1 — V5 which are equal to---— for x = o. , 1 — V2 V2 , f —----------— ■** + ..., 2 4 ^ /" = --— + —** + ... ^ 4 2GO URSA T: HYPERGEOMETRIC SERIES. 343 (63) EF(a, « + £,£, x) ¡2 — A4° = [—2—) F(4<*> 2a + b 4« + h 0 (2 — t"Ya + [—2—/ ^^4ar’ 2a + b 4« + i, ?% (64) Gx*F(a + i, « + f, f, *) / 2 — /Va = (,—-—j f{4 2a + i, 4a + £> H t tf, t" are those two roots of the equation (¿2 + 4* 4)* — (2 — /)\r = o which are equal to 2 V2 — 2 for .r = o. t' = 2 Vz - 2 + (3 Y2 - 4)x* -f. . . . , t" = 2 i7! — 2 — (3 V2 - 4)j4 . (65) HF(a, £ — <*,!, ■*) = //(I — *)*-F(î — or, a + I, jr) = fit - 0“JP(4«, i, 2« + I, t') + /"·(! _ i”)aF(4a, h 2a + f, O + /'"‘(i - t">)aF(4a, i, 2a + I, f") + 'lv“(i - tSyTF(4a, h 2a + Î, r), (66) + *>*-«,*> *) = A**(i — x)kF(i — a, a + f, J, *) = /'“(I - 0^(4*- 2« + b f) - ^,,a(i - oa^(4«, i 2«+1, n - t'"\l - t'")aF{Aa, h 2a + I, t">) + 4^ ¿lv“(i - naF{4a, h 2 a + f, F*) ;344 LINEAR DIFFERENTIAL EQUATIONS, t", t'", tu are the four roots of the equation (21 — 1 y -f- 16t (1 — t) x = o, taken in the order in which they present themselves when the vari- able x describes successive loops round the origin in the positive sense. /, = i + Tj(C°S4 +*/Z71 siniH + * * · » = i + ^F-(cos^- +1/^1 sin—+ . . . , =i+V2 (C0S 4^ + 1 sin V)·*4 + · * · ’ ^ = i+^(cos^ + /=lsin^ + · · · , 7J fAU- 4r(i)r(2* (cos ? + ^sm 4) r(«)r(« + i) · (67) HF{a, a + 4, J, *) = f1 ~^4^4 4? ) ^(4flr, i, 2a + j, /') + (‘ + ^'1 4,"T-F<«“· *■ 2« + b '") + ( i ·+· 4i'" - 4t",x \ 2a j J?(4a,i,2«+bn + (1 + 4tu - 4tl")"F{4a, h 2« + ¿lv), 2« + *,0 . r I W// . 4-f f2\ „„ 4^7 ( * x /1 -f-4/jr—4//a\2Ct (68) Lx*F (or + J,«+f, f. ■*) = ^---------------j F{4a, -(■ —) ^(4«, i, 2« + b n i+4^-4 < "'»\2a —j ^(4», J, 2a + J, t"') + ,----/1 + 4^iT — 4*lv,V« 4^7 ( 4~~J ^(4«, 4, 2a +1, r) 4GO US SAT: HYPERGEOMETRIC SERIES. 1', t", t"’, ¿lv are the four roots of the equation (21 — i)4 — x{i + 4t — 4?)* — O. 345 Thus t + · · · » «"=*+iWi‘,+···- r' - i + · · ·» V— I *lv=* ■** + ···» r ,,Al i6r(|)r(2or + j) ™ !»/> + £) ' (69) /?(«, a + i,2a + |,^)=(i £ 2ot + f,^) = (^—-------“) F(4a> 2a + b 4^ + i- O; tf being one of those two roots of the equation 16/3 (1 — t) -\- x if + At — 4)a = o which are zero for x = o. (70) -F(«, « + i, 2ar + f, *) = (1 — x)i F(a + or + i 2« + t>*) (2 ƒ/A\ 2a ——) F(4a, 2a + i + £· *’); / is one of those two roots of the equation i6/a (1 — /) — x{2 — t)* = o which are zero for ;r = o.346 LINEAR DIFFERENTIAL EQUATIONS. (71) F{a, a + h 2a “J~ I. ■*■) = (* — *)* F(a + a 4" 1' 2a + l> ■*■) = (f — 6/ + 1)“ ir(4a, 2a + i, 2a + f, /) t denotes that root of the equation 16/(1 — /’) + x{f — 6/ + i)’ = o which is zero for x = o. (72) F{a, a + i, 2a -f f, *) = (1 — *)*-F(« + i. « + f» 2« + f, *) = (1 + 4/ - 4?yaF(4a, i, 2a + £,/); / denotes that root of the equation 16/(1 — /) — (1 + 4/ — 4*yx = o which is zero for x = O. (73) F{a, a + i, 2or + |, x) = (1 — xyF(a-{- £, « + £, 2a-\-\,x) = (1 + ty*F(4a, 2a + i, 2a + £, /); / denotes that root of the equation 16/(1 — tf — x(i -f- ty = o which is zero for x — o. (74) F{a, « + £, 2«+ £,*) = ( 1 — *)* .F(«+i, «+£, 2or + £, x) = (1 — 2/)«« i?(4«, 2« + £,/); / denotes that root of the equation 16/(1 — t) x{2t — i)4 = o which is zero for x — o. Formulae furnished hy the Inverse Transformations. (75) F{4a, 2a + £, 2a + f, x) = (1 — x)*~*aF(% — 2a, £, 2a + f, ·*) r — i6x(i — x)9~] O’ - 6x + l)~™Fya, a + h 2a + f, > r i6;r(i — (x + « + i, 2« + |,GO URSA T : HYPERGEOMETRIC SERIES. 347 (76) F{4a, 2a + 4a + £, x) — ( i — x)* - 4a + £, x) (4_^)-V[^ + ij2a + }-i^|]i (77) 7^(4», i, 2a + £, *) = (i — x)l~**F(2a-\-\, £ — 2a, 2a + £, *) (1 + 4X - 4x')~ ™F^a, a + £, 2a + £, j , = ' 0 - « + *, 2. + 1 ■ Formula furnished by the Rational Transformations of the Third Degree. (78) A1F(a, % — a, i, x) = At(I — *)LF(£ — a, a + £, i, x) = /,zo( I — t'yF{$a, 3 a -f £, 4a + f, 7') + /"“(I - i"TF(3a, 3« + i, 4« + f, 7"), (79) BlxiF(a + i, |—a, f, *) = ^,(i — *>^(1 — a, a + £, f, x) = *'“(1 - Oa^(3«, 3« + h 4« + f- *') ~ t"'\ 1 - n“^(3«, 3« + i> 4« + f, 7"); f, tn denote those two roots of the equation (91 — 8)2 + 27^(I — t)x = o which are equal to -f for jtr = o: l‘ = i + Yr~'^1*'+· r=t-V=ls-£*‘+ . 2 Vhr (2a + r(« + i)rHP|)’ = V'— Krr)« 4 Vxr(2a + I·) r (a) r (a + 4)348 linear differential equations. >(8o) AtF(a, x)= A,(i - x)iFQ - a, a + i, x) = ¿'“C1 - 0“^(3«, 3« + i, 2a + f /') +1"\i - 3«+*, 2« +i> n, (8l) B1xiF(a + i, 1 — a, f, *) = ^^*(1 — a, «-)-|, f, *) = /'“(i — tJaF(T,a, 3a -f 2a + t') - t"\i - ¿"r^(3«, 3« + i, 2a + b t")\ ·/', t" are those two roots of the equation (i — g/)’ + 27/(1 — tfx = o "which are equal to \ for x = o: ( = t + V=1 ?As. * + (82) AJF(a, £ — a, $, x) = At(i — x)*F(i — a, a + fa x) = (— t'YF^a, i- a, 2a + 4, /') + (- t"jF(s*, i-«, 2« + f n; (83) BlxiF(a + i, a, f, *) = B^{i — x)^F{\ — a, a+|, |> x) = (~ ¿W3*, i - 2a+ 1» O - (- t'TF(3a, | - a, 2a + I- *"); /, /" are those two roots of the equation (1 + 8/)’(i — /) — 27/* = o which are equal to — i for x = o: • · ·GOURSA T: HYPERGEOMETRIC SERIES. 349' (84) F{oc, i — a, i, x) — (1 — xfF($ — a, a + i, x) = (1 — t)'F(ia, i — a, $, t), (85) F(ot + 4, f — a, I, *) = (1 — x)iF(i — a, a -f f, f, *) = (1 - *)a + *9-j-g^(3« + i, 56 - «, f, /);. t being that root of the equation (9 — 8 tft — 27(1 — t)x = o which is zero for # = o. (86) F{a, i — a, 4, x) = (i — — «, a +1, h x) = (l — 02a-^(3«. « + & £> *)>■ 87) F{a +i, f — a, *) = (1 — *)iF(l — a, « + !»!> ■*) = (! -^a+,-^r>^(3«+i, « + *, f 0; 9 * / being that root of the equation (t — gft + 27(1 — t)*x — o which is zero for x — o. (88) C,F{a, a i, i, x) = (—) ^(3», 3« + i, 4« + f» O + (4 ) ^(3«> 3« + i. 4« + I. f'), (89) «+$> !>■*)= (4 43t) F(l«, 3« + i, 4« + & O· — (“ ^ ) F(3a> 3« + 4« + ■§» O £ tt" are those two roots of the equation (9t ~ 8)9 + (3^ ~ 4)%x = o350 LINEAR DIFFERENTIAL EQUATIONS. which are equal to f for x = o. = S + · · · * t'' = §-~L*i+ ..., 2 4/^r(2or + f) _ 4 V~nr(2a -f- f) C> ~ r(a + i)r(« + *) ■’ ^ “ r(a)r{a + |) * (90) ^) = (r + 303a-^r(3<*» 3<* + i> 2a + f> 0 + (1 + Zfy^F^a, 3a + i, 2a + £, t"), (91) f, x) = (1 + 303“ A3«, 3« + i. 2a+f, ¿') - (I + 3t"Y F{ia, 3« + *, 2a + £, t") ; /, t" are those two roots of the equation (I - 90’ - *(3* + 0’ = o which are equal to ^ for x = o : (92) C.A». « + £, I, x) — (1 — 403“-^(3«, i ~ a, 2« + £, f) + (l - 403“ F(ia, i - a, 2a + f, t"), (93) A** + 4. « + f> h *) = 0 — 4*')3“ A3«> £—«, 2a+f, ¿') — (1 — 4*")3“ A3«, i - a, 2a + |> /"), t" are those two roots of the equation (1 + %tf (1 — t) — (1 — 4 ff x = oGO UR SAT: HYPERGEOMETRIC SERIES. 25> -which are equal to (— ■§·) for x = o: VT /' — — 1 — J 1-i 4- i — -g- g · · · i +------ (94) F(a, a +i, x) = (1 — y) F(^a, i — a, t). (95) i> x)—^\l o) f 1» ¿); t denoting that root of the equation (9 — 8/)V + (3 — 4*)3* = o which is zero for x — o. (96) F(«, « + i, h X) = (1 + j) -F(3«> « + |, i, 0. / ^\3a+5 Q (97) F{a+h «+*, f, ^) = (^1+ 3J ~^(3« + h « + f, f *) ; t being that root of the equation 0 — 9)7 - (* + 3)"^ = o which is zero for x — o. (98) £,/?■(«, £ — a, f, *) = £,(l — *)*^(* — a, a + f, x) = (1 — i - oc, i, f) +(1 - n*F(3a,i- *,i,n + (1 -ryF(ia,i-a,h n. (99) G,x^F{a -f- ¡, i — a, x) = Gx(i x)txiF(i a, ar-|-j·, ■§> x') = (1 — t’)°-F{la, i — a, h t’) +/(I - t"YF(3a, i - a, i, t") +A*-nmF(3a,i-a,t,n; t\ t"y trn are the three roots of the equation (3 - V)' ~ 27(1 - t)x = o,352 LINEAR DIFFERENTIAL EQUATIONS. taken in the order in which they present themselves when the vari- able x describes successive loops round the origin in the direct sense: !' = i-+W···. j = -i + V=-l^, 7y~2 f" = t- ^p/**»+ p _ 3 v*r{\) _ - 9 1 - r(i - a)r (a + i) ’ ^ “ r{a)r{\ - a) * (loo) HrF{a, i — a, f, x) = (i — x)*F(f — a, a + f, x) - ?a F(3a, £ — a, 2a + t') + ¿"“-F(3«, i — a, 2a + 4, t") + t'"*F(ia, i - a, 2a + i t"0, (ioi) isT1;ri.F(a+£, £—a, f, x) — ^xi^ — x)*F(i—a, a + |, *) =.· ¿,aF(3a, £ — a, 2a + ¿') +j*t"mF($a, * - a, 2a + f t") +/('"* F(3a, i - a, 2a+ t', t", t'" are the three roots of the equation (1 — +)* + 2 ^tx = o: 3 V2 jrJ + ¿" = + 3 ,*/2 0+ ^ = (*)* 3 V2 ^ = i+ig-y^+ ... 3r(i)r(2q + 4) _ r(« + +r(a + |)’ 1 w 9r(|)r(2a+4) J»r(a + i) ■GOURSAT: HYPERGEOMETRIC SERIES. 353 (102) Hfipc, i — «, f, x) = H& — x)iF(§ — « + i. I> ■*) = (—Ot1 — 02ai?(3«. 3«+i, 2a +1, 0 + (- Oa(i - 02a^(3«- 3« + i, 2« + », t") + (i _ ¿'")a(i - ¿"')2a^(3«, 3a + i, 2a + f t"% (103) Ktx*F (a + i, i — a, f, x) = ^(1 — *fF{\ — a + |, f *) = (- 0‘(i - ^)2a^(3«, 3« + i, 2a + f 0 _!_ƒ»(_ _ t"YaF(3a, 3a + *, 2a + /") + /(-0“(i -t"y*F(la, 3a + it2a + f t'\ ttn are the three roots of the equation (3* + i)3 — 27/(1 —(fx — o : 2?V 2 ' = -4-4, >+..., ^///_ t 2^2 * = — i — ——J **+ ■ · · . (104) L,F{a, a + i, f, jr) = (9 _ 8*')20*,a^X3a, | — a, /') + (9 - i - a, *, *") + (9 ~ St"yat"'aF(3a, l-a, l, (105) M^F(a + a+f f x)= (9-8/')^'^ (3a, |-a, £, *') +/2(9 - 8^V"^(3a, I - a, I, *") +/(9- 8/"02^"^(3«, * - a, h f") ; are the three roots of the equation (4* 3)3 — (9 — Stytx =z o:354 LINEAR DIFFERENTIAL EQUATIONS. A = (27)“ 3 ^r{\) /’(i-a)r(a+i)’ Mx - (2 7)“ 9 (I) J»r(| - «)· (106) PF{a, a + f, x) = (I + 8*0"(I ~ t’YFila, * - a, 2a + f /) + (I + 80~(l - 0“^(3*. i ~ a, 2a + f /") + (I + 8'"')2a(l - f"YF{la, i - a, 2a + *, O- (107) QxiF(a + i, a + I, f *) = (i + 8/')2a(' ~ i’YF^a, i - a, 2a + f, /') + >2(I + 8/")2a(i - ¿")^(3a, i - a, 2a + /") + ƒ(! + 8/"')2a(i - t”'YF(3a, $-a,2a + l f’J. /", are the three roots of the equation (1 - 4ty - (1 + 8/)2(i-/> = o: 2 4/ 2 ' + fir _ 1 3 V2 · 1 . t — i — -g— ƒ**+..., 3V2 + . . . , p _ 3^(i)r(2a + I) n _ _ 9 r(|)r(2a+ *) r (a + i)r(a + *)’ y - 7(^7(¡ThT ‘ (108) « + i, f, -*·) = (1 - 90“^(3a. 3« +i, 2a + f, /') + (I ~ 9*"W3*. 3a + |, 2a + *, t") + 0 ~ ^"'Y’-F^a, 3 a + 2a + -f, f"),GO UR SAT: HYPERGEOMETRIC SERIES. 355 <109) Qx^F{a + i, a + I, f x) = (1 — Qty*F(3a, 3a + h 2a + f, /') + /*(l - gt"yaF(3a, 3« + £, 2a + f, *") + /(i - 9*"02a^(3«, 3« + i, 2a + I, are the three roots of the equation (3* + 0s — 0 — ¥f* = °: 2 = - 1 + — *» + · · ., *"'= - *+ 3 2V2 j** + · · ·. 2 V2 /’*»+----- <110) -F(a, a+i, 2a+$, *) = (1 — *)*.F(a + £, a+f, 2a + |, x) — (1 — %~) F (3a, 3a + 2> 4a + f> 0> t' is one of those two roots of the equation 2’]f{i — f) (9^ — 8)*x = o which are zero for x = o. (ill) F(a, a + i, 2a + |, x) = (1 —xyF{a-\-%, a+|·, 2a + f, x) = (l +92a(! - tyF(la, «+i,4a + |, /'); if is one of those two roots of the equation 27/* + (/ + 8)2 (1 — t)x = o which are zero for x — o. <112) F(a, a + h 2a + £, x) = (1 — x)l F(aa + 2a + |, x) = (i — Fi^a , 3a -f- 2a -|- f, t) ;3$6 LINEAR DIFFERENTIAL EQUATIONS, t being that root of the equation 27/(1 — /)’ -j- (1 — 9tfx = o which is zero for x = o. (113) F(a, a + i, 2a + 4, x) — (1 — xfF(a + b a + & 2« + b *)' = (1 + 8/'/“ (1 - t)*F(3a, i - a, 2a; + f-, /) t being that root of the equation 2jt — (1 + 8^)a(i — t)x = o which is zero for x = o. (114) F(a, a-\-$,2a + %,x) = (i - xfF(a + a -f4, 2« -j-4, *), f 3 f \ 3a = l1 --j) F&a> 3« + i, 4« + t' is one of those two roots of the equation 27/* (1 — /) — (4 — 3/)*^ = o which are zero for x = o. (115) /?■(«, ar + i, 2a + 4, *) = (l — *)l.F(« + £, a + |, 2^+4, *)'. / A3a = V1 - 4/ F(3«, a + b 4« + i /'); /' is one of those two roots of the equation 27/’ — (4 — tfx = o which are zero for x = o. (116) F(a, a + 2a ~l· b x) = i1 ~ x)k F(a + i, a + f , 2a 4, = (1 + 3*>a ^(3«, 3« + i, 2« + 4, ,) t being that root of the equation 27/(1 — if — (3/ + 1 fx = Q.GOURSAT: HYPERGEOMETRIC SERIES. 357 •which is zero for x = o. <117) F{a, a-f£, 2a + £, x) = (l-x)lF{a-{-b, « + £, 2a+ |, *) = (1 — 403°-^(3«. i - a, 2« +; t being that root of the equation 2 *]t —{— (i — 4^) % —— O which is zero for x = o. Formula furnished by the Inverse Transformations. >(i 18) /''(3a, 3a + £·, 4« + i. ■*) = (1 — ^)4-ja.F(a + |, a + 1, 4« + I. ·*) (■ -f) ’*-,i[“·"+*’ m+*> · <119) F(sa, 3 a+ i, 2a + f, *) = (i — x)i~4aF(i — a, I — a, 2a +1·, x) (1 — gx)~*aF £a, a + £, 2a * - 27* (1 -*y T’ i1 ~ 9XY 2?x(l — .r)2 l· (I + 3*)_3a^ [a, «+ *» 2a + f, J ; <120) ^(3a, a + 4a + $, x) = (1 — x)*F(a + |, 3« + £> 4« + h x) i x\~2a T — 27x2 ~l (l +3) i1 - x)-°-F\_ 2a + f, ! (122) F{$a, i — a, i, x) = (i - ,r)S-2a.F(£ — 3a, Ì + a, i, x) T , (a — 8;tr)2;tfl (i-^-^La, i-a, 4^\-3a f , ! ! (9 - !+yj "+t« 2- (4*·- 3y_ (123) F{3 2 ’ 0 I-®*-1 F « + i> t— a> l> O - 9)^ ' -27(1-x)\ (j-D(*+D^[«+*· -+*■GOURSAT: HYPERGEOMETRIC SERIES. 359 Formula furnished by the Inverse Transformations of the Other Rational Transformations. (126) F{^a, 4«+ J, 6a-\-£, x) = (1 — x)*~**F(2a+l·, 2«+i> x) (27 — 36·*·+ 8-r2\_2“c-r_. , ! , 5 —64^3(i - *) “| i — 27 8*\ -3“ ) Fa+i’2a+6’ (8^_ 36^+27/J ’ F\_a’ "+ *’ 2« + i : (127) F(4«, 4a 2« + I, *) = (1 — x)*-6aF(i — 2a, £ — 2a, 2a + £, x) (! 20x &0— ^ [«, « + i, 2« + f, (I _6tox - siy] ’ r . , 64^r( 1 — xf~\ (1 -)- 8x)~3*F |^ar, a + £, 2a-\- £, _|_ g^s-J ; (128) F{4a, 2a + £, 6a -f- i, x) = (i — xY‘F(4«+ b 2a + £, 6a -f- J, *) ,27— l8;r— ^\-2a„r . , , 64X3 "I -j ^|_a, a+ i, 2 a+ £, 27ÿ J - 2 7 tr\ —3“ r 6zpf3 1 -j (I - x)-/· L«, «+ i. 2«+ f ix_ 9y{l_x)j ; (129) ^(4«, I — 2a, 2a -f- f, x) = (1 — xYF(4a + £, £ - 2a, 2a-\-£, x) (1 + i%x — 2Tx'Y^F^a, a + £, 2a + £, > T 64.x ~~ (I - cpr)-3“(i - x)—F\ji, a + b 2a+ f fe—yfjiXx) 64X (130) F{4a, £ — 2a, f, x) (1 -*)-ƒ■ [a, *- a, Î, (8 — gxfx~ 64(1 — x) J’ /8 - i6x + 27*y f A x -(8 - 9*)·* ~| V 8 / + 2’ 8’ (8 — 36x + 27xy J ’360 LINEAR DIFFERENTIAL EQUATIONS. (131) F(¿a, 2a + i, I, x) -(* + 8)V 8 — 20X — x2' (132) F(4« + b f - 2a, A *) (ï — *)-“-έ^ι — i — a, t, /8 — 36* + 2yx^~M-ì ^ 9* j 4 (8 - gx)3xi F a + i, a +|, $, ^ ^ 27^/] » (133) -F(4« + i, 2« + i, f, *) (I _ xy 3a —1 (i + |)f [a + $, * - a, f, ~4(^18J/] > (-——-) (1 + fV [“Ή’ «+T- f> (134) F {6a, 2a + b 4a + I, *) = (1 — x)i-*aF{2a + i, I — 2a, 4a + *) ' (2— sx— ir (? +2Jr3\-2a^r , , ,« —27^(1—^)* η ) F\a, a + i, 2a+v, (2χ*-3χ*-3χ+2)'] Γ , . 27jt’(i — xf η (ï — ·* + λ·5) ^/^a, a + i, 2a + i, 4^> __ ^j^sJ ', (135) F{6a, § — 2a, 2a + |, x) = (1 — Λτ)*-2β/?(4α + i, i — 4«, 2a +f, x) i + 30X — g6x* + 64+)-2a Γ , , , „ — 108* (1 — x) ”| F |^a, a + i, 2 a + T, ^3 _ g6je> + 3Q;tr + ^J- Γ , . io8jt(i — x) “I (I - l6x + 16x')- 3*F [a, a + *, 2a + f, (l _ l6^+Ì6+)3J;GOURSAT: HYPERGEOMETRIC SERIES. 36l <136) F(6a, 4a + £, 2a +1, x) = (1 — jtr)*-8a/'’(■§■ — 4«, I — 2a, 2a + £, x) (1-33*—33x'+*Y™F[a, a+\, 2«+*. > r , — Io8jt(i — jr)n (I + 14* + ^)-3“Jp[«r, « + *, 2« + *, -(^+-i4jr_jT75iJ ; (137) -F(6«, 4« + |, 8« + *) = i1 — + b 4« + £> 8« + i> *) ¡64 — g6x -(- 3ox* -(- x 64 i)- (16 — Îôx-j-x3]-^ _r ' . I 1 1 r . 108^(1-.r) a,a+i,2a + T> ^ ^ + 30^+^ 0· 16 "\_3a zrT , , , * — io8*4(i — x)-\ v F La’01+¥> 2a+(16 -16^+^j)sJ · The transformations which we can effect when two of the three elements ar, /?, y are arbitrary have been completely given by Kummer. In the case where a single element is arbitrary, he has indicated some particular cases of the rational transformation of the fourth degree and a certain number of irrational transformations. The other rational transformations and the greater part of the irra- tional transformations above seem to be new.CHAPTER Vili. IRREDUCIBLE LINEAR DIFFERENTIAL EQUATIONS. SOME properties of these equations have already been noted in Chapter IV, but we shall study the question in a rather more gen- eral manner. Before entering into the study of these equations from the modern point of view which requires a knowledge of the group of substitutions belonging to a given linear differential equa- tion, we will give some general theorems concerning irreducible differential equations taken from the memoir by Frobenius* pre- viously referred to. In all that follows we will assume m< n. Denote by P the operator dn dxn d*~l +A^r7 + · · -+A' and by Q the operator dm dm-x dxm ^ q* dx”-1 qm ’ where p and q are uniform functions of x. The differential equation (0 _ dy , d"~y , p*=d?+p'd^ + " -+^y = ° * Frobenius : Ueber den Begriff der Irreductibilitat in der Theorie der linearen Differentialgleichungen. Creile, voi. 76, p. 256. Frobenius refers to the memoir by Libri in Creile, voi. io, p. 193, and also to the “ Note” by Brassinne in the Appen- dix to Voi. II of Sturm’s Cours d’Analyse, 362IRREDUCIBLE EQUATIONS. 365. is reducible or irreducible according as it has or has not integrals in common with an equation of lower order, say dmv dm~1v (2) Qy = ^ + q'd^' + ' · " + ^ = a Let n — ;« = /, and for brevity write Q instead of Qy; no inconven- ience can arise from this abbreviation, as it will always be clear whether we mean by Q the operator or the differential quantic: which is the left-hand member of (2). Form now the derivatives dQ dU2 dx’ dx2 ’ * ’ dxl’ and from the equations so obtained find the values of dmy dm+ly dny dx™’ dxm^’ ' ' ’ ~dx" and substitute these in (1); we have then an expression of the form (3) = + r^dbfi + + · · · + r& + r«R = °> where r09 rx, r2, . . . , rt are uniform functions of x9 and R is a linear function of dxy dx y dy dxx ’ dxx~1 9 ’ dx9 ^ (\-m— 1), having for coefficients uniform functions ©f x. From this equation it is at once evident that all functions which are at the same time integrals of P = o and <2 = 0 must also be integrals of R — o. For convenience we will employ a notation borrowed from the Theory of Numbers and write equation (3) in the form (4) P = R mod Q.364 LINEAR DIFFERENTIAL EQUATIONS, In this congruence P, Q, and denote differential expressions in which the coefficient of the highest derivative is unity, and, denoting by n, m, 1 the orders of P, Q, and 7? respectively, we have the inequalities n ~ > A. Suppose that #// the integrals of 0 = o are also integrals of .P= o; then it follows from (4) that they must also be integrals of R = o; but the independent integrals of Q = o are ^ in number, and the order of = o is A, which is less than m, and as 7? = o can only have A. independent integrals, it follows in this case that R is identically zero. Therefore : If the differential equation P — o has among its integrals all of the integrals of Q— o, then P can be put in the form P = d‘Q d'-'Q dx1 • + *iQ> >or P = 0 mod Q, Again, suppose Q = o to be an irreducible equation, and suppose the equation P = o has an integral Y which is also an integral of Q = o; then the order of Q cannot be higher than that of P; if then P= R mod Q, Y must also be an integral of R = o; but the irreducible equation Q — O can have no integral in common with an equation of lower order, so that R must be identically zero ; that is, we must have P=o mod Q. It follows therefore at once that If a linear differential equation has among its integrals one which is also an integral of an irreducible linear differential equation, then all thè integrals of the latter equation are integrals of the first. Suppose we have two equations P = o of order n and Pl = o of •order the integrals, if any, which satisfy these two equations will be integrals of a third equation which can be found by a processIRRED UCIBLE EQUA TIONS. 365; quite analogous to that for finding the greatest common divisor.. We have, viz., P~P^ mod Px, Px = P3 mod , Pi-i = mod -P.· · Denoting by nk the order of Pk, we must have n = nx > n% > n% > . . . , and must vanish at the latest when k — 1 -f- nx. Suppose ni+l = o but Ui not zero; then Pi+1 either reduces to merely y or is, zero. In the first case P = o and Px = o have no integral in com- mon except y — o; that is, they have no integral in common. In· the second case the integrals common to P — o and Px = o are integrals of P{ = o, and all the integrals of Pi = o are integrals of P — o and Px — o. If, therefore, a linear differential equation is reducible, there exists a linear differential equation of lower order all of whose integrals are also integrals of the given equation. If P = o is a reducible linear differential equation, and Q = o is a linear differential equation of lower order all of whose integrals are integrals of P — o, then, as we have seen, we have P~ o mod Q; that is, the left-hand member of a reducible linear differential equa- tion is of the form dlQ , dl 'Q dx1 dxl + + rtQ. We will revert now to the original definition of P and Q as operators,, and add the operator R defined by366 LINEAR DIFFERENTIAL EQUATIONS. Instead of the congruence P=o mod Q we can now write Py = R (Qy). We recall that the orders of P, Qy R respectively are n, my /, and n — m -f- /. Suppose w to be the general integral of the equation Ry = o, and v an integral of Qy = w; then v is also an integral of Py = o, and, from what has been assumed concerning this equation, is a regular integral. The function w = Qv is then also a regular func- tion, and consequently Ry = o is an equation of the same form as Py = o. The differential equation Qy — w containing the / arbi- trary constants belonging to the function w is therefore an integral equation for the given reducible equation Py — o, and this last equa- tion is deducible from Qy = w by differentiation and elimination of the l arbitrary constants. If then a linear differential equation of order n has among its integrals all the integrals of an equation Qy — o of order my each of the integrals of the given equation sat- isfies a differential equation Qy = w, in which w is an integral of a determinate equation of order n — m. Conversely, a linear differ- ential equation is reducible when it has for an integral a differential equation of the form Qy = w. This form of the integral equation is the characteristic property of reducible linear differential equations. So far we have only explicitly considered equations of the type studied by Fuchs in his first two memoirs, viz., equations with uni- form coefficients, a finite number of critical points, and having only regular integrals. We may consider more generally linear differen- tial equations of which we assert merely that they have uniform coefficients. An irreducible equation of this sort is defined as one which has no integrals in common with a differential equation of lower order having also uniform coefficients. The theorems already proved can be readily seen to hold for this more extended class of equations; but as the whole matter will be taken up presently from a different point of view, it is not necessary to dwell longer on it here. One general theorem, however, it is desirable to give, viz.,IRREDUCIBLE EQUATIONS. 367 that if of two distinct integrals of a linear differential equation one is equal to a differential expression with uniform coefficients of the other, then the differential equation is reducible. Suppose ^ dmy , dm~ly , ~ ^ dxm~l where qoy qiy . . . , qm are uniform functions of x. Denote by ya •andjy1 two distinct integrals of the equation dnv dn~y Py ~ ^ dx* dxn~' + · · · + P«y = 0 ’ where p0, piy . . . , pH are uniform functions of x, and suppose Ji = Qy0 · If now, contrary to the statement in the above theorem, Py — o is irreducible, then, since one integral of Py — o is an integral of P{Qy) = o, all of the integrals of Py = o must be integrals of P{Qy) = o ; if then y is an integral of Py — o, Qy is also an integral— that is, Py — o will be satisfied by the functions y0, Qy0, Qy*> Qzy« · · · ; but as Py — o can only have n independent integrals, we must arrive at a function, say Qky, which is linearly expressible in terms of the preceding ones, or we must have an equation of the form (t»y» + ^,Qy.+ · · · + <**Qky = o, where ak is not zero, and between the functions y», Qy», · · · Q^'y, there is no such linear relation with constant coefficients. The number k must, of course, be greater than unity, since by hypothesis y0 and y1, = Qy0, are distinct integrals. If k < n, then we have still368 LINEAR DIFFERENTIAL EQUATIONS. to find n — k integrals which, together with those written above, will constitute a complete independent system. Write now fir) = a„ + ay + . . . + akt*, and = fir) +f{r)s + . . . r — s and consequently fir) = a, + ay + . . . + a#*'1, fif) = «·, + a*r + · · . + akr>i ~2> f„ir) = ak. Assume further Ry = fir)y +fir)Qy + . . . + fifQ^y, then R{Qy) =f{r)Qy + · · · + f{r)Qky. We have, however, fir)Qky. = «*-% = r[f(r)y, -j- fir)Qy<> + · · · + fkir)Qk !?·] — f ir)y. · If therefore r is a root of fir) = °> we ^ave RiQy.) = r'Ry« > and consequently each integral of the irreducible equation Py = o must satisfy the equation RiQy) = rRy- Denote by/, the integral and Ry0 will become (co ~i~ cir ck~yk~1)Ry0 ckRyk + · · · + cn-iRyn-i, and consequently all the branches of this function are expressible as linear functions (constant coefficient understood) of Ry*, Ryk, ···, Ry»-^ It follows therefore that Ry0 satisfies a linear differential equation with uniform coefficients whose order is at the most n—k-\- 1. The linear differential equation Py — o has therefore an integral Ry, = /,<>>„ + flr)y, + . . . +fk{r)}’k-l in common with a differential equation of lower order, and in con- sequence cannot be irreducible. We will now take up the subject of reducibility of linear differ- ential equations with uniform coefficients from the point of view of the groups to which such equations belong. Suppose P = o to be an equation of order ny and Q — o an equation of order m < n all of whose integrals are integrals of P = o. Denote by Yx . . . Ym a system of fundamental integrals of Q— o; then any substitution belonging to the group of P= o can by hypothesis only change Yx . . . Ym into linear functions of themselves. Let jyw+I . . . yn de- note the functions which with Yx . . . Ym form a fundamental system of P— o; then the substitutions of the group of P will change these functions in general into linear functions of themselves and of Yx ... Ym; that is, all the substitutions belonging to the group of P will be of the form IF Y ' f f j y1 * · · yn * · · · fn370 LINEAR DIFFERENTIAL EQUATIONS. where f · . . fm are linear functions of Yx . . . Ym alone, and /W+I . . . fn are linear functions of Y1 . . . Ym, ym+1 . . . yn. Reciprocally, if we can choose y1 . . . ym such that the group G contains only substitutions of this form, then these functions will satisfy the equation dy d”‘y dx’n yu dy, dy, ~dx' ' ' dxm ym, dym dmym ~dx' ' ‘ dxm where the coefficients of y and its derivatives have for ratios only uniform functions. Suppose x to describe an arbitrary closed con- tour containing one or more critical points ; then yx . . . ym will change into + · · · + c™ym, cnu y l + · · · + ^mm and consequently all the coefficients of this equation will be multi- plied by one and the same factor, viz., the determinant I c* of the substitution corresponding to the path described by x ; the ratios of the coefficients are consequently unaltered, and these are therefore uniform functions. The question of the determination of the group of a given linear differential equation will not be taken up here, but we can investigate the following question, viz.: Having given a group G composed of linear substitutions S, Sl9 S9 9 . . . among n variables, required to determine whether or not the linear differential equation which has G for its group is satisfied by the integrals of analogous equations of order lower than ny and to determine the groups of these equations.IRREDUCIBLE EQUATIONS. 371 From what precedes this can be stated as follows: Required to determine in all possible manners a system of linear Junctions Yx, . . . , Ym of the variables yx, . . . , yn such that each of the substitutions S, Sx, . . . of which G is composed shall change Yx y ... , Ym into linear functions of themselves. This enunciation may still be slightly modified, and in order to do so it is necessary to introduce a new term which shall replace the French word faisceau employed by Jordan. The author has not been able to find any single English word for faisceau which is not already employed for other purposes, or which would be at all appropriate, and therefore suggests the term function-group. We will therefore say that a system of functions forms a function- group when every linear combination of these functions forms a part of the system. If Yx, . . . , Ym forms one of the systems of functions which we have to determine as above described, then each substitution of the group G will transform the different elements of the function-group corresponding to Yx, . . . , Ym into other elements of the same function-group. In analogy with the theory of groups of substitutions, we see that each function- group will contain a certain number of linearly distinct functions in terms of which every other function in the function-group is linearly expressible. We can say then that the function-group is derived from these linearly distinct functions. Suppose now that in the function-group corresponding to Yx, . . . , Ym we choose any other m linearly distinct functions; then (using for brevity F. G. to denote function-group) every other function of the F. G. is linearly expressi- ble in terms of these new m functions, and these latter will further form a system possessing precisely the same properties as the given system Yx, . . . , Ym. If now we consider the F. G. instead of the particular system Yx, . . . , Ym, we shall have the advantage of a greater freedom of choice of the functions from which the F. G. is derived. Our problem may therefore be stated as follows: Required to determine all the function-groups such that each of the transformations S, Sx, . . . shall transform any element of one into some other element of the same one. There will of course always be one such F. G. formed by the aggregate of the linear functions of yx, . . . , yn. If there is only one the group G will be said to be prime, and the corresponding372 LINEAR DIFFERENTIAL EQUATIONS. differential equation will be irreducible. If, on the contrary, there^ should be more than one, then to each of the F. G/s derived from the linearly distinct functions Y1, . . . , Ym there will correspond; a reduced differential equation whose group will be formed by the. changes which the substitutions of G impose upon Yx, . . . , Ym. We have now first to indicate a means of ascertaining whether the group G is or is not prime, and in the latter case to determine a function-group containing less than n linearly distinct elements.. This question will be taken up in Chapter XI. All that immedi- ately precedes concerning the group of an equation, etc., is taken directly from a memoir by Jordan.* Chapter XI. is also taken from, this memoir, the only one that the author has knowledge of which deals directly with the subject. * Mémoire sur une application de la théorie des substitutions à Vétude des équations; différentielles linéaires : Bull, de la Société Mathématique de France, t. 2,.p. ioo.CHAPTER IX. LINEAR DIFFERENTIAL EQUATIONS SOME OF WHOSE INTEGRALS ARE REGULAR. We will now resume the direct study of the integrals of linear 'differential equations with uniform coefficients, and will treat par- ticularly the cases where not all of the integrals are regular, and show how to determine the number of such integrals in each case. The method employed is due to Frobenius and in part to Thom6. The equation is , v „ dny , dn-y ^ “ lïx*Jrt>'dxn-'~ dn~ 'dx' rr + +Pny = o; the critical points of the coefficients px, . . . , pn are to be poles, and therefore in the development of any p in, say, the region of x — o we shall have only a finite number of negative powers of xy ;and so for this region each p may be written in the form (2) P iM x“ ’ where II (x) contains only positive powers of x. Denote by œx the degree of x in the denominator of px, by the degree of x in the .denominator of etc. We will for convenience call the order of the coefficient pi. Form now the series of numbers (<*?0 = o) (3) + ny <», + «— 1, go* + n — 2, . . . , oon, which denote respectively by I20, £lx, A,, . . . , Hn . Let g be the greatest of these numbers ; obviously, there may be more than one 12 equal to g. Supposing, however, that these numbers are arranged in ascending order of their subscripts, •(4) y I2\ y 12^, y y 373374 LINEAR DIFFERENTIAL EQUATIONS. we shall consider in particular the first £ly counting from the left*, which is equal to g. Suppose this is 72t·; then (5) A = ooi + n — iy =g; A will be called the characteristic index of the given differential equation. This notion of the “characteristic index” is due to* Thome. In the case studied already, i.e. where all the integrals are reg- ular, it is obvious that we have as the greatest values of gd1 , . . . , oony oox = 1, = 2, . . . , and the numbers i20, £lx, . . . , £ln have n for their maximum value, and so i20 = g — n. The theorem proved by Fuchs in this case may now be stated in the form : In order that the differential equation P — o shall have all of its integrals regular in the region of x — o it is necessary that its coeffi- cients contam in their developments only a finite number of negative powers of xy and, further, that its characteristic index shall be zero. And the converse theorem is: If the coefficients of P — o contain only a finite number of negative powers of xy and if further, the characteristic index of the equation is zeroy then in the region of x — o all the integrals of the equation are regular. These integrals are, as we know, each of the form •*-p[ + 0. log * + · · · + 0.log»*], where the functions 0 are holomorphic in the region of x = o, and therefore contain in their developments only positive powers of x; and further, 0O, . . . , 0a do not all vanish for x — o. The expo- nents Pj , p2, . . . , pn are roots of a certain algebraic equation—the indicial equation. The method of arriving at this equation may be briefly recalled. The differential equation is of the form (6) dny .IIJx) dM~y dxn x dxn "1 + · ■ . + nH(x) xH y = o ; where the functions 77 contain only positive powers of x. Now make = xp, and substitute in this equation ; we have (7) x>~n[p(p- 1) . . . (p —«+ I) + n,{x)p(p — 1) . . . {p — n — 2) + . . . + nn(x)J.EQUATIONS SOME OF WHOSE INTEGRALS A EE REGULAR. 375 Multiply this result by x~*>, and equate to zero the coefficient of x~n, and we have the indicial equation (8) p(p—i)...(p — n+i) + n>(o)p(p — i) · · · (p — n-f- 2) -f- · · · + njo) = o. The following theorems may be easily proved, and the reader is advised to consult Thome’s memoirs in vols. 74 and 75 of Crelle concerning them. They are given here without proof, partly to economize space but mainly because they are in great part included in the more general theorems which follow and of which complete proofs are given. When the coefficients piy , . . . , ps of the differential equation P — o contain in their developments only a finite number of negative powers of x, and if the equation has at least n — s regular linearly in- dependent integrals, the remaining coefficients, /S+I . . . pn, “will contain in their developments only a finite number of negative powers of x, and the characteristic index of the equation will be at most equal to s. As consequences of this we have: 1. If the 5—1 first coefficients px . . . ps_x ofP— o contain in their developments only a limited number of negative powers of x, and if ps does not satisfy this condition, then the equation has at most n — s regular linearly independent integrals. 2. If all the coefficients px . . . pn contain only a finite number of negative powers of x, the equation has at most n — i regular linearly independent integrals where i is the characteristic index of the equation. If i — o (Fuchs’s case), we know that the equation always has n regular integrals ; if, however, / > o, it may be that the equation will not have n — / such integrals. For example : Write and consider the two differential equations (9) (9Ï y=0, Y -f- k = o, where h is an arbitrary function of x.376 LINEAR DIFFERENTIAL EQUATIONS. Let us form the equation (10) , dY „dh h~dx~Ydx~°’ that is, (II) d'y , dy dS+P'di+P'y-°' where dlogh dk d logk p' = k dx ’ *' = dz k dx All the integrals of (9) and (9)' satisfy (4); and conversely, as (10) gives by integration Y = Ch y if y is a solution of (11) or if y verifies equation (9), — will verify equation (9)'. Therefore, if (9) and (9)' have no solutions pre- senting the character of regular integrals, equation (11) will have no regular integrals. Let pi and h be taken arbitrarily, whence = that is, for x = o, k is infinite of the order 4. This order being superior to 1, equation (9) has no regular integral. Equation (9)' has now no solution of the nature of regular integrals, for its general integral is (12) y = e-fMx{Cf — f kefkdxdx)\ or, replacing h and k by their values and effecting the integration within the brackets, 03) y = r7"' [©(■*■) + C4 log x\,EQUATIONS SOME OF WHOSE INTEGRALS ARE REGULAR. 377 Cx being a constant, and the function &(x) containing in its develop- ment an unlimited number of powers of x~xy as well as e~fkdx. Therefore, none of the integrals of (9) and (9)' being regular, equa- tion (11) has no regular integrals. It is of the form +a^-1)^-(,°-,,+2> + · · · +/*-i ~ + A«J· We see at once that the product x~pP(xp) can be developed in inte- ger powers of xy and that the development will only contain a finite number of negative powers of x; the coefficients in the development378 LINEAR DIFFERENTIAL EQUATIONS. will contain only positive powers of p, and pn is the highest power that can occur. When the differential quantic P is given, its characteristic function P(xp) is of course known. Suppose, how- ever, a characteristic function xpf(x, p) to be given where f(xy p) is an integral function of p whose coefficients are functions of x : what is the corresponding differential quantic ? Write We know now that the integral function f(xy p) of p can be placed in one way, and in one way only, in the form (18) f(x, p) = U„p{p — i) . . . (p — n — i) (16) f{x, p+ i) — fix, p) = fix, p) and (17) W fi*, P)]p=o + £4-i pip — 0 · · · ÌP — n + 2) + . . . + Utp -)- U„ ; it follows then that (19) xef(x, p) — xp Unxn is the characteristic function of the differential quantic We know that the product (20) x~pP(xp) = pjp—l) . . . ip — «+1) jr* + / PÌP ~ 1) · · · (p -n+2) + · · · + Pn-1J + Pn can be developed in a series of ascending powers of x, containing only a finite number of negative powers and in which the coefficients are integral functions of p of degrees at most = n. We wish nowEQUATIONS SOME OF WHOSE INTEGRALS ARE REGULAR. 379 to determine the first term of this series. The exponents of x in the denominators of the expansion of x~f>P(xf>) are obviously the num- bers £10, £liy £ln above defined. If g is the greatest of the numbers £ly the first term of the series will clearly^be of the form S~+f \ where G(p) is an integral function of p and does not contain x. If we denote by y the degree of G{p)y we have obviously (21) y — n — iy where i is the characteristic index of the equation, i.e.y i is the index of the first £1 which is = g. In the particular case of i = o the equation G(p) = o is the indicial equation already defined in the case where all the integrals of P = o are regular. In the case of z > o we can now generalize this notion of the indicial equation and say that G{p) = o is the indicial equation of our present differential equation P=oor of the differential quantic P. The function G(p) will be called the indicial function. Thus to obtain the indicial function of a differential quantic, we form its characteristic function and, after multiplying by x~^y develop the product in ascending powers of x ; the coefficient of the first term is the indicial function. It is to be noted that, knowing gy it is sufficient to multiply the characteristic function by x^~p and then make x = o; thus [x'-oPM]^ is the indicial function of P(y). Some properties of the indicial function of the differential quan- tic P will now be given. First: if pn is identically zero, the characteristic function and consequently the indicial function is divisible by p. Effecting this division and changing p into p -f- 1 in the quotient, we obtain, by equating it to zero, the indicial equation of the differential equation dy of order n — 1 obtained by taking ^ for the unknown variable. Second : if in the equation P = o we put (22) y — xf>*wy38 o LINEAR DIFFERENTIAL EQUATIONS. the indicial equation of the differential equation in w thus obtained will have for its roots those of the indicial equation of P = o di- minished by p0. For the characteristic function of P(x^w) = o is P(xpo+p). It is therefore deduced from the characteristic function of the equation in y, by changing p to p + p0; and, consequently, the same is true of the indicial equations. Finally : if in the equation P = o we place (23) y = f{x)w, being a holomorphic function in the region of the point zero and not vanishing for x — o, the indicial equation of the equation in w thus obtained will be the same as that of the equation in^. For it is easily seen that, the equation in w being of the form / dnw dn~Iw ^ ~dx* + An -2„£. + (A-h^)^ + + (A + p«)w = o, •its characteristic function is the sum of two terms. The first term, <25) ^[^-rl)"^(p~" + 1> +A p(o — 1) . . . (p — n + 2) • · · APr, is the characteristic function of the equation in yt and in the second term, (26) 2>[p0,-|) · -<<■-» + ■> + pA-')--.(f-’ + 2)+ ■ ■ · +e.]. the highest exponent of x in the denominator is inferior to the high- est power, gy in the denominator of the first term. Whence it follows that upon multiplying by xs-? and then making to obtain the indicial function of the equation in w, the second term will vanish and the result will be the same as if the operation had beenEQUATIONS SOME OF WHOSE INTEGRALS ARE REGULAR. 38T performed upon the first term alone, that is, it will be the indicia! function of the equation in y. It follows from these propositions, px, pa, . . . , py being the y roots of the indicial equation of P: 1. That if in P = o we put (27) y=y,w, where yx is an integral of the form x^tp^x), ip(x) being holomorphic in the region of the point o and not zero for x — o, the homogeneous differential equation in w thus obtained will have an indicial equa- tion whose roots are Pi Po y Pa Po f · · · > Py Po f and, this equation being divisible by />, one of the quantities px, pif . . . , py must be equal to p0; let px — p%. 2. If in the equation in w we now put w —f zdx, the indicial function of the equation in z will be of degree y — 1, and will have for roots Pa Po Ps Po If · · · f Py Po 3. Consequently, if in P = o we make the substitution y = yifzdx, the equation in zy of the order n — 1, thus obtained will have for in- dicial equation the equation of degree y — 1 which admits the roots Pa Po · · · > Py Po I· The simple relation (28) t —y = ft between the order of the differential equation, its characteristic index, and the degree of the indicial equation, enables us to replace the notion of the characteristic index by the more rational consid- eration of the indicial equation. The characteristic index of an.382 LINEAR DIFFERENTIAL EQUATIONS. equation is merely the difference between the degree of the equation and that of its indicial equation. Whence it is evident that all propositions concerning the characteristic index and, consequently, relating to equations whose coefficients are infinite of a finite order for x — o can be expressed by aid of the indicial equation. Thus: The number of linearly independent regular integrals of the equa- tion P — o is at most equal to the degree of its indicial equation. To obtain a clearer idea of the characteristic and indicial func- tions of P — o, the equation may be put in a certain normal form. Thus we may write i dny <29) A dn^y r 1 ~n-1 __-A dx"-1 +ù?*%+*-’=°· I px pn-1 Reducing the fractions — , , pn to the least com- mon denominator Xs and multiplying by xs, the equation becomes dny dn~xy (3°) T^ = ^ndi« + ^a~ld^ • · — Oy where the functions 4 > tl, ··· 7 Ifl contain only positive powers of x and do not all vanish for x = o. This form of the first member of a linear differential equation will be called the normal form. The characteristic function of the differential quantic T is (31) T(x>) = x^[iap{p — i) . . . (p — m+i) · · · (P — nt-\-2)-\- . . . + ¿„-.p-f-A]; hence the product x~pT(xp) contains only positive powers of x and does not vanish for x — o; its constant term is the indicial function. Conversely, it is easy to see that if a linear differential equation has a characteristic function fulfilling these conditions, it is in the normal form. Hence an examination of the characteristic function is sufficient to determine whether the differential equation is or is not in the normal form.EQUATIONS SOME OF WHOSE INTEGRALS ARE REGULAR. 383 Our next step will be to define a composite differential quantic and prove one of its important properties. In the differential expression dav da~zv (32) A(y) = A^+A,^z-1 + dy • + A-'dï + A·* the letter^ will be considered a symbol of operation such that A(y) indicates the definite operation to be performed upon y: <33) Also, A(B), or simply AB, will indicate that the same operation is to be performed upon B. If then B is a differential quantic, AB will also represent a differential quantic Cy and we shall say that the quantic AB = C is composed of the qualities A and B taken in that order. The same definition holds for a quantic composed of more than two quantics. If the coefficients of the component differential quantics con- tain only a limited number of negative powers of x as is here sup- posed, it is clear that it is the same with the composite quantics. Let AB = Cy and let « <34) ' A (x?) = x*h{x, p) = 2lih^p)xi‘+>', B(xp) = x<·k{x, p) = 2,kv(p)x<‘+’', - C(■*rP) = P) = 2Jk{p)xf+K, be the characteristic functions of the quantics A, B, and C. We have (35) CO'*) = AB(xf) — A \j2vkv{p)xr+'r] — '2vky(p)A(x*+'’); that is, (36) = ^A(p + ^pK+*·· From this equality it follows that if A and B have the normal form, since in that case h^p) and kv(p) vanish for negative values of p and v but not for the value zero, 4 (p)xk will contain only posi- tive powers of xy and will not vanish for x = o. Therefore, the char- acteristic function of C divided by xp fulfilling these conditions, C384 LINEAR DIFFERENTIAL EQUATIONS. will itself have the normal form. Moreover, making x = o in the same equation, we have the following simple relation between the indicial functions of A, B, and C: (37) hip) = Kip)Kip)· We may therefore state the following proposition: If a differential quantic is composed of several differential quan- tics each of which is in the normal form, it has itself the normal form, and its indicial function is the product of the indicial functions of the component quantics. The degree of /0(p) is consequently the sum of the degrees of KiP) and K(p). More generally, we may remark that if two of the three differen- tial quantics A, B, and C have the normal form, the third has also the normal form. The notion of the component factors of a differential quantic leads directly to that of reducibility; but as this subject has been treated elsewhere, it will not be resumed in this conhection. Resuming now the question of regular integrals, we see at once that if the equation P = o has a regular integral, then P — o is a re- ducible equation. For, if P = o has a regular integral, it necessarily has one of the form xPtp(x), where ip(x) is a uniform function which does not vanish for x = o. Now obviously the function (38) j. = ti*) is an integral of the equation of the first order (39) dy dx where (40) Pi*) X p 1 Ki*) Px{x) containing only positive powers of x. Therefore P = o is reducible and has among its integrals all those of Bx dy dx PI*) o. (41) XEQUATIONS SOME OF WHOSE INTEGRALS ARE REGULAR. 385 We can now place P — o in the form QXBX — o, where Qx is of order n — 1. If, again, Qx = o has a regular integral, we can write (42) <2, = <2A, where Qa is of order « — 2. Continuing this process, we have finally (43) ip(x), (47)EQUATIONS SOME OF WHOSE INTEGRALS ARE REGULAR. 387 which is not a regular integral. The function tp(x) contains only positive powers of x and does not vanish for x = o. Suppose u0 and wQ to be the values of u and w for x = o. The characteristic function of the considered equation is xP(up -f- w), and the indicial function is (48) M„p + W0 . In the first case, then, where u0 is not zero, and consequently where the equation has its integrals regular, the indicial function is an integral function of p of the first degree; and in the second case where u0= o and the equation has no regular integral the indicial function is a constant. The converse of this is readily shown. It is obvious from what precedes that if the equation P — o has all of its integrals regular, then the degree of its indicial equation is equal to its order; and also that the number of linearly independent regular integrals of P = o is at most equal to the degree of its indicial equation. These results have, however, been previously arrived at. We will now obtain the precise condition which the differential equation P — o must satisfy in order that the number of its linearly independent regular integrals may be exactly equal to the degree of its indicial function. We suppose the differential quantics/5, Qy D to be in the normal form. If the number of linearly independent regular integrals of P = o is y, where y is the degree of the indicial function of P, then P can be placed in the form P= QD, where Q is of the order n — y and has a constant for its indicial function. For, since P= o has y regular integrals, it can be placed in the form P = QD, where Q is of order n — y and has no regular integrals, and where D is of order y and has all of its integrals regu- lar. The indicial function of P is then the product of the indicial functions of Q and D ; but since D has all of its integrals regular, its indicial function is of degree y, and this is also the degree of the indicial function of P; hence the indicial function of Q is a constant. Conversely, if the differential expression P having an indicial function of degree y can be placed in the form P — QD, where Q is388 LINEAR DIFFERENTIAL EQUATIONS. a differential expression of order n — y having a constant as its in- dicial function, then P = o will have exactly y regular integrals* For, since P is of order n and Q of order n — y, D is of order y ; now the indicial function of Q being of order zero, that of D is of order y, and since the indicial function of Q is a constant,. Q can have no regular integrals, and so the equation P — QD = o will have for regular integrals those of D = o; that is, P— o has ex- actly y regular integrals. Therefore, finally : In order that P = o having an indicial function of degree y shall have y regular integrals, it must be possible to place P in the form P — QD, zvhere Q is of order n — y and has a constant for its indicial function. Irregular integrals of the form (47) have been called by Thome normal integrals. It is not the intention here to go at all into the theory of these integrals, indeed that theory is in a very imperfect state owing principally to the, in many cases, impossibility of ascer- taining anything definite concerning the convergence of the series involved. The reader is referred to Thomé’s papers in Crelle, but particularly to the following papers by Poincaré : “ Sur les équations linéaires aux différentielles ordinaires et aux différences finies ' (Amer- ican Journal of Mathematics, vol. 7, pp. 208-258), “ Sur les intégrais irrégulières des équations linéaires" (Acta Mathematica, vol. 8, pp. 295— 344); see also a not» by Poincaré in the Acta, vol. 10, p. 310, en- titled “ Remarques sur les intégrales irrégulières des équations liné- aires.'' A Thesis by M. E. Fabry (Paris, 1885) entitled u Sur les in- tégrales des équations différentielles linéaires à coefficients rationnels" may also be advantageously consulted. The first paper by Poincaré in the Acta Math, is by far the most important of these references ; the author regrets to be unable to give an account of it here, but limits of space prevent.CHAPTER X. DECOMPOSITION OF A LINEAR DIFFERENTIAL EQUATION INTO SYMBOLIC PRIME FACTORS. The coefficientspx, ...,ƒ* of the differential equation dny dtt~Iy =?W = ^+A^+---+A; = o are, with the exception of certain isolated critical points, holomor- phic functions of x in a portion of the plane which is limited by a simple (t.e.y non-crossing) contour. P can be put in the form <2) P = AnAH_l . . . Alf where (3) Ai = ^-aiy is called a symbolic prime factor of P. For convenience the word “ symbolic” may be omitted, as it is always understood. We can now show that P is decomposable into prime factors. Let (4) = »,, yt — vjvjlx, . . . , yn = vjvjlxf v%dx . . . ƒ vndx denote a set of fundamental integrals of P = o. Form now the system of differential equations /x . dy (5) Ai = dx~ aiy = °> * = 1, 2, . . . , 389390 LINEAR DIFFERENTIAL EQUATIONS. having for integrals vxv% . . . vn ; then (6) It is obvious now that the composite expression AnAn_z . . . At is identically equal to P\ that is, the coefficients of the derivatives of y of the same order are the same in both expressions, a fact already established. It is easy to see now that every decomposition of P into prime factors is obtainable by this process, each such decomposition corre- sponding to a chosen system of fundamental integrals. We see at once an analogy between algebraic equations and linear differential equations. When we know the roots of the algebraic equation, its decomposition into prime factors enables us at once to write out the equation; so, knowing the decomposition into prime factors of a differential equation, we are enabled to form the differential equation which possesses a given system of fundamental integrals. If y1 = vx, y? = vjvjlx, . . . , y„ — vjvylx . . . f vndx are the integrals, the differential equation will be It is to be noticed, however, that while the arrangement of the factors in the algebraic equation is arbitrary (ue.y the factors are commutative), the arrangement of the symbolic factors in the differ- ential equation is not in general arbitrary. When we know a system of values of the functions vl . . . vn we know also all possible systems of fundamental integrals of P = o, and so can in all possible ways decompose P into its prime factors by aid of the formulae (7) where (8) ai = ~^c log (yxv^ . . . v,) i = i, 2, . . . , n. = S tog (<%.». Vn)' 1 — I» . · ■ , ft. (9)DECOMPOSITION INTO SYMBOLIC PRIME FACTORS. 391 Conversely, if we know the coefficients a of the prime factors, we can integrate P = o ; for we have , v Jaxdx fia2—ax)dx f{an — an-i)dx (10) vx = e , = e , . . · , vn = e and a system of fundamental integrals of P — o is then , v faYdx faxdx f{a%—ax)dx (n) y, = e , y,= e feJ , . . .. If now we wish to decompose the expression into its prime factors, we have one of two things to do: either cal- culate a system of values of the functions vx1 v2, . . . , vM9 or evalu- ate directly a system of values of the coefficients al9 . . . , an. In the first case we should have first to find a particular solution vx of the equation P(zx) = o, then a particular solution v2 of P{vxfz^dx) = o, etc. The equations in zx, z2, . . . , zn will be linear and of orders n, n — 1, . . . , 2, 1 respectively. In the second case we should have to find first a particular solution ax of P(efUxdx') — ot then a. particular solution of p\e^Uxdx fe^^~~u^dx dx\ =0, etc. The equations in ux, &2, . . . , will be of orders n, n — 1, . . . , 2, I, but will obviously not be linear. The formulae giving the coefficients px .. . pn of . . dny , dn ^ ~dx*~^^' dx"" + ··■·+ P*y = 0 in terms of at, ait . . . , an are easily found. Suppose (12) P = AnA„.t . . . Alt and further suppose A.= dn-y dxn~l dx~y + + · · · + Ç»-,?· (13) AnAn. I.392 LINE A R DIFFERE NTIAL EQUA TIONS. In this last equation replace ƒ by and so form *he expres- sion AmAn., . . . AtA,, or P(jy); we have then dn v dn~lV r ~\dn~2y (H)P^) = -¿n + (ft-ft)+ [t-(*-0“ftftj r dn~'a dn~2ax dax 1 + · · · + [ “ - q' 'dN7 ~ ' ' · ~ ~ ?-AJ * ; from which we find (l5) A = ft — , x dax A = ?. — (»— 0 — ?*"· ’ (« — i)(« — 2)d2al da, A = ?. -1—ld?' - *·<" ~2)^ -ftft. I . 2 P" — ~ ~d^x ~ Ql dxn~2 ' dx dax ~ dx ~ V*-1**1' If now we start from the expression An and form successively the expressions An , AnAn_x y AnAn_lAn_2, · · · > d4nAn^lAn_2 . . . Ax, we readily find, by aid of the preceding equations, the following values for px, p2, . . . , pn , viz.: (i6) ƒ, = —(#! + #a + ... + dan da, da„ V(n—\){n — 2)d'lax dx ’ (»-2)(« — L 1.2 ¿r2 1 i . 2 ¿¿r2 1 ' a^dx +(*_3Xft+ · · · +ft)^· + · · x da, "1 da, "1 3)^ + ··· J + "* L(*-3)^+...]+DECOMPOSITION INTO SYMBOLIC PRIME FACTORS. 393 These formulae give the coefficients / in terms of the coefficients > · · · > > now, since A, A, have the same values whatever be the chosen system of fundamental integrals of the differential equation, and consequently whatever be the chosen system of the functions v, it follows that the functions f are invariants. If ax, , . . . , an are constants, we have <18) A — — 2ait A = A = — 'Sa&Pk* · · · > and therefore A > A » · · · are the coefficients of the algebraic equa- tion 0 — ¿jX* — . (* — a,) = o, having ^ for roots. A transformation which is frequently useful consists in changing P = o into an equation whose integrals are those of P9 each multi- plied by the same factor, say N. If z be the dependent variable in the new equation, then <19) z = Ny and y — ~z, and the new equation is therefore <”> p{lfs) = 0· Let <21) y, = yt - v.fv^dx, . . . , yK — vjv^dx . . . ƒ vjx denote a system of fundamental integrals of P = o, and let P = A„An^ . . . Ax denote the corresponding decomposition of P into prime factors. In order to multiply all these integrals by N it is obviously only394 LINEAR DIFFERENTIAL EQUATIONS. necessary to change vx into Nvx ; with this change of v, the general coefficient at becomes (22) «i + ~ log N, and A( = O is (23) S-[“‘ + SlogAr]J’ = 0· We have thus the identity (24) NP[~y) = Ah' Ah'_, . . . A/. Suppose N —-, then a[ is W (25) a’ = ai + ±\ogV-^\ogW. In particular let (26) then (27) and we have (28) where (29) V=vlt W = i, d_ dx log V — a,, = a:a: ,.. .a/, w cl( — di —j— aj. Other special cases are readily found. We have, in what precedes, lowered the order of the equation P — o by unity by assuming that we knew a solution, yx = vx, of Ax = o; that is, a particular solution jy, of P — o. By aid of this solution we find the equation (30) A„A„.Z. . . A, — + · · · + Q«-iz — 0DECOMPOSITION INTO SYMBOLIC PRIME FACTORS. 395 of order n— i, and from equations (15) have its coefficients ql... qn-x given as functions of px . . . pn and ax by (31) ?i = *i+A> ^2 = Çiai + A + (n — 0 > q^a . A , (« — i)(# — 2) d*ax , , x dax ■ + A + ........■ .2 — IE? + <*■* ~ 2) Eli dy Since in Ax, = ^ yx, we have from (11) (32) „ —±dl+ a'~ y\dx’ the order of the equation P = o has been lowered by unity by the transformation (33) dy _! dh _ „ whereas the ordinary formula for the lowering of the order of the equation is (34) y = yjzdx. There is, however, a simple relation connecting these two pro- cesses. From the equation y = yxf zdx we derive (35) II and from (36) dy_ dx y^dxy ’ we have (37) I II396 LINEAR DIFFERENTIAL EQUATIONS. It follows now at once that the integrals of {38) AnAn_i ... A, = 0 obtained by one of the two methods are equal to those of P{yifZ(^x) obtained by the other when each of the latter integrals are multi- plied by yx. If we multiply the solutions of <39) = o by yx, which is done by changing z into — , we have, since the first y 1 coefficient of P J'-- dxj is unity, the identity (40) P(,,fldx) = AxA„.t ... A,. If now we divide the solutions of AnAn_t ... A9 — o by yx, which, by a formula analogous to (29), is done by changing a{ into a{ — al, i =: 2, 3, . . . , n, we have a second identity, (41) ~ P(yJ'ydx) = AJA.'., . . . A’, where (42) a[ = at — ax. Suppose the algebraic equation = o has a root y = yx; in f dF order that y — yx shall be a double root we must have = o for y ~ yx. A similar property exists for the linear differential equation dny dn~xy +Ay = o., Suppose /, a solution of ƒ* = o obtained by equating At to zero in P = A„An^ A In order that/, shall be a solution ofDECOMPOSITION INTO SYMBOLIC PRIME FACTORS. 397 where the factor Ax has been dropped, it is necessary and sufficient that yx satisfy the equation (43) dn~'y ' dxn ~1 ■(* x dn~*y • · Pn-iy — °· The proof of this is very simple. The identity (40) shows that the equation AnAn.x . . . A, = o is satisfied by y = yx if (44) P{y'fy\dx) = °; that is, if (45) P{xy,) = o· This means that P = o must have y — xyx as an integral. Substi- tuting this value of y in P = o, and remembering that P(y^ = o* we have , dn~ly dn~2y (46) «^rr + («- · · · +P»-,f = o, which proves the proposition. It is clear now that if the equation (47) AnAn-i ... A, = 0 has again the integral yx, it is necessary and sufficient that we have (48) p{y>f ~y~dx) - p (*>.) = 0; and in general the necessary and sufficient condition that yx shall be a solution of P = o, and of the n — I equations derived from P = o by the substitution39^ LINEAR DIFFERENTIAL EQUATIONS. is that P=o must admit of the n solutions (50) ƒ„ xyt, x'y,, · · · ,xK 'y*· These solutions are called conjugate solutions of the differential equation, and they are analogous to the equal roots of an algebraic equation. Suppose the differential equation P= o to admit of k conjugate solutions: we can readily show that P can be decomposed into n prime factors of which the last k shall be equal. To prove this, let (50 y, = xv>, y* = x\, · · · ,yk = xk~lvl, ƒ*+.··· denote a fundamental system of integrals of P — o, and let (52) P— A„A„_1 ... A, be the corresponding decomposition of P. If now we calculate , v3, ... , vk by the ordinary formulae, viz., (53) y, = Vi, y^ — vjv^dx, . . . , yk = vj"z\dx . . . fvkdx, we find at once (54) v, = 1, vs = 2, . . . , vk = k — 1, and consequently (55) d = j log (vtvt . . . v,) = j log (*'— *K) = j log v,, or (56) a,i — ax for i = 2, 3, . . . , k. Conversely, if P is decomposable into n prime factors of which the last k are equal, then the equation P — o admits of k conjugate solutions. If the system of integrals (57) yi = vl, y.t = vjv^dx, . . . , yn = vjvylx . . . fvjx corresponds to the decomposition AnAn.x . . . AxDECOMPOSITION INTO SYMBOLIC PRIME PAC TORS. 399 «of P, we see at once that vz, . . . , vk given by equations (io) (viz., d\)dx ^ __ gfiflk ak--i)dx . where ax — = v1, y2 — xvxi . . . , 7* = xk~lvx. It is easy to see what takes place when in the equation AnAn_x . . . Ax = o we have any k consecutive factors equal. Suppose, for example, (59) Ak+X — Ak — . . . — A2\ the equation (60) AnAn_x . . . A% = o admits now the k conjugate solutions (61) vxv2, xvxv„ *?vxv%, . . . , Xk-Jvxv2, and the proposed equation has the k integrals (62) y% = vjvylx, y% = vjxv.dx, . . . , yk+1 = vjxk~lv,tdx, satisfying the relations (63) d Vi d y, -z--=V9, -----= XV.2 ax vx ax vx 2 d yk+1 ’ dx vx that is, the derivatives of the solutions are conjugate, and not the solutions themselves. f We have already remarked that the order of the factors A in the decomposition of dny dn~ly pm=7£+a3?4+...+p.7=o is not arbitrary. Suppose, however, that for any arrangement of these factors we find the same expression P’ that is, we find the40° LINEAR DIFFERENTIAL EQUATIONS. same values of the coefficients p ; the factors are then commutative» We have now to find the conditions which must be satisfied in order that P can be decomposed into a system of commutative prime factors. To do this compare first the two differential expressions (64) and « A„A„.t . . ■ A,A, (6S) An-An ■ ■ ■ ■ a2a2 Call for brevity the common part of these expressions A, that is,. (66) An.2An.z . . . A,A, = A, and we have at once (67) AnAn.2A d'A — («*-: + ^ dan _ j dx' dx (68) A„.2AnA d'A ~~ dx' - («*., + a») («A-. dan\ dx) In order that these may be identical, or that (64) and (65) ma y give the same set of values of the coefficients /, we have (69) dan dan_x dx dx ° ’ that is, an — an_z == constant. We have then that, in order that in a composite differential expression we may be able to change the order of the first two prime factors, the difference of the coefficients a of these factors must be constant. Again, compare (70) AnAn.zAn.2An^An., . . . A, and (71) AKAn.tA..sA9_,4„_t ... A,.DECOMPOSITION INTO SYMBOLIC PRIME FACTORS. 401 By aid of the preceding theorem it is clear that if (72) = S, and (73) ¿„-*¿«-*¿.-4 . · · A,, = T, are identical, we must have (74) a*-* — a«-3 ~ const. In this case (70) and (71) are obviously also identical. Conversely, if (70) and (71) are identical, then (72) and (73) are also identical. In fact, we have, if (70) and (71) are identical, (75) AxAn.,S-AxAx.lT=AxAx.1(S-T) identically zero; but (76) A„A„.l(S — T) = d- {Sd~ T) - + «„) + («»-.«* T)‘ Suppose this is not identically zero, that is, suppose 5 and T are dky not identical, and let R -j--k be the first term in 5— T which does not vanish ; then in (76) we shall have the term R dk+2y d^2 which will not cancel with any other term, which is contrary to hypothesis. Hence S and T are identical. We see at once now that if in a differential expression we can change the order of any two consecutive factors, we must have that the coefficients of these factors differ only by a constant. We are thus led to the general result: In order that the factors of a differential expression which is com- posed of prime symbolic factors may be commutative, it is necessary and sufficient that the differences of the coefficients of the factors taken in pairs shall be constants.402 LINEAR DIFFERENTIAL EQUATIONS. Consider now the two equations Ak = o and — o; that is, (77) dy — aky = o, of which yk and yL are the general integrals. In the first of these write y = yLz and we have (78) if now ak — at is a constant, we have for z the value (79) z = Ceax, where C and a are constants ; conversely, if z is of this form, ak — at is a constant. The condition, therefore, that the difference ak — at shall be a constant gives for the ratio of the integrals of equations (77) the value factors are commutative, it is necessary and sufficient that the ratios of pairs of integrals of corresponding factors shall be of the form Ce°-X, where C and a are constants. The special forms of this result for equations with constant coefficients are readily found; this will, however, be left as an ex- ercise. It is evident that a differential expression composed of prime commutative factors vanishes if an expression composed of one or several of its factors vanishes. By aid of this remark we will proceed to determine the form of the differential expression (80) From this results the following theorem : If the factors of a differential expression composed of prime symbolic dnv dn~l/v when it is decomposable into commutative prime factors. We remark first that the coefficients p are symmetric functions of theDECOMPOSITION INTO SYMBOLIC PRIME FACTORS. 403 coefficients a of the commutative factors. These coefficients are found from equations (16) when we write in the latter dax data dan dx~ dx ~~ ‘ dx 9 and so for the higher derivatives. Suppose now P to be decomposable into commutative prime factors ; then (82) P = AnAn.x . . . Alf where — ak = const, for i, k — 1, 2, . . . , n. Consider a solution yk of Ak — o; yk is also a solution of P= o. Now we know that (83) Ip{yky)=A'nA'K_x . . . A/ Sk where (84) a( = ^ — ak, i — 1, 2, . . . , n. It follows then that the expression has its coefficients constants, and so 1 dn v dn~lv . dy (85) yP{y*y), = Q{y), =^nJrql +··_·+ ft- ^ * y 'wh^e q1, . . . , qn_x are constants. Changing y into —- , we have m P{y)=y*Q[j)· Therefore : Every differential expression P{y) which is decomposable into commutative prime factors is of the form <87) P(y)=ykQ[l)·,404 LINEAR DIFFERENTIAL EQUATIONS. where yk is a function of ^ and Q(y) is a linear differential expression with constant coefficients. Conversely: Every linear differential expression of the form. PO)=?.Q{f) is decomposable into commutative prime factors. We have in fact (88) Q(y) = BnBn_t . . . Blt where the coefficients b are constants; now we know that (89) ykQ[^j = BJBn'.x. . . Bfr where the coefficients b' have the form (90) and so the differences 6/ — b/f are constants,, and consequently the factors Br are commutative. It is therefore necessary and sufficient that P{y) should have the formykQ in 'yd order that it may be decomposed into commutative prime factors. It is now easy to see what in the supposed case is the form of the integrals of P = o. Let w19 . . . , wn denote a system of in- tegrals of Q = o; one of these is of course a constant, since Q com tains no term in y, and the others are of the form xaeP*., The equa- tion Q (—j = o, or P(y) = o, will then have yk yw, · · , y’kW*. as a system of linearly independent integrals; Finally, then, if P is decomposable into a system of commutative prime factors, and if yk is a function which causes one of these factors to vanish, P = oDECOMPOSITION INTO SYMBOLIC PRIME FACTORS. 405 has a system of linearly independent integrals of the form , the integral yk being included in this system. As an example, let us find the condition to be satisfied in order that the differential ex- pression of the second order may be decomposed into commutative factors, say <90 (92) We have directly P— A%Al = A,At. (93) giving (94) \ai “h ^!l)> A == aiai 3 with the condition (95) dax da<% dx dx Now <96) dax _ da3 1 d dx dx 2 dx and (97) consequently axa^ (98) (*» — a,J il^ii PL 4 - A+2^+4 ’ but ax — #a = const., therefore — == const., 4 (99)406 LINEAR DIFFERENTIAL EQUATIONS,. which is the required necessary and sufficient condition. If the two commutative factors are equal, then (ioo) dp, Pi 2 dx o. If we transform the equation (ioi) d'y dy dx2 +/.ƒ = <> by removing in the usual way its second term, we have (102) where (103) d'y , , S; + « = °. T_ . Idp,_Pl 2 dx 4 then if the seminvariant / is a constant we can always decompose dy , dy dx* + A dx ~ ° into commutative prime factors. We will proceed now to apply the results arrived at concerning decomposition into prime factors to the subject of the regular in- tegrals of a linear differential equation. We will recall first the properties of the integrals of the equation of the first order, / x dy (104) dx~ay = 0’ where a is a uniform function of x containing both negative and positive powers of x, and where the double series so formed is con- vergent in the region of x — o. In order that this equation shall have only regular integrals it is necessary and sufficient that a shall ot be of the form — , where a is a uniform function of x containing only positive powers of jr, and which may contain a power of x as aDECOMPOSITION INTO SYMBOLIC PRIME FACTORS. 407 factor, that is, may vanish for x = o. If these conditions are satis- fied, the integrals of (104) are of the form y — xPip(x), where p is the single root of the indicial equation, and ip(x) is a uniform function of x which does not, vanish for x = o. If a does not contain only positive powers of x, and if a becomes infinite of order v -f- 1 for x — o, the integrals are of the form £+p+· · · + * y = cx x Xv x<> ip (x), p and ip(x) being characterized as before. When a contains only positive powers of x we will call dy dx a —y xy a regular prime factor of a differential expression. It is easy to see from equations (16) that a differential expression P which is made up of n regular prime factors is of the form (105) dy pt{x)d-y pn{x) r dxn ' x dx"'1 “r" xn y’ where Px{x), . . . , Pn{x) are holomorphic functions of x in the region of x = o, and may vanish for x = o; and further, that such an expression is put into its normal form when we multiply it by xn. We have shown that if a given differential expression is com- posed of differential expressions which are in the normal form, the given differential expression will itself be in the normal form, and its indicial function will be the product of the indicial functions of its components. Bearing this result in mind, we will proceed to the consideration of expressions of the form dny dn~'y d*+p'dF=i+ · · · +A-r> which is not in the normal form, but where the coefficients p are developable in series going according to integral powers of x and containing only a finite number of negative powers of x. Suppose,408 LINEAR DIFFERENTIAL EQUATIONS. first, that a differential expression P is composed uniquely of regular factors, say (i°6) where the factors ^■uAn-1 · · · Al9 (107) . dy cti e% * Ai ~~dx~ ~xy (* - l> 2» · · · » ") are all regular, and let ƒ,=»,, jv, = vjv^dx, . . . , yH = vjv^dx . . .fvjx denote the fundamental system of integrals corresponding to this decomposition of P. How shall the factors A of (106) be modified in order that their new expressions shall be in the normal form, and that at the same time they will give P in its normal form, viz. xnP ? This question is readily answered. Multiply each factor A by xy and let A" denote the product; A” is of course in the normal form, and since the equations (108) . A” = o, A" = vtvt, A^A'f = vxv%vt9 . . . admit the integrals (109) vlt v1 vlj'V~dxj'V£-dx, ... we see that (no) p” = a:*a:*... a” = o has the same integrals for a fundamental system. Suppose then we (X- add to the coefficients the quantities (III) d , , . i — i ~r log x S =-------s ax x or, what is the same thing, increase the functions by the numbers i — 1; then, as already shown, the functions v%, v9, . . . , vn will beDECOMPOSITION INTO SYMBOLIC PRIME FACTORS. 409 •each multiplied by x\ the factors A" are now changed into factors Af of the form where Q and D are linear homogeneous differential quantics of orders n — s and s respectively, each of which has unity for its first coefficient and rational functions of x for the remaining coef- ficients. P is then of the same form as its components Q and D, and is of order n. Let x sy xky xh denote the powers of x by which we must multiply the differential expressions P, Q, D> respectively, in order to put them into their normal forms. We will now find4io LINEAR DIFFERENTIAL EQUATIONS. the relation connecting the indicial functions g(p)y k(p)y k{p) of P, Q, D. Make in D the substitution the resultant expression D' has the normal form, a fact which is seen at once when we recall the law of formation of the coefficients in Df. The expression P(y) now becomes QDor P(xhz). Now put Q into its normal form Qr y by multiplying it by xk, and we have The expression xkP[p^y) is now in the normal form, and, denoting by g'(p\ b'(P)> h (p) t^e indicial functions of xkP(xhy)y Q'y D' respec- tively, we have We have of course the relation g = k -f- h among these exponents- We have already shown that the indicial functions of xkP{xhy), or P(xhy)y and D(xhy) are immediately deducible from those of P(y) and D{y) by changing p into p -j- h ; consequently y = *** Ï (Ii6) xkP(xhy) = Q'D\ (”7) g'(fi) = k'(p)A'(p). (i 18) g'(p) = gip + k), h’{p) = hip + h), and, since k\p) = k{p)y (119) gip + h) = k (p)h{p + h). Changing now p into p — hy we have the identity (120) gip) = h (p)k(p — h). In particular, if D is of the form (121) we have (122) gip) = h{p)k{p — s),DECOMPOSITION INTO SYMBOLIC PRIME FACTORS. 411 since h is now equal to s. By proceeding to the case where P is of the form P = QDE, then to the case where P = QDEF, etc., we will easily arrive at the following general theorem : If P denote a differential expression of order n composed of 6 differential expres- sions, viz., P = DqD^-x . . . Dx, where the components D are linear homogeneous differential expres- sions whose first coefficients are each unity arid whose remaining coef- ficients are rational functions of x, and if x11* is the power of x by which it is necessary to multiply in order to reduce it to the normal form: then, denoting by g (p) the indicial function of P, and by hffi) the indicial function of D{, we have the identity (123) gif) = Kip)Kip - K)Kip — K - K) · · · Kip — K — K — · · · — K)y and also K + ^2 + · · · + hn — g, where g is the exponent of the factor xz, by which we have to multiply P in order that the product may have the normal form. If all the Dys have all their factors regular, we have (124) k = k = · ..=/*„= 1, and consequently (125) g(p) = K{p)K(p — 1) . . . Kip — n + 1). We deduce at once from this general theorem the corollary: The degree of the indicial function of P is equal to the sum of the degrees of the component's functions D. We will now take up the general question of the decomposition of the differential expression „ d"v , dK~y P ~ dx” + A dP^ + · · · + where the coefficients p are developable in double series proceeding according to positive and negative integer powers of The re--412 LINEAR DIFFERENTIAL EQUATIONS. striction to a finite number of negative powers of x will not be here applied. We know that P = o always admits an integral of the form y1 — xr

The forms of the remaining coefficients are easily found. It is easy to see that these coefficients q are, like the coefficients p, uniform in the region of x = o. In fact the coefficients q are made up from the f s and sums of the form a — h 2 cax~a a = o tb(k-a) T (x) where a is an integer. Each of these sums is then uniform and continuous in the region of x — o, but of course having x — o diS a critical point. The same conclusion then holds for the coefficients q. It is important to remark here that though this result has been arrived at by assuming tp(x) to contain only positive powers of x, this restriction is not necessary: it suffices that tp(x) be developable in a double series proceeding according to positive and negativeDECOMPOSITION JNTO SYMBOLIC PRIME FACTORS. 4IJ powers of x and convergent in the region of x = o, and further that tp(x) shall have no zeros infinitely near x = o. The necessity for this last remark is easily shown as follows: Suppose f(x) to be a uniform function of x in the region of x = o, and suppose the point ^ = o to be an isolated essential singular point for f(x); i.e., there must exist no pole or essential singular point infinitely near the point x = o. In this case the function f(x) is developable in a double series proceeding according to positive and negative powers of x and convergent in the region of x — o. The derivatives fr,fn> . .. are of course developable in the same way. We require for our purpose, however, that this shall also be true for the loga- ƒ' /w rithmic derivative y- and for the functions —y. If f(x) has no zeros infinitely near x = o, then IX*) /(*) can have no point of discontinuity infinitely near x = o, and is consequently developable in a conver- gent double series in this region. If, however, f(x) = o has an in- finite number of solutions in any region however small of x = o* fix) then will have in this region an infinite number of poles, and consequently will not be developable in the same way as f(x). This shows us at once the necessity for the above-mentioned restriction upon the function tp(x)- Suppose now (128) and let P = AnAn Ai9 (129) yx vx, y2 — vx f ,V'ldxj . . ., yn — vx f v^dx . . . f vndx denote the system of fundamental integrals corresponding to> this decomposition of P. As the auxiliary equations of which vx, v2, . . ., vn are integrals are of the same form as Py we can, and will, choose vx, , . . . , vn of the forms (I3°) vx = xT* x(x), = xr% Jx\ . . . , vn — xr* have no zeros infinitely near the point x — o, it follows that the coefficients a are, in the region of x = o, continuous and monogenic functions having this point as a critical point; they are also uniform: for, when the variable turns round the point x — o, the quantity under the logarithmic sign is multiplied by tf2*r*(rH-'*a+· · *+r«), and therefore its logarithmic derivative is un- altered. The coefficients a have therefore the same properties as the coefficients p. We have thus the result: If the functions cp(x) admit of no zeros infinitely near the point x~o, it is possible to decom- pose the differential expression P into prime factors having uniform coefficients of the form +00 2 CiX\ where i is an integer. Resume now the particular case where the coefficients p contain only a finite number of negative powers of x. The decomposition of P into prime factors of the form dy dx + oo — y 2 CiXt — 00 leads to two important propositions : (A) If Pis decomposable into prime factors of the form dy dx + °° y 2 Cyf, the degree y of its indicial function is equal to the total number j of regular factors which enter into this decomposition. To prove this, we will consider the decomposition P — · · · ^1 9 which contains j regular factors. Let us neglect for a moment, in factors which contain an infinite number of negative powers of x, all of the powers of x~x which, in absolute value, are greater than a given arbitrary number and let P’y = BnBn_x. . . Bl9DECOMPOSITION INTO SYMBOLIC PRIME FACTORS. 415 -denote the modified form of P. The expression Pr is of the same character as P, and the regular factors in Pf of course coincide with the j regular factors in P. If now yf denote the degree of the in- dicial function of P\ we know that y' is equal to the sum of the degrees of the indicial functions of the factors B. Now the ir- regular factors B have constants as their indicial functions ; the regular factors B are all of their first order, and so their indicial functions are of the first degree, consequently we have yr — j\ This equality evidently exists however great k may be ; we may then increase k indefinitely, and since y' has y for its limit we have, finally, y — j. The proposition A is thus proved. As a corollary it may be remarked that the number of regular factors which enter into a decomposition of P into prime factors of the form dy i -----y SC#' dx — « is constant whatever be the chosen method of decomposition. In a decomposition P— AnAn_x . . . Ax let cr denote the greatest possible number of consecutive regular factors which such a decomposition can have when we count back from the factor Ax; that is, Aa, A#—x, . . . , Ax are all the possible consecutive regular factors which can appear at the right-hand end of the decomposition AnAn.x . . . Ax. Our sec- ond proposition is now : (B) The number, s, of linearly independent regular integrals of the equation P — o is equal to the greatest number, = vx, yt = vjv^dx, . . . , yn = vjv^dx . . .fvHdx. Let P — A„An.t . . . Ax denote this decomposition, where dy dx + 00 — CO and where the equations A, = o, = o, . . . , As = o admit the regular integrals ^ i, = xP'+toiprf,, . . . , . . . Vs = xPi+Pa+- · •+Psi/?iif?t . . . 7ps. Since these last s factors are regular, we have a = s. It is easy to* see, however, that we will not have cr > s. Suppose in fact that it is possible to so decompose P into prime factors of the form dy dx + °° — 00 that the last s -f- t factors shall be regular,—say P — BnBn_l . . . Bs+( . . . BJBX, Denote by w. s+t wl, , * ♦ J wxw% .DECOMPOSITION INTO SYMBOLIC PRIME FACTORS. 417 the respective solutions of the equations Bx = o, B9 = o, B3 = o, . . . , Bs+t = o. These solutions are regular and of the form x?tp{x), where tf?(x) is a holomorphic function of x in the region of x = O and tp(o) does not vanish. It follows then that the ratios, WjW2 . . . wi_1wi . . . Wi_x 9 that is, the integrals, w,, . . . , «Vu, of the auxiliary equations, deduced the one from the other by the ordinary substitutions, are regular integrals of the form x?ip(x). Now since the equation which gives wsM has at least one regular integral, that which gives ws+t_x must have at least two and that which gives ws+t_2 must have at least three regular integrals, etc., finally, then, the equation which gives wx, that is, P= o, must have at least s -f-1 regular integrals ; therefore, since this is contrary to our hypothesis, we must have exactly cr — s. It is easy to see that proposition (B) is still true when the prime factors are not restricted to be of the form dy dx + 00 y 2 CiX\ — 00 From (A) and (B) we derive at once the following theorems: {a) IfP—O has all of its integrals regular, P is decomposable into n regular prime factors. {b) If P is decomposable into n regular factors, P — o has all of its integrals regular. (c) If P —o has all of its integrals regular, the degree of its in- diciai equation is equal to its order. (d) If the degree of the indiciai equation of P is equal to the order of P, then the equation P = o has all of its integrals regular. The following theorem is easily established :4ï8 linear differential equations. (e) If P — o has all of its integrals regular, it has a fundamental system of integrals belonging to exponents which are roots of its indicial equation. If p — o has all of its integrals regular, P is decomposable into n regular factors, viz., P = AnAn_x . . . Al9 and from (125) we have the relation g(p) — K(p)K{p — I) · · · hn{p — « + i), connecting the indicial functions g and h of P and its factors A. If now we write Ax — o, A2 — o, · · . , An — o, we see that the integrals • vx, vxv%, . . . , vxv% . . . vn of these equations belong respectively to the exponents Pi> p2— Pa — 2> ···> Pn — n+ I, where Pi » Pa > ··· 9 Pn are the roots of the indicial equation g{p) = o. It is obvious then that the integrals y, = ^ , y*=vlfvjlx, . . . , yn — vjvylx . . .fvjx oi P=o belong to the exponents p,, pt, , p„ respectively. In the case where the integrals of the differential equation are not all regular, propositions (A) and (B) give the following theorem : The number of linearly independent integrals ofP — oisat most equal to the degree of its indicial equation. The form which the expression P must have in order that the equation P — o shall have s regular integrals is given in the follow- ing theorem : In order that P = o shall have s linearly independent regular inte- grals it is necessary and sufficient that P can be put in the form P — QD,DECOMPOSITION INTO SYMBOLIC PRIME FACTORS. Al9 'where Q and D are respectively of the orders n — s and sy and are ex- pressions of the same kind as P (that is, the first coefficie?it in each is unity and the remaining coefficients are rational functions of x), and where D — o has all of its integrals regular while Q — o has no such integral. We will show first that this condition is necessary. We have shown that if P= o has s regular integrals, it admits of a decom- position P=A„A„.I . . . AS+,AS ... A, into prime factors of the form dy + 00 A=z;-'?.cs· where the last s factors are regular. Write now As . . . A,A, = D, AnAn.x . . . As+1 = Q; then P= Q£>- Since D is of order s and contains only regular factors, we have dsy Px{f) ds~y D = + dxs dxs' 1 + PAx) fs the form of the P's being known, and D — o has then all of its integrals regular. The expression Q is of order n — s. Effecting now the operation QD and identifying the result with P, we obtain a system of equations which show (under the above-mentioned restriction as to the functions 0) that the coefficients of Q are, like those of P and D, rational functions. Further, Q has no regular integral ; for, if it had, we could terminate a decomposition of Q by at least one regular factor, and so P, = QD, would have more than s regular integrals, which is impossible by proposition (B). Second, this condition is necessary. Suppose P — QD, where D — o has all of its integrals regular and Q = o has no regular integral. Every solution of P — o which satisfies D = o is regular. Every solution420 LINEAR DIFFERENTIAL EQUATIONS, of P = o which does not satisfy D = o will satisfy one of the equa- tions D = u, where u denotes some one of the integrals of Q = o.. Now by hypothesis none of the integrals of Q — o are regular, and if in D we replace y by a regular function, the result is a regular ex- pression, and therefore no regular value of y can make D equal to u. It follows then that D — u can have no regular solution, and conse- quently that the only regular integrals of P = o are the s regular integrals of D — o. It is easy to see the cause of the difference y — s between the* degree y of the· indicial equation of P — o, and the number s of linearly independent regular integrals of this equation. Suppose in, fact that P is decomposed into prime factors of the form dy dx + CO — y 2 CiX1. This decomposition, whatever it may be, will, as we know, contain: y regular factors ; of these regular factors a group of s at most can be placed at the right-hand end of the decomposition, and the remaining y — s factors cannot come into this group (of course in particular cases we will have y = s). The difference y — s then arises from the presence of regular factors in Q which cannot be placed as consecutive factors at the right-hand end of the decom- position of Q, Observing that the indicial functions of D and Q are respectively of the degrees s and y — sy we can enunciate the two following; theorems : (o') In order that the linear differential equation P = o of order n possessing an indicial equation of order y shall have y — y linearly independent regular integrals, it is necessary and sufficient that P shall have the form P =,QDy Q and D being of the same form as Py and that <2 — 0, being of order ?i — y -f yy shall have no regular integrals· and shall have an indicial equation of order jn. (¡3) In order that the linear differential equation P — o of order n and having an indicial equation of degree y shall have y linearly in- dependent regular integralsy it is necessary and sufficient that P shall be of the form P — QDy Q and D being of the same form as Py Q being of order n — y and having a constant as its indicial function.DECOMPOSITION INTO SYMBOLIC PRIME FACTORS. 42 1 We can further show again the truth of the theorem: (y) The s linearly independent regular integrals of P — o belong to exponents which are certain s of the roots of the indicial equation for P = o. This is immediately seen, since if we put P in the form p= QD, we have the relation g(p) = k(p)k(p - s) between the indicial functions. Now the s regular integrals of D = o, that is, all the linearly independent regular integrals of P — o, belong, as we have seen, to exponents which are the roots -of h(p) = o. It follows then, from the above relation connecting the indicial functions, that the regular integrals of P = o belong to •exponents which are certain s of the roots of g{p) = o. The following theorems concerning the adjoint equation are ^easily seen to be true; their verification will, however, be left as ^exercises. I. The indicial functions g (p) and Q {p) of the adjoint expressions P and p are derived the one from the other by changing p into — p + fi — i, where x& is the power ofx by which we must multiply P in order to put it in the normal form. II. If the linear differential equation P = o has all of its integrals regular, the same is true of its adjoint equation p = o. The theorem (y) above is here changed into the following: III. In order that the linear differential equation P — o of order n ¡having an indicial equation of degree y shall have exactly y linearly in- dependent regular integrals, it is necessary and sufficient that its adjoint equation p = o shall have among its integrals all the integrals of a linear differential equation of order n — y which has a constant for its Indicial function. It is easy to see from the preceding investigations that all the properties of the regular integrals of linear differential equations can be arrived at by aid of the decomposition into prime symbolic factors.CHAPTER XI. APPLICATION OF THE THEORY OF SUBSTITUTIONS TO LINEAR DIFFERENTIAL EQUATIONS. I. We may recall the definition of the product of two substitu- tions ; viz., the product 55, of the two substitutions 5 and Sx is the new substitution which produces the same effect as if the substitu- tions S and Sj were applied successively. A similar definition applies to the product of any number of substitutions. In particular* we speak of the power of a substitution, say Sa, which means that the substitution 5 has been made a times. In Chapter III, equation (2), we have the general type of the substitution which corresponds to any critical point of the linear differential equation with uniform coefficients; viz., if yl, . . . , yn denote a system of fundamental integrals in the region of the assumed critical point, we have (0 C\\y 1 ~h” ^12^2 “H · · · + ^xn yn c2iyx + c22y2 + · ♦ · + c2nyn yn; Cmyi + cn2y2 + · · · + cnnyn It was also shown how by aid of the characteristic equation (2) C11 S) C2l . . . CnT C12, C22 s . . . cn2 = o C\n y C2n . . . Cnn S this could be reduced to the canonical form given in equation (80')* of the same chapter. The equation (So') may, by a slight change in notation, be written in the form 422.APPLICATION OF THE THEORY OF SUBSTITUTIONS. 423 ƒ/. • · y*1 ; • -Vi*1 y:. ··ƒ/*; ·?,(ƒ,' +y/) ·· • sXy*h +y**) y't · ■ •If·, -f.0*+ƒ>-,)· · • si{ykf +yk/-d z;. ■ · */’ ; -v/ • •v/* <. . • · · · where the integers k, /, . . . satisfy the inequalities > ^2 * · · > t A > K · · · y and where kx + -f- . . . + kp, lx + K + · · · » · · · are the re- spective degrees of multiplicity of the roots sx, s2, . . . of equation (2). We shall speak of the functions _y, z, . . . as forming classes cor- responding to the roots s19 s9, ... , respectively, of the characteristic equation ; thus, we shall speak of the functionsy as being of the first class, the functions z as being of the second class, etc. The group of the differential equation has been already defined, but it will be convenient to repeat the definition here. Suppose a9 b, cy . . . to be the critical points of the equation ; when the variable travels round any one of these points, the integrals submit to a sub- stitution of the form (1), or, when the integrals are properly chosen, of the canonical form (3). Any path going round only one critical point can of course be replaced by the loop (lacet) belonging to this point, and any path424 LINE AX DIFFERENTIAL EQUATIONS. going round more than one of these points may be replaced by the loops, taken in proper order, belonging to these points. Corresponding to each of these paths there is a certain substitu- tion ; denote by A the substitution corresponding to the critical point a, by B the substitution corresponding to b, etc. If in Fig. i we start from O and travel along the curved line back to <9, we shall have gone round the two points a and c. Now we can in the usual manner (by aid of the dotted line OM) see that this path is equivalent to the two loops a and y; we therefore apply to the functions first the substitution A and then the substitution C, or simply the substitution AC. Again, if in Fig. 2 we start from 0 and go round the curved line and come back to 0, it is easy to see that we have applied the substitution A CE. All possible substitutions of this sort that we can form from the simple substitutions A, B, Cy . . . , their powers and products, form the group G of the equation. Denote by F a function-group of the equation such that all the substitutions of G transform the functions y, z, u, . . . among them- selves. It is clear that this function-group can be regarded as con- taining certain other function-groups. For example, suppose we consider the single function y and its. transforms by A and the powers of A ; these will form a function-group which is obviously contained in F; and so in many different ways we may form func- tion-groups which will all be contained in F. We will call such func- tion-groups minors of F. We will show now that F can be considered as derived from a certain number of distinct functions, each of which contains onlyAPPLICATION OF THE THEORY OF SUBSTITUTIONS. 425 variables of a single class. Any function whatever in F is made up of the sum of a certain linear function of the variables y (say F), a linear function of the variables z (say Z), a linear function of the variables u (say £/), etc. If then f denote a function contained in F, we can write (4) /=r+z+<7+------------- Let denote the function-group derived from ƒ and its trans- forms f', f ”, . . . which are obtained by operating on f by the different powers of A ; again, let denote the function-group formed from the partial functions F, Z, . . . , and their transforms by A, A*, .... We may write <5) <&'y being made up of y and its transforms, % of Z and its trans- forms, etc. It is obvious that the function-group # is a minor of Let now p denote the greatest of the indices k of those of the func- tions y$ which enter in F; cr the greatest of the indices / of those of the functions ^ which enter in Z; etc. We can now establish the following theorem: The function-group # contains each of the partial functions F, Z, . . . and is derived from p + cr-\- . . . distinct functions, of which p are formed exclusively with the variables y\, cr are formed exclu- sively with the variables z\, etc. We have to show first that the number of distinct functions in is not greater than p -j- a -f- . . . We will determine first how many distinct functions will be given by F and its transforms. Denote by Yp_i a linear function of thej/s whose index is less than or equal to p — i. Now it is obvious that <&' contains the function (6) Y=\'y;jrr%"+ . . . +Y„-t, and it also contains the transform A F, and consequently it contains the function (7) Y' = I AY— Y, S | which has obviously the form <8) Y = A'/,-, + + · · · + .42^ LINEAR DIFFERENTIAL EQUATIONS. The functions F and Yr are by definition elements of the sub- function-group -3; ^1 (io) Fo- = Fo- - F<— = Jiy' + V>," + . . . . This last function gives identically (i i) —A Y?-1 — Yp~x = o. The functions Y, Y, . . . , Fp_I are evidently linearly independent^ as each of them contains certain of the variables y which are not contained in the others. Again, we have from the above equations (12) A Y = s,{Y+ Y'), AY' =s(Y'+ Y"), . . . , A Fp-1 = s, Fp-1 ; that is, the substitution A changes these functions into linear func- tions of themselves alone. The different powers of A will obviously do the same thing, and consequently the number of linearly inde- pendent functions in 0'y is exactly equal to p. In the same way we see that the number of distinct functions in 0'z is equal to cr, and so for the other sub-function-groups. It follows then that 0' con- tains exactly p -f- a -{- . . . distinct functions, and consequently that 0y which is a minor of 0\ can contain no more than p + , A'fp, . . . are all contained in 0, and of the set 0, A(p, A*(p, . . . there are evidently p distinct functions. In the same way we can show that 0 contains cr distinct functions involving only the variables z, etc.—that is, 0 contains in all p -f- cr -f- · · · distinct functions; and since 0 is a minor of 0', which contains the same number of distinct functions, it follows that 0 and 0' are identical, and therefore 0 contains the functions F, Z, U, . . . above defined. Now 0 is a minor of the function-group F; there- fore i7 contains Y, Zy . . . among its elements. We have then, finally, that if the function-group F contains the function (4) f=Y + Z+U+..., it contains also the functions F, Z, U, . . . Let now Yl9 Fa, . . . denote the linearly independent functions contained in F which contain only the variables y; Ziy Z^, . . . the analogous functions of the variables z ; etc. The function-group F is derived from the elements Y1, F2, . . . , Zx> 2T2, . . . , etc. For, de- noting by f, = F+Z+ . .428 LINEAR DIFFERENTIAL EQUATIONS. any one of the functions contained in Fy we know that Yy which depends only on the variables yy is a linear function of Yxy y . . and that Zy containing only the variables zy is a linear function of Zx, Z^y . . . , etc. We have thus established the above proposition concerning the function-group Fy viz. : that F can be considered as de- rived from a certain number of distinct functions Yx, Fa, . . . y Zx, Z„ . . . , etc., each of which contains only variables of a single class. Suppose that in F we have the function Y of the variables yy say The different transforms of ƒ are also contained in F. If we per- form the substitution A any number of times in Yy we obtain func- tions which only contain the variables y; but if we perform the sub- stitutions By Cy etc., we get new functions which are linear on the one hand with respect to the variables yy zyuy . . . , and on the other hand with respect to the constants A/, A/', . . . Denote by Px, Pt, . . . the different products that can be formed by multiplying one of the constants A. by one of the variables yyzyuy. . . ; as both variables and constants are finite in number, there will be only a finite number of the products P. Let rjiy rj^y . . . denote linear functions of the ys .alone, Zx, £a, . . . linear functions of the ^’s alone, etc.; then we can write From what we have above proved it is clear now that F contains (17) <18) (19) Bf — Vi + Ci + · · · > Cf = r), -f- Ç, -j- . . . , V„V*......Ci. C,, · · · , and also <20) vt = — %> vi =-Ath' — Vi' · · · > S, o.APPLICATION OF THE THEORY OF SUBSTITUTIONS, 429* each of which contains only variables of a single class. These dif- ferent functions tjl9 77/, 77/', . . . , · · are of course linear func- tions of Px, , . . . It may be, however, that some of these functions 77, C, · · · , rf i · · · are linearly expressible in terms of the remain- ing ones and f; if such functions exist we will discard them, and so have finally a certain set of distinct functions, say ƒ, ƒ', ƒ", . . ., ƒ % of which ƒ'...ƒ r have been derived by operating upon f by By Cy . . . and by powers of A. We may take now each of these new functions f', ƒ", . . . , fry and by precisely similar operations arrive at still other functions which will be contained in F and each of which will contain only the variables of a single class. These new functions are again linear functions of Pi9Pi9 . . . ; neglect, as be- fore, all of them that are linear functions of the remaining ones and of fy f\ . . . , fry and say the new set of distinct functions is /-+« Treat the functions frJtl . . . fr as before, and neglect all the new- functions which are linearly expressible in terms of the remaining ones and of f9 f’9 . . . , f5. We will continue this process until we can find no new function which is not linearly expressible in terms of those already found. That the series of operations is limited is clear from the fact that each of the functions which we form is a linear function of the quantities P19 P9, . . . which are limited in number. Suppose the functions finally obtained are f ƒ', . . . , f*9 each of which contains only the variables of a single class (of course there will in general be several functions which contain the variables of the same class). Each of the functions ƒ, f', . . . , f* is transformed by each of the substitutions of the group G into a linear function of f9 ffy . . . , In order to prove this it is only necessary to show its truth for each of the substitutions Ay By C, . . . from which G is derived. Take, for example, the function f9 = 77/. The transform, of this by the substitution A is (21) An: = slrh" + Vl')·, operating with By C9 . . . , we have (22) Brjx' — 77, + Ci + · · · (23) C77/ = 77a 4" C*+ · · ·430 LINEAR DIFFERENTIAL EQUATIONS. Z> · . · containing respectively only the variables but each of these partial functions 77, C, ... is a linear function of f> fr, . From the functions /, ƒ', ...,ƒ* we can form a func- tion-group ^ which possesses the property of having its different functions transformed into one another by the substitutions of the group G. Suppose . . to be the functions which depend only on the variables/, and let Wy be the function-group derived from f9fx, , . . . , or, what amounts to the same thing, derived from ƒ, 0,, 02, . . . , where (24) 0, =/i+ <»,ƒ, 0,=/, + «»,/, ···; , G72 , . . . being constants each of which is subject to only one condition, viz., that it shall not be a root of a certain algebraic equa- tion. The function ƒ is of the form (25) and fx of the form (26) fx = [ay/ + by” + · · . + + · · · W + tyy/ + b'y” -f- . . . + */'/'« + · . · + . . . ; therefore 0, is of the form (27) 0j — [(# + oox)y/ + by” + . . . + dy/ + . . . ]A/ + Wy:+^+oo/)y”+ . . . ]Jl"+--- We will now determine gjx in such a manner that the determinant (28) a -j— gox , b , d . . . a9 , V + 07,, d9 a” , b” , d” + œl . . . shall not vanish. It is obviously only necessary to choose for qd1 any value that is not a root of the equation Ax = o. The constants g72 , /+w + · · · with the substitution 2,= (a + (»>/ + by” + . . . a'y/ + (P' + + · · · . The functions 0a, 03, . . . will be obtained by operating upon ƒ with similar substitutions , . . . . It follows, therefore, that the function-group Wy will be identical with the function-group derived from the transforms of ƒ by the substitutions of a substitution-group r which is derived from 2l9 299 . . . Let us assume now that we are able to ascertain whether the group T is prime or not, and in this second case we will suppose that we can determine a function-group ©which is derived from a number of distinct functions, this number being less than the number of the variables y; further, © is to be such that the substitutions 219 . . . shall replace its functions the one by another. Suppose, first, that r is not prime, and denote by ¿z any one of the functions in ©. Choose the constants A', A", . . . , such that (29) * = *>/+*'>" +----------- The function S and all of its transforms by the substitutions , 2i9 . . . will form a function-group contained in ©, and consequently depending upon a number, say p, of distinct functions, at most equal to the number of distinct functions in ©. Now among the distinct functions from which W is derived, and each of which contains only one class of variables, there will be only p containing the variables y9 and the number of these variables is by hypothesis greater than p. In the case of each of the other classes there can of course be no more functions than the number of variables of the corresponding class. The function-group W will then contain less than n distinct functions, which will be obtained by substituting in f9f,, . . , f* the Special values of A', A", . . . and, after effecting the substitutions,432 LINEAR DIFFERENTIAL EQUATIONS. neglecting those functions which are linearly expressible in terms of the others. In this case, then, the group G is not prime, and we have deter- mined a function-group containing less than n distinct functions. In the second case, suppose F to be prime. Whatever be the system of values chosen for A', A", . . . , the function-group Wy ob- tained from Afyxf -f- Anyf by the substitutions of the group r will depend upon a number of distinct functions equal to the number of the variables y. It follows then that Wy will contain among its functions all those that can be formed from these varia- bles. Suppose y' one of these functions; then y/ is necessarily contained in F\ Since A', A", . . . are no longer restricted in value, we may write X' = i, A" = X'" == . . . = o, and ascertain whether or not the function-group W formed under this hypothesis contains n or less than n distinct functions. If W contains n distinct func- tions, then F, of which W is a minor, will also contain n distinct functions, and G will therefore be prime. If W contains less than n distinct functions, G will not be prime, and the system of distinct functions upon which W depends will have to be determined. Suppose now that F contains no function of the variables y of the first class, but contains a function Z of the variables z of the second class. By a procedure identically the same as the one just described we can determine a function-group W (if such exists) which shall be a minor of F depending upon less than n distinct functions. And so in like manner we can proceed if ^contains no function of the variables y and no function of the variables z, but a function U of the variables u of the third class, etc. By continu- ing this process we see that we can always determine a function- group W containing less than n distinct functions, unless indeed F is known to contain always n distinct functions, in which case G is prime by definition. The question proposed in Chapter VIII concerning the group G was, “ Having given a group G composed of linear substitutions A, B, C, . . . among n variables, required to determine whether or not the linear differential equation which has G for its group is satisfied by the integrals of analogous differential equations of orders lower than ny and to determine the groups of these equations.” In what precedes we have supposed that we knew how to answerAPPLICATION OF THE THEORY OF SUBSTITUTIONS. 433 this question for the group T and all similar groups. The number of variables in T is less than those in G (since T only contains the variables of a single class), and the problem is therefore reduced to a simpler form, and may be considered as solved when we examine the case so far excluded; viz., the case when the characteristic equa- tion corresponding to the substitution 5 has all of its roots equal— or, say, has but one root. We will suppose (changing very slightly the previous notation) that the characteristic equation corresponding to the substitution A has the single root a. In order that we may have a new problem to solve it is necessary that the characteristic equation corresponding to each and every substitution in G shall have but one root; if, for example, there existed in G a substitution, say Sy corresponding to which there was more than one root, then we might reason with .S as we have already with A and effect the required reduction by the processes described above. We must therefore suppose that there is no substitution in G whose equation has more than one ropt. We will, in this case, first show that G cannot be prime, and will give the canonical form of its substitutions which shall bring this fact into evidence. This being done, we will show how to ascertain whether or not the assumed hypothesis is admissible, and, if it is, give the canonical form of the substitutions of G. In case the hypothesis is not cor- rect we will show either how to obtain directly a function-group containing less than n distinct functions, or a substitution, say 5, corresponding to which the characteristic equation has several dis- tinct roots, and so be conducted to the case already investigated. Denote by a, d, .. . the single roots of the characteristic equa- tions corresponding to the substitutions A, B, . . . We may write A = a&, B = i®, ... Here a, b, . . . represent substitutions which multiply all the vari- ables by a, b, . . . ; and H, · · · substitutions whose determinant is unity and corresponding to each of which we have the character- istic equation (s — i)n = o.434 LINEAR DIFFERENTIAL EQUATIONS. We have now a group (5 derived from S, J3, . . . as G is derived from At By . . . Consider any substitution, say T, of G, and let T=aP. . . UaW . . .; then XT = and this is a substitution contained in (¡5. Let (30) A = A. — jx.......... K> Mi — s · · · = o be the characteristic equation corresponding to XT. The roots of the characteristic equation corresponding to T are evidently the roots of A = o multiplied by the constant (a*#3 . . . )w; but by hypothesis this last equation has only one root, therefore A = o has only one root. We will now show that the single root of A = o is unity. In order to do this, we will show that if this statement is true for any two substitutions 5 and XE, it is true for their product SXC; then as we know that the proposition is true for the substitutions 35, . · . , from which (5 is derived, it must also be true for their products taken two and two, three and three, etc., and consequently for all the substitutions of (5. We may evidently suppose the variables^, . . . so chosen that the substitution S shall be in the canonical form; viz., by the above hypothesis, (31) y/, y^· ■ - jr Zs] + ƒ. · · · Ut ; Ux-f- The substitution X£ cannot, of course (with these variables), be sup posed to be in its canonical form, so we may write yy> «.ƒ.+ ·· • + + · . . + clul + . . · (32) z = «1 ; + . • · + + . APPLICATION OF THE THEORY OF SUBSTITUTIONS. 435 Prom these last two equations we have (A. denoting an integer) y,; [ƒ. 4- *4 + X(~\ 2·1-+ · · ·] J'» + · · · +(^1+^1+ · · · · · · +0i+ · · 0^1+ · · · (33) sAxr = y, + · · · ux ; [ƒ, + A^ + --1 /x + . . +(*i+Vi+ · * -H+ · · · +(/i+ · · *)UÆ · · · The characteristic equation corresponding to this substitution may be written in the form (34) A = ^ + sn'1 + + ... + (- i)*D = o, where D is the determinant of 5ÎE, and where 0,0^... are inte- gral functions ; but since 5 and TE are products formed with the substitutions H, 3B, ... each of whose determinants is unity, it fol- lows that D = i. The single root of J = o will then be an nth root of unity, since the nth power of this root is (to sign près) the last term of equation (32)—i.e., unity. We have then (35) A = {s- oy = o, where 6 is an nth root of unity. By comparing (34) and (35), we see that we must have (36) and consequently „ n(n — 1) 0 = — nO*’1, 0, = \ ^ ■■ ti"-2, i. 2 (37) 0» = (_ ny, 4>y = n(n — i)~ L i . 2 -i These equations are of finite degree in A and are satisfied by hypothesis for all values of A ; they are therefore identities. The coefficients 0, (px, . . . are therefore constants, and A — o is inde- pendent of A. Suppose now A = o; then by hypothesis again we have A — (s — if. It follows then that all the substitutions436 LINEAR DIFFERENTIAL EQUATIONS. have for the corresponding characteristic equation simply (s — i)w = o, and so finally every substitution in © has for its characteristic equation (s — i)n = o. We will now suppose the variables so chosen that the substitu- t tion H is in its canonical form, and for simplicity we will assume (38) yy*, y*> y<; y^ y*> y*> y* ^ , ^3 ; zx+yx, ^2 + , z9+yt Any other substitution, say XT, of © may be written in the form (39) XC = yp ; tfpiji + · · * + ^P4y4~l· 'Wi 4~ · · · +^3^3 z? ; cp*yi 4- · · ■ 4~ cP4y4 4~ 4~ · · · +<3WS3 We have now (40) = yp ; (tfPx4“AA»I)jv4" · · · + i^Pi^^b^y^a^y^b^z^ . . . ZP 5 (Cpi +^pi) Ji+ · * · 4~ (rP3+^^P3)^3 4" <:P4^/44"4>i'Sri“l· « * * The characteristic equation corresponding to this is obviously (4i) aM + Ain — #12 + ............ ^21 I a^21 , ¿?22 [ a^22 j . = o, which, as already shown, is independent of X. In expanding this we find certain terms, viz., those contained in the expression Ai„, A^21 > A.&j2 5, > M.,. — i (42)APPLICATION OF THE THEORY OF SUBSTITUTIONS. 437 Involving only the constants b119 blif . . . , b33 and not cancelling with any other terms. The terms containing these constants b must therefore vanish identically, and so, equating to zero the coefficients of A/, AV, AY, we must have (43) (44) (45) ¿n + ¿22 + &»3 — ° ; ¿11 y ¿13 + ¿22 y b„ + ^33 » K ¿21 9 ¿22 ¿32 y ¿33 b, 3, b» ¿221 ¿32 y K K, = o. We may remark here that since we have only the three independent variables zx, z99 z3, we may, without altering the substitution H, replace jy,, y3, by arbitrary linear functions of themselves, pro vided that we make an analogous change in zx, z^y By aid of this very obvious remark we may obtain a simpler form for the sub- stitution xc. From (45) we can clearly determine three constants, say , /a, /3, such that ^SPwZ\ ~j"" ^12*^2 ~f” ^13^3) (46) + + b^t + ba>z,) I ^3(^31'^! | ^32^2 + bi2^a) O* Supposing then that /,, /2, /3 are determined so as to satisfy (46) ; if now we replace one of the variables yx, y3, y3 by kyi-\-ky*-\-ky*> it is easy to see that the corresponding transform of this variable in the substitution XT will not contain z; if y3 be the variable so replaced, the effect in X£ will be the same as if the coefficients b3l > ¿32 9 ¿33 were made zero. Suppose then that this operation has been performed and that we have <47) ^ai — b%i — bsa o,438 LINEAR DIFFERENTIAL EQUATIONS. it follows now that (44) reduces to (48) K b„ = o. By aid of this equation we can determine two constants miy m,r such that (49) which will not affect a, but will make the resulting coefficient to Ti which is analo- gous to the original biZ equal to zero. If blz = o, we shall arrive at the same result by permuting the variables yx, zx with _y2, z9. (2) Suppose bzz = o: if blz is not zero, we can cause the new coefficient in Tt which is analogous to the original b12 to vanish by taking for new variables blzy2 -f- bizyz, b12^2 -f- bX3za in place of y9 and z%; if bn = o, the same result will be obtained by permuting yz, z% with yz, z%. Denote now by (so xa, - Jfp f ft pi.yI 4“ * * * 4” ^ P4-^4 4~ ^ Pi^i 4- · · · 4“ b P3^3 z? ; c'piyi 4" · · · 4" ^paya 4~ dr?xzx + · · · 4· d' P3Z?> any substitution of the group (5 ; write, for brevity, (52) b pa —- {OL px —|— jxb px^bjfj —}“ (ft p2 4“ p2)&2 c P*y* -b · • · + c"hyA + 4- · · . . +*v3 As already shown, the coefficients blx", . . . , bj' must satisfy equa- tions of the same form as (43), (44), and (45), and in particular (54) iu" + i»,, + V, = o. Substituting here the values of bx", bj\ and bj' from (52), and equating to zero the coefficients of p, we have, on taking account of equations (50), (55) + ^3 As + ^32^23 = °- This equation must necessarily be satisfied whatever be the substi- tution m of the group 0; it must therefore be satisfied when we replace the coefficients b'21, b'zi, ¿'32 of XU by the corresponding coefficients btl", bj'y bj' of \); consequently, whatever may be the value of p, we must have (56) = °· If now in this equation we replace bj\ bj'y bjf by their values and equate to zero the terms multiplied by p, we shall have, by aid of (50), (57) = o. This equation can be satisfied in two different ways: first, by making (58) vn = O; second, by making 4 A 3 = o. But we have shown that if either bn or b23 is zero, the other may also be made zero, and so the equation bxJ?^ = o is equivalent to (59) blt — o, bt 3 = o,440 LINEAR DIFFERENTIAL EQUATIONS. and finally, from (55), (60) bn = o. It is now easy to establish the first of our theorems, viz., that under the assumed hypothesis as to the roots of the characteristic equations corresponding to the substitutions of the group (3, this group cannot be prime. There are two cases to be considered according as b\x is or is not equal to zero. Suppose first b\x not zero. We know now that each of the quantities ^11 9 ^12 > b\z 9 ^21 > ^22 9 ^23 9 ^31 y ^32 y ^33 is zero, and consequently that the substitution X£, which is any sub- stitution whatever of the group (3, gives as the transforms of yx, y^, yz linear functions of yx, . . . , yA only. Denote by XT, X£' . . . the different substitutions of (3 ; by F, Fr, . . . the corresponding trans- forms of yx. Each of these functions depends only onyx ,ya, y3,y4f and consequently the function-group formed by the linear functions aY a' Y' -f- . . . can contain at the most but 4 distinct functions. Let Yx, . . . , Yk denote these functions, where k =4. Further, we remark that substitution in (3 changes the functions of the series F, Y\ . . . among themselves. Suppose J£x changes F into Y1; we know that XE changes yx into F, and consequently X^XL changes yx in Yx ; and since XTXT, is a substitution of the group (3, it follows that Yx belongs to the series F, Y\ . . . . The same reason- ing of course applies to the general case, and in this case we see that the substitutions of (3 transform into one another the functions of the function-group a Y -f- a' Y' -f- . . . . The number of distinct functions Yx, . . . , Yk contained in this function-group being less than the number of variables, it follows that (3 cannot be prime. The substitutions of (3 can obviously be put in the form Fi ... F,; axYx+... + aiYk,...,/31Y1+...+/3kYk ^*+1 · · · * Vi ^1 · · · ~f" YkYk “l· Yk+i Yk+i + · · · 9 ··· (61)APPLICATION OF THE THEORY OF SUBSTITUTIONS. 441 In the second case, suppose b'2l = o, and let F, F', . . . be the trans- forms of yz by the substitutions of i,and so, by a proper choice of the independent variables, the substitutions of this group can be put in the form Yx . . . Yt ; ^(F, . . . F<), . . . , ; ^+I(Ft. . · Yt), ... , <&(F, . . . Yk) ’ where / < k and 0 denotes a linear function of the quantities in parenthesis. If l > 1, we can again so choose the variables Yx ... Yt that the partial substitutions (64) | Yt ■ ■ ■ Yt; 0,(F; ...Y7), . . ., MY, · · · y,) I shall take the simpler form (65) vt. . .vm ; MY, · · · Ym), · · ·, MY, ■ · · Y.) Ym+I . . . Y,; fm+l(Y, . · · Yi), . . MY, ■ ■ . Yd ’ where m < L By continuing this process we can obviously so choose the independent variables F that the first one, Yx, shall, for each substitution in (5, be changed into itself multiplied by a constant;442 LINEAR DIFFERENTIAL EQUATIONS. and since the characteristic equation corresponding to each substi- tution in (5 is a power of s — i, it is obvious that this factor reduces to unity. We have then the theorem : If each substitution in (3 has (s — i)n = o for its characteristic equation, there must exist at least one function of the variables yx . . . yn. which is not altered by any substitution. There may obviously be more than one such function. Suppose Yx . . . Yp arep distinct functions possessing this property; then the substitutions of (5 can be put in the form (66) Fi . . . F, */ + i · · ai + · · · + &»Ynt · · · > A^I + · · · + A* K Now the first member of the characteristic equation of such a sub- stitution is equal to the product of (s — i)p by the first member of the characteristic equation belonging to (67) | J'iH-i · · · Y* > aA+i YpH · · * ”l· an Yn y · · · > A+i · · · ~\~ fi*Yn | - Substitutions of this form corresponding to the different substitu- tions of (3 will then have ($ — i)n~p = o for their characteristic equa- tion. We conclude, then, that we may so choose the variables as to make these substitutions of the form (68) Yph ■ ■ ■ r>H-i 1 p-Yq r/+?+I . Yn ; Yp+I... +a'nYn... Y^ + .. .+/»', Yn By a continuation of this process we see that all of the substitu- tions of (5 can be put in the form (69) y> · · · Y> ; Y.---Y, Yfi+i · ■ ■ K- Y*+i-\~ft+i · · · Y,+f' Yq+1 · ■ · Yr; Fi + I+/? +-Yr+fr where /¿+I . . . fq are linear functions of Vlt. . . ,Yy, /i+I . . ./rare linear functions of F,, . . . , Y · etc. As every substitution in (5 has now this form, the elementaryAPPLICATION OP THE THEORY OF SUBSTITUTIONS. 443 substitutions H, 35, . . . , from which (5 is derived must also have the form of (69). We arrive now at the second part of our problem. So far we have assumed that all the substitutions in (5 have unity as the single root of all their characteristic equations, and on this hypo- thesis have found the form (69) for each of the substitutions. We have now to show how, for any group (3, we can ascertain whether or not the substitutions B, B,. . . are of this form. To do this we will endeavor by the method of. indeterminate coefficients to find out whether or not there exist linear functions of the inde- pendent variables y1, . . . , yn which are unaltered by the substitu- tions H, 35 ,. . . We will assume certain linear functions of yx, . . . , yn of the form, say, a'lJj + * * · + anfny and operate upon them by the substitutions fi, 33, ... If the trans- forms of such functions are the same as the functions themselves, we will arrive at a series of linear homogeneous equations for the determination of the constants al9 . . . , an, the number of these equa- tions being, of course, in general greater than n. It may happen that these equations can all be satisfied (as certain determinants may vanish) and still leave a number, say py of the coefficients a arbi- trary. If, however, we have p arbitrary constants, we will also have p functions Y19 . . . , Yp possessing the required property; viz., the transforms of Yiy . . . , Yp by each substitution in (3 will be again Yx, . . . , Yp. The next step is to ascertain whether or not there exist functions YP+1,...,Yq whose transforms by the substitutions of (5 are equal to themselves increased by linear functions of Ylf . . . , Yp. We thus arrive at another system of linear homogeneous equations of the first degree, etc. If in continuing this process we never arrive at an incompatible system of linear homogeneous equations we may take Yx ,Yt, Yq,■ ■ . for the system of variables which shall throw the substitutions a, b, ... , and consequently all the substitutions of (5, in the form (69). The transition to the substitu- tions A, = aiXy B = b\S3, ... of the group G is at once effected by multiplying the transforms in <5 by a, b, . . . Since now each of the substitutions of G multiplies each of the functions Y1, . . . , Yp by a constant factor, the problem proposed concerning this group is solved.444 LINEAR DIFFERENTIAL EQUATIONS. We will now take up the case where, in following the above in- dicated process, we arrive at a system of incompatible linear equa- tions for the determination of the coefficients a; if this case presents itself, we know at once that <5 contains at least one substitution whose characteristic equation has several distinct roots. To solve the problem which thus presents itself, we have to do one of two things ; viz., we may either determine a function-group W containing less than n distinct functions, or we may determine a substitution, say Sy whose characteristic equation contains several distinct roots. We will assume for our independent variables those which shall give the substitution B in its canonical form, and, for simplicity, will take the special case above considered, where , n _ yx, y*, y* > y*; y, > y* > y3 > y4 iyo) B = *3, ; *i+yi> *,+y*> We will now determine the transforms of yx by S, 3B, . . . ; if among these transforms there are any which are linear functions of the others and of yx, we will discard them ; suppose the remaining distinct functions to bejy/, . . . , ytr. Operate now on ƒ/, . . . , yxr by B, . . . and obtain new functions, from which discard again all those that are linear functions of the others and of yx, yx9 . . . 9y*\ if, again, there remain new independent functions y9r+1, . . . , y*, we will transform them by B, 3B, . . . and discard as before. By continuing this process, which is necessarily limited, since the total number of distinct functions cannot exceed the number of the in- dependent variables yx, , yA, we shall finally come to a point where every new function obtained by the operation of the sub- stitutions H, . . . is a linear function of the preceding ones. The linearly distinct functions so obtained, say yx, yx9 . . . are evi- dently changed into linear functions of themselves by B, 38, . . . (and consequently by all the substitutions of (5). The problem now divides into the two cases above mentioned. First: Suppose that none of the functions yx, y/, . . . contain any of the variables zx, ¿r2, ; then the number of these functions is at most equal to the number of the functions yx, y^, y3, y4. We can at once proceed to the general case of n independent variables yy and so see at once that we can obtain a system of less than n functions such that the sub-APPLICATION OF THE THEORY OF SUBSTITUTIONS. 445 stitutions of (5, and consequently those of G, transform these func- tions into linear functions of themselves without introducing any new functions. The function-group W formed from yx, yx9 . . . gives us the solution of our problem. Secondly: Suppose that among the functions y1, yx9 . . . there is one, say yf, which contains z., , z%; we will then stop the pre- ceding series of operations when we arrive at yf, and by retracing our steps find the substitution, say XT, which changes intoyy. This substitution will of course be of the form (39), and the coefficients bxx, . . . , ¿33 may or may not satisfy equations (43), (44), and (45). Suppose the coefficients do not satisfy these equations; we can then determine a substitution whose characteristic equation shall have several distinct roots. The first member of this equation will be the same as the first member of (41), and on being developed will be of the form + cps*'1 + fas*-* + . . . = o. It is only necessary here to assign a value to X which shall not satisfy equations (37). This can be done by a number of trials, at most equal to nr, where r is the degree in X of one of the coefficients 0> 0i > · · · If equations (43), (44), and (45) are satisfied, we can choose the in- dependent variables so that equations (50) shall also be satisfied. If, further, we have bxJ?33 = o, we can, as shown above, so choose the variables that both bxi and ¿23 shall be zero, but ¿13 shall not be zero. We now proceed to determine the transforms of y3 by the substitu- tions H, S3, . . . , neglecting all transforms which are linear functions of the remaining ones and of y3. Repeat this process on the new functions so obtained, and continue in the manner already several times described. We will finally arrive at a, necessarily limited, series of functions y3, y/9 . . . , which the transformations of G will transform into linear functions of themselves. Suppose none of the functions y3, y3, . . . contain zx; then their number is at most = n — 1, this being the number of independent variables when zx is omitted. We will thus have a system of less than n functions form- ing a function-group and such that the transformations of G simply interchange these functions among themselves. Suppose now that one of the functions, say yf9 contains zx, and446 LINEAR DIFFERENTIAL EQUATIONS. let TU be the transformation which changes into yf ; then TH will be of the form (50), and obviously the coefficient b\x will not be zero. Since neither b13 nor b\x are zero, it follows that equations (55) and (57) cannot be simultaneously satisfied. If TH does not satisfy (55), then v = will be of the form (53), and we can determine jjl so that (54) shall not be satisfied, and then determine X so that the substitu- tion aAlt) shall have a characteristic equation other than (s— i)n =0. If TH does not satisfy (57), we will determine jli in such a way that (56), which is analogous to (55), shall not be satisfied, and then reason with ID as we have with in the previous case. In what precedes we have shown how to determine whether or not the group G is prime, and, if it is not, how we can choose its variables so as to throw its substitutions in the form y, ■ · · K ; MY, ■ ■ ■ Yk), --------------MY, · · · Yk) {?1 Y>+, . . . Y„; MY, · · · YÒ Y1+1 . . . Y>; M,(Y, · ■ · Y»), ■ ■ ■ , MY; ■ · · Yk) We can continue this process and finally get for the form of the substitutions in G (74) Yx ... Ym ; O', Yx + * · · + am Ym y · · · y fii Yi + · · · “f* fit* F„ Ym + 1 ... Yn ; y x F, -f- . . . + Yti Yn y * · · y $1 Y\ + · · · “1” the group formed by the partial substitutions (75) I Yx... Ym; ^1F1+... + ^wFw, ..., fixYx +.. . + fiMYm being prime. If now the group formed by the substitutions (76) -j-1 * · · Yn ? Ym -f-1 Ym 1 | · · · | Y n Fn y + i Ym + ! -f- · · · + §n F«APPLICATION OF THE THEORY OF SUBSTITUTIONS. 447 is not prime, we can choose the variables Ym+I, . . . , Y„ in such a (77) to throw these substitutions in the form Ym -f-1 · · • Ym' ; a *» +1 F» + i + · • · -j- l & m -j- 1 ¥in + i ""f” · · · —f~ ^ m' Ym* > fi'm + i ^« + 1+ · · · + firn* Ym· forming a prime group. By continuing this process we can finally throw all the substitutions of G in the form (79) Oix Fj —(— ... -j— OLm Yfft y · · · fi\Y\ + · · * + PmYm a'm+1 Ym+i + ··· Y Y V V . · m-\-1 · · · * m’ i + a'm,Ym,+ /,„+,( Yt... Y,„), + F„.) F^+1 · · · Ym" 1 OL Ym'-j-i “f” · aUmnYmn-\-fmlJrX(Yx . . . YJ)9 + fi”«'Ym»+MYl... Ym) 9 fi m-\-iYm~\-i “f" * * * • > fi"m’+i Ym'+1 + · · · the partial substitutions <80) V Y flTi Fj + . . . + a Y **■191 * in 1 ··· 9 + * · · + 6 Y r'm m (81) y , Y , ■ »*-+-i · · · m' : t & tri-f-1 -J- i ~f" · • · 1 ** m’ Ym' » fi'm + x Ym + x “(“ · • · + (3'm'Ym' forming prime groups.448 LINEAR DIFFERENTIAL EQUATIONS1 We can now show how to determine all possible function-groups which are such that the substitutions of G simply change among themselves the functions of a given function-group. Denote by F such a function-group. We will say that Fis of the first class if the functions of which it is composed depend only on the variables y[, . . . , Ym ; of the second class if some of its functions depend on the variables Ym+I, . . . , Ymf without containing any of the following variables, etc. Since the group formed by the substitutions (80) is prime, there can exist but one function-group of the first class which will be formed by all the linear combinations of YJ . . . Ym. We will now give the means of determining the function-groups of class k when we know all those of classes inferior to k. Knowing the function-group of the first class, we can then proceed to build up all of the function-groups of higher classes. To fix the ideas, sup- pose k = 3, and denote by F, F'y . . . the different function-groups of the first and second classes, and by # one of the unknown func- tion-groups of the third class. Denote by F, Px, . . . linear functions of Fw/+I, . . . , Ymn; by 0, 0, ... , linear functions of Y1, . . . , Ym,; then the different functions of $ will be of the form P + Q, Px -f- Qx, . . . Now the substitutions of G (when G is in the form (79)) trans- form among themselves the functions P-\-Q, P1 -f- Qx, . . . . In order that this may be so it is evidently necessary that the substi- tutions Ym'-^x . . . Ym" J OL m’-\-1 Ytn,~f-i + ··· + «" mn Ym" j I . . . , ft"m'+x Ym>.J_j + · . . “f- m" Ym" transform among themselves the partial functions Py P1, . . . . But by hypothesis these substitutions form a prime group, say F, and consequently the function-group formed by P, Px, . . . will contain all the linear functions of Ym>+X, . . . , Ym», and in particular will contain Ym>.|_,. It follows then that 0 will contain a function of the form (83) f=Y+ Q = Ym'+i XxYx . . . -f- \m> Ym>. Form now the transforms /', ... of this function by the substi- tutions Ay By . . . , considering X1, . . . , as indeterminates, and discard all transforms which are linear functions of the remainingAPPLICATION OF THE THEORY OF SUBSTITUTIONS. 449 ones and of f. We will continue this process in the manner which has already been several times described, and finally arrive at a series of independent functions ƒ, ƒ. . . ,f* · That the series of operations which give rise to new functions which are linearly inde- pendent of the preceding ones is limited is easy to see; for the transforms which are obtained by the successive substitutions are linear functions of Ym>+X, . . . , Ymn and the m'a products of A, Yx, . . . , Amf Ym>. The number of the functions f9 f', . . . , fl is thus at most = mn — m* -f- m> *. By combining fy f\ . . . , fl linearly we shall obtain a function-group whose functions are transformed the one into the other by all the substitutions of G. These functions have the forms P -f- Q, Pf + Q!> · · · , Pl + Q\ where P^ Pf, . . . , Pt are linear functions of Ym’+X, . . . , Ym»y and Q, Q\ . . . , Q are similar functions of Yl, . . . , Ymf. The substitutions of T permute among themselves the functions of the function-group formed by the linear combinations of P\ P', . . . , Pt; but r is prime by hypothesis. There- fore, among the functions P\ F\ . . . , P* which are formed by aid of the m,r — m! variables Fw/+I, . . . , YM", there will be m!' — m' distinct functions. Suppose that Py P'9 . . . , p**"-™’-1 are these distinct functions, and let 0 = IP + CJ be any one of the functions of the function- group 0. The function p of the variables Ym'+T, . . . , Ym" can be written as a linear function of the mn — m! distinct functions Py Fy . . . , P**"-™'- I. For example, let p = dP+ . . . + dm"-m'-xPm"-m’-'i'y then evidently where CJ, depends only on the variables Yx, . . . , Ym>. The function- group 0 will now be obtained by combining the mn% — m! functions Py ... , jP**"-*»'-! with the functions dj,2, . . . which depend only on the variables Yx, . . . , Ym>. It may happen that all the functions (Si,, di2, . . . reduce to zero; but if they do not, they will form a function-group whose elements are permuted among them- selves by the substitutions of G. The function-group so formed will then be one of the assumed function-groups Fy Ff, . . . Among the450 LINEAR DIFFERENTIAL EQUATIONS. functions , Q2, . . . it is easy to determine those which belong to the function-group derived from /, ƒ', . . . , this function-group being obviously a minor of In order to obtain the required functions, we remark that by subtracting properly chosen linear func- tions of ƒ, ƒ', . . . 1 from ƒ**"-»*', . . . , ƒ* we obtain func- tions (SJ**"-**', . . . , which contain only the variables Yl9 ... 9 Ym>\ and further, the coefficients of the variables in these functions will be linear in \19 ... , \m>. If we wish to have (¡Ü,, . reduce to zero, it will be à fortiori necessary that the functions . . . , which form a part of the series (S^, Q2, . . . shall vanish. In order that this may happen, it is obviously necessary that all of the coefficients of F,, . . . , Ym> shall vanish in each of the functions . . . , ^ ; we will then have for \x, . . . , \m, a system of linear equations in general greater in number than mr. If these equations are not incompatible, then to each system of values of the Vs which satisfy them there will correspond a function- group 0 derived from the functions f f \ . . . , m If we require the functions <2^, (Si2, . . .to form by their linear combinations a function-group included among those already known, i.e.9 F, F\ . . . , it is necessary that , . . . , (2^ shall belong to this function-group, say F. Now let x, x\ . . . be the dis- tinct functions in F; all the functions of .Fwill then be of the form MX + M'X'+- · and we must have Q""-*' = MX + M'x' + · · · , Replacing in these equations . . . , (St, x, x'9 . . . by their values in Yx, . . . , Yin> and equating to zero the coefficients of each variable, we shall have a system of linear equations for the determination of \l, . . . , \m>, /¿, ¡jl , . . . , //,, /¿/, ........ If these equations are incompatible, there will be no function-group $ of the third class containing F; if the equations are compatible, then to each system of solutions there will correspond a function-group We may remark, however, that if in the expression /= . . .+\nt'Ym,APPLICATION OF THE THEORY OF SUBSTITUTIONS. 451 , \m> have had values given them which permit us to deter- mine a function-group $ of which F is a minor, we can obtain an infinite number of such systems of values of these constants each of which will give rise to the same function-group by simply consider- ing its functions in the form f+rX+r'x' + y"x"+- · ·»' where v, v\ v", . . . are arbitrary constants.CHAPTER XII. EQUATIONS WHOSE GENERAL INTEGRALS ARE RATIONAL. HALPHEN’S EQUATIONS. A SPECIAL class of Fuchs’s regular equations, that is, equations all of whose integrals are regular, is the class of equations all of whose integrals are algebraic, and a still more special class is that in which the general integral is rational. The investigation of equa- tions whose general integrals are algebraic is reserved for Vol. II, but a brief account will be given here of the equation whose general integral is rational. Write the equation in the form d*y d1t~ly 0) F[,) = j£+P^,· 'dx'-1' • + Pn? = O, and let xy, . . ., xp denote its finite critical points; then, since the equation is to be in Fuchs’s form, we have p _ F*(x) k [>K*)r where Fk is a polynomial in x of degree k (/> — i) at most, and f(x) = (x — x\)(x -X,). . . (x - x„). As by hypothesis the integrals of (i) can only have poles as critical points, it follows that the roots of the indicia! equation for each of the critical points xlf x2, . . . , xp must all be integers, and, further, that no logarithms can appear in the expressions for the integrals. For any given equation we can find at once whether the first of these conditions is or is not satisfied by simply forming the indi- cial equation corresponding to each critical point and obtaining its roots. That the second condition may also be satisfied the equation must be such that equations (47) of Chapter IV are satisfied for all the groups of integrals belonging to each finite singular point. 452EQUATIONS WHOSE GENERAL INTEGRALS ARE RATIONAL. 453 Supposing all these conditions fulfilled, we see that the general integral is uniform throughout the plane since its singular points at a finite distance are all poles. To show that it is not only uni- form but also rational, it is only necessary to show that the point infinity is also a pole. To show this we form the indicial equation for the point x — oo . Divide the roots of this indicial equation into groups, the roots in each group differing from each other only by integers. Suppose the smallest root in each group to be denoted by a, a'y . . . respectively, and the number of roots in each group to be denoted by /?, respectively. For very large values of x the general integral will be of the form ¿[0» + 0> l°g^+ 0, log*^+ . . · + 0—t + ^[0o' + 0/ loS^+ 0/ log"\ + · · · +0'«'-.log“'_Ij] +................................................ the functions 0, 0', 0", . . . being series going according to ascend- ing integral powers of As already seen, however, this expression must be uniform, and so the logarithms must disappear and the ex- ponents ay a', . . . must be integers. It follows, therefore, that the general integral has the point infinity as a pole, and so is a rational function of the form P and Q being polynomials in x. These polynomials are very readily found; the denominator Q is known at once, since from the differential equation and the various indicial equations we know the finite poles of the general integral and their respective orders of multiplicity. Suppose the poles xx, x^, . . . , x? to be of orders of multiplicity alt asf . . ., ap respectively; then ob- viously Q = A(x — x,)ai{x — x2)a*. . . (x — Xp)apy where A is a constant. The order of multiplicity of x = 00 is known from the development according to powers of - , and so the degree454 LINEAR DIFFERENTIAL EQUATIONS. of the numerator P is known ; to find its coefficients we have sim- P I ply to identify the development of -p^r according to powers of — with (j x the corresponding development furnished by the differential equa- tion itself. In connection with the preceding the reader is referred to a note by Mittag-Leffler in the Comptes Rendus, vol. xc. p. 218. Halphen’s Equations. A rather more general class of equations than the preceding has: been studied by Halphen. Consider those equations whose inte~ grals are regular and uniform in every region of the plane that does not contain the point xx. The other critical points, ;r9, x%, . . ., xpr must now be merely poles of the integrals, and so the roots of the indicial equations corresponding to these points must all be integers; and further, no logarithms can appear in the developed forms of the corresponding integrals. Let us suppose these conditions all satisfied for the points x^, x^, . . . , x9; we can now find the general integral. Consider the region of the point x19 and let y0, yt, . . . , yk denote one of the groups of the system of fundamental integrals belonging to this point. If r denote the corresponding root of the indicia! equation, we have y· — (x — xòru> - y, = (x~ + «J, yi iX Xl)’\^kUa + + . · . + Uk\, where u0, uy, . . . , uk are uniform in the region of xiy and _ log (x xi) 0,(0,— 1) . · · k-\-1) 1 _ 27ti ’ · ’ ~ k\ We can make a further assertion concerning the functions u, viz., they are rational functions ; in fact, the points x^, xz, . . . , xp being^ ordinary points for the functions (x - xiYr and log (x — x,),HALPHEN1 S EQUATIONS. 455 and poles for y0, yx, . . . , yk, must also be poles for u0, ux, . . · , uk; these functions are therefore uniform not only in the region of xx, but throughout the plane. Again, the functions (—,r, =¿(.-5)", and log (x — xt), = — log ^ + log — —'j, are regular expressions for x = oo ; the same is true for jp0, , . . ., yk9 and consequently for uoy ulf . . . , These two properties of the functions n taken together show that they are rational functions, say _/>o _P, _Pk -- Q , UX -- Q , . . . , Uk - Q. The polynomials P and Q are found just as in the preceding case; we know the poles x^, xz, . . . , xp and their respective orders of multiplicity, and so the denominator Q is formed at once; a develop- ment of the general integral for very great values of x will give us the degrees of the numerators PQ, Px> . . . , Pk; to obtain their coeffi- cients we have only to substitute the preceding expressions in the differential equation and identify the result with zero. Another very interesting class of equations also due to Halphen * is the class where the general integral is of the form y = c^flx) + cte*>*ft(x) + · · · + cne°-«*xfn(x), where . . . , fH are rational fractions. Halphen’s investiga- tion involves certain properties of differential equations whose co- efficients are doubly periodic functions of x. The following inves- tigation is due to Jordan: Consider the differential equation dny dn Jy dn~2y (2) J\y) + P'd^i + P>d^*+ · · -+pnf = O, * Comptes Rendus, vol. ioi, p. 123S.456 LINEAR DIFFERENTIAL EQUATIONS. where PofPiy . . . , Pn are polynomials the degree of any one of which is at most equal to the degree of the first one, PQ; this is of course equivalent to saying that the developments of P P P J i x 2 M n P 9 J>> - p ·*■ 0 ■L 0 -*0 according to decreasing powers of x contain no positive powers of x. We will suppose that the integral of (2) contains as critical points at a finite distance only poles; the preceding considerations will of course enable us to make sure of this fact in any particular case. Write now (3) y = Rty where R is a rational function of x. Equation (2) now takes the form (4) PaR~ + nRR> + P.R dn-'t A»-')»,?. dxn~1 2 + («-0 ^ + PJi dn~2t dxnI~2 . =0, and this is of the same form as (2) ; for, after clearing of fractions, its coefficients will be rational polynomials in x, and its integrals possess, by (3), only polar singularities ; finally, if we admit that in the development of R according to descending powers of x the first term is Axp, then the first terms in the developments of RRr\ . . . will be, to a constant factor près, x to the powers / — 1, / — 2, . . . respectively. It is easy to see, then, that after dividing by P0R the coefficients of (4) can contain in their developments no positive pow- dn~zt ers of x ; for example, take the coefficient of —r—— , viz., nP+ PXR Rf . Px PaR ’ ~ n R P„’ from what has been said it is clear that no positive power of x can appear in the development of this. A particular form of the preceding transformation will now be applied to equation (2); we can of course determine in advanceHALPHEN'S EQUATIONS. 457 the poles, say xx, , . . . , xp, of the general integral of this equation, and also their respective degrees of multiplicity; say these are Mu j · · · y Mp· Now transform (2) by the relation _ t · y ~ {x — xÿ'{x — -O'1’ . . . {x — x9Yo ’ the result of the transformation, say (5) dnt dn~lt ~dxn~■*" ®xdxn~l + · · · + Qtt — °y is of the same type as (2), but its integrals have no poles. Again, make (6) t — dxv; the new transformed equation, viz., <7) (8) dnv + a dH-'v dxn~x I" · · · + ^nQ0 + . . . + £?« v = o, 0 ¿¿r + ¿7” V + = o, is obviously of the same type as the original ; we will suppose X so determined that the coefficient of the term of highest degree in Rn is made to vanish ; Rn will then be a polynomial of lower degree than R0. Denote by , £2, . . . the roots of R0 = o ; by decomposition into partial fractions we have, remembering that the degree of Rk is at most equal to the degree of R0, (9) § = Ak + 2 k,l Bikl where A and B are constants, and in particular An = o. The points £ being ordinary points, in the region of which the integrals are458 LINEAR DIFFERENTIAL EQUATIONS. regular, the index l can, in the enumeration, only take the values i, 2, . . . , k. The indicial equation relative to Si is (io) r(r — i) . . . (r - n + i) + Binr(r - i) . . . (r - n + 2) + Bi22r(r — 1) . . . (r — n + 3) + . . . = o; the sum of the roots of this is (II) n(n — 1) Bi II. Now since Si is an ordinary point for the integrals, these roots are necessarily unequal non-negative integers; the least values which they can have, therefore, are given by the series o, 1, 2, . . . , n — 1 ; their sum is, therefore, at least equal to o -j— 1 “I” 2 —|— . . · ti — 1 n(n — 1) 2 It follows from this and (11) that £ilx is either zero or a negative in- teger, and a fortiori the sum (12) S=2Bilt taken over all the points S is zero or a negative integer. dv Let us suppose first that Rn is not zero, and take v' = for a new variable. The equation in v thus becomes (13) J? dK~JV' I R dn~*V> I 0 dx*-'^ 1 dx"~2 ~t • · - + Kv = o ; differentiating this, then eliminating v between (13) and the new equation, we have (14) KRn % + m1 + + [(ie'.-x + R*)R« - R»~JR»y = o.HALPHEN'S EQUATIONS. 459' This is obviously of the same type as the equation in v ; the ratio, of the first two coefficients, which in the ^-equation is is now R, , R/_RS K R«' Now let R0 = {x- *,)-(* - Sf* . . . , Rh — (x — Tf^ix — ... ; we have then r: 1 1 . R.~ 1 1 L ... y r: fix , A , Rn~ x— Vi h 1 H L . . . , the sum S' formed for the ^'-equation in the same way that 5 was; formed for the ^-equation is now obviously (15) S' = S + 2a- 2/3. Since 2a, the degree of , is by hypothesis greater than 2/3, the degree of Rn, it follows that S' is greater than 5*. In like manner, if we form a z/'-equation in the same way as we formed the ^'-equa- tion, we should find S" > S'. If we could continue this process indefinitely, we would form an unlimited series of increasing integers S, S', S", . . . , none of which, however, are positive, which is absurd. It must be then that there is an equation in the series the coefficient of whose last term is zero. Suppose this to be the case for the ^-equa- tion ; this equation admits a constant as one of its integrals, and so the equation in v admits a polynomial 7t{x) of degree m as an integral, and finally the /-equation has a particular integral (x). Make now t — ^x7t f tydx; /, satisfies an equation of order n — 1 which is of the same type as the /-equation. This new equation therefore admits a particular in- tegral of the form ^n^x), n^x) being a polynomial. Again, write t1 = eK'xnJ'tjlx,4^0 LINEAR DIFFERENTIAL EQUATIONS. and continue the above process. We of course come at last to an equation of the first order whose integral is of the form tn-1 = cnex*-'x7tn—i{x)y where cn is an arbitrary constant. The general value of t is now immediately seen; we have merely to retrace our steps from this last equation, performing successively the indicated integrations. As we know how to effect the integrations, we can see immediately what the final form of t is; it is, viz., t — 2ckea*xWkt the c s being arbitrary constants and the W's polynomials. Now, since ___________________/ ___________________ ^ ~ (x — X^Mi^X — x^f** . . . (x — Xp)t*p 9 we have finally, for the general integral of the equation in y, (16) y = cyoc^f^x) 4- c^xflx) + ... + cnea«xfn(x), , fn being rational fractions. Reciprocally, every differential equation whose general integral is of this form belongs to the type considered. To show this elimi- nate the constants ck between (16) and its derivatives, and suppress the common exponential factors ; we will thus obtain an equation with rational coefficients which, by clearing of fractions, can be made integral. Suppose the equation is then Po dny ~dxn + P* dn~zy dxn~i • H- Pny — o. As we know, its integral has, at a finite distance, only polar singu- larities. It remains now to show that the degrees of Px, Pa, . . . , Pn are not greater than that of P0. This last, however, will be left as an exercise for the student. Halphen, in his celebrated “Mémoire sur la réduction des équations différentielles linéaires aux formes in- tegrables ,” * and also in his paper in the Comptes Rendus, gives the * Savants Étrangère, t. xxviii. pp. in, 180, 273.HALPHEN'S EQUATIONS. 461 following three examples of this class of equations, viz. (n is an integer throughout): (I) d'y dx' 0 n(n -f- 1) + a = o, which is a very well known equation ; n2 dy (II) d'y , I dx* “·“ X3 ¿/.T —in this n must be prime to 3 ; d'y 2n(n-\-i)d2y 4 n(n-\-i)dy [1-N + «]y = o, (III) dx' dx' 1 x3 dx n[n + i)(« + 3 )(« — 2) + 0 arjy — O- In the Comptes Rendus paper the following examples are also given, viz.: (IV) d*y 2 (n-\-\)d2y f6n dF x ¿F + VF 2 a -y — O; this has one solution^/ = ax* — 2(2n — 1), and two solutions of the form *±v **/(*)> where /is rational. ,Vv d'y ** dy \ 2a \ n(n+l) dx2 x2 — i dx L*2 - I ^ ** + (a — n){a + n + 1)] y — o ; for this, supposing n positive, there are two solutions of the form where f(x) is a polynomial of degree n -f- i. When the general equation is restricted to have the single critical point x — o in the region of which the integrals belong to the ex-.462 LINEAR DIFFERENTIAL EQUATIONS. ponents o, 1, 2, . . . , n — 2, n, Halphen shows that the form of the equation is dny dxn dn~'y dxn~Y — Axa AAdn'2y x / dxn~2 2a A.-,\ x ! n — \OC o. The solutions of this are exponentials where a is a root of the equation f{a) — an~x -f- Axan~2 —f- ... —|— An-X — o ; and the remaining solution is y — ÇO.X /V)1 /(«) J ’CHAPTER XIII. TRANSFORMATION OF A LINEAR DIFFERENTIAL EQUATION. FORSYTH’S CANONICAL FORM. ASSOCIATE EQUATIONS. Although, as previously stated, it is not intended to go into the theory of Invariants in the present volume, it is nevertheless de- sirable to give an account of the transformation of the differential equation and its reduction to the canonical form adopted by Forsyth.* The differential equation may be written in the form __ dnY dn~xY R»dx" +nR'^^~ n(n — i) dn~2Y — -R* 1*2 dxn . . +RnY= o. This, by the familiar transformation Y = ye~I^\dJC and subsequent division throughout by R0, is changed into /•\ dny | «I u dn~2y ( _^| n dn~3y ( ( n ^ dxK ‘ 2| n — 2| 2 dxn~2 3| n — 3)R% dxn~3 * # * Rfl^ °‘ To the letters P^, P9i . . .we may add P0 and PxJ understanding, of course, that P0=i, Px= o. The numerical coefficients are obviously equal to n{n — 1) n(n — 1 )(n — 2) n{n — \){n — 2)(n —- 3) —3 ’ 3 *·” and are written in these forms by Laguerre, f but it is desirable for * Invariants, Covariants, and Quotient-derivatives associated with Linear Differ- ential Equations, by A. R. Forsyth. Phil. Trans, of the Royal Society, vol. 179 (1888), pp. 377-489. f Sut quelques invariants des équations différentielles linéaires. Comptes Rendus, t. 88 (1879), PP* 224-227. 463464 .INEAR DIFFERENTIAL EQUA LIONS. the present purpose to write them in the form chosen by Forsyth. We find readily, P2 = r0r2 - r; - (RqR/ - R/RJ, P% = R:RZ - 3R.RXR% + 2R: - (.R0Rx" - RS'RJ, From these equations we see that the functions P are independent of any particular choice of the dependent variable y, and are there- fore seminvaricints of the original equation. Suppose now that (1) y — u\, where A is a function of x, and suppose that u satisfies the equation (ü) dnu n\ dn~2u^ n\ dn~*u ate*' 21 n — 2| dzn~2 31 » — 3| ®sdzn~* + * * ‘ + @*u ~ 0 in order that (i) may be transformable into (ii), z must be some func- tion of x, and when this is the case there will be n equations con- necting A, z9 Xj and the two sets of coefficients P and Q. These equations are obtained in the following manner: Making the sub- stitution y — uX in (i), we have dnu dk dn ~1u , (a) A -\-n —+ -----4- ' J dxn /V ■>* 1 ^«-1 1 d2X dn~*ii + 2 \n — 2 dx P. dxn d*'2u ----X —1— in — 2) dx*-2 ^x ’ 21 n — 2| dx2 dxn~2 dXdn-*u dx dx”~3 + . = o. In this equation we have to change the independent variable from x to z in order to compare it with (ii). Write (2) z = (Kx) ; then, by a formula due to Schlomilch, we have where dmn = s d5u dxm ~ s=i 4 dz* A^s = Limit, when p — o, d °f dp"TRANSFORMATION : FORSYTH'S CANONICAL FORM. 4^5 Writing mj Cm<, = AKtS, it follows from this last that Cm,, = coefficient of pm in (o0' + ^ p'. . ·^|¿, i)**; then the coefficient of ^n-r-t,s n — r — t\ in the foregoing expression is Pr d*\ 2r\ i\ d?’ the summation extending to those values of r and t that leave the sum r t unchanged throughout—that is, the coefficient is Wr+t r + ty and therefore ^Qn-s A„-»,SW« ¡E3TT Changing s into n — s and introducing the quantities C from (3), and this becomesTRANSFORMATION : FORSYTH'S CANONICAL FORM. 467 If it were desirable, the summation in this might extend to the value 6 — n, for Cm^ m> vanishes if m < mr. Writing (iii) in detail for the lowest values of s and giving Px and Qx their zero values, we have in succession n equations in all. There is a first invariant of the differential equation which is readily derived from these last equations. The invariant might properly be called Brioschi’s invariant, as he first showed that it existed in the case of differential equations of the third and fourth orders. Forsyth’s deduction of this invariant (which follows) from equations (5)', (6)', (7)', involves an important modification of the forms of these equations. We have from (4) +>,c n-3,n-z> — À, Wt = X‘ and from (3) cn Cn.2t„.3 are similarly found to be C„, *_3 = AO — 3K”-3] 2Z" + (4n — 10)ZZ + (n - 2)(« — 3)Z* \, Cn-m = AO - 3+ (3* - 8)Z^, 0,-2, «-3 = £0 — 3>'"_3^· By means of these and the above values of A', A.", A/", (7)' changes into (7) Z" - 3Z'Z +Z> = 1±ri{P>- Q,z'’) -Ptz. Differentiating (6) with respect to x, and remembering that Q2 is a function of z, and that ^V' = ^/2^, we have 12 6 (dp* _,3 ^6 A________ « -f-1 Wj; dz, ) n-[- I TRANSFORMATION : FORSYTH'S CANONICAL FORM. 469 subtracting (7) from this gives 2ZZ’ -Z% = (Ç* - z'■ ~) - ¿'ZQ% n-\-i\dx dz 1 n + i 12 ^v’.-s'a+^p.z·. multiply (6) by Z and subtract from this last, and we have 6 (dP% ,tdQ. n -f- ï ta’--'•D-ïtî or, finally, (3^-2a)y = 3g-^., the result of the elimination of Z between (6) and (7). This last re- sult is BrioschTs invariant. It appears from the preceding investigation that there exist rational integral functions of the coefficients of a linear differential equation and their derivatives such that, when the same function is formed for the transformed equation, the two functions are equal to a factor près, which factor is a positive integral power of z\ These functions are the invariants of the differential equation, and the ex- ponent of z' is the index of the invariant. Reverting now to equation (5) and integrating it, we have \z'^n r) = constant ; since equations (iii) are homogeneous in the dimensions of this constant may have any arbitrary value other than zero; taking then the value unity, we have (iv) \ = z'. This establishes only one relation between the quantities z and A, and we may suppose z arbitrary so far. We may now impose any other470 LINEAR DIFFERENTIAL EQUATIONS. condition we please upon z and X which does not violate (iv) (or (5))^ say such a condition as will make Q,B = o. By (6) we must then have where, by (8), If we write 2Z = Z1 + 12 p ~ n + 1 *’ zn _z_ 2 X' z* “ n — i X 0f z=-2~e the equation which determines Z becomes (9) and so we may write (10) d*0 dx8 ' n-f-1 P.d = o, X = On-1, zf = 0-2 Hence by a solution of a linear differential equation of the second order we can remove the second and third terms of the general linear differential equation of any order. This result was first given by Laguerre.* The modified form of the equation to which we have now come is Forsyth's canonical formy\ and for the future we shall speak of a linear differential equation of order n as being in its canonical form when the terms involving the derivatives of order (n — 1) and (n — 2) of the dependent variable are lacking. d*0 Since 6 = P ~i, we have — i-s' ~*s'"; substituting in (9), we get 3*"' , 3 P, 1 _n 4 A ’ * Comptes Rendus, t. 88 (1879), p. 226. t Forsyth’s canonical form differs very little from Halphen’s, but is more conven- ient in computing the invariants.TRANSFORMATION : ASSOCIATE EQUATIONS. 471 or, on reducing, \2 6P. n+i’ 3 2 W J * Associate Equations. \z, x\ being the Schwarzian derivative — — In Chapter I, equation (42), we defined Lagrange’s “ équation adjointe,” and there spoke of it as the “ adjunct equation for the future, however, we shall use the term associate equation. * The differential equation being given in the form /. T\ d*y d*-'y (II) d^ + R' R. * dxn~ its Lagrangian associate is dnv . dn~2 , _ dn~3 dn~3y dxn~3 • · · + R«y = o, (I2> Hx” + {vR'] ~ dx^{vR^ +···+(- lYvR· = o. Let y1, , . . . , yH be a system of fundamental integrals of (11); a selection of any # — 1 of them will suffice to determine the n — 1 coefficients R. Say we take ylf , . . in (11) and solving, we have ■ > y·*-i> substituting these d*-y, dx”-2 d”-i~yl dx”-·-1 dny. dxn d”-'+Iy1 dx*-*+» ·· - y, Rt = d~yt dxn~2 dn~i~ 2yt dx”-1'-1 dyt dx" d”~i+Iy^ dx”-'^1 ...y, rfn-2yn-i dx"-2 d—*-'y»-x dx”-'-1 dy„-r dx” d”~i+Iy„. dxn-i+y *When Chapter I was written, and indeed when an earlier form of the present chapter was written, I had not seen Forsyth’s memoir, and had not been able to find an adopted English term for Lagrange’s “équation adjointe,” so I used the word adjunct, suggested by the German “ adjungirte,” and notunlike the French “adjointe.” It seems better now, however, to employ the word associate, or, when speaking sim- ply of Lagrange’s “équation adjointethe word adjoint. It is unfortunately too late to make this change in Chapter I.472 LINEAR DIFFERENTIAL EQUATIONS. where the subscript n in vn indicates that the function yn is absent, and where d*-*yx dn-*yx dx”~* dx”—i • yx d—y% d»~y, dx*~2 dx»~3 ' ‘ dK~yn_i d*-*yH-r dxn~2 dxn~* • y% Substituting these values in (n) and multiplying through byvn, this equation becomes dny d”-y dy dxn dx*~* dx y dy i dn-y, dy dxn ¿V"-2 dx yi dy, d»~y, tty. dx* dx”-* dx yn dyn-, a 1 dyn., dx* dx!— dxn~I yn-x d»-y dy dx*-1· dx y d*-y1 dn~*yx dyx dx"-1 dxn~2 dx yx d»-y, d"~y, dy. & II dxn~2 ' ' ‘ dx yn dn~'y„-1 dn~yn-, dy„-x dx*—1 dx»~· dx yn-xTRANSFORMATION : ASSOCIATE EQUATIONS. 473 It follows therefore that vn is an integrating factor for (n), and con- sequently that vlf vi9 . . . , vn are integrals of (12); that is, the in- tegrals of (12) are the n determinants of dn~2y„ d”-2yn-r d”-y, dn~2y1 dx”~2 ’ dx*-* ’ ’ dx’ dx”~2 d”~3y„ d”~3y„^ ' d”~3yt d”-3yt dxn~3 ’ dx”~3 ’ • · * ’ dxn-i > dx”- 3 y«, yn-1, y*> [It is clear from this that if all of the integrals of the given dif- ferential equation are regular, then all the integrals of the adjoint equation are also regular. This remark applies to all the associate equations.] It is known that if (12) is the Lagrangian associate of (n), then reciprocally (11) is the Lagrangian associate of (12); and it is evi- dent that if either be in its canonical form, the other will also be in its canonical form. Forsyth shows now that the dependent variable v of Lagrange’s associate equation is merely the last one of a set of dependent variables associated with the dependent variable of the given equation. These variables are all transformable by a substitution similar to that which transforms the original dependent dz variable, viz., multiplication by some power of —; and they possess dx the property that all combinations of them, similar to those by which they are constructed, are expressible explicitly in terms of the varia- bles of the set. The following is, Forsyth’s account of these new variables: Let yx, , . . . , yn be a set of fundamental integrals of the given equation; they are of course linearly independent, and we will further assume concerning them (an assumption justifiable in the general case, but not necessarily so in a particular case) that there exists no linear function of them with constant coefficients which is equal to a polynomial of degree less than n — 1. Then of course the linear independence of the functions will hold for their derivatives up to the (n — i)th inclusive. Any other set of funda-474 LINEAR DIFFERENTIAL EQUATIONS. mental integrals Yt, Fa, . . . , Yn are linear functions with constant coefficients olyx,yt, . . . , yn. We may write (Fiy . ., Yn) = S(yiyy„. . .jj, where as before S denotes a substitution with non-vanishing deter- minant. We have also (drYx drY2 drYn\ _ c.fdryx dry2 dryn\ \dxr ’ dx" 9 m m '9 dxr)~^ \dxr 9 drr ' ' ' ' ' dxr) for any value of r. If we retain this last equation for values of r equal o, i, 2, . . . , n — I, we shall have n sets of variables subject to the same linear transformation; and these variables are linearly in- dependent of one another, since for the satisfaction of the differen- tial equation we need the nth differential coefficients of the functions yy but these have been specially excluded. Since the n quantities y are linearly independent they may be looked upon as the co-ordi- nates of a point in a manifoldness of n — I dimensions; similarly, under the hypothesis made concerning the derivatives up to order n — i, each of the n — i sets of derivatives, each set being made up of derivatives of the same order, may be looked upon as represent- ing the co-ordinates of a point in a manifoldness of n — i dimen- sions. And, since the law of linear transformation is the same for all the sets, all these points may be taken as belonging to the same manifoldness. There are thus n different and independent sets of cogredient variables connected with the single manifoldness of n — i dimensions. In the theory of the concomitants of algebraical quantities of any order in the variables of a manifoldness of n — i dimensions, it is necessary to consider all the possible classes of variables which can enter into the expressions of these concomitants. Clebsch * has proved that there are in all n — i different classes of varia- bles which thus need to be considered, and that if xiy x2, . . . , xn; y,, yt, . . . , yn; zx, ^, . . . , zn; . . . be n sets of cogredient varia- bles, the several classes are constituted by minors of varying orders of the determinant (itself an identical covariant) * “ Ueber eine Fundamentalaufgabe der Invariantentheoriey Gottingen, Abhandlun- gen> voi. 17, 1872.TRANSFORMATION : ASSOCIATE EQUATIONS. 475 X%, · · · , *» ƒ.» Jj) ··· 9 yn *1 > . . . those of one class being minors of one and the same order. The variables of any class are linearly, but not algebraically, independent of one another, except in the case of the first class, constituted by minors of order unity, and the last class, constituted by minors of order n — i (the complementaries of those of the first class), in each of which classes the n variables are quite independent of one another. And all similar combinations of variables are expressible in terms of variables actually included in the classes. In connection with our differential equation we have obtained n different and algebraically independent sets of cogredient variables; the functional derivation of the sets, one from another in succession, by the process of differentiation has been excluded from any inter- ference with their algebraical independence. We already have one class of variables, viz., yx, y9, . . . , yn, analogous to the first class of algebraical variables, and another class of variables, viz., vx, , .. ., vnr analogous to the (n — i)th class of algebraical variables; and the re- lation yxvx + y%v% + · · · + yj>« = o, which is satisfied, is precisely the same as the corresponding relation between the similar variables helping to define the higher class (Clebsch, /. c., p. 4). Hence, from the point of view of purely alge- braical forms, we infer that the suitable algebraical combinations of the sets of variables, which have arisen in connection with the differ- ential equation, are the minors of varying orders of the determinant y*> ■ , y* dyx dy, dyn dx ’ dx’ ' ’ dx d”-% dn"y. d"-y„ | dxn~l 9 dxn~l ’ dxn~'476 LINEAR DIFFERENTIAL EQUATIONS. which, since Rx = o, is a non-evanescent constant. These variables may be arranged in classes which may be called linear, bilinear, tri- linear, and so on. In the case of the algebraical quantities it is a matter of indifference which set of minors of a given order be taken to constitute the variables of a class corresponding to that order. Thus for the second class the same kind of variable is obtained by taking the (x, y) minors, the (x, z) minors, the (y, z) minors, and so on. A difference, however, arises in the case of the variables con- nected with the differential equation. There are n sets of linear variables distinct in character from one another, as the variables of any one set, say yf yf . . . , yj> though submitting to the same substitution as y19 y,if . . . , yn, satisfy an entirely different differen- tial equation. There are ------—J sets of bilinear variables distinct in character; thus y,, y, yr, y,' yr, y, yi> y* j yr, y: 1 yl, y, are three distinct variables of this class, subject to the same law of linear transformation; and so on for the higher classes. Most of these must, however, be excluded, and of the foregoing algebraical combinations we must for our purpose select only those which possess, what we may call, the functional invariantive property, that is, those which have the invariantive property of reproducing dz themselves, save as to a power of ^, after the transformation. Of the n sets of linear variables constituted by the several sets of n quantities y, n quantities yr, and so on, only the first set possesses the functional invariantive property, and we already know by (iv), if n be the new dependent variable, that we have the relation (13) y — uzf — u\. Of the \n(n — 1) sets of bilinear variables, each set containing -&i{n — 1) variables, only one possesses the functional invariantiveTRAN SFORMA TION : ASSOCIATE EQUATIONS. 477* property, viz., the set constituted by the \n{n — i) variables of the: type fay fa yd* y? This statement is readily verified by applying the substitution (13). Suppose denotes the original bilinear variable, say the one just written, and v% the transformed bilinear variable, dua ~dz’ chip Hz' We have, since A = ua m fa =uaxf ffi= Up\, fa' = —«'A—4(« — i)«aA*'~V', yi' = — i^A/"1/',. giving f«> fa f/y ft or dua . y°· II *■ V. dUa y» or, finally, -~zfA - £(« - i>aA/"V', dUa ~r'^'A — £(# — i)upXz' V', (14) = IV», = (”· 2) __ ua A It is obvious from this that the functional invariantive property does, not hold for any bilinear variable of the form V. Jy V. JNS where y and d are respectively different from a and jff.478 LINEAI? DIFFERENTIAL EQUATIONS,. In precisely the same way we can show that of the \n{n— i)(#—2) sets of trilinear variables, each set being constituted by correspond- ing minors of the third order, there is only one set of which each variable possesses the functional invariantive property, viz., the set of which the typical variable is ya n y«> y« t. = yl> yy"> y?', ye ■ yy> yy The relation of transformation in this case is easily seen to be (15) where vz is the corresponding transformed trilinear variable. In general, of the /] n-p\ sets of /-linear variables, each set being constituted by minors of the /th order, there is only one set which has variables possessed of the functional invariantive property, viz., the set of which y1 > y/> y>", ■ ■ . , y*> y/> y· · • , yj*-* yp> y/> y*"> · · 1 • ^ is a typical variable. If vp denote the same /-linear variable asso- ciated with the transformed equation, the law of transformation is (16) ^ = ^v,+2 + ---+J>-1 __ „f -\pin- i) + */(/-i) — VpZ The last set of variables is that for which / = n — 1 ; and the typical variable of the set is the variable of the Lagrangian adjoint equation.TRANSFORMATION : ASSOCIATE EQUATIONS. 479 We see now that there are in all n — i sets of variables ; all the variables in any one set are particular and linearly independent solutions of a differential equation the dependent variable of which is a typical variable of the set. Hence, connected with the given differential equation, there are n — 2 other differential equations ; these are the associate equations. The n — 2 new dependent varia- bles, derived by definite laws of formation, mayT>e called the asso- ciate dependent variables ; and, calling them in turns the associate variables of the first, second, . . . , (n — 2)th rank, the differential equation of which the dependent variable is the associate of the n\ (p — i)th rank is linear and of order ——----------. For the functional l\ transformation of the original dependent variable given by (13) the law of transformation of the associate variable of the (ƒ — i)th rank is given by (16) ; and if we call two ranks complementary when the sum of their orders is n — 2, their associate variables of complemen- tary rank are transformed by the same relation, since for such varia- bles the index of the factor power of zf has the same value. For example : in t% the power of zf is — (n — 2), and in tn_2 the power of z' is — (n — 2), and so 12 and ¿*_2, whose ranks (or orders) are re- spectively 1 and n — 3, are complementary. The associate variables may therefore be arranged in pairs of complementary rank; in the case of n even there is one dependent variable of self-complementary rank. Each pair has the index of the factor power of zf different from that for any other pair. The simplest case of this arrangement is that which combines in a pair the original variable y and the variable tn_T of Lagrange’s adjoint equation ; and the two depend- ent variables have the same functional transformation. Leaving for the present the general subject of associate equa- tions, we will derive some properties of the Lagrangian associate, or adjoint, equation, due in part to Thome and in part to Floquet. Let the given differential equation be , dn~*y dnv dn~1v <.7> = ^ + * dxn~2 . + Pny = o. This can be placed in a determinate composite form in the fol- lowing manner : Writing dy480 linear differential equations. construct the series of linear differential equations of the first order Ax = o, A2 = o, . . . , An = o, admitting respectively as integrals VAL VA), where, if yx, , . . . , yn are fundamental integrals of (17), we have y, = , A = . . . , = v.fv^dxfv^dx . . . fvKdx. For brevity write F; = vxv2 . . . vj9 j= 1, 2, The value of ^ is then given by „ 1 dVS d ^ jr K. = T - —- = — log Vis J Vj dx dx 0 J We have identically (18) P = AnAn_x. . .At. In fact, the expression on the right-hand side of this equation is an- nulled by the general integrals of the equations A j — O, A j — ^1^21 A2Aj — 9 · · · y An-j An-2 · · · -^i ^1^2 · · · ·" Now Ax = o is satisfied for 7 ^ is satisfied by y — vxfz\dx, A2AX = ^,^3 is satisfied by ƒ = vxfvjlxfvzdx, and so on ; thus the two equations AnAu_x . . . Ax = o and P = o, each of order nx have a system of fundamental integrals in common, dny and consequently, as the coefficient of is 1 in both cases, the first members of these equations must be identical, for, if they were not, their difference, which is at most of order n — 1, would be annulled by n linearly independent functions, which is impossible. It followsTRANSFORMATION : ASSOCIATE EQUATIONS. 4-81 then that (18) is an identity. In the same way we see that the equation of order n — i, admits the n — i linearly independent integrals yXJ yif . . . , yn.If and consequently has Cly1 + C2y2 + . . . + Cn„1yn.I as its general integral. Again, the equation Ay) = Vn admits the particular solution yn, and consequently the equation where Cn is an arbitrary constant, admits the particular solution Cnyn, and so the general integral of this last equation of order n — i is which is the same as that of the equation P — o of order n. It fol- lows therefore that (19) is a first integral of P = o. Writing (19) in the form A(y) = An_xAn_2 . . . Ax = o, (19) A(y) = CuVn, Cih + + · · · + Cnyui Vt~IA(y) = Cn and differentiating, we have dny Make the coefficient of in this unity by multiplying by Vn, and we have the differential equation (20) whose first member is identical with the first member of P = cl Writing Vn~x = v, this identity is482 LINEAR DIFFERENTIAL EQUATIONS. and we see that (22) » = Vs' = . vn is an integrating factor of P = o, and consequently is the dependent variable of the adjoint equation (23) m = £-^(/>)+^(/>) + · · ·+(- iyp,v=o. Consider in general two differential quantics • · · + s°y> (24) d*v d ~1v sw=s.g+s,:‘/-’’ 1 da?-1 • + ; these will be said to be adjoint differential quantics, or simply ad- joint quantics, when the following relations exist, viz., (25) S(y) = (- i)‘ ^(SjO dx° ' ’ dx°-' ^ * SO) = d(S„-,y) dx d(S / s dnv , . , , Pn + ~pVi it follows therefore that the adjoint to is pd«f, n d—y 0 dxn ~ 1 dxn~' + · · · + P» y dxn dxtl a result of course immediately obtained by Lagrange’s method. Letting A still denote a function of x, it is easily shown that the / k 4/\ adjoint of the quantic 5 is (- ')‘^[A(S7)]. The differential quantic <- (dky \ o d°+ky ( ( , c dky 0\dx») ~ ^ dx~** "f" 'dx°+*~‘ + · · * -r ^dxk has (- i)'+* d°+k(Sy) dx°+k d°+k-'(S,y) dx) , d-HAv) — dx* dxdxn~ — 1)HPhV — O. Suppose the expression 2H(v) to be formed for the adjoint equation in the same way that A(y) is formed for the given equation; we have then the two identities JWP) = [>H(w)], vP^=dx giving (29) vP(y) -y&p) = j- \_vA(y) -ƒ»(»)];TRANSFORMATION : ASSOCIATE EQUATIONS. 487 that is, the difference vP(y) — yp(^) is the derivative of a differential quantic which is linear and homo- geneous in y and v and whose coefficients are linear homogeneous functions of P1, P%, . . ., Pn and their derivatives. We may write (29) in the form (30) vP{y) - jp(i') — ^-B{y, v). The differential quantic B(y, v) is called by Frobenius * the beglei- tende bilineare Differentialausdruck to P ( y) ; we shall call it the associate bilinear differential quantic, or, when there can be no ambiguity, simply the associate quantic. It is easy to find now the condition to be satisfied in order that a given differential quantic shall be self-adjoint. Let P{y) be the given quantic and f>(z/) its adjoint ; if the coefficient of the highest derivative of y in P{y) is P0, then the coefficient of the highest derivative of v in p(z>) will be (— i)*P0 ; in order then that the quantics P(y) and p(tf) shall be the same (save of course as to the letter which is used to denote the dependent variable in each) we must first have that the order n of the quantic is even, say n = 2v. This being granted, equation (30) must have the form (31) vP{y) - yP{v) = ~ B(y, v). If now we interchange y and v, the left-hand member of this equa- tion changes sign, and so we have or (32) B(y, v) = — B(y, y). * Uebet adjungirie lineare Differentialausdrücket G. Frobenius, Creile, vol. 85, p. 185.488 LINEAR DIFFERENTIAL EQUATIONS. Suppose P(y) is the negative of p (iv), then we have (33) +yPty) = ») ; interchanging ƒ and does not alter the left-hand member of this equation, and therefore In this case, of course, n is odd. We can state the general result as follows: (1) If a differential quantic is self-adjoint it must be of even order, and its associate quantic must change sign when its two dependent variables are interchanged. (2) If the adjoint of a given differential quantic is the negative of that quantic, then the common order of the two must be odd, and the associate quantic must be symmetric in its two dependent variables. It is obvious that the restriction of evenness is unnecessary in the case of a differential equation. The following remarks on adjoint expressions, due to Halphen,* though reproducing to some extent what has already been said, will be useful for future reference. Halphen’s notation is retained, at a slight sacrifice of uniformity of notation in this chapter, in order to facilitate reference to his important memoir. Denote by gQ, gx, . . . , y%, yx, . . . given functions of the inde- pendent variable x; byy and rf indeterminate functions of x\ con- sider the differential quantics These two linear quantics are said to be adjoint to one another if there exists a third bilinear quantic B(yf rj) which is linear and homogeneous both with respect to y and its derivatives up to the * Sur un problème concernant les équations différentielles linéaires. Par M. G.-H. Halphen. Journal de Mathématiques pures et appliquées, 4me Série, t. i. p. 11. B{y, v) = B(v, y).TRANSFORMATION : ASSOCIATE EQUATIONS. 489 order n — 1, and with respect to tf and its derivatives up to the same order, and which is such that we have identically, that is, whatever be the functions y and rj> (34) vG{y) + (- Iy-yrfr) = B\y, rj), accents as usual denoting differentiation with respect to x. This relation completely determines one of the quantics G(y), r (7) when the other is arbitrarily given. The relation between these quantics is expressed by either of the following systems of equations : <35) - y,=g* y* = —g*+g·' Yi=gi~2gi +g" Y, = —g,+3g*'—&"+&'" go = Yc g* = — Y, + Yo g* = y* — 2Yi + yl g* = —y,+3y,'—3Yi"+y/" The law of these equations is obvious, and the two adjoint quantics have the forms ' G{y) = (ytyjn) - n(yty)(n~l) + (y, ƒ)<”-=) + . . . <36) "i n(n-i) r(v) = {g*vV - ■ i.2-- (^v)(”~2)+ · · · + (-1 )n+lg»v· The two quantics G(y) and r{rj) being formed in this manner, (34) is satisfied when the associate quantic B{y, rj) has either of the forms B - pQ/n~') - Pi/·—* + A,/”“3) + · · · + (- I)”/?„-,/ + (- I)”+'Pn-iy, — v)”-'B = — b,rfH~^ -f- b,?/”-3) + . . · “I- (— i)nb„-ji -|- (— (37) (490 LINEAR DIFFERENTIAL EQUATIONS. where (38) 0» = gJh A = (gcV)' — ngS1> 0.={&vY - n(gi>y+n-n~2 A = (a?)"' - «(¿i?)" + - y--·1- U&Y - t—i)(n—2) 1-2*3 (39) = (y.y)' ~ K— (y>y)" — n(y>y)' + . 2 ^ i / v„ , v/ I «(«“I)/ v »(»—1)(»-2) ^, = (r.^) -«(x.>0 H------yy~ (r^)---------yy-—nA In order that two quantics, say P(y) and may be self- adjoint, we have seen that they must be of even order, and that on interchanging y and v, B(yf v) must change sign. As illustrations take the cases of n = 4 and n = 6, and use (37). For n = 4 the quantic P is («) and its adjoint expanded is ( > dx' p' dxi + r 1 3 dx* 1 dx‘ V' 2 dx + 3 dx1 J dx ~r \ * dx dx‘ dx31 The conditions that (a) and (a)' shall be the same are at once found,, by comparing coefficients, to be dP P — a p — p p — _a_ P — P r 1 - --------- J 2 f * 3 - » -*4 -- 4 ·TRANSFORMATION : ASSOCIATE EQUATIONS. 491 i he associate quantic is (fi) B(y, v) — vy'" — [7/ — Pyo\y" fi- [v" — Py' + (P, — Pfiv\y' - \y"’ - py + (P3 - 2P;y - (P% - p; + p/» ; arranged according to v this is (py - v'"y + v"[P,y +/] - *'[(P, - 2p,')y + PJ +/'] + v[(p, - p; fi- pfi)y + (Pt - p/y + pyy f'\ In (fi) interchange y and v and we have (fif V'y - v"[y> - p,y] fi- v'\y" - PJ + (P, - P/)y] - _py + (/» _ 2P/y _ (y _ P’ + p/y].. That (fi)' and (fi)" may have opposite signs we must obviously have y - p*y = P*y +/> ·'· pi = ° ; using this, we have next y+pj = p.y+y, ··· p* = p*'> again, - /" - py+(p, - p.y = - /" - p,y - (p, - p,y giving and finally PA = PA. These are the same results as found by direct comparison of (a) and (a)'. Take the case now of n = 6. From the general form of the ad- joint equation it is obvious à priori that the coefficient of the second highest derivative must be zero, i.e., P1 = o ; the sextic can then be written (v\ €y.y I pù-y p ^ dx° + ^ dxi + Ps dx3 "r P* dx3 + p° dx + P'y '492 LINEAR DIFFERENTIAL EQUATIONS. the adjoint sextic is (rY -[«-*§+3 "I" [”^6 dP2- dx. ¿2P3 ¿r2 d*PP\ dv ·]■ d_P, dx d“P. d‘P, d'P, dx* d'P,~] dx' J In order that (y) and (y)’ shall be the same we must have, as is easily seen by comparing coefficients, P-J3 r'~ dx' 6 ~ dx dx% ' Pa, /^jandPg remaining arbitrary. The same results of course must be obtained by considering the associate quantic B(y, v) ; this, which is easily found, is (, - 2P;y + (pt - p: + y >] y - ov + py- (y - ip>" + (p< - y + 3 y>' — {P* — P<' + P” — P,'")v\y ; arranged according to v this is (d)' _ vy + _ V’"\p^y +ƒ'] + V"[(PZ - zp;)y + /> y +y "] - 7/[(/> - 2P; + 3y"0 + (y - 2A0/ + yy' + ƒv] + »[(/>. - y' + y " - y">+(y - p: + y")y + (y-y>" + yy"+y]. In (S) interchange y and v and we have (d)" vy - zpy + v"\y" + P,y] - v"[/" + P,y - (y - P2')y] +»'[ yv+yy' - (y - zp:y+(p* - y' + y">] - ®[y+yy" - (y - 3yoy' + (y - 2/v + 3y'oy -(y-y, + y"-y'")^].TRANSFORMATION: ASSOCIA TF EQUATIONS. 493 In order that (d)' and (£)" shall have opposite signs we must have, as is readily seen, dP% dx ’ d*P, dx3 In the case of the quartic (assuming always Px — o) replace Pz, Pz by 6PZ, 4P%, and in the case of the sextic replace PzyPz,PAi P by 15P7, 2oPz, 15/^4,6Pb ; that is, multiply the coefficients of the two quantics by the corresponding binomial coefficients. The conditions that these quantics shall each be self-adjoint are, for the quartic, (40) 2 dx o; for the sextic, (41) 2 dx P — 5 dP4 5 d'P, Forsyth in his memoir already referred to gives the forms of certain invariants connected with the general linear differential equa- tion ; among these are two, denoted by ©3 and ©6 respectively, which have the forms ©s (42) i __p _3 dP% “ 8 2 dx ®> = P> . 15^s 2 dx * 7 dx3 5 d'P% iQ7n+ii 7 dxz 7 n+ i 2 Comparing these with (40) and (41), it is seen at once that the con- ditions to be satisfied in order that a quartic or sextic linear differ- ential quantic shall be self-adjoint are, for the quartic, for the sextic, ©s = o; ©3 = 0, ©6 = o.494 LINEAR DIFFERENTIAL EQUATIONS. In the case of the octic (■· o· P-’p-·· P-\Î'')r the conditions for self-adjointness are or and _ 3 dP.t 3 2 dx ’ P_i^ + 1^ = 0 6 2 dx ~ 3 dx‘ u’ 6 2 iÙrT2 dxa — °’ p 7 ¿P« | 35 daP, 21 d*Pt ’ 2 ¿¿r '1~ 4 dxa 2 ¿¿r5 ~ a In terms of Forsyth’s invariants these are ©3 = o, ©5 = o, ©T = o. The general form of ©7 is = p 7 dP» 105^ 35^ | 35^, 7^, 7 7 2 ' 22 ¿£r2 ii 33 dxr4 44 - u i+ll f + 3,)(2-P·- sS) + 505» +41)^· dsP -iS(*- + S)^ 7 3« + 4 L ¿V», f „ I ¿P,\ , dP, ? il /Z -f- I ( ^ ¿¿r2 \ 3 ' dx/ ^ dx dx ) 115$^ + 6048% + 6909 22(11 + i)3TRANSFORMATION : ASSOCIATE EQUATIONS. 495 Mr. G. F. Metzler has put this invariant in the following form, which, for the present purpose at least, is rather more desirable than the above: ^ _ p _7dP 105 d*P, 3Sd*P< 3Sd*Pz 7 7 7 2 dx ' 22 dx1 11 dx% ‘ 33 dx* 44 dx* 7_ 11 ^7(33^+93)©,- _9_ 385 «* + 17^n + 1919 22 3 3 (« + if (3« + 4) ”(«+i) ¿80, d.X - - 35 i/x dx + 21 ©. ■]· The subject of adjoint quantics will be taken up again after an ac- count of the invariantive theory has been given ; enough has already been said, however, to show that a differential quantic is self-adjoint when certain of its invariants vanish, and it is easy to see that in any case three of these invariants must be 03, 0B, and ©7.* An interesting theorem due to Appell, and closely related to the invariantive theory, may be merely stated here without proof; the reader is referred to Appell’s paper in vol. 90 of the Comptes Rendus, p. 1477. Given the equation dy d*-y d”-y dx” ' 1 dx”'1 8 dx”'3 "· • · + P»y = 0 ; let j/j , ja, . . . , be a set of fundamental integrals; Appell’s theorem is: Every integral algebraic function, F, of y1, , . . . , yn and the derivatives of these functions, which reproduces itself multiplied by a constant factor other than zero when we replace y19 y9t . · · , yn by the elements of another fundamental system, is equal to an integral alge- braic function of the coefficients of the differential equation and their derivatives multiplied by a power of e~Sp\dx. * Mr. Metzler has proved that the condition for self-adjointness in general is that the invariants 0 with odd suffixes must all vanish. The proof is rather too long and complicated to give here.CHAPTER XIV. LINEAR DIFFERENTIAL EQUATIONS WITH UNIFORM DOUBLY- PERIODIC COEFFICIENTS. Before taking up the subject of the integrals of these equations it will be convenient to recall a few points in the theory of doubly- periodic functions. These functions are all constructed by aid of the element function 6x{x) (or, if a be a constant, 6x(x — a)). If go and go' denote the periods of the doubly-periodic function, then we know that 2ir ix irioi' 6,{x + go) = — 6x{x), 8x{x —}— Go') = — € « ¿r ex(x\ Another element function which plays an important part in the theory is the logarithmic derivative of 6X, viz., ¿1*) = * For x — o 6x{x) has a simple zero, and so Z(x) has a simple pole. Inside each parallelogram of periods the doubly-periodic function will have a certain number of zeros and a certain number of poles (account being taken of the orders of multiplicity). Now it is a known theorem that the number of zeros inside a parallelogram of periods is equal to the number of poles. Let this number be m, and let ax, . . . , am denote the zeros and ax, . . . , am the poles inside a parallelogram of periods; then by another theorem we have m m (i) — 2 cti — /AGO + //g/ , I I where /a and ja' are integers. Now a doubly-periodic function hav- ing go and go> as periods is given by fl.O-O . . . ejx-a„) ^ 0t{x — a) .. . e,{x — a„) ‘ 496DOUBL Y-PERIODIC COEFFICIENTS. 49 7 This obviously has ax ,-· . . , am as zeros and a1, . . . , aM as poles. Further, a? is a period ; in fact the exponential is unaltered by the change of x into x -j- go, and each function 0X merely changes its sign to minus—there are, however, an even number of these functions, and so the sign remains unaltered. Again, go' is a period : changing x into x -\-go' reproduces the exponential multiplied by the factor zfjL'iriui' € « ; 6^{x — ax) is changed into — q~xe w * ai> Bx{x — ax), and so for all of the other functions 0X. The whole function is then multiplied by +*1 + . · .+am-ax~. . .-am) _ ^ 27rz> _ ^ Another representation of the doubly-periodic function is due to M. Hermite. Suppose a, b, . . .to be the poles of the function in- side a given parallelogram of periods, and a, j3> . . .to be their respective orders of multiplicity; the doubly-periodic function f(x) is then given by the formula (3) f{x) = A1Z(x-a)-AX'(x-a) + (—Y~1 H----------------rZ(-»>(*· - a) 1 l .2 ... (a — 1) v ’ + B,Z{x -b)- B£\x - b) + . . . H------( ----rZ*~*{x - b) 1 I . 2 . . . (JS — i) V ’ +.............................................. + C, where C is an arbitrary constant, and where the sum of the residues Alt B1 , . . .is zero, viz., (4) ¿, + 2?,+ ...= o. This need not be verified here. Another and very important class of doubly-periodic functions are the doubly-periodic functions of the second kind, as they are called by Hermite. Denote by © such a function with periods go and go' as before. Then © satisfies the rela- tions @(x + go) = sx©(x), @(x + go') = s1'G(x)J498 LINEAR DIFFERENTIAL EQUATIONS. where s, and s/ are constants. It is easy to form such a function as this by aid of the function ^,{x\ and at the same time to let the con- stants 5, and st' take any value we please. Write, viz., (5) ^=‘,JJwr· where p and q are arbitrary constants. We have now (6) G{x + od) = e^Q{x), G(x + go') = e* G(x). Now since p and q are arbitrary constants, we may write (7) e** — sY, , / I 21Tt /co'A-----q e " = s. where s1 and are equally arbitrary constants. Suppose now that among the infinite number of values of the logarithms of s, and s/ we choose arbitrarily two values which we will denote by Lgsx and Lgs/ ; denoting now by m and m' two arbitrary integers, we have, from (7), ipCsD — L gS, + 2Mf7Zt, poat + —= L gs/ + 2 mni, GO from which follow (9) P = Lg^ 2 m'ni q — — -. [û?Lgi/ — (w'Lg-yJ —|— moo — m'go'. Among the systems of values of p and q given by these equa- tions there is one only for which the point q lies inside the parallel- ogram of periods having the origin as one vertex and otherwise determined by the periods go and go! ; this system of values of p and q will be the one employed in all that follows. The function G(x) so formed admits the multipliers s1 and s/ with respect to the periods go and go', and inside the parallelogram of periods considered has the simple zero x — q and the simple pole x — o. The case of q — o must of course be excepted, as in thisDO UBL Y-PERIODIC COEFFICIENTS. 499 case @(x) reduces to the exponential e*x and has no zero or pole in the parallelogram. A doubly-periodic function, which is to be con- sidered presently, has periods go and go' and poles a -j- q, a, by . . . of orders of multiplicity respectively equal to i, a + i — 2, /?, . . . Denoting such a function by we have, from (3), (10) $ikix) =A1Z(x—a)-\-A^Z'(x—d)-\- . · . -\-Aa+i_2Z(a+*'-3)(x—a) +BxZ{x-b)+B,Z\x-b)+ . . . -f- . . . -|— MZ{x—a—q) —}- Cf where A, B, . . . C are constants and (n) A, + Bx + . . . + M = o. This function can be given in terms of elliptic functions by aid of the relation (12) Z (x—a) — Z(x) = sn a sn x sn (x—a) (4-) The truth of (12) is easily seen: in fact both members of the equation have the same periods go and go', the same simple poles o and a, and the same * residues; finally, they are identical for go' x = We can obtain from (12) (or can verify directly) the rela- tion (,» +*■(!)· From this last equation by differentiation we get the values of Z"{x — a), Z"\x-a\ ... Form in the same way the values of Z{x-b), Z'{x — b), ...5oo LINEAR DIFFERENTIAL EQUATIONS. Eliminating now all of these quantities from (io), and denoting by C' the new additive constant, we have (14) $a(x) = Al sn a sn x sn(x — a) sn\x—a) da+£~* i A a-{-£ — 2 dxa+*-4 sn2(^r — a) sn b sn x sn(;r — b) + M- sn (a + t) sn x sn(x — a — q) + C', For the complete determination of the function ^(^r)in any particular case it is only necessary to determine the constants. C. One more preliminary remark may be made before proceeding directly to the investigation of the integrals of the given differential equation. The function Z(x) satisfies the relations (15) Z{x 4- go) = Z(x), Z{x-\-a>') =-----------h Z(x). œ Form now the function (16) mx -f- m!Z{x — a), — M* = fKf* — !) I . 2 — 1) · · · (M — n + 1) 1.2 ... « M» = .., _ /(/- 0 A “ 1.2 .......... , _ /(/ — 0 · · · O' — » + x) Mk ~ I .2 .... « ’ we have obviously <21) j » = O + O^O — i) · · . O — « + 2) * 1.2...» _ /> — I) · . . o — n + 1) 1.2...» Af}xn — o, . ¿Mn' = O, = M'»-1 , a series of relations similar to those obtained in Chapter III (equa- tions 84-86) for the functions 61, , . . . , 0k, . . . . Every poly- nomial, integral in both y and //, can, considered as a function of y9 be written in one way only in the form A oMo, + ^iMi + · · · > where the coefficients A are polynomials in y' having the form Ai = Bi0y0' B{1y/ + · · · > Bi0, . . . being constants. Any such polynomial then, say 77, can in one way only be placed in the form II = 2 Baa, y ay'a>. <22)502 LINEAR DIFFERENTIAL EQUATIONS. By aid of (21) we derive at once (23) All = 2 Baa'fa—il^a' , a, a' A'TI = 2Baa.'fa^a>-l Consider for a moment the polynomials (24) l A xk — ^ Xaa> f** a'y B'Tk— a a' = A. — I — ly which are of the same form as U. We have at once (25) r a/ -vr* / " ^ X*oS fta ft a' —1 9 a, a' ^ t-ff Im ^ t f dbi — 2 X aa' fta-xft a'· The expressions for JHu and A'S'Tk are not needed; their forms are, however, obvious. The expressions in (25) are again polynomi- als in fx and ft!. Suppose the condition (26) A'BTk = jb'5 is to be satisfied; from (25) we must then have (27) Xa— 1, a' = X a, a'-i · We can now determine a polynomial of order A. — l in ft and ft'r say (28) n% = 2 Baa’fafii'*’, « + «'=/, a, a' such that its variations ' ^5* 2 Baa't*a-rf*'a' , i, a' 2 Baa' /^a a'—i » (29)DO UBL Y-PERIODIC COEFFICIENTS. 503 shall be respectively equal to S**, H'i*; for, in order that these re- lations may hold we must have (30) Baat —— Xa—i, a' y B aaj = X a,a' — i · Now if the functions 3, S' have been determined in such a way that equation (26) is satisfied, it follows immediately from equation (27) that equations (30) can also be satisfied. We will take up now the investigation of the integrals of the linear differential equation with uniform doubly-periodic coefficients. We have seen in the case of the linear differential equations already studied that there always exists at least one integral of the equa- tion which, when the variable travels round a critical point, is changed into itself multiplied by a constant factor, the factor being a root of the characteristic equation corresponding to the particular critical point; or, in other words, the effect of imposing upon this integral the substitution corresponding to the critical point considered is to multiply the integral by a constant which is a root of the character- istic equation of the substitution. The uniform coefficients of the equation are unaltered when the variable turns round the critical point. In the equation with uniform doubly-periodic coefficients these coefficients are equally unaltered when the variable x is changed into x -f· go or into x -f- go'. The question now naturally arises in this case as to whether or not the equation possesses an integral which is multiplied by a certain constant, say s, when x changes into x + coy and by a certain constant, say /, when x changes into x -f- 00'. Corresponding to the changes of x into x -(- go and x -f- go respectively, we have the substitutions 5 and S'; and so, if such an integral as we have described exists, we might expect by analogy to find the multipliers s and s' as roots of the characteristic equations corresponding to the substitutions .S and S'. Picard has shown that every linear differential equation with uniform doubly-periodic co- efficients and possessing only uniform integrals has always at least one integral which is a doubly-periodic function of the second kind whose multipliers s and s' are roots of the characteristic equations corresponding to the substitutions S and S'. That is, the equation always has at least one integral such that when x is changed into x -f- go the integral is multiplied by s, and when x is changed into504 LINEAR DIFFERENTIAL EQUATIONS. x + go' the integral is multiplied by s'.* It has been supposed that the integrals of the differential equation are all uniform ; whether or not the integrals possess this property can always be ascertained by developing them in the form of series. In what follows we will assume that the integrals always satisfy the condition of being uni- form. We have seen in Chapter III that it is possible to determine a system of fundamental integrals jynf · · · > y^\y · · · > y 21^9 · · · > · · · > y\i\9 (3 0 -S’il) · · · 9 ) Z21 9 · · · > ^2»J> * · · 9 £(119 · · · 9 such that the substitutions 5 and S' shall take the canonical forms (32) 5 = Vlk9 · 9 y ik 9 · · * j ; s, yI*, Zlk 9 · . . 9 Zik 9 · · * ! > 9 S10'.·*+ Yik)y · · S*(zik + Zik)i · · (33) S' = y iky ··· 9 ytky · · · 9 $iyikj ··· 9 sx (yik —j— r«),. . . Z\k 9 · · · 9 Zik y · · · y *^2 Zik 9 · · · 9 *^2 iffik I ^ ik) 9 · · · where sl9 si9 . . . are distinct roots of the characteristic equation cor- responding to the substitution S, and 5/, s^', . . . are distinct roots of the characteristic equation corresponding to S', and where Yik, Y'ik are linear functions of the integrals y whose first suffix is less than i9 etc. The integrals now being supposed to be chosen so that (32) and (33) are satisfied, let us consider the class y119 . . . , yik9 ... If now we change x into x œ and x + od'9 we have the partial sub- stitutions (34) liki · · · ; s/y,*, · · ·. s,'(yik + Y'ik),... |, * Picard: Sur les équations différentielles linéaij-es à coefficients doublement périodiques ^ Crelle, vol. 90, p. 281. See also Floquet: Sur les équations différentielles linéaires à coefficients doublement périodiques. Annales de l’Ecole Normale Supérieure, 1884. The student should also consult Hermite: Sur quelques applications des Fonctions Ellip- tiques. Gauthier-Villars, Paris, 1885.DO UBL Y-PERIODIC COEFFICIENTS. 5°S where, by hypothesis, (36) Yik = 2al7kylm, Y'ik = 2 b‘7kylm, l, m l, m where the summations extend to all values of the first index / which are less than i, and to the corresponding values of the second index m. Suppose, for example, / = vy where v is less than /; then the corresponding values of m will be, from (31), m = 1, 2, . . .4. Since we have the relation SS' = S'S, we must also have, from <34) and (35), <37) co-' = m>, the system of relationsLINEAR DIFFERENTIAL EQUATIONS. 5°6 the summations extending over all values of / which are less than i and greater than and over the corresponding values of m. We will now proceed to form functions y a > · · · > y*k > · · · which submit to the substitutions (34) and (35), satisfying the con- dition (37) or, what is the same thing, the system of conditions (41). Denoting by a a constant, and recalling the definition given above of the functions ©, we have (42) @(x — a -f- go) = s1 @(x — a), Q(x — a -f- od') = sl'®{x — a). Write now (43) >.* = — a)&,k, .............................. yik = @(x - «)[*« + 2 If- 9tJ. v l, m In these equations the functions 77** are determinate polynomials of degree i — / in jj. and pi' ; the functions are ordinary doubly- periodic functions with periods gj and go', i.e., ${x —I- œ) = where the coefficients p are uniform and doubly periodic, having g> and go' for periods. From equations (42) we see that all of the functions y in (43) whose first suffix is unity satisfy the conditions of (34) and (35). To investigate the remaining ones we will write (44) y* = ®(x — a)zit.DO UBL Y-PERIODIC COEFFICIENTS. 5 or Now since the change of x into x -f- go multiplies Q(x — a) by ,. and the change of x into.# -f- g/ multiplies this same function by si, it is clear, from (34) and (35), that these changes must cause the functions z to submit to the substitutions (45) ^ I ^ikf ··· 9 %ik 9 · · * 1 %lk 9 · * · 9 %ik ” I ^ik 9 ···(>■ (4^) I | Z, . . . , Zfo , · · · 9 %ik 9 * · * 9 %ik 1 Z ¿fc , . . · [ , where the functions Zik, Zrik are deduced from the functions Yik, Yfik. of (34) and (35) by replacing in the latter the functions y by the new functions z. We have of course, from (45) and (46), the relation (47) rr' = r'r. As all the functions z whose first suffix is unity remain unaltered by the substitutions r and t', they are ordinary doubly-periodic functions, and we can therefore write (48) *1* = ; and consequently, so far as the functions ylk are concerned, all the re- quired conditions are satisfied. Let us assume that we have succeeded step by step in constructing all of the functions z whose first suffix is less than A, and that the general form of these functions is given by (49) ** = 2 n%$lm {i < A). a, a' We have now to construct the functions Z\k such that (5°) Zkkiy -)-&?) = Z\k -f- Z\k9 -f- gX) — Z\k “f~ Z\k · The functions Z and Zf being linear functions of the z s whose first suffix is less than A, we have, by hypothesis, (51) z» = 2z'„ = 2 g'S#*, l, m l, m where anc^ S' are polynomials of order X — 1 — / in and // and depend linearly on the coefficients of ZXk and Z'm, and where the summations extend to all systems of values of l and m for which / < A.508 LINEAR DIFFERENTIAL EQUATIONS. The substitutions rr' and r'r applied to z^ give ( tr'zxk = Zhk + Zhk + Z\k + d'Zkkj | Tfrzkk = Z\k + ZKk + Zrkk + §Zrxk\ SrZkk denoting the increment received by ZKk from the substitution t', and dZ'xk denoting the increment received by Zf\k from the substitution r. Now since rr' = r'r, we must have <53) ô'Zm = ôZ'v We have to bear in mind now that we have to form functions in general which are such that the results of the substitutions r and r' shall respectively be equal to the results arising from changing x into x + oo and x + co'. Now by hypothesis these equalities hold for the zik{i < A) already constructed, and consequently they hold for Zxki Z\ky which are linear functions of the same. It follows then that (54) rzu = J'Zu = l, m ' âZ'» = AZ'u = 2AM,5*im. (Since 0 is an ordinary doubly-periodic function, we have A

n and © = G(x — a), = e*[x~a) v v J Qx\x — a) The logarithmic derivative of © is (65) ^ =p + Z(x — a — q) — Z{x - a), or, from equation (12), ©' sn a tool (66) — = p + -7---------r—f-----------X + Z ( — v 7 © sn(^r — a)sn(x —a—q) \2 Since go and go' are the periods of the elliptic function sn t, the second member of this equation is a doubly-periodic function which we will denote by 12, that is, (g7) /+- sn q sn (x — a) sn(;r ------\ + Z(: —« - 9) ' too 2 or (68) From this last equation we have (69) ©' = ©n, and consequently (70) + ©$' = ®\n$ + //2 — 0^= ©j.Q[ii90+fin-x\n$ + *'\ +A-A LO· + £>'] * + 2X2^' + f +------ The coefficients pt are doubly-periodic functions which by hypothesis have the points a, b, . . .as poles of orders of multiplicity at most = /, and so from (14) they are of the form (72) p,=A‘ sn a + B‘ snxsn(x—a) sn b -A, snxsn(x — b) -B, s vl(x—o) I -...-A} dl~2 sn \x—a) dxl~2 sna(^r— a) • · · + C\ In equation (14) we have a similar form for 0 which is linear and homogeneous in the as yet unknown constant coefficients At, A, Bif B2) M, C. From (14) we deduce by differentiation the values of &"9 . . . ^ Finally, if we write, for brevity, (73) + we have (74) D. — p’ sn q 1 sn (x—a) sn (x — a — q)’ Z' (x — a — q) — Z'(x — a), I i + sn2(^r — a — q) sn2(^r — a) and from this last equation we derive at once the values of nr nf Substituting these values in (71), we obtain the final expression for the doubly-periodic function A. If now jn is an integral of the equation, A must vanish, and by a known theorem A will vanish if we can show that it has more zeros than it has poles.DOUBL Y-PERIODIC COEFFICIENTS,. 513 # The function ƒ n has the points a, b, ... as poles with orders of dxy multiplicity at most equal to ar, ¡3, . . ., and consequently has these same points as poles of orders of multiplicity at most equal to diy „ a i: /3 i9 ... . If now we multiply r by the coefficient pn-i > we obtain as the product an expression which has the points a, b, ... as poles of orders of multiplicity at most equal to a n, § -f- .... It follows then that OA, which is a sum of such expressions, pos- sesses the same property. Further, 0 never has more than one pole and one zero, and so the total number of poles of A, when we take account of their orders of multiplicity, is at most equal to «+»+ /* +«+ · · · · In order then that A may vanish identically it is only necessary to show that it admits of arbitrarily chosen zeros the sum of whose orders of multiplicity is greater than or, in other words, if the sum of the orders of multiplicity of the poles of A is less than d, there must exist . . . — d + 1 zeros. The system of equations so obtained is generally superabundant, but we nevertheless know à priori that they admit of solutions. These equations are linear and homogeneous with respect to the constants A,, A,, . . . , BB„ . . . , M, C of 0. If now we eliminate these constants, which may be done in different ways, we obtain algebraic relations connecting p\ sn q and its derivative cn^dn^. To each such system of values of pr and q corresponds for p a value (75)514 LINEAR DIFFERENTIAL EQUATIONS. and for sx and sx the values · (76) i, = e*·, st'= eM + *^T. Substituting the values of p' and q in the equations of condition, these will determine certain of the coefficients Ax9 Ait . . . , Bx9 B%9 . . . , M9 C' in terms of the remaining ones, and if these remaining ones are v in number, we shall have v particular integrals yxl, yI2, . . . , yivt which are doubly-periodic functions of the second kind admitting sx and sx as multipliers. We will thus have obtained all of the integrals which are doubly-periodic functions of the second kind. To another system of values of pf and q correspond other values of the mul- tipliers, say s9 and etc. We determine then in the above manner all the pairs •*,,■*1, *^<2 > *^a » *^$» *^s » · · · of multipliers and the corresponding integrals J^n y · * · y y ik y · · · y Zlx y . . . , , . . . , which are doubly periodic of the second kind. If the number of these integrals is equal to n (which is generally the case), their linear combination will give the general integral; if this number is less than ft, we have still to find some integrals. Let us suppose that we have constructed all of the integrals yik, zik, . . . whose first index is less than X, and that we have determined the corresponding linear functions Yik , Yikfy .... We seek now to determine the integrals yxk (if such exist) and the corresponding functions Y\ky Y\k> · · · · Now we know that (77) ƒ« = ©[$**+2125**.], where, remembering the limits of the summation, everything is known save the indeterminate coefficients A**9 A^y . . . , M**9 C,KkDO UBL Y-PERIODIC COEFFICIEN TS. SIS of and the coefficients of YKki Y\k which enter linearly into the polynomials ZZ**. Substituting the above expression for yxk in the given differen- tial equation, we obtain as the result 0AKk, where Axk is a function such that the sum of the orders of multiplicity of its poles is not greater than . . . . Further, AXk is doubly periodic: to show this it is only necessary to change x into x -f- go ; the result of this change is obviously the same whether it be made in ykk before the substitution in the differential equation or be made in QAkk after the substitution. The result of changing into ^ + go is to change ykk into sx{ykk + Ykk); and since Ykk is a linear function of the integrals y already found, it fol- lows that on substituting sx{ykk -f- Y\k) in the differential equation the result will be s^Am. But 0{x go) = sx0{x), and consequently A\k(x -|- GJ) — A\&(x). A similar result will obviously be obtained if we change x into x + w'* that is, we should find A\k(x -f- go ) = Axk{x)> and therefore AXk is a doubly-periodic function. In order now that Ay& may vanish (as it must if yxk is an integral) it is only necessary to show, in the manner already described, that the sum of the orders of multiplicity of its zeros is greater than ^ ~j~ fi —{— ti —[- .... We will thus determine a system of linear homogeneous equations for the determination of the unknown coefficients. If this system is compatible, there will still exist, and we can determine, new inte- grals yik; if it be not compatible, we will know that the integrals yik of the first class have been all determined, and in order to obtain new integrals we shall have to start with those of the second class,516 LINEAR DIFFERENTIAL EQUATIONS. viz., the integrals ziky etc. This process being continued, we will finally arrive at a system of n linearly independent integrals. Lame's equation — [m(m — i)>fe2snajr -f- h\y — O (m a positive integer, h an arbitrary constant, and k the modulus of the elliptic function sn x) is probably the most important known equation of the type above considered. It is not possible, however,, in the limits of this treatise to investigate this equation, and so the reader is referred to the various papers on the subject by M. Her- mite in the Comptes Rendus of the Academy of Sciences (Paris) in Crelle’s Journal, and particularly to M. Hermite’s treatise “Sur quelques applications des Fonctions Elliptiques" (Paris : Gauthier-Villars, 1885). Two memoirs by Count de Sparre in Voi. Ill of the Acta Mathematica should also be consulted. The common title of de Sparre's two memoirs is “ Sur FEquation d'y dx 2 + ^sn^cnjr , sn x dn x cn x dn x~l dy 2V----;-------\~2V1l---------2Y^--------- \ -f 1 en x en x J dx dn x Lsn x 1 K — OK + ^ + 1) + K — OK + »'. + i) en2 x + & ¿^.K ~ v^n' + v + 0 + &sna x(n + v + + o (n — v — r, — vt -j- 1) -}- lt\y- Équation où vy v1, désignent des nombres quelconques, n, niy n2J nz des nombres entiers positive ou négative, et h une constante arbitraire.” The subject of equations with doubly-periodic coefficients will be resumed later in connection with the invariantive theory ; the reader is here, however, advised to consult Halphen’s “ Mémoire sur la réduction des équations différentielles linéaires aux Formes intégrables" (Savants Etrangères, vol. xxviii), and also Chapter XIII of the second volume of his “ Traité des Fonctions ElliptiquesT END OF VOL. I.