CORNEUL iUNIVERSITV LIBRARIES Mathematics Library Whfte Hal^ 3 1924 059 551 022 DATE DUE CAVLORD rRINTEDINU.S.A. Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed 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 Standard Z39. 48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. 1990. Ml XI Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924059551022 Cornell Hmvevsiitg Jilratg BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF X891 h..k,X.Q-l^ llA/l'^ A PRESENTATION OF THE THEORY OF HERMITE'S FORM OF LAMP'S EQUATION WITH A DETEMINATION OF THE EXPLICIT FORMS IN TERMS OF THE J. FUNCTION FOR THE CASE n EQUAL TO THREE. CANDIDATES THESIS FOB THE DEGKEE OF DOCTOR OF PHILOSOPHY PBESENTED BY J. BRACE CHITTENDEN, A.M., PABKER FELLOW OF HABVAKO UHIV., IHSTBUCIOR IS PBIKCETON COLLEOE. TO THE PHILOSOPHICAL FACULTY OF THE ALBERTUS-UNIVERSITAT OF KONIGSBEEG in PR. PRINTED BY B. G. TEUBNER, LEIPZIG. 1893. en ...iv^fcfli^i:.- DEDICATED TO THE FIEST OF MY MA.NY TEACHERS, MT MOTHER WHO MORE THAN ALL OTHERS HAS RENDERED THE REALI- ZATIONS OF MY STUDENT LIFE POSSIBLE, FOR WHOM NO SACRIFICE HAS BEEN TO GKEAT IN FURTHERING THE INTERESTS OF HER SONS. Introduction. The following thesis is practically a presentation of the general analytical theory of Lame's differential equation of the form known as Hermite's. The underlying principles and also the general solutions are therefore necessarily based upon the original work of M. Hermite, published for the first time in Paris in 1877 in the Comptes rendiis under the title "Sur quelqties applications des fonctions elliptiques" and on a later treatment of the subject by Halphen in his work entitled "Traite des Fonctions Elliptiques et leur applications", Vol. II, Paris 1888. M. Hermite has employed the older Jacobian functions while Halphen has used in every case the Weierstrass p function, and not only the notation but the ultimate forms as well as the complex functions in which they are expressed are in the two works iatirely different. As far as I know, no attempt has before been made to establish the absolute relations of these different functions. In attempting to do this, I have developed the intire theory in a new presentation, working out the results of M. Hermite in terms of the p function, having principly in view a determination of the explicit values of all the forms for the special case n equal to three. I may add that owing to the exceptional privilege granted by the Minister of Education and the Philosophical Faculty of the Albertus-Universitat allovdng the publishing of this thesis in English, 6 Introduction. I am not without hope that this general presentation of the theory of Lame's Functions may prove a welcome addition to the literature of the subject where in English Todhunter's "Lame's and Bessel's Functions" is the only representative. Finally I must acknowledge my indebtedness to Prof Lindemann not only for the direction of a most valuable course of reading but for a general although, owing to a lack of time, a by no means detailed review of the work. Contents. page Introduction . . . 5 Part 1. History and Definitions. The Problem of Lame 11 The Problem of Hermite 13 Definitions 15 Part 2. Hermlte's Integral as a Sum. The Function of the Second Species 17 Transformation of Hermite's Equationv ... 20 Development of the Integral 21 Development of the Eliment of the Function of the Second Species . 23 Determination of the Integral 25 Part 3. The Integral as a Product. Indirect Solution . ... . 28 Solution for 77 = 2 . . . 30 The Product Y of the Two Solutions 32 Direct Solution 37 Determination oi Y ioi n == S 40 Part 4. The Special Functions of Iiame. Functions of the First Sort . . . .42 Functions of the Second Sort . . . ... . . 43 Functions of the Third Sort 44 Part 5. Beduction of the Forms " jt = 3 ". Identity of Solutions . . . . .45 Determination of x and v. First Method . . ... 47 X as function of (P . . 48 Factors of * ... .... 49 Case * = .... 49 Definition of W and ^j(j') as Function of ^P . . ... 50 Definition of % and p' (v) sis Function of j; 51 Reduction of Lamp's Functions = ....'....51 Integral x == ^ ^2 Case D = 52 8 Contents. page Relation of Y and C to the Special Functions of Lamd . . 62 Analytic Form of Y and y . . 63 Condition 0=0. Special Functions of Lam^ 63 Condition P = 0. Functions of First Sort . . 54 Condition Q = 0. Functions of Second Sort .55 Absolute Eelations of Q^ and 0^ . . . 55 Determination of 56 The Integrals (?, = 0, §. = 0, §3 = 0. . . . 66 The Discriminant of F . . . . . . . .57 Resultant of Y and *(«). . . 67 Discriminant in terms of this Resultant . 58 Discriminant in terms of P and Q . . . . .... 68 Special Results, n = 3 . ... .59 Determination of x and v. Second Method . . 60 Reduction of the General Function . 60 Development of *(« ^ 3) . . . ... 62 Development of ^(n ^ 3) . . . . . .64 Development of iE(n = 3) . . . . .65 Reduction of x and v from these Forms . ... 66 General Forms for a", p (*) and js '(*)... . ^ . . . .66 Determination of Forms (n = 3) 68 Reduction to the First Forms .... . . .69 Determination of v. Third Method . . 70 Value of the Constant tj ..... 70 General Form as Product of *i , *a > *3 ■ '^^ The Functions F,, F^, i\ . .72 Forms for p {v) and p' (v) in terms of F^ and *^ . . 72 Relation of F^^^ to x ^i^d the Factors of ;f 73 Reduction to the Forms of M. Hermite . 73 General Discussion . . ... 73 Review of the Theory . 73 General Integral P = 74 Integral Q = 0, v = w^, x = .74 Integral F;; = or j; = , * = toj , a; =(= 74 Case r = . . .76 Functions of M. Mittag-Leffler . . ... ,76 Relation to the Case x = ^ ■ .76 Definition of the Functions . . 75 Determination as a Special Case of the Doubly Periodic Function of the Second Species . . •. . 76 Determination of the Bliment, v = . . . .77 Integral (^ = 0) 78 Table of Forms and Relations (h = 3) . . 79 Thesis. Part I. Historical Development and Definition of the Equation of Lam6. The Problem of Iiame. In order to arrive at an understanding of the highly gener- alized forms that have taken the name of Lame it is adivisable to return for the moment to the original problem of the potential in which they claim a common origin. Lagrange and Laplace (1782) in their researches with respect to the earth regarded as a solid sphere developed the potential function*) which led to the development of the theory of the Kugel- function. From this date until 1839 the only name that need be mentioned is that of Fourier (1822) who, in developing his theory of heat solved the problem with reference to a'right angled cylinder discovering the series named after him. In the following decade**) however Lame***) generalized the work of his predicessors by solving the problem for an ellipsoid with three unequal axes thus laying the foundation for the develop- ment of functions of which the former are but special cases. He used to this end the inductive method arriving at special solutions through a study of the problem already solved with reference to the sphere. The problem of Lame may be stated thus: Let the surface of an ellipsoid be given by the equation m = «„; it is required to find a function T which will satisfy the equation of the potential and which for the value u = u^ ivill reduce to a given *) See note Heine, Handbuch der Kugelfunctionen, p. 2, Berlin 1878, and Heine, 2* vol. Zusatze zum ersten Bande. **) See also reference to Green Heine p. 1. ***) Memoire sur les axes des surfaces isothermes du second degree con- sid^r^s comme des fonctions de la temperature. Journal des Matb^matiques pares et appliques. 1« s&ie. t. IV, p. 103. 1839. 12 Part I. function of v and w, where T is the temperature at a point whose elliptic Coordinates are u, v and w. The working eliments are then, the potential function, generally written w 2£|=» a a or transformed in terms of the p function [2]. . {pv~pu)-^, + {pu—pv)^+{Tpu-pv)^ = the relation, [3] T^f{u)f{v)f{w) and the equation [4] g = [Apu + B]y where y = f(u) and A and B are constants. If T is developed by Maclaurin's theorem with respect to the rectangular coordinates, we may write:*) t^^ T=2; + 7;+T, + ... + T„ + --- where T„ in general is an intire homogenious polynomial of the w* degree, it is observed that each of the functions Tn will also satisfy [1], the equation of the potential, in which case [1] would be an intire homogeneous polynomial of the (m — 2)* degree. This polynomial must be identically zero which will impose -j{n — 1)m linear conditions. The quantities T„ will have in all y (m + 1) (w + 2) constants, which leaves the difference 2n -\- \ equal to the number of constants that may be considered arbitrary. Now the general expression for x^ in terms of p is known to be T being a constant, from which we see that by a change of variable Tn may become an intire homogeneous function of the w'^ degree with respect to the variables [7] Vpu — Ci , Ypu — e^, Ypu — e^ quantities proportional to the axes of the ellipsoid, and of the 1'* degree, pu being of the second and p'u of the third. We have then that T, the function sought, is composed of similar functions T„, where T„ is of the w* degree, is symmetrical *) see Halphen. Vol. II p. 466. Historical Development and Definition of tbe Equation of Lam^. 13 with respect to u, v and w and having 2n -\- \ arbitrary constants^ is capable of satisfying the equation [2] of the potential. From the above relations we derive i^^■ • ■ ■ = fufvfw = ^9^ = [Apu + B]T with corresponding equations for v and w. If then one can find 2n -\- 1 systems of constants A and B of such sort that for each of these systems there exists a solution y^fu of equation [4J where y is an intire function of the n"' degree each of the corresponding products fu fv fw will furnish a term T„ of T and the problem of Lame will he solved. The value of A for all of these systems is n{n -\- 1) where n may be considered as always positive, since the substitution n ~ — (w -f- 1) does not alter the value of A. The Problem of Hermite. Continuing our review we find that one of the original forms of Lame's equation expressed in terms of the Jacobiau function is corresponding to the form [4]*) [10] S? - [^"('^ + 1)J'" + -S]2/ = where h is an arbitrary constant and n a positive whole number. Lame succeeded in finding the requisite number of values of h to complete his solution for the ellipsoid and the solutions of [4] corresponding to these values are known as the original special functions of Lame. The problem then arose: Required to determine a solution of Lame's original equation which shall hold for any values of h and n. Except for the special values w = 1 and w = 2 no advance was made towards a solution until M. Hermite**), making use of the progress in the theory of functions inaugurated by Cauchy, arrived at the solution and by so doing opened a new field for •) See tranefonnation p. 20. **) Sur quelques applications des fonctions elliptiquea. Comptes rendus de I'acad^mie des sciences de Paris. 1877. 14 Part I. .the application of the elliptic functions and leading later to the integration of a large class of differential equations.*) In this connection M. Hermite introduces the functions called by him doubly periodic of the second species, which have the special property, that save for a constant factor they remain unaltered upon the addition to the argument of the fundimental periods. The solution of M. Hermite developed in terms of snu and for n odd may be .v?ritten in the form [11]. • y^F{u)=-^,{-'^ + —^,^_^-\--- + K-,f(u) where n = 2v — 1, with a corresponding form for n even, where f(u) is a doubly periodic function of the second species, namely, f(u) = e^(«-'*'>x(M) where . _ E'iO)Hiu + + ¥^ X{u) — ©(„) 0(^) e That this shall be a solution the quantities co and A must be determined to correspond with definite conditions and herein lies the chief difficulty when explicit values of the functions are sought. Moreover the above development fails as we shall find when seeking to deduce the special functions of M. Mittag-Leffler from the general form. M. Hermite was thus led to a new presentation of the general solution in the form of a product, namely na (u -\- a) „!■„ aaau a form of solution suited to every case. The general theory based upon the latter solution has been lately perfected by Halphen**), who, confining himself in the main to the use of the p function, presents the subject in an excellent but highly condensed form. *) Equations of M. Simile Picard. Comptes rendus, t. XC, p. 128 and 293. — Prof. Fnchs, Ueber eine Claase von Differenzialgleichungen, welche dnrch Abelsche oder elliptische Functionen integrirbar aind. Nachrichten von GSttingen 1878, and Hermite: Annali di Matematica, aerie II, Bd. IX, 1878. **) Traits des Fonctions Elliptiques et leur applications. B. II. Paris 1888. Historical Development and Definition of the Equation of Lam^. 15 Definitions. Returning to form [9] of Lame's equation we observe that it lias the following properties: It has a coefficient w (w 4" 1) l^^sn^x -\- h that is doubly periodic and has only one infinite x = iK' and its congruents, and it is known to have an integral which is a ratio- nal function of the variable. Conforming with these peculiarities M. Mittag-Leffler*) defines the general Hermite's form of Lame's equation of tJie n"' order as a linear Jiomogenious differential equation of the order n having coefficients that are douNy periodic functions, having tJie fundimental periods 2K and 2iK' and everywhere finite save in the point x ==iK' and its congruents which alone are infinite and whose general integral is a rational function of the variable. The general theory of Herrn Fuchs**) then gives the form, namely [12] Y«+^,{x)y^''-'>+-- + 0„(x)y = O where ^aC^) = «o + a^sn^x *s(^) ■■= /3o + ^isn'x + ^,Dlsn^x ^ii^) = yo + yiSn^x + y^DxSn^x -\- y^BHn^x But a better generalization based upon a full representation of the singular points is given by Prof. Klein***) and later stated as follows by Dr. Bocherf). First the ordinary form of the equation of Lame may through transformation become ff) ..„. ^ + -i(-^- + -^— + - l-\ ^ = -^^ + ^ y LJ-'^J daj* ' ^ \x — e, ' x — e^ ' x — ej dx 4 (x — e^) (x — e^) (a; — e^) ^ where the exponents of the zeros e^, e^, Cg are and y and that of the infinites — ~ . From this generalizing by the intro- duction of n zeros we have the following definition: *) Annali di Matematica, tomo XI, 1882. **j Comptes rendus etc. 1880. p. 64. ***) Math. Annal. Bd. 38. t) Deber die Reihentwickelungen der Potentialtheorie. Gottingen 1891. tf) See also transformation p. 20. 16 Part I. Hiatorical Development and Definition of the Equation of Lam^. „Mit dem Namen Lamesche Gleichung ieseichnen wir eine iiberall regulare homogene Differentialgleichtmg eweiter Ordmtng mit rationalen Coefficienten, deren im Endlichen gelegene singulare Punkie e^, Cg • • e" sdmmtlich die Exponenten 0, y besetzen, und in unendlichen nur einen uneigmtlich singularen Punkt aufweist." Lame's equation becomes in accordance with this definition and freed from the possibility of a logarithmic irrationality through a determination of the coefficient of x"~^: rui i^.n^dy L^ -' da;' ~ 2fx dx where fix) = (x — Ci) {x — e^ ■■■{x — e„). It .is further evident that this form, like the Hermite form and as previously developed by Prof. Heine, is but a special case of a general equation of a higher order. In speaking of Lame's equation we will understand an equation c(mforming with the above definition whose general form is given by [14] and, if the order is higiisr than the second, distinguish by mentionr ing the order. Forms [9] and [10] will then be called Hermite's forms of Lame's equation or simply Hermite's equation, where again the order need be mentioned only if it be other than the second. Any solution of any form of Lame's equation will be a function of Lame and if the doubly periodic functions fiLrst deter- mined by Lame are mentioned they will be designated as the special functions of Lame. [15] Part II. Hermite's Integral as a 8am. The Function of the Second Species. We have the problem required the integral y of the equation ■j^^-=[n{n-\-\)¥sn^u + h'\y where h is taken arbitrarily, n is any intire positive nurnber and k is the modulus of the elliptic function. M. Hermite introduces to this end a function which he names doubly periodic of the second species, which may be defined as a product of a quotient composed of a functions, the number of zeros being equal to the number of the infinites, and an exponential, having the property of reproducing itself multiplied by an exponential factor when the variable is increased by tlie periods 2K and 2iK. It is defined then in general by the relations: F{u + 2K) = (iF{u) [16] F(u + 2iK') = il'F(u) ^ ■' a{u — ft,) c(m — ftj) . . . a(u — 6„_i) The factors (i and (i are called Multiplicators. M. Hermite might have been led to the employment of this function by the following analysis which is essentially that given by Halphen.*) Consider for the moment that y be such a function of the second species but having instead of the n different poles but one pole M ^ of the n^ order in which case the function will have n roots. Upon developing the properties of this function one finds that its second derivative has the same multiplicator as the function *) Bd. II p. 495. 18 Part II. itself and that therefore the quotient y" : y will not only be doubly periodic but will have a single pole u^o of the second order. This function then satisfies the necessary conditions and the corresponding quotient - may then be written equal to n(n -j- 1) sn^ x -\-h where h is a constant. But we have taken this function with the condition that it have but one pole of the order n subject to the above conditions which affords n arbitrary constants and employing also an arbitrary constant factor we obtain {n -\- V) arbitraries in all. That is sufficient to satisfy all the conditions and leave A to be chosen at will. Hence we must conclude that there is no reason why y should not be a doubly periodic function of the second species and our problem reduces to the determination of a function whose general form and properties [16] are known. From this standpoint we have: Required a function such that F(u + i2) == iiF{u), a =2K F(u + £1') = (i'F{u), Si' = 2iK'. Define: [17] /•W = ^^T^^"*) which function we will speak of as the Eliment the general form [16] being a product of similar aliments. We have the fundimental relations: 6{u + £l')= - 6(M)e2l'«+ ■)'■«' Whence f(u -\-£l) = A ^±1) ^(.+n)+^,v ^ ' ' e{u) Choosing then v and X correctly we may write ft = ei.f2 + 2.;v *) Hermite, in the following analysis, employs the function given on p. 11, namely the function x expressed in terms of the & function. Hermite's Integral as a Sum. 19 with a corresponding value for ft' and we may then write f{u) ^ ^ where ^ is a doubly periodic function, that is $(« + m£l-{- n£l') = 0{u). Again f{u - a.) =-jfiu) and f{u - PJ) = A /■(«). Whence f{u — s — Sl) = -f{u — g) where F(s + ^) = (iF(z) and we derive [18J 0{z) = F{2)f{u- z) where is doubly periodic. From this point the development of F{u) depends upon the theory of Cauchy, as it is obtained by calculating the residuals of ^ for the values of the argument that render it infinite and equating the sum to zero as follows. First f(u) becomes infinite for the value m ^ whence its residual E^^^fiu) = [MfM]„=o = A — = ^-1^ = A6(y) and becomes equal to unity if we take e{y) Whence [19] /•(„)= ^(?L±lle^«. '-■' J ' '^ J a{u)a{v) Again E^H^) = ,^r„ (^ - ^) ^ W = . = . (^ - u)F{i)f{u - z) and developing f(u — s) we have E,9{s) = - F{u) Again let a be any pole of F{u) in which case, developing by the function theory, we may write F{a + £).=o = Ab-^ + ^,Z),£-i + A^BIe-' + • • -\- AaD"e-^ + «o + "i^ + «2«" + • ■ 20 Part II. and f{u — a — s) = f(u — a) — \Duf{u — a) -\-^Blf{u — a) where We have then -Ba <& = /^o *-^(« + *)/(« - « - = Af{u -a) + A^Bufiu - a) + A^Dlf(u - a) -\ + ^„D"/(m — a) with similar expressions for E/,, JEc . . . But $ being a doubly periodic function we know that the sum of its residuals with respect to m, a, 6 . . equals zero whence [20] F{u) = 2* Un^ -a) + A, Bufiu - a) + A^Dlf(ti - a) + ■ ■ + A^Blfiu - a)] where Aj is determined from the first development. This important formula still further narrows our problem to a consideration of f{ti) in terms of which and its derivatives under conditions to be determined it is now evident that y = F^ (u) may be expressed. Transformation of Hermite's Equation. We have written Hermite's equation in its original form [21] g = [n(« + l)Ps«^a; + ;t]. That this is however but a special case of a more general form is seen as follows. Take the integral _ r dx _ hdx J ]/(i - x^{i - kn^) J yA We have dy dy dx dy 1 dX dx dX dx y2 or dx ^ dX Hermite's Integral as a Sum. 21 whence d^y d^y 1 x A' dy dX' dx^ A i Adx or da;« "~ dX* "•" 2 ^■' dX' Substituting we derive the ordinary form of Lame's equation [22]. . . .A^^, + \A'^^-in{n + l)¥sn'x + h-\ = 0*) The value of A gives as singular points + 1 i i T ^^"^ °®- For our present purpose however we need the equation ex- pressed in terms of u and pu which is derived from (21) by means of the relations and making the substitutions: we obtain: Define [23] JB = /i(ei - 63) - n(w + 1)^3 . Whence our equation may be written: [24] y"=[n{n+l)pu + B]y. Development of the Integral. We observe, since snx reduces to zero only for the value a; == 0, that we have but one pole of the second order in Hermite's equation and that we may therefore develop y within a cercle whose radius is less than SI', the form being y = M' [2/0 + 2/iM + 2^2"^ -\ ] whence y' = vu—% + (v + l) y^u' + (v + 2)M'+i2/j + {v + 3)m''+22/3 + .. y" = v{v-l) u^-'y^ ^v{v+l) y,u^-' + (v + 1) (^ + 2) u^y, + • ■ *) Compair general form [14] p. 16. 22 Part II. We have also P («) = -i+ Ci «^ + CgM*. + • • These values in [24] give: v{v—l) y,u^-^ -\ = n(n+ l)2/o"'~' H whence: v(v — 1)=m(m+1) or v= — n. This value gives since the uneven powers fall out from which we again derive y"= n{n + 1) -A_+ (^ _ 2)(« _ l)-\ + (^ - 4)(« - 3)-^, + („_6)(w-5)-^,, + ..- + (n-2i)(n-2* + l)— ^V+2 " 1 h^ \ h- -\ n ' 71 — 2 ' n — 4 ' * ' * "I m — 2i "• _M M W M J [" (" + ij (^ + c,«^ + cX + ...) + 5] 1 7j, h.n (n + 1) 1 B7ii Bh^ Bh. , +n(w+ l)q^^ + w(w+ l)ci/t, -^H + W(W + 1) Ci/»,- „_^,,,3 H h « (m + 1) C2 -^ H Equating the coefficients of like powers of u in this identity one finds — 1 (w - 2)(w — l)/ii = n(n +\)h^+B _1_ |(„ _ 4)(,j _ 3)/,^ = ^(^ ^. i)[7,^ + cj + h,B u -^::::j | (w — 6) (w — 5) h^ = w (n + 1) (feg + c^h, + cj + fcjB Hermite's Integral as a Sam. 23 1 (« - 8) (n - 7) ^, = n {n + 1) Qi^ + c^h, + qh, + Cg) + /^s^ -^=27418 I ('*~2*)(**—2i+l)7<,= w(M + !)[/*, + Ci/<,_2 + C2/i;_3 -I Whence we obtain all the coefficients in the development for y by means of the recurring formula. [26] 2 J (2 i — 2n — 1) h = « (w + I) [(hh-2 + Cs/j.-s H + Ci-2hi + C;_iJ + hi-iB. Since then fe, is determined we have when n is even and equal to 2v 1 ^1 '')■ — 1 2/ = —r + -^i;i=2 -^ \ f- + ^»- and if n be odd and equal to 2v — 1 1 fil '*v — 1 2/ = ^V^ + "2r-2 -I H -T" + ''^" M it " where lii is given by (26). Development of the Eliment f{u). Having now a development of y we can, if we develop f{ti) and substitute in the development of F^u), find by comparison the conditions necessary that y = F^ (u) be a solution. We have then to determine the development of since [«/"(w)]a=o = 1 To this end we develop first the function [27J 9'W = /'We^^+^^"'"=^'^«-"^^- We have: pu = — = — -— . i — = — -4- c.u' 4- CjM* + ■^ du du Qu u- ' '^ ' " ' whence GU U 3 1 5 ^ 7 a 24 Part n. By Taylor's theorem: d-{v) , d» - {v) = t{v) — up (v) — :j-^p (V) - YTY^P W • • ■ Passing now to logarithms we derive: -47^ W-Flb' W-5J Integrating we have: log 9) = — log M + ^2 |y + ^3 |y + -^4 ly H whence LJ8J 9) = - e -i[i+p,j;+p.i;+p4+..] where Pj = ^j = - ij(v) ; P3 = ^8 == — p'W; P4 = - 3/(f) + Ui = A+^A' Pj = — Spvp'v = A^ -\- 10 A^A^ etc. showing that the coefficients Pi are intire functions of pr andp'v.*) *) The functions Pi correspond to the functions SI in Hermite's treatis, for example 1 4- »'■' P2 = — p(y) = SI = k'sw'm ^ — P3 = — J3'm = Bj = Jt'STOW cnM dwM see p. 126 development of %■ Eermite'g Integral as a Sum. 25 From these forms we pass immediately to [29] /•(«) = (m)[i + (A + tv) + (A + tvf i^ + • • -J = i j[l + (A + §w)M + (A + (^ + tuf) ^] + [Ps + 3P, (A + g«) + (A + gw)'] n^ + • • •) = i + flo +B^,M + H,u^-\-Hy + • ■ • Take A == a; — gv whence [30] . . ^(„) = ^+;^^e(-«^)" ■- -" ' \ ^ e (m) o (v) 1 I 1 / 9 1 -n N M = ^ + X + (^^ + P^) I + (:c« + 3P,a; + P3) : /» + (o;^ + 6 P, a.^ + 4 P3 a. + PJ ^-^ + • ■ • = i + J3o + ITi (m) + ^^ {uf + ^3M^ Where in Hermite's Notation [31] ^, = i-(23 + 3P,x+P3) ^3 = ^ (^* + 6P,a;^ + 4P3a; + P,) Detenuinatiou of the Integral. We are now enabled to determine the exacb expression for F(u) and the conditions necessary that it become equal to y by a process of comparison of the several developments obtained. 26 Part n. First we have: f{u) = i + So + ^iM + HiU^ -\ \-Hiu'-\ f'(u) = — ^ + fii + 2S2M + ^HiU' -I 1- iHiU'-^ H f"{tC) = + A _|_ 2B2 + 2 • SlTsM H 1- i (i — 1) HiU*~^ + i{i — 1) {i — 2) Hiu'-^ H C:ai; = + ^-^ + 2-3...(n-l)5„_. + ... + i(i— !)• •■(«■ — «+ l)fl;M'-»+i + --- Again 1 \ ^v — \ yn=i^-i = ;j^27zri + ^^^7^ H 1- -^ + A.M 1 ''1 ^v — l yn=ir =-rr + 2v-2 H H T + ^*''- And in general y = FiU = J„/-W + ^a-l/^'"-" +•■ + /■ = ^^Z-t"-!) + ^^-aff"-') H h /" (w odd). Now substituting the values f^"^ found above and ordering the coefficients so that the residual with respect to u will be unity we find by comparison that we may write [32J ■ -y-F^ («) = j^ ^(-») + ^^ A./'c-B) + . . . K_,f (71 odd and =2r — 1) provided x and v be so taken that the constant term equal zero and the coefficient of the next term equal K and 166 \ y -^aW (n-1)!' (w — 3)!' (n — 6)!' hy if (" eyen and = 2 r) provided x and v be so taken that the constant term equal h„ and the coefficient of the next term equal zero or in general [34] (- ly-^y = ^ /•("->) + (^ /./("-»> + ^ hfi^-^^ + . . . (n odd) Hermite's Integral as a Sum. 27 where the last terms are obtained to accord with the above con- ditions. Substituting the values /"<"> we find the conditions to be IH2V-2 + I^Hiy-i + hiffir-e + • • • + K-iHo = [35] (2V - l)ir2,_i + (2v - 3);JiJ32r-3 + (21/ - 0)hH2r-i -j (..even) ^2 — 1 + ^^H^v-Z + k^H^r-, + (" K-lH, + K = These conditions being satisfied y = jF'(«) and we have two forms If F{u) — [n (n + l)pu -\-B]F{u) = since finite for u = ik' = — • A second solution being likewise obtained by making the sub- stitution w ~ — w the general integral may be written: [37] y=^cF{u) + c'F{—u). Part III. Integral as a Product. Indirect Solution. It will be shown in developing the forms for the case « = 3 that the original solution of M. Hermite as a sum will not be applicable in the forms given in the last chapter, when B is so taken as to give a value, v equal to zero, which leads to a second development in the form of a product, the eliments being as in the first case doubly periodic functions of the second species. Assume that [38] y^^ffJ^fi^uta where the product is composed of n factors obtained by taking a, 1), c in place of a. The derivative of the logarithm is f =2'[s(- + «) - ? w - m =2^j^-i^ while a second differentiation gives From the first equation f— T= ^^— / P'" ~ p'^ Y [ i_ '^ pu — p'a p'u — p'b lyj ^J * \pu — pa) ' 2 ^j pu — pa ' pu — pb ' But the addition theorem gives: 1 p u — p a , . , , , whence y x- ' ,^; -f I 2,^ pu — pa pu — pb Integral as a Product. 29 In order to decompose the last term in this expression we write: 1 p'u — p'a p'u—p'b „, . 1 7\ + ^r=jF [S (« + «)- g (« + fc) - 5»* + m- Take the value m = (a + 6), remembering the relations Writing then f{a + 6) for the right hand member of the above equation under these conditions we get f(a + 6) = 2wjp(a + 6) + 2pa -{- 2p(a + h) -\- pa + pi = 2(n + l)pu+ 2Epa. from which we see that in general we would have ^ = w (» + \)pu + (2w — 1) Upa the quantity in brackets being equal to zero. If now we reunite the terms g(M + a) — Sa, %{u + 6) — gfc etc. in the general expression and make equal to zero the sum of their coefficients we obtain n equations of condition, namely, writing j)a = a; p'a = tt'\ ph ^ ^\ p'b = ^'; f' + §' I «' + y' I i^' + d' ■ _„ y'+«' [ y'+ P' I y'+ t^ _j_ ... == q y — a ' y — § y — fll If then we can solve the equations considered as simultanious a'^ = 4a* — g2a — gs together with the relation (2„ _ 1) (a + ^ + y + . . .) = ^ we will satisfy the necessary conditions to enable us to write: — = m(w + l)pu + B. 30 Part HI. That is 7~T 6(u 4- a) a (u) a (a) is a solution of Hermite's equation whatever be the value of B, provided a, b, c . . fulfil the above conditions. Solution'for w = 2. It is clear that, save for small values of n, an attempt to solve the above equations by the ordinary methods would give rise to insurmountable difficulties. The case w = 2 however, which is famous as affording a solution to the problem of a pendulum, constrained to move upon a sphere, can be readily solved as follows: Given n = 2: we have the conditions o:'2 =p'^a = 4a'' — g^a — g^ [41] r=p"'b = 4^'-g,^-g, pa+pb = ^B or «'2 + /S'^ = 0. Observe that by designating ph by — ^ the above relations remain unaltered and that we may therefore write 5-3 ia'-g,a-g, = -4:fi'+g,^. or A(a^ + n-g,{a + P) = whence 'Rut «'-«^ + /3'-T^2 = 0. ±JlHi ^ = «-|-£ whence the equation that determines the values of S. [42]- • • |B2-|«5 + a^- 1^2 = and also ^ = 0.*) If then n = 2 and a and b, the arguments of a = ( — pa^), are so taken that B shall have the values of the roots of equation (42) Hermite's equation will have the solution *) If in this result we take JB = — f we obtain the formula r-«f + a'-^?, = 0, see Halphen II p. 131. Integral as a Prodnct. 31 [43] y = c— i ffXa-.,b)u du \_ au J dui.au J where v ^ a + &.*) That our solution given above be complete we must obtain the corresponding values of x and v as follows: ^ OM L CM J We have also [441. . . ■ ,,+,a+,i-'^-^±^ = {J^^ since p'a = p'b = a'. Again we have or Hence whence g^-ag + a2-|^, = 5 = f + |y<72-3«^ [45] pa - -pi=V92- -3a^ pa -\- ph = a p'a. = _ 4a3 _|_ 5'2«- ^3' These values in (44) give: • p(v) = — icc^ + g^a - 9i - 3a» -^3 c' + 9. 3a' — g. *) The last is the form given for the expreasion cos CX + tcos CY in the solation of the pendulum problem in the direct investigation of which one arrives at the expressions d'X ,^^ dT ^^^ d'z ^^„ , where N is found to be 3r'(2pM — po,) which causes the solution to depend upon Lamp's functions. 32 Part III. If we take a = 2b we have where 95 = 463 — g^j, _ g^ and q)' = 126"^ — g^ For X we have: [47] . = g(a-6) + ga-£6 = l^^f^^±^^ 1 p'(6 — a) — p'(a) p'(6 — <')+p'6 — p' , (?'(<-") , y'(t-") I 20 « -r (f _ p) -r t _ y -r (« _ p) (« _ y) . . . whence making < = a we have 2C [51] («-(J)(«-y)--. and in a similar manner we find P (p_„)(p_y)... *) see theory of p and c functions. 34 Part m. These values of a' and /3'... determine the constants a, h ... provided we can find the value of the constant C. It is also clear that C must be a constant involved in the relation Y=f and we are thus led first to a development of Y according to the powers of t and to the finding of the relation between the coeffi- cients. Thus y becomes available in a practical form and C being determined as a function of T and its derivatives we have our relation in a new form [52] y = ±VT. I expand these principles of M. Hermit e*) (Annali di math.) and Halphen**) as follows: Lame's equation may be written f53] y"= Py where P = n{n+ l)pu + B and y^VY. Seeking the equation in terms of Y we write r= 2yy' whence r'= 2/2+ 2yy"= 2y''+ 2Py'^ 2y''-\-.2P¥, also {y"- 2PT)'= 4y'y"=iPyy'=: 2PY whence [54] r"-4pr'— 2P'r=o a linear differential equation in Y of the third order. From the theory of the linear differential equation, if y and s are solutions of (53) yy -\- qe will also be a solution y and q being arbitrary constants, and we derive also as distinct solutions of the transformed (54) y^, yg and e^ obtained from the complex form P = n{n -{- l)p u and the transformed may be written: [55]- • Y"'-4:[n{n+l)pu + B]Y'—2n(n+l)p'uY=0 where This value indicates that (55) has n solutions in terms of p (m) *) Bd. n. p. 498. **) Bd. II. p. 498. Integral as a Product. 35 from which it follows also that Y may be written as an intire polynomial of the n*'^ degree in t = pu. That is [56] ■ . • . r= <» + oji"-! + a^t"-^ -\ 1- an-it + a„. Equation [55] is written in terms of derivatives with respect to u whence to determine the coefficients in (56) we must express (55) also in terms of derivatives oi t = pu and equate the coefficients of like powers in the two identities thus obtained. Take 9 = 9{t) = 4<^— g^t — g3=p'h( whence 1 3 _5^ 3 and 1 D^Y=DiYDj = (p^D^Y DmB^.Y — D.YD^.u DIY = (D,uf ^^"^ — "^ {D,uy = '3l Y- -\ 9'^ 9"Dc Y These substitutions give: [57] {^f-9,t-g,)^+3(6t'-\9,)^^-A\{n'-\-n-3)t-\-By^ — 2n{n + l)Y=0. From [56] we obtain the values of these derivatives, namely ^ = Mi"-! + a, (w — 1)^-2 + 02 (w - 2) f-^ + ag (w - 3) t"-* -\. ajn — 4) 1?'-^ + ■ ■ ■ ^ = w(« - 1) t—^+a,{n - l)(n — 2) t—^+a,(n-'2)(n-?,)t''-^ -\-a.,(n — S)(n — 4) f-^ -\ ^ = w(w — 1) (w - 2) t"-^ + a, {n -l)(n- 2) (n — ?,) p-* -\- a^ (« — 2) (w — 3) (« — 4) f-5 + ... and equating the coefficients to zero we have: 36 Part 111. w — 3: ia^ in — 3) (« — 4) (w — 5) — g^a^ (n — 1) (w — 2) (« — 3) — g^n (n— 1) (w — 2) + ISog (w — 3) (w — 4) -ig,a,{n-l){n-2)- 4{n' + n-3){n-3)a, — ABa^ (« — 2) — 2n (w + 1) aj = n — 4: 4a(w — 3)(w — 4)(w — 5) — 5'2«2('* — 2)(m — 3)(w — 4) - gTgai (w — 1) (w — 2) {n — 3) + 18a^ (w — 4)(w - 5) - I =« — 1, w — 2,.., or k = 1 .2 ... These results are simplified by employing the notation intro- duced by Brioschi, namely: by means of which the above forms are expressed as follows: [59] [45f' + I 9," S^ + 9,' S + 9,]g: + (l85^ + 1 ,," fi + | ,,') ^ _[4(n^+„_3)S + '^%"]g-2«(« + 1)7=0 [60] Y = S" + A^S^-^ + A^S^-^ -\ \-An Integral as a Product. 37 [61] 2{n~(i){2fi+l){(i+n + 1) A^-, = \2((i + I) {(I + 1 - n) ill + 1 + n) hAn-,-y + |(^ + 1) (^ + 2) (2ft + 3) - I- log^^^^ °1 1 eu du ^^ 3/2 Whence T— 20 iY^ du^^^Ll a{u) ^ = ^ = -^lo'g77^±^>e-»t« or 2r f o (m + a' log 2/ = log/7^^^ e-f" - log C C = n0a. Whence the value of y is obtained directly, namely [661 • . • ■ ■ y - JJ- faOu + a) „f„ The third method of integration is then the following: Calculate the polynomial Y by the aid of the relation [56] or [61] from which derive the Constant C^ by means of equation [64] extracting tlie square root to obtain C and finally obtain the constants r 2 G / -m 2 ivhen a = ya, B = yb . . . are the roots of Y. These relations determine the arguments a ■ b • c . . ., having which tlie solution is If we take the second root of C^ we obtain the integral s obtained also from y by changing u into — u. 40 Part III. Determination of Y for w = 3. The foregoing solution while complete and rigid from a theo- retical standpoint needs to be greatly perfected before it becomes practically applicable. It is indeed but another example, the in- variant theory being a second of the fact that it is often an easier task to obtain a general than an explicit form. Having determined the explicit forms for w === 2 let us attempt to apply the above rule to the next case w = 3. From (60) and (61) we obtain. Given n = 3 Yn=3 = S' + A,S+A, where ==-T(446'-%2* + ^a)- Again S = t — 6 = 4S' + 126S« + 12b^S + W — g-S - hg^ — g^ = iS' + 1265^ + (126^ -g,)S+ 4¥ - Ig, - g, = 4^" + 126^2 + tp'S + + • • = By that is a function of the first sort will be a root of Hermite's equation provided fli — A^ = B : 02 — -^2 = Ba^ : aj — A.^== ^Og etc. Where the quantities (A) are linear functions of the quantities (a). But since the number of these condition equations is greater by unity than the number of unknown (a) it follows that upon their ellimination we obtain an equation in B whose degree will equal the number of equations, that is jw + 1 if w is even and y(m — 1) if n is uneven: For example take w = 2, whence y = p -\- a^ and y"= p" and we derive p" — 6(p + a^p = Bp — Ba^ or Ba^ + -3-^2 = 0, also Qn^ -j_ ^ = The Special Punctions of Lame. • 43 whence and we find y=p — jB where B'^ — 3g^ = 0. Again let n = 3 in which case the equation iu B would be of degree y (w — 1) = 1, that is B = 0, for which value we have at once y=.p'{u). Substituting indeed this value in Hermite's equation for n = 3 we derive at once p'"— I2p'p = a well known identity. Define (P = 0) equal to the equation in B of degree -j {n — 1) that in any case determines the values of B giving rise to an integral of the first sort. We have then that when P == the general solution of Her- mite as a sum has in place of f(u) the p(u) and may be written [70]. ■ . (_l)»y = ^-p("-«)(M) + ^_7»,p('.-*)(M)* + (-n-^5)] ^^^P''-''(--) +■■■ the coefficients being the same as in the corresponding general development.*) Functions of the Second Sort. To attain a function of the second sort assume that n is odd aud that the solution has the form [71] y = Sl/pU — ea a = 1.2.3 where e may be developed in the form s =|)("-3) 4- ttip^"-'"^ + a^p^"-" an equation in p differing from (70) in the degree of the deri- vatives only. Proceeding as in the former case by substituting in Hermite's equation one finds that the solution holds provided B be now taken equal to any one of the roots of a perfectly deter- mined equation of degree j(m-{- 1), the right hand member of which we will define as Qa which is equal to zero. *) see (34) and (26). 44 • Part IV. The Special Functions of Lamd. Writing for conrenience Hermite's equation in terms of the derivatives of g with respect to pu by aid of the identity jp2 = 4p^ — g^p — g^ we Iiave*) [72]. . (4^,3_^^^_^^)^^+(io/ + 4e„i, + 4e^-|^,)fj = [(« -\){n + 2)p-\-B- ea\e. Take now for example w = 3. whence and (72) becomes lOi,^ + 4eap + Ael - If,, = (lOp + B - e.) (p -^ a,) and differentiating we have 4ea = lOa^ -\- B — e,. or whence 1 1 T) ai = yea — jgii and [73] <2„ = B2-6e„B + 45e^-15^, = an equation whose degree is 4-(«+l) = 2, and as a may have the values 1, 2 or 3 we have in all six values of B giving a doubly periodic solution of the second sort and determined by an equation of the sixth degree defined as [74] Q=Q,Q2Qz- Functions of the Third Sort. We have finally solutions that are doubly periodic of a third sort the integral being written in the form: • [75] y==sy(pu — efi)(jati — ea) where w is restricted to an even member and s has the form and a similar analysis to the former cases shows that this solution holds when B is the root of a determinate equation whose degree 18 jn. *) compair transformation p. 36. Part V. Reduction of the Forms when n equals tliree. Identity of Solutions. Having developed in the foregoing the necessary underlying principles we return to the case where n equals three, that is to a determination of the integral of the equation [76] • y"=\\2p{u) + B^y where B is to be arbitrarily chosen. The first form obtain from (32) is y-y + Kf and from the first of equations (26) we have \= — -T ' where B = 15fo Hence disregarding the constant the integral is [77] y=.f'-Uf where and X and v satisfy the conditions (35) •- -■ 13^3+^1^1=^2 where x = t,v — ia — t^ — Ic , * , (p. 17 and p. 16.) V = a -\- b -\- c. a o o 02 o CZ2 46 • Part V. The second form .obtained from (66) is [791. . . «=7T-^"-"t-^e-"f« = rT-'^'*— — ^'-> I'yj y 11 aacu " lle{a)B{v)^ ^ c (M - g) c(M — &) c(M - c) ^(^^„ ^ f j^ _.^) „ (7(a) c(?>) (t(c) c^oo where «=i'(«)=(„_p) („_,)- ^ and C = + Vy"; r= S' + ^aS + ^3 (p. 40.) The transformation of form (79) to form (77) may be accom- plished as follows. Taking the eliments we have c (m + o) .,►„ 1 u . a(u)ab ^ u 2 ■'^ ^ e(M + c) „-. 1 u . whence _ e{ u + a) (s(u + b) a(u + c) g_,^^„+5j+^^,)„ ^ 0(a) <;(6) o(c) ©"it = [^-l(^«+i'^)+J(i-X«)+) Take /■= gfr + «jf j_+iO e-„(ta+C*-l-M = g(« + '') c<—^-'>" ' o(a + * + c) oM auev = ^ - Y (2>« + IJ?" + Pc) H r= - ^ — I (i^o +pb-\-pc)-\ f"= L 4- . . . Whence we observe that we may write 2/ = I Cr'M — (pa + P?* + P^) fill But Reduction of the Forms when » equals three. 47 and, disregarding the factor j, we obtain the first form: y = f"-Sif. Having then a method of reduction the determination of a:h:c is involved in the determinate of v. Determmation of x and v. First Method. To this end v^e have from (31) and (26) n, = x-, B, = I (a;^ + P,); H, = \ {x' + iP,x + P,) and also ^*i = - lo "'^ ~ 120 20 whence relations (78) become I {x' + 6P,x' +^Ps^ + PJ - T (^' + ^2) = It' - ^,^ [80] set I = I B or /*, = — - and take from (p. 24) P, = pv- P3 = - iJT/; P4 = - Sp'v + li/. Which values reduce our relations to the form (a) (aj' — 3p(v)x — p'{v) — 3lx = (b) Lb* - 6p{v)x^ — 4p' {v)x - 3/ (v) — 2Z + 2Zi) (v) = ^ — ^/^ which are reduced forms of the equations of condition that y = Fi (x) be a solution in addition to which we have the identity p'^vf = 4p\v) — g^p{v) — g^ and the useful relation ^, = I (^' - P(y)) , or p {v) = x'- 2H,. The product of equations (80) is an equation of the seventh degree in x the roots of which are functions of v and B and hence the values of S that will reduce x to zero are in number not more than seven. But when x equals zero (and v = wx), y is in general a doubly periodic function and the doubly periodic special functions of Lame 48 Part V. are in all seven in number for n equals three one being of the first sort and six of the second. It follows then that by elliminating p(v) and p'(v), we should obtain a; as a function of ^ where ^ is a function of B the vanishing of which will be the condition for the special functions of Lame. This complicated ellimination , suggesting the practical use- lesness of . this method for any higher value of n is performed as follows. Multiplying the first equation by four and subtracting we obtain 3a)* - 6p{v)x^ — lOlx^ — 2lp{v) + 3p\v) = — -' + g, whence the relation p{v) = a;2 — 2ifj gives (c) 36Hl — 36bx^ + 12lH,+5P-3g,=^(). Again from (6) and the identity p'(vf = (36a; + 3p{v)x — x^ = UH^ + Qp^{v)x^ + ^r" + \Up{v)x^ — G&a;'' — Qp{y)x* = 'dVx^ + 9a;2(a;* — 4.x^Hi + 4Si) + x" + \Ux^{x^ — 2H^) — 66a;* — 6a:*(a:2 — 2fl;) = 4(a;« - 6a;*^, + 12x^H\ - 8i?0 - 9i{x^ - ^S^) - g, or multiplying by 9 (d) BlZ^a;^ - 108x^ Hi + 108 ia;* — 9 • miH.x^ + 9 • 32 IT? + 9g,x? - 18g,H, + 9g, = 0. From (a), (b) and the value for p{v) oi^-6x\x^-2H^)-4:x{x'-dp{v)x-Sbx)-S{a^-4:X^H^-^4:Hl) -2lx' + 2l{x'-2H,) = ^-^-g, or (e) I2lx^ — 12 H^ - 4b H^ = ^' - g, and multiplying (e) by 3 and 85, it becomes (f) 36 ■ 8Za;''H, - 36 • 8JS? - 96lHl = 40/ If, - 24^^^, whence from (c) elliminating H\ (g) 8HV _ 1083'Hl + lOSlx' - miHy - 9&IH\ = mVH, — e^^jiT, — 9g^x^ — 9g^. Reduction of the Forms when n equals three. 49 Whence a further combination with (c) gives (h) 12Px^' - 121H\ — 32PHi + 6g^H^ + ^' + 9^3 - ^^92 = and again (i) 8PH, Whence [81] I - 65r,^, 40 i^ 3 - ^ffz + 8i^2 F,= 10?^ - ( •7 j^ 2? gIp -!^.) lOl' — i iaj-b. ■ 6{r' - a,) a, 4 and , 2Z where From this value of H^ we have by substituting in (c) 12528 — ilOa.l* — 22h.l^ + 93o?Z^ + 18o, 6, ? + 6? — 4a? ^ 4(;'' — a,)^ 4- (U;» — 9a. Z — 6,)' _ *(0 where $(0 = 125Z« — 210a, Z* - 226i?' + OSa^Z^ _f- iSa&i? + &^ — 4al S = 36l, D = {l' — a,), Z = 1^ = 36 0(1) = is then the condition for the existence of the special functions of Lame the seventh value oi B, as we have already . seen (p. 43), being .6 = 0. 0(l) must then be Q(l) times a constant and as we have seen that Q is separable into three factors of the second degree it follows that 0(Z) is a reducable equation of the sixth degree.**) Moreover if we make the transformation *) The expressions used here are essentially the same as those of M. Hermite in his celebrated Memoir. The following redaction of the function (p{l) is also indicated by Hermite. **) It is interesting to note that it is not given under the head of reducable forms of the sixth degree by either Clebsch or Gordan. 4 50 Part V. the coefficients of 9 all reduce to functions of the absolute in- variant of the fourth degree c-"-^ and c3 _ ''i _ 1 g2 _ (1 - fc^ + kr _ ^^ ana ^, ^^^ ^, ^^ _l_ ^,^, ^^ _ ^,y ^^ _ g,.,^, and we have the form: .1. [83] ■ *(y = -^-^^ = 125^ - 210cr — 22|« + 93c'r +' 18c| + 1 — 4c3 = 0. If then this equation be written in its expanded form in terms of the modulus h it will not be difficult to see by inspection (for rigorous proof see p. 56) that if we write [84] $ = $^0^®3 these factors of $ corresponding to the special functions of the second sort are, as given by M. Hermite: $j = bV — 2{¥ - 2)1 - 2,¥ [85] ■ • $2 = 5^' — 2(1 - 2k^)l — 3 $3 = bV — 2(1 + h^)l — 3(1 — hy. When O = we have x ==Q whence, as before stated, ^ = is a necessary condition for the existence of a doubly periodic function. But in order to be a sufficient condition it must involve a definite value of v, that is v must be a half-period. That this is the case, although the reverse as we shall find later does not hold, is seen by a determination of v as follows: We have (p. 47) p(y) = x- — 2H^ = ^ffl — 12 Z(Z- — ai )(10Z» — 9,a^l — &,) ■ ~~ 362 Ifi — a,)« Define ^(l) = ^{t) — 121 {P — ai) (10^5 — 8aJ — h^) = 5i« -f 6aJ— lOfeiZ^— 3aJZ^ + Ga^bJ + 6f — 4a^. Whence we write t86] i'W = ^^-^«-'-^' = 36y(i^- Returning to (80, a) we have 2)'(v) = x{3? — 323V — 30 ^ 36K«'-a,)* ^ a; • X 18?(i' — o,)* ■ Reduction of the Forms when n equals three. 51 Where we define % = I [*G> - 3^(0 — 108P(P - a,y\ = V — QaJ" + A\l^ — ^a\l — hl + Aal Where ^ = ?' — (1 -\-h^)l — 3k^ [87] C=l^ — {¥ — 2)l — ^{\ — k^). Refering then to note (p. 24) we have: [88] ■ • ■ . p'{v) = — ¥suH ■ cn^v ■ dn^v = ^J^y , ■ That is p'(v) vanishes where x vanishes which gives v ^ Wz a semi-period, and in consequence, when ^ := 0, f reduces to f = '^ tL g— uH'(b) = are conditions for one and the same function of Lame. In this case p(v) and also the p'(v) become infinite which gives v = or the congruent values 2mw •\-2m'w'. The general form of our integral will not hold for this exceptional case and we are obliged to return to the treatment of the subject from the standpoint of a product. Halation of Y and C to the Special Functions of Lame. Keturning first to (Part IV, p. 42), the elimentary determina- tion of the special functions of Lame, we there found with reference to B that, first, if n be odd, it is determined by two sorts of equations, one of degree -^ {n — 1) giving rise to functions of the Reduction of the Forms when w equals three. 53 first sort, and the other, three in all, of degree j (n -{- 1) giving rise to functions of the second sort; whence combining we have, n being odd, B determined by an equation of degree j (n -\- 1) -{- j (n — 1) = 2n + 1. If w is even we find but one equation, degree -^n-\-l, for functions of the first sort and three equations, degree —n, for those of the second sort making a single equation whose degree as in the first case is 2w + !• If then these roots are all different we have in all 2n -\- \ special functions of Lame. Returning now to the forms (65) 2C = «'(«-/3)(«-y)--- we have the half periods or values of the roots a, /3 that will reduce them to zero. Moreover they will not be double roots, for consider t ^ ti as a double root of Y in which case all the terms of equation (57) will reduce to zero save the second which will be identically zero, which is a condition that the root be tripple. Differentiating we find an analogous equation and a similar course of reasoning shows that the root must be quadruple and so on which is absurde. Hence the roots that are half-periods are not double. On the other hand any other root of Y may be double but as a similar course of reasoning shows it could not be tripple. If then C = all the roots will be double unless they are semi-periods and we may_ write [89] • • r = {jgu — gj)' (j9M — 63)'' {pu — 63)'" n(j)u — paf whence [90] . . . y = Y(pu — e,)^ {pu — e^Y {pu — 63)'" n{pu —pa) where E, E , f" = or 1 . But this form we observe at once is that assumed in every case by the special functions of Lame where we found y always equal to a polynomial in p{ii) times some one or more of the factors {pu — ej)'/'- TJiat is C = is a condition tliat the integrals he tJte special double periodic functions of Lame. By a transformation similar to that on p. 35 we may write equation (64, p. 38) in the form: 54 Part V. 4c^^i4f-g,t- g,) [(f )^ - 2 Y^-^] - (12^^ - 9.) Y '-^ + 4[m(w+ l)t-\- B]Y'' and we have (62, p. 37) (- iTB" [3 • 5 . 7 • • 2n— I]'' "■" from which relations we see that the highest power of B in c^ is 2n -\- \ and that the condition (7=0 gives rise to an equation of the 2w + 1°* degree in B which is as the number of the special functions of Lame. Refering to (68, p. 40) we see that C'- = has been found- as an equation of the seventh degree in B as required by the above theory. Functions of the First Sort. Following the notation of M. Halphen designate by P the first member of the equation that determines B corresponding to func- tions of the first sort. Refering again to (Part IV) we observe that if n is odd each of these functions contains the factor p'u. For example we have: w = 3 •.y=p' where B = Q , the degree in B being unity. n = h:y = f' -^Bp =p' {\2p — ~B) where B^—21g^ = the degree being two, etc. But p' (m) = 4 {pu — Ci) {p%i — e^) {pu — e,) whence for n odd or equal to three, £, e', s" are all equal to unity. Moreover we have obtained Y (67, p. 40) expressed as a poly- nomial in t and h in the form and since p'{ex) = t'{e,)==0 we derive [91] Y„=,(e,) = -b[ Qit ^^^ Qs- Refering again to Lame's special functions we see that if ^j = the function of Lame corresponding contains the factor Yjou — e^ if w is odd and the tvFO corresponding factors Ypii — gg , Ypu — 63 if M is even. In the first case ^1 is a factor of Y(e^) and in the second case of ¥(€2) and of ^(63), vfhile in the second case we have also Y(e^) contains the factor Q^Q^. Returning to w = 3 we have (see (73) p. 44) [Q,]n=B ==B'-6e^B + 45e,'- Ibg^ [92] . . . [<2,]„=3 ^B'-Ge^B + Abe,'- Ibg^ [<23]"=3 = B'-6e,B-{-4be,'- Ibg, or in general writing B = Ibb and ip = g)(b) = 4¥ — g^b — g.^ [93J [Qi]n=z = -i'-b[/(A;2 — 2)2+"l5ifc* 2/ = {P + 4 e^ - 1 (3e, + y3 (5^3 - 12ep) } |/^-=i; = {i' + ^(^^-2;±i>/(F="2Fri5li}V^-l(F-2) Reduction of the Forms when n equals three. 57 riOO] Q, = 0:B = 3e^±ys {5g, - 12e,) = 1 - 2k' ± ]/{! — 2ky + 15 = {p+^(l-2fc^) + 5^V'(l-2Fy + 15}l/p-f(l-2F) (^3 = :\B = 363 + ys (5^2 - 12e^) = 1 + /;'•' + 2 1/(2 — A;^ - 3A; 2/ = {P + 1^3 - ^ (3^3 + 1/3(5^7=12^) ) yp=i; = ll) + ^ (1 + 2P) + i- y{2^WT-'6k \ yp-\{l^) all of which are special functions of Lame of the second species, the general form being y = gypU — Ca where and as given (p. 43) the general form for n = S including the above is [101] y = {p + \e^-roi^)Vp'^^^ where -B = 3e„ + y3(5^2- 12e2j. The Discriminant of Y. From (65) p. 38 we have 2C=a'(a - ^) (a - y) . . . = ^'(^ - «) (^ - y) • • • = ■ • • = l/^(a - ^)(«- y) ■ • -l/^iKiS- «)(/3 - y) ... = .. . where -

'+ 3 (^3 - 6)^] = S^[(p'+ 3{C, — hf] [(p'+ 3(62 — f>f] [?>'+ 3(ei - &)T (=ee (94)) which for the special case w = 3 furnishes the interesting relation, Q differs only by a constant factor from the discriminant of Y. Remembering that A has been determined equal to ( — 1) we have from (97) Q = b^[i{l^ - a,y + (lll^ — 9al — if] and the relations: I = 3b : a^ = ^g^, h-T9B--A-\ (12&^ -g,):A, = - | (446' - 'dg,b + g,) 4(12 _ a^y ^ 4 . 274 : Ul' — 9aJ — \f = — 27^3 [106]- • • (2 = 33-55[44 + 27J^] [107] A = M + 27^I] which latter value we would have derived directly from the form Y=8' + A,S + A,. Writing A.^=-^(p' and A^ = -^

^-8.276«p«p'+16-276V']. Again we find 2 #(?) 4^3^A __\_^']/E.Y [109]- ■ • ■ X — -36"?(?2"_ a,) ~ 36 3*6(p'^ U^i'K b] 1 2 Ul/ 6 from which value we again see that the vanishing of 9)' is equi- valent to the vanishing of D. (oompalr p. 49 and 52.) 60 Part V. Determination of x and v. Second Method. We have the general theorem: every rational function of j)U and ji'u can be written in the form: J.ff (m — *,) o (m — I'j) ■ • • ff (m — a-J ^' ^ ' e (m — r/) c (m — v^) ■ ■ ■ a {u — v'Jj where the number of functions in the numerator equals the number in the denominator, making the number of zeros equal to the number of infinites. The reverse theorem is also known and we may write: r-n/^T/ i\»7 << (m — a)Glu — b)a(u — c)e(u-\-v) -, . 1 , ,„ / \ where $ and W are intire polynomials in pi( and p'u, \ a con- stant to be determined and the relation exists a -\- h ■]- c = v. Also, from the general theory, the degree of the right hand member is four, p (m) being considered as of the second degree and p' (?f) of the third. The degree of 9 and 'J*' are thus determined as follows: Q W n odd: ~ (w + 1) y (w — 3) n even y w j n — 1 w = 3 2 0. The n roots of the first member in the general case being a, 6, c ... we have: [111] 0{a)-^a'W{a) = O where a'==p'{a), a=p(ce). From (p. 38) Tt =-.=(«- ^) (« - y) •• • whence J_ , _ / dt \ 2C" ~\dY/t^a and [111] becomes [112] [^^-^l . =0- But a, ^,y, . . . are also roots of Y, whence the. relation [113] 9'^-W=^EY where E is also in general an intire polynomial in t whence Reduction of the Forms when n equals three. 61 We have also etc. for the other roots of Y. The degrees of [114] are n odd: Y * 1 („ _ 3) - n = - I (n + 3) ' ± (n + 3) - 1 n even : \ (n) — 1 — w = — (y w + 1 ) , (y " + l) — 1- We have Y = f^ + a^f^-^ + a^P'-^ -\ \-a„-it-\-an r = w^-i + {n — 1) aj"-^ + (m — 2) a^i"-* H h ««-i and r^ _ nt"-^ + g, (w — 1) t"-'^ -\ _ *q 1 ^ I ^ I or w^-i + tti (w — 1) ^"-2 + 02 (m — 2) ^»-» + ••• = 60 (<"-' + «1^-' + • ■■■) + ^1 i^"-' + «1*"~' + ■ • ■) and equating the corresponding coefficients we obtain: tti (w — 1) = wai + tj or 61 ^ — «! [115] h^ = — 2a^ + al 63 = — Sag + fflifla — ''^i etc. — — — — — Proceeding in like manner we vrrite:

\ '_!_ 2l- 4_ ga I g3 I U ] U U U whence 1 1261 rf lo-%.,.) «"+' "^ Cto" "^ «("-'> "^ Cto"-2 "^ ■ Reductiou of the Forms when n equals three. 67 and From developments [126] and [127] we find [128]- • • ^1 = 0^; ?2=^'; q.i=^, n being odd,, and from developments [125] and [126] [129] ?i = 6'S„ a + /? + y -I 1- i^v) = q^' — 2q,, i)'i. = 2 (3 q,q, - 3 g, + q,') = ^A 2Bj Bo 2 g V "1" 6'3B„^ These forms are transformed by the aid of the relations 0= cV^ (p. 56); (2« - 1)(« + ^ + y + . . .) = 2? (p. 29) (2»i-l)a, = -7?(p.36) whence (a + |3 + y + ••■) = -a. =2,,^, giving as result: Qy_ _ _y_\/9. CB, c'B, V P Qy'- 2 73, E [130] X n odd. ^^ =c'PB,^ 2 ( §y ' 2 12 p and from these the combined forms arise 2a; c*P4* • B„ y 3y. ^ B„ p'v B, y 2w — 1 These formules are perfectly general for n odd and the corre- sponding forms n even obtained in like manner are [131] ^ ~ Py ~" y K P pv p V Py y Pv' Y tQBl Py' 2P, n evm. -2c^ j 2n — 1 3J?oy. + if 1 1^1 2a; ' -^ y 2« — 1 The superiority of these forms over those first derived, showing as they do at a glance the synthetic relations, is unquestionable 5* 68 Part V. and the explicit forms for our case n equals three and also for n equals four and to some extent for yet higher values, are obtainable with greater easy than by the first method. Even here however the forms increase in complexity so rapidly that n is practically restricted to the lowest values. For case n = 3. We have found all the eliments except y^ which is derived from development of @, or more easily as follows. From (106, p. 59) and from (p. 65) W„ = , = Qy --= {3S' + A){—GA,S' + 9A,8 - 4 .1^) + 9C2A,S - 3A,)iS' + A,S-^A,) = -(4Al + '21Al) and a comparison gives immediately 1132] .... J' = -(ii)V The other values for the eliments have been found, namely: P=16h A^Tf' l = 3b We have then for n equals three _2_| A.! 3q>'V - . (compair 109, p. 69.) ^ J_ |qp'''+ 27 qp'^ -.8(27) fee;, g?'+ 16 (27)fe''y' 'l T Squaring we have: I134J- [135] Reduction of the Forms when n equals three. 69 ml}'-— «i)« * w ggj/ja ^, \i> (oompair 8a, p. -lu.) Again we have: _ 4[4^^ + 274] , 4(|9-66 .p-) 9' = ^mb'-2VU'g,->r2Wgl-2\mg,-\-21gl + bAg,g,b. - \QShq)(p' = -bl84:¥ + \12Wg^ — lO%¥g\-\-\2QQWy.,-\0%hg.,g.^ whence 21606« +.2166*5-, + ^Omb^g, - 9,b^g\ - 546sr,ir3 - ff* + 27fli^ [137] pi; = 366(1446^- 24b*(/j + f,^)^- Again from the first method „ 1 + fc* ,2 ., it(3&) 1086(96* — a,)'' where ^ = 5(36)« + 6a,(36)*— 10? = C^^i ■^= 631 ^— 36Z, 2)==^ (J*— a), a = a, . 70 Part V. It is, finally, evident from the general forms that if it be required to determine p'v it will be easier first to find pV = ?3 _ ill _ ^ ^^ -r pv g^ y 2*1 — 1 ^ jqp - 6bq>- _ g^ _ 2!»'^- Determinatioii of v. Third Method. The formulae may be obtained by a third method and in yet different forms as follows: Starting anew with equation [110] we write [139J (— 1)"^, -^ ^n+i — = ^iP^) - ^i' «« 'P^ (Pw)- Also r 0(M - 0) 1 ^ _ J L aa ■Ju=^Q whence it follows that the left hand member of (139) depends for its value on the terms (- ')"fci o{m)'' + ^ But we have again r^l =1 whence we may write, taking n odd L ' ff(a)«(6)- . • c(r)(eM)'' + »J„=u m"+' and from p. 66 „ = ^L_4. ^ J__i_ That is n being odd *i = -Bo- And a similar investigation gives n being even 1^1 — -C-- Beductiou of the Forms when n equals three. 71 Since v ^ u -{- h -\- c we may write and multiplying by this factor we can separate the left hand member into factors of the form I140J g(« + »^)g-,,. ^gia all 0(1 a a for U = itJj. But for this value p'{w,) = and our relation becomes [141JAud we obtain in a similar manner and Recalling the known relation p u = — 2 - — I 2_ we have upon taking the product of the above equations [142] • • ¥p'ap"b .■■p'v = {— 2)"+' *i®,(p3. Again from the relations (65) ,_ 2C « - (a - P) (a - y) (a - *) . . . ^^■ to n terms and we obtain the product 2"c»^_ i)»'2--"-'>= (_ i)-!r»<"-'\2)"cr'' [143]a'^'y'...= (_ 1)4^-1) (2)" C" A A being the discriminant of Y. Substituting this value in [142] we derive [144]. . . (-l)^"'"-''2"CpV = (-l)''+'2^.*,03A. Again squaring we get or (see [89]) [145] {—\rh^Y{e,){pv-e^)^^\e,) and we have also the two corresponding expressions. 72 Part V. We have shown (see p. 58) that when t = e^ we have Y(ej)= — c^PQ whence it follows from this and relation [145] that ^(Cj) is divisable by Q^ and in general ^(ei) by Qx. We thus derive the relations [146]. . . . O^^Q.F, : *, = V,F, : 0, = Q,F,: We have also found n being odd: l = B^:G= c'YFQ : z/ = (- l)~^ c^n-ipY^"-^^ q2 <"-'> These values in [144] give (- l)^'"~""5Jg2„py^Tp'^ or _ and from [145] ByF(^i (pv — e^) = 01 = Q\Fl n odd. Whence we have in general r-iAO-i ^^^^ [148] P''-'' = c^B^- The corresponding expressions for n even are _ _ 2c^^.0,<^3 -l/l [149]. ^V — Cx = y'P Again from [130] P^ e lc*PB„= B„' \ V p c^B„»p y p = _ ^?__ ! £z! _ ay^.g^-B o ^'\ i/y t-'B^'P I c« c'B/P I K P — 2y ( (gY'-3.B„B.PcM l/^ c^Bo'^ I c" j K P' Compairing the second and fourth forms we have [150] F,F,F, = -^,iQf-3B,B,Fc% Reduction of the Forms when n equals three. 73 Substituting the values w = 3 (p. 68) and refering to the value *^f X (P- 51) we find the relation [160]- . . . F„=, = [I\F,F,]„=,= ^^-,x = ~ABC. It follows then that Xi if expressed in terms of the modulus k and 6 or as a function of 6, e^, ^fg and g^, will be separable into three factors which from the expressions for O are seen to be of the same degree in b, namely, the second. The factors of x which we before obtained by. inspection (see p. 51 [87]) are [161] • • ■ B = P — {\ — 2¥) I + S{¥ — h*) C=V — Qi' — 2) Z - 3(1 — k^) and we find the relations: [162] F, = ^A- F, = I^B; F, = l^C. Taking now S = 361 and D = P — a^^ l^ - 1 -{- k^ — /c* we find the following relations of M. Hermite 0(1) PQB [163] a;^ = S6l[P—a,) SD' —p'v = iij = k^ SHU cnu dnu = — X'.hx AliCx k^sn^c3 = i + k^ If 36l(P— a,) _12l{P—aJ''{l + k'')—ti,{l) FA^ whence 7c* en'' a = 3 3&l{P — aJ' •iQliP-^a^y l-il{l'— a, )■-' (-2 /;■■'— 1) + i/) {I) Q B" SI)' dn^a = ■Aiil(l'—a,y HD'' ~" SI)' S6l(l'— any- where X = k and fflj = a and ca = v. (see also note p. 6D) General Discussion. Reviewing the foregoing theory we have found that when w = 3 y = f"-3hf and that in general y is a function of f where we write the one exception occurring where v equals zero. 74 Part V. We find further, that where ^ or ^ vanish in which case x and p'v also vanish, our integrals, six in number (» = 3), become doubly periodic and are in fact the original special functions of Lame of the second and third sort. We have found for x the general value |/| from which form we see that x will be zero when y and Q vanish and will be infinite where B or F vanish. But from the form ^ Qy^ 2B, B_ P'^ c^PB^' B 2w+l we observe that pv is also infinite where x becomes infinite through the vanishing of B^. We have further that in case P vanish the integral becomes a function of Lame of the first sort in which p takes the place of f in the general solution the form being [1 64] (- 1)"!/ = j^,p'-''u + j^, q.p^-^^u + ^^ g,2,(-6)M + . . . the values of B conforming with the above cases being roots of the equations P = 0, Qi = 0, ^^ = 0, Q^ = 0. Moreover when Q vanishes x and p'v will vanish simultaniously which makes i' one of the semi-periods mi, and f may be written fl«5] /•.=« = + ^- Again, observing the last forms obtained, we see that v can also be a half period ii Fx, n being odd, or ^x, n being even, vanish, but it does not follow that x will also reduce to zero. That is the integral will in general have the form [166] _ oju+j^ e(^-q»0)« =.'Jl^u ■ 1 /r«i trnt. when Fi = 0, or ^1 = 0, or X = 0, or ^ = 0, or B^ = 0, or C = 0. In this case as in general two distinct integrals exist which are doubly periodic of the second species the second integral being a form which does not differ from f, a peculiarity which does not appear in the special functions of Lame. Reduction of the Forms when n equals three. 75 We have fiually but one more case to consider, namely when V = 0, a condition arising when B^ or y, common to the functions X, pv andp'v, vanish, in which case the integrals become functions named after their discoverer.*) Functions of M. Mittag-Leffler. As M. Hermite observes (p. 28) the vanishing of A, B, C and D are necessary conditions that the integrals shall be functions which he first called functions of M. Mittag-LefiFler, but they are not sufficient conditions. The functions are in fact special cases of /i and /j having the additional property that the logarithms of the so called multiplicators are proportional to the corresponding periods. In this case the integrals assume a special form where the elimentary function is a function of p and p' multiplied by a determinate exponential having the above property. We can show that these are but special cases of the general doubly periodic function of the second species of M. Hermite as follows: We have as the general form a{u — ffli) o(m — a,) . . . . and f} and ij' are constants. The factors (i and "ft' are general and we may if we choose take them at pleasure and then seek the corresponding function. Doing this we have ft and ft' given and also q to determine B — A from the relations [170]. Solving we have log ft =2r](B — A) + 2qw ]og(i' = 2,]'{B- A) + 2pm;' See Mittag-Leffler, Comptea rendus t. XC, 1880, p. 178. 76 Part V. whence ,.n..V logf*' — '2'logf* = ^QirjW' — iq'w) = gni; (■rjtv' — ri'm^-^) w' log ;* — w log fi = 2(JB — A) {r/w' — n' w) = {B — A) ni. This solution however becomes indeterminate when F(x) becomes doubly periodic, for then p = and B — A = 2mw + 2m' w'. This gives i7i(2mw + 2m' w') = w' log (i — w log ^' whence w loK a — 2inm w' log ft' + 2i7tm' which means that the logs of the multiplicators are proportional to their corresponding periods. Returning to the form a(u) we observe that when V = Oi + a^ + • = we have V = 2mw -\- 2m' w' and f vanishes showing that this elimeut can not be utilized in this case. Written as a product however and for m = 3 we have [1721 • • • • y = "(" + '') "(" + ^) "''■'* + ") c-"^'^''+'^* + "^ where and our eliment may be taken as a rational function of pu and p'u multiplied by a factor of the form e9". It is moreover known that any function f{u) of p and p' may be resolved in the form f{u) = L + P where L = 1^^(U — V,) + l^tiu — V^) + ?3§(4t — Vj) + • ■ P = c + SmF^'^{n — v) where This property being general, we have, f being doubly periodic, but to multiply by e?" to find a development for the eliment required in [172) namely [173J 0(«) = e«?»g(M) Reduction of the Forms when n equals three. 77 We have then g(re) = 0(uje-'-"' g'(tt) .^ 0'(u)e-'-"'— Q0{n)t~V" i"{u) = 0"{ti)c-'-"'— 2p0'(7j)c-^" + Q^0(jt)c-(-"' §"(3)(|,)= 0"'(M)e-i'«— 3p*"(M)e-s" + 3p^ $'(")<'"'■'"— P'^OOe"""- Whence [174]eP"gw(M) = ^^(m) - ^ p$(''-i)(m) + !i(!L=:l) p2${«-2)(„) -| We have then a decomposition in the form [175]- ■ • • /i(M) = cey«+y Va. ,*«(«-"») where v„ stands for the several infinites of f^ (u) and 0^^'> for the derivatives where v must be of an order one degree less than the multiplicity of the infinites. The coefficients A will be determined in general by developing /i (m) according to the powers of (u — v„) while c will be a fixed value depending upon the given conditions. In our* case then we may write [176] /i(M)ce^'' + g^t-ef". This function when v is zero, in which case $ = and D = 0, takes the place of f{u) and hence the general solution is Vi =fi"{u) — 3bfiU = (ce?" + gM • e?")" — 3 6 (g?( • fi?" + ce?") /;'(«) = QQ^>^ -f g'ttet'" + p£(?t)e?'' /;"(m) = p2ce*"+ rwef" + 2p§'(M)e^" + (>'£^0')e^" whence and we have [177] • • ■ 2/i = (gw ■ ef" — 3ig«cC'" + c'e^" = e^" [§"« + 2pr« + (p' — 3?>)ei' + '■]. But from the foregoing theory in this case we have the coeffi- cients of §(m) equal to zero, i. e. or p2_3J = [178]- ■ • Q' = 3h. IS i^art V. Reduction of the Forms when n equals three To find c we proceed as follows: — in = 1 Jfj m' u 20 3 * M" JO e^M* (!■'«■' in, II I V "■ V ** I Hence ,y = [i + .« + ^ + i^ + -.]ie-S"-l -^'•[ff+lo' «' + ■•■] + «+ •■( and taking c so that the constant terui equal zero we have [179] c = |-p' = 2p6. The general solution (v = 0) is then: «/, = (gweP")" — 3b{iu-e^") + 2Qbe<~'" wh ere p = VTb . Finis. Table of Forms w = 3. Forms for w = 3. The complete Integral is y, = CFiu) + C'F{- u) where y = Fiu)=f"{u-)-3bf(u) and ' ^ ' auav the ordinary form of the equation of Hermite for w = 3 being: g = [12i>(«) + 5]t/. A second form of the integral is: — e(u — a) „!■„ JL A Gaou J. J. oa = aiu — a)a(u—b)e(u — c) g(^'„+^j+f,)„ <>aff6 oc(0m)' •where a; = gv — ga — g& — £;c v = a-+ & + c and B ^ I5b which is intirely arbitrary and is originally expressed in the form -B = A (ci — fig) — w (n -f- 1) 63 in which case the equation of Hermite is We have also the general form: — y = + VY = y{pu — ej' (jati - e^Yipu — e,)'" ]J^ (pu —pa) e, e, e" ^ or 1 . 82 Table of Forms n = S. The functions developed in the general theory have values as follows:

'' ~ 3Gb'«6^ G* 84 Table of Forms w = 3. , (p'^+ 21(f''— lOSmqp' nv — =i ~ ~ ~ pv — e;.= -^-„- ['^f 36()p'^6 f- pv = V -, 2a; ^ 2 ()p where i>{l) = 0{l)— 121 (P- o,){lOl^- 8aJ — h,) = 5Z«+ 6aJ- lOb^P- 3a,^i^+ Ga^bJ + h^^ - 4^ ;t(0 =4[$(0 - 3^(0 - 108i^(Z^ - a,f = V- Qa^l" + 4^1 Z" -. 3a,2? — 6,^ + 4ai8 = ^ . j5 . C j5 = ^2 _ (1 _ 27^2) Z + ^(¥ - ¥) = f Fa C=V—(]!?—2)l - 3(1 — ifc2) = fF3 F= F,F,F3 = 3,^,;^ = jA.^ . ^ . C. Case 1. P = 0. Integral a special function of Lame of the first sort. y=p'. B = 0. Case 2. Q = 0; ®(Zj = 0; Q^ = 0; ^^ = 0; ^3 = 0; a; = 0; 25'v = 0; V = Wi. Integrals, six in number, of the second sort. a,u efu 4- w,) , __, 9 , ^__ 1 _L__._!^ L^J^ (>-ui:(wx) a — 1,2,3 ' ^^ CM euoiWj^ ?(";.) = '«■ where (a) Q,==0 i? = 3e, + }/3(- 12e^ + by,) = h^ — 2 ± -[/{k^ — 2f + I5k* Forme for w = 3. 85 (b) (?, = B = 3e, + >/3 (5p, - 12e,) = 1 - 2F + l/(l - 2A^) + 15 2/ = {i' + |e, - f„ (3e, + v'3(5^::ri2^)) } y^"^ = {p + i (1 - 2P) + ^ y(T^2Ff + 15 } 1/7-1(1-2^) (c) «3 = £ = 363 + yS (5^2 — 12e^ = 1 + F + 2 ■)/(2 - Ff'-^Jc y = {p + Y^^-ro (3^3 + VH^92^- 12«|))} y^^ = {i'+s(l + 2/^0±il/(2^^^-3Ml/p-l(l+^- Case 5. Fi^O; x = 0; A = 0; B = 0; C = 0; v = ra^; a;=|=0 ffjM