«l 4:^ LIBRARY OF THE University of California. GIF^T OF Class \J\f\JJL^^<^>^. ®l|e MttiDetBiti) of Cliicago FOUNDED BY JOHN D. ROCKEFELLER. On the Reduction of Hyperelliptic Functions (p=2) to Elliptic "Functions by a Transformation of the Second Degree. A Dissertation Submitted to the Faculties of the Graduate Schools of Arts, Literature, and Science, in candidacy for the degree of IDoctor of P*liilosopliy (Department of Mathematics) by J. I. Hutchinson. i UNIVEP.SITY ) or / or Gottingen 1897. Druck der Dieterich'schen Univ.-Buchdruckerei (W. Fr. Kaestner.) ,^^ , OF THE X ( UNJVERSfTY ] On the Reduction of Hyperelliptic Functions (j) = 2) to Elliptic Functions by a Transformation of the Second Degree. Introduction. The problem that is the subject of the following paper is not a new one. As early as 1832 Legendre^) had shown that the two hyperelliptic integrals J Sixil- x') a - x^x') ^^ J\lx(l- X dx six (1 - x^) (1 - x^ X') J\/x{l- x') (1 - x' x') are each expressible in terms of two elliptic integrals of the first kind by means of a quadratic transformation. Immediately after, Jacobi'-^) in a review of Legendre's work pointed ont that this property belongs to two linearly independent integrals of a more general type than those discussed by Le- gendre, viz. /dx , C xdx -==:- and /- \lH{x) J\jR{x) where Mix) = x{l-x)(i~k]ix)il + Jcx){l-i-Xx). In 1867 Konigsberger^) proposed and solved the problem, to determine the nescessary and sufficient condition on the roots of the sextic in order that the associated hyperelliptic integrals may be reducible to elliptic integrals by a transformation of the 1) Legendre— Traits des Fonctions Elliptiques, t. 3, pg. 333. 2) Jacobi — Werke, Bd. I, pg. 380. 3) Kouigsberger — Crelle, Bd. 67, pg. 58. 1 _ 2 — second degree. Denoting these roots hy a^, . . . a^, oo, he found the required condition to be This relation expresses the fact that the roots are in involution, and shows that Jacobi's integrals are the most general integrals having the property proposed. In obtaining the above result, Konigsberger made use of a theorem communicated to him by "Weiers trass to the effect that, whenever two hyperelliptic integrals of the first kind, linearly independent, can be found, which are reducible to elliptic integrals by an algebraic substitution it is always possible to transform the corresponding ^-functions into a new system such that the modulus TJg ^0- In a later paper *) Konigsberger gave a new proof of the same result, and extended to the problem of redu- cibility the algebraic transformation method employed by Jacobi with reference to elliptic integrals. In 1869 G r d a n ^) , in connection with his investigations on the binary sextic, gave the algebraic condition for reducibility in invariant form. He showed that when the skew invariant of the sextic is zero, every associated hyperelliptic integral may be expressed as an aggregate of elliptic integrals by substituting as new variables the quadratic co variants I and m. In 1876 the inversion problem was taken up by Her mite') who expressed the odd hyperelliptic functions belonging to the irrationality \Ui{x) in terms of elliptic functions. Immediately following him Cayley*) accomplished the like result for the even functions. The solution of the inversion - problem is also contained implicitly in the work of Pringsheim^), who, while studying the general quadratic transformation of the •d-- functions, derived as a special case the formulae connecting the elliptic and hyper- elliptic ^-functions, which he made use of to obtain the reduction of the integrals, previously effected by Jacobi by algebraic methods. 1) Konigsberger — fJeber die Reduction hyperelliptische Integrale auf elliptische. Crelle Bd. 85, pg. 273. 2) Gordan — Applicazioni di alcuni risultati etc., Annali di Mathematica, Serie II. — Tomo II., pag. 346. 3) Hermite — Sur un exemple de reduction d'int^grales abdliennes , aux fonctions elliptiques. Annales de la Soci^t^ scientifique de Bruxelles, t. I (1876). 4) Cay ley — Comptes Rendus, t. 85, pp. 266, 426, 472 (1877). 5) Fringsheim — Mathematiscbe Anaalen, Bd. 9, pg. 445 (1875). I — 3 — In 1882 Picard ') proved that the relation among the •8- moduli previously found by Weierstrass may, by a proper trans- formation , be expressed in the form ^[2 = l] ^'^d also that the reduction from the hyperelliptic to the elliptic integrals can be eifected by an algebraic transformation of the k*'' degree. This is a special case of a theorem announced a little later by Weier- strass^). In the following pages I have attempted, at the suggestion and with the kind assistance of Professor Oskar Bolza, a new treatment of the reduction problem taking as my point of view the CO variant character of the problem, first hinted at by Gordan. This suggests the systematic use of another normal form for the sextic, other than that employed by Jacobi, and adhered to by most of the subsequent writers, and the solution of the problem is greatly simplified thereby. In the first part of the paper, the fact that the roots of the sextic form an involution is dwelt upon , and some theorems regarding involution, bearing directly on the subject in hand, are deduced. Besides the two reducible integrals usually mentioned, a complete list is given of those special cases of reducible integrals characterized by more than one involution on the roots of the sextic. Following this, I enter upon a detailed examination of what I may call the general case of reducibility (one involution) , be- ginning with a study of the correspondence between the hyper- elliptic and elliptic Riemann surfaces , and the derivation of the reduction - formulae for integrals of the second and third kinds. The remainder of the paper is chiefly devoted to a solution of the „ inversion -problem". Conforming to our point of view, the 0- functions introduced by Klein are used, and the three simple reduction - formulae of Theorem VI, are among the most important new results given in the present paper. They unite in an implicit form the results of both Hermite and Cayley, and contain even more, since the factors of proportionality which disappear when the quotients are formed are here preserved. The simplicity of these results and the ease with which they are derived clearly show the advantage of the present mode of treatment ^). 1) Picard — Sur la reduction etc., Bulletin d. 1. Soc. Math. d. France, t. 11, pg. 25. ; 2) Kowalevski — Acta Matliematica, Bd. 4, pg. 400 (1884). 3) Compare Cayley's remark : „Les reductions pour obtenir les values des dix fonctions sent trds-penibles". Comptes Rendus, t. 85, pg. 426. 1* _ 4 _ Finally, the Weber and the Gopel relations satisfied by the (3-fanctions are derived, and attention is called to the way in which the Kummer surfaces , represented by them , specialize for the case of reducibility before us. § 1. Algebraic Reduction by JacoWs MetJiod. Jacobi's method for the transformation of elliptic integrals ^), can be extended at once to the investigation of the reducibility of hyperelliptic integrals *) to elliptic integrals by a transformation of degree n. If R {x^, x^ denote a binary sextic, and if P(?7, F) =: {Y.U +f.' V){\V + I' V){)^U+)^' V) iy U -\-v' V), then this method, for n = 2, leads directly to the result : Theorem I, In order that the hyperelliptic integral of the first Jcind, '{A^x,-\-A^ x^) (x, dx^ - x^ dx^ r SlR{x,, x^) may be reducible to the elliptic integrcd UdV-VdU M f \IP{U,V) ' M being a constant, by the quadratic transformation, U z= a^xl + 2a^x^x^ + a^xl V = b,xl + 2b^x^x, + b,xl it is necessary and sufficient that R{x^, x^ be decomposable into three quadratic factors, (p, t^, x, such that

, / in the form (6) ^ _ ^^(^2_p.^a) while the sextic l?(iP,, a?,) may, in consequence, be written in the form (7) f{z, , z^) = a^e\-\- Sa,z\el + Sa^ z\ + a^z\ , By the linear transformation (5), the integrals (4) are changed into such as differ by constant factors only from z^{zdz) r z^{zdz) (8) (^0^' f VA^x,^.) ' J^^fi^r,^^) ' 1) See Reye — Geometric der Lage, dritte Auflage, p. 139. 2) Clebsch — Binare Formen, p. 208. — 7 — By the quadratic transformation Ci = -s'J, C2 = ^l we obtain where -R. (C, , CJ = 4C, K C^ + 3a, CI C, + 3a, C, Q + a, Q ^ B, (C, , Q = 4C, K C: + 3a, C: C, + 3a, C, C, + a, Q . The normal form, (7), for the sextic, suggested as it is by the properties of the reducible integrals, offers decided advantages over the Richelot's normal form employed by Jacobi, and adhered to by Hermite , Cayley , and Konigsberger , the Richelot's form having no special relation to the reduction problem before us. As already remarked in the introduction, one of the principle objects of the present paper is a simplification of the problem by a systematic use of this normal form ^). As a first example of this simplification, we are now in a position to establish as once a general theorem. For the greater convenience we pass to non-homogeneous notation. Theorem TTT. If F{g,^f{z)) denote any rational function of the two arguments, s, \/f{z), then the most general hyperelliptic inte- gral, jF{z,\lf{z)d3^ is expressible in terms of two elliptic integrals, together ivith algebraic and logarithmic functions. For, it is evident that F{z,^f{z)) can be separated into the terms G^{z') + z ■ G,{z') + \^f{z)- H,{z') + z\Jf{i) ■ H,{z'), where G,, G^, H^, H^ are rational functions of -£^^ When we make the sub- stitution z^ == C, and integrate, the first two terms lead to algebraic and logarithmic functions, while the last two are elliptic integrals. By combining this theorem with the addition - theorem for elliptic integrals it follows at once that the sum Slf{g)dz' is expressible in terms of two elliptic integrals together with 1) This normal form has already been employed by Pi card (Bulletin de la Socidtd Mathematique de France, t. 11 (1883), pg. 45), and by Bolza (Inaugural- Dissertation, Gottingen, 1886, p. 32). — 8 — logaritlimic functions , a theorem obtained by Hermite by means of Abel's theorem ^). The substitution (5) contains two arbitrary constants, Jc,7c'. For our further purposes, however, it is convenient to determine these by the conditions, (1) Determinant of transformation =1, (2) a, = a,. The coefficients in (7) are now invariants of the sextic , and we proceed to show how they may be expressed in terms of the simultaneous invariants of 9, ^, /. These are -) (11) A,^,^ = - 2 ^: [3^ ^ZT = - ^0 ^" (^' + *'>> A^.^ = -2xW ^r^ = - ?o '!>. (a' + r ) • The following useful combinations of these invariants are easily derived: Ai.^-A^^A.^^ = foylir-fy (12) A'y^ - A.^^A^^ = x>: if-a'f (13) A^^A^^Ayy = Sal A^yAy^A^,,^ = 9a,a^-al (A^^A^y — A^y) (AyyA^^ — Ay,^){A^^A^^ — A^^) = 27[at + 4a,{a\+al)-eala,a,-Sa\al]. A word is perhaps necessary in regard to the derivation of this last equation. Comparing the left hand member with (12), we observe that it is the negative of the discriminant of the cubic form, a„ Cf + da^ C* C, + 3a, C^ Cl + a, C,. Referring to Clebsch's Binare Formen, pg. 114, we see that it is equal to the bracketed ex- pression in the right member of (13), except for a numerical 1) Annales d. 1. Soc. Scien. d. Bruxelles, 1876, pg. 10. 2) The notation is the same as that used by G r d a n , Invariantentheorie, Bd. II, p. 147. ^ 9 — factor. This factor is easily determined by assigning particular values to the roots, say, a^ == i, [3^ = — i, -^^ ^ — 1, From the equations (13) we can obtain ci,l,a,a^,a^{al+ al) rationally in terms of A^,,, etc. Hence cil,al are the roots of a quadratic equation whose coefficients are rationally expressible in terms of V^^ A^,^, Vl A^,^, \J^ A^^, A^.^, A.^,^, A^,i, . By combining equations (11) and (12) we obtain invariant expressions for (p^^ (J^q, y^^, viz., while ({)o, ^0 are obtained directly from cp^ by the cyclic permu- tations (x^) (?X'^) respectively. The ambiguity arising from the cube root is due to the separation of the sextic into three qua- dratic factors. The cube root of unity may be chosen arbitrarily for one of the functions, say (p^. The others are then determined. For, by eliminating a", P", y'^ from equations (11), the resulting equations in tp^, (j>„, y^^ involve only the ratios of these quantities. The signs of the radicals, Vi^ma; Vi^M? Viyv > ^^e entirely arbitrary. To determine the signs of the remaining radicals, we consider the three functional determinants of ?,4'>X> which we will denote by ■6-x,., *y,., ■6- x. Thy satisfy the relation ^) We find for their explicit expressions X? ^^ *^ XT -^5(y-^cpcp ■ ^1^2 It is evident that the sign of one of the radicals may be arbitrary, white the signs of the others must be chosen so as to satisfy the above invariant relation. If 9ii9z-i and g^^g^ be the invariants of i?i and i?2 respectively, these may be expressed directly in terms of the coefficients a,,, Oj, ttg \ and hence, by means of (13), in terms of ^ „, etc. Simil- 1) Clebsch — Biniire Formen, pg. 204. 2) See Klein - Theorie der EUiptische Modulfunctionen, Bd. I, pg. 15. - 10 - arly, if e^, e^, e^, and e^, e^, e^ be the roots of the two quartics, when put in the Weierstrass normal form, they may be expressed in terms of the roots, and hence in terms of the irrational invariants of the sextic as follows : e.-e, = ao(P'-f) = ?o^ \-\ = a,a'{^'-f) = ^^E (15) e,-e, = a,if-a') = ^,F \-\ = a,^\f-a^) = ^!^^F e-e, = a„(a*-p^) = x„G^ ~e-~e, = a„f (a'-p^ = X,fG gj + e^ + eg = e^ + e^ + ~e^ = where F = \/A'y^-AyyA^^ ^ — ^A'i^^—A^^A^^ § 3. Special Cases in which there exist more than Two Beducible Integrals. It might happen that the roots of the fundamental sextic were in involution in more than one way. In such cases there would exist more than two hyperelliptic integrals of the first kind, each of which could he directly expressed by a single elliptic integral of the first kind. The determination of these can he reduced to the study of those sextics that admit linear transformations into themselves. For this purpose we make use of the following : Lemma, A. If three quadratic forms, having no common factors J whose discriminants are not zero, he in involution, then there exists one and hut one binary coUineation ^) of period 2 which trans- forms the three simultaneonsJy, each into itself, with a change of sign and an interchange of the roots in each. B. Conversely: When three quadratic forms are simultaneonsly invariant for a binary coUineation of period 2, which interchanges the roots of each^ then the three forms are in involution. 1) By a „binary coUineation" is meant a homogeneous linear transformation, pZ[ = (MT, + bz.^ pg'i = czi + dZi in which the factor of proportionality, p, is considered as non-essential, so that two substitutions differing only by the factor p are regarded as identical. - 11 - The truth of A appears immediately when we consider that the three forms may he expressed as the sums of squares by a linear transformation. Then, if the coefficients of the general linear transformation be determined on condition that it shall leave the three forms invariant , it will he found that the collineation (16) '' ^ '^' is the only one having the required property. It will also he found that this operation interchanges the roots, and reverses the sign of each form. In order to prove B, let a, a' ; p, p' ; y> y' ^^ ^^^ three pairs of roots. There exists only one linear transformation that in- terchanges a and a', p and p'. Written in the non-homogeneous form for convenience, it is 0z' (a + a'- p - p') + (^ + ^') (PP- aa') + aa' (p + p') - pp' (a + a') = 0. According to our assumption this equation must be satisfied by 4f = Y) ^' = f'j Hence YY' (a + a' - P - P') + (y + y') (PP'- aa') + aa' (p + p') - pp' (a + a') = 0. In determinant notation this relation may be written 1 a + a' aa' 1 p + p' PP' =0, which is the well-known condition for involution. From this lemma follows: To every decomposition of the sextic f into three quadratic factors , «p, ^, X , belonging to an involution, there corresponds a collineation of period 2 ivhich transforms f into itself at the same time interchanging its roots in pairs. And vice versa : To every collineation of period 2 which transforms f into itself, accompanied by an interchange of all the roots in pairs, corresponds a decomposition of f into three quadratic factors forming an involution. Hence, in order to solve the problem proposed at the be- ginning of this section, we have (1) to determine all sextics with linear transformations into themselves, and (2) to select from the automorphic transformations, belonging to each case, those of period 2, having the properties mentioned - n - in the lemma. To eacli such transformation correspond two reducible integrals. The solution of the first part of this problem has already been given by Bolza in the American Journal, Vol. 10, pg. 50. The result there enunciated is : Any binary sextic, f, with linear transformations into itself, (except of course the transformations, ^/= ± i^J, ^.^ = ± i^'^) , and wJiose roots are all distinct, is reducible by a linear substitution, of determinant 1, to one of the following canonical forms ^), I /• = a^^l + 3a, ^\ ^l + 3a, ^\ s\ + a, 4 {Cyclic group, w = 2) n. f= a,z,{z\^z\) {Cyclic group, n = 6) in. f = z,z^ {a, z* + 2 a, z\ z\ + a„ z"^ {Bihedron group, « = 2) IV. /■= a,z\ + 2aX< + a,z\ {Dihcdron group ^ w = 3) V. /•= ao(^x + ^") {Dihcdron group, w = 6) VI. f = a^z,z^{z\ + z\) {Octahedron group). The six roots of the sextic may be interpreted as points on the surface of a sphere of radius unity, and those substitutions that transform it into itself, as rotations of the sphere which leave the configuration of points as a whole unchanged ^). "We have, then, merely to select those rotations through the angle tt, whose axes do not pass through any of the six points. To each such rotation corresponds a pair of reducible integrals whose numerators are represented by the two points on the sphere through which the axis of rotation passes. The total number of pairs of integrals for each case is respectively 1,0, 2, 3, 4, 6. Case I is identical with the canonical form of § 2. For the remaining cases, the hyperelliptic , together with the equivalent elliptic integrals and the reduction formulae are given in the following tables. 1) The invariant criteria that characterize each case are given in the paper just referred to, pg. 70. 2) See Klein — Vorlesungen iiber das Ikosaeder, pg. 31. - 13 - o I— t + CO •* e" I + + CO iO + CO + 1 ^ ^ «sr 1 «»" 6i ^r tsT 1. ^>- + § 1 t» 15 - Case V, z^ {zds) ^2) ^2) {2^+Z^){0d0 ) ^f^a m) VauC2(C!+CD m) (C!+q) / \/2 (WC) V2(WC) VaoC,(C, + CJ(C: + 14C,C, + CD Co = ^2 C. = (^x-^J C2 = (^x+^2)' / / {z^-^ pz^){zdz) (WC) 9 J V«oC2(c.+g(c!+i4c,c,+q) (C^C) V2 P J V^oC.C, + y(q+i4c,c, + CD Cx = (^.-P^2)^ C2 = (^I+P^J 2iTi 6 {z,-fz^{zdz) (WC) ya r pV V«oC.(c,+g(C!+i4c,C3+a) / vf(^^J p^ i VaoC.(C. m) (^-P^^J + g(q+i4c,c.+CD 16 — VJ> ij> + «4 «4 iW^ 1 + tH 1—1 •'^i. ■<^ + + OJ 1 l(M o 1 CO ■^ a + CO + CO + *♦ ,** "^ ... — ,^ '^ t4 «si v^ ^ e< «* ^Js*, «4 1 •■+>. + ,.»r > ,**"■ ^ ^<^ ^> ^ ^°^ ^ ^^^ paired iu the inTolution. — 19 — A Ti r* Three involutions , t) j^ ^ > ABC ABC /TTi- TTT^ FED' EDF' ^^^^' -^^^• V. The sextic consists of the six vertices of a re- gular hexagon inscribed in a circle. Four involutions j^y^ , ABC ABC ABC ,^. ^^ FED' EDF' DFE' ^^^^- ^'' VI. The sextic consists of three mutually harmonic point-pairs, and is represented by the vertices of a square inscribed in a circle, together with the two imaginary cir- cular points at infinity, Coo, F oo. Six involutions , T\nv> > ABC BAD BAD CAD DFE ' ECF ' EEC ' FBE ' ^^g, (Fig. VI). Fig. IV Fig. V Fig. VI § 4. Periods of the Integrals of the First Kind. Introducing non- homogeneous notation, we proceed to a more detailed consideration of Case I, beginning with a study of the periods of the integrals, dz and their corresponding paths in the Riemann surface represented by the function (19) V/S) = Va„^«+3a,^*+3a,^^ + a„ = \JcLjz'-a.'){z'-^'){z'-f). If B,{0 = ^%^' + Sa,C + 3a,C + a J = 4a,(C - a^) (C - ^ (C - f) (20) R,{0 = 4C(a„C-f 3a,C+ 3a,C+aJ = 4a,C(C- a^) (C-p^) i^-f) 2* 20 - ihen, by the transformations (21) (a)j .jri,Y= '' ^^1 V^(c) = 2^V7?) we have (22) r 2ds _ r d: r a^ ^ r d: \lR,{r) J \Jaz) V^.(C) By these two transformations , the hyperelliptic Riemann surface T is brought into a two to one correspondence with each of the elliptic surfaces belonging to the functions , \jli^{ C) and SjRJJl.) , which may be designated respectively by T and T. Lines radiating from the origin (i. e. the point .s- = 0) , and concentric circles about the origin in P correspond to similar paths in T and T. Fig. VII Fiff. VIII Pig. IX — 21 - Let the two - leaved surface , T (Fig. VII) , be regarded as cut into pieces along the straight lines joining the origin with the branch-points, a, p, y, etc., and extending to infinity. In the elliptic surface T, the branch - points are a'*, [3^, Y^ °°. We choose the branch-lines so as to join a^ to p^ and f to oo. If the branch- line a? [3^ be the conform representation of the line a p in T, and if, further, we agree that the point at the origin in the upper sheet of T shall correspond to the like point in T, Then the strip a^ p^, extending to infinity , corresponds exactly with the strip aOp , upper sheet with upper , and lower with lower. A similar correspondence holds between the strips P^Oy' and pOy- The upper sheet of -fOct.^ corresponds to the strip between yO — a , beginning with the upper sheet at and passing into the lower sheet beyond the branch - line. The remaining sheet of yO— a corresponds to the lower sheet of •f — OcL^. In a similar manner, the upper sheet of — aO — p corresponds to the continuous sheet in a^ [3^ that lies above from to the branch - line , and below beyond it. The remaining sheets then correspond. The upper sheet of [3^0y'^ corresponds to the sheet of — pO — y that is above from to the branch -line, and below beyond it. The remaining sheets then correspond. Finally, the strip — YOa cor- responds to -fOcf? without any inversion in the order of the sheets. In the surface j > the branch - point replaces oo of T , and, accordingly, the second branch -line runs from to y^- A.s in the previous case, the strips aOp, POy , —YOa correspond respectively to a'Op', P'Oy', fOcf.\ But that sheet of YO-a which is above at 0, and below beyond the branch -line, corresponds to the lower sheet of y^O^^- The upper sheet of — aO — p corresponds to that which is below at , and above beyond the branch - line in the strip a'^Op^; while a similar correspondence of sheets holds in regard to the strips — pO — y and p'*OY^ Having established this correspondence between the hyper- elliptic , and the two elliptic Riemann surfaces , it is easily seen that to a path in the former corresponds unambiguously a certain path in either of the latter ; but to a path in an elliptic surface corresponds simultaneously two paths in the hyperelliptic. Of particular interest are the period -paths. In Fig. VII is represented a canonical system of such paths , so chosen as to represent the reduction properties of the periods in the simplest form. The images of K^, K^, K^ may be made to coincide with 2K (i.e. Z" twice traversed), K',K respectively, by simple deformations which respect the connectivity of the surface, while - 22 -^ the image of K^ can be contracted into a point. Likewise , in T the images of K^,K^,K^ can be made to coincide with 2K,K,^', while the path corresponding to K^^ can be contracted to a point. By means of these correspondences we may at once express the hyperelliptic periods in terms of the elliptic. Let 2(o, — — -" — along _V^i(C) the paths K and K' respectively. Also let 2w, 2(o' be similar results for the integral u .=/, /^~77\ ? when referred to the paths K, K'. Then we find for w^ , w^ the following table of periods : If we put Kr K, ^3 K, w. 4(1) 2(0' 2(0 w. 4(0 2w 2w' ^1 = r- ^1 * 4(0 ^ 4(0 then Vj, ^2 are normal integrals of the first kind with the table of periods, K K ■^3 K, ^l 1 1(0' 1 2(0 2 v^ 1 1 1(.^' 2 2a^ — 23 — This table establishes the truth of the Picard-Weierstrass theorem^) for our case, viz., t,2 = ^. § 6. Integrals of the Second and Third Kind. In the present section we propose to discuss the reduction to elliptic integrals of Klein's commutative integral ^) of the third kind x'y>^ r" r'' \lf{^) \lf{z[,z[) + a'X ' {^d,){z'd,') xy V -^y' 2(^^') • \Jf{,^,z,) V//K,<) ' and Weierstrass' „ Normal integrals of the second kind", w^, w^, associated ^) with the two reducible integrals of the first kind and x'y' with the integral Q^ „ . A. Integrals of the second hind. The explicit expressions for W3, w^ are given by Wiltheiss (1. c. pg. 276). For our canonical form (7) they are - -A /- ^Wa^.,^j '6a^z\){zdz) T Similarly, in the two elliptic fields we find that to the system of normal integrals of the first and second kind. (27) 1) Picard — Bull. d. 1. Soc. Math. d. F., 1. 11, pg. 43; Kowalevski — Acta Mathematica, Bd. 4, pg. 400. 2) Klein — Hyperelliptische Sigmafunctionen, Mathematische Annalen, Bd. 27, pg. 443. 3) Cf. Wiltheiss — Ueber eine partielle Differentialgleichung etc., Math. Annalen, Bd. 29, pg. 275; and Bolza — On Weierstrass' Systems of Hyper elliptic Integrals etc. (A paper read before the Mathematical Congress, Chi- cago, 1893). 24 belongs the covariant commutative integral of the third kind, (28) ^.,^ J J 2(-T ^B:^Q^IiMA)^ 7) 7) and to the system, M, = (WC) (29) belongs «^ = _ ri^^o C^ + 3a, C. C, + ^3 CD (CdC) qvJ2.(c„g (30) v.^ j^j^, 2(ccT V^MgyV^^g' where -^^(Ci, Cal C'„ Q and -F, (Ci, Ca; Ci, Q are the second polars of i?i (Ci, C2) and B,^ (Ci, C2) respectively. By inspection we are easily led to the result: Theorem IV. The hyperelliptic integrals w^, w^, w^, w^ admit of reduction to elliptic integrals as follows : (31) w, = u, «^4 = iW2 + ?«2Wl If the periods of Wj along the paths K, K' are 2-/], 2"/)', and the periode of u^ along the paths K, K' are 27], 27]', then the integrals w?, , w^ have the following table of iutegrals ^^ ^. ^. Jfi w. 27j + ta,w, V+I«i<^'. 7] + fa,o) w^ , 2^ + fa,w ?+|«aW, 7]'+|a,a>' J5. Integrals of the third hind. Passing to non - homogeneous — 26 - notation for convenience, and introducing the integrals J^ J^, «' V^ (C-C')v/iJ.(C) '' an easy calculation establishes the relation, (32) S, + S, = 2S+log^-f^ + 2mni where m is an integer depending for its value upon the paths of integration xy, x'y'. Returning to homogeneous notation, the integrals x'y' rri' l'r{ xy St) 5t] are expressible in terms of 5, 5^, S^ by the formulae*) Q = S — ^ w '^w ^ xy a=l,2 " 2 + a From this follows at once : Theorem V, The covariant commutative integral Q satisfies xy the reduction-formula, (33) I ^y ^i«/2+^2«/i Vi^l+vX ^n % 1) Wiltheiss — Math. Annalen, Bd. 33, pg. 269—270; Bolza — Oq Weierstrass' Systems etc., pg. 11. — 26 — The hyperelliptlc integrals of the second kind can now be readily expressed in terms of elliptic c-functions. Fori) o referring to ^r^, ^3, and o referring to ^„ g^. Hence, from (31) follows Also, since ^) ^'^ '-o(»^')<«f)' we 'obtain from (33) s^'^'^'+ioff^i^^^t^ . Vii^+y^y'i. = 1 q(^ )qW ) ' ^2/ ^ ^1 2/2 + ^2 2/1 2/1 ^; + 2/2 ^i (wj'^') a (uf) (34) { __ _ ' ' a(w )a(Wi ) § 6. The Hyperelliptic a- Functions. "We now propose to express Klein's hyperelliptic 0- functions in terms of elliptic o - functions , and for this purpose we make 1) See, for instance, B 1 z a — On the First and Second Logarithmic De- rivatives of the Hyperelliptic c-functions, American Journal, Vol. XVII, pg. 11. 2) Klein — Hyperelliptische Sigmafanctionen , Math. Ann. Bd. 27, pg. 466. — 27 — use of their definition in terms of the integral of the third kind ^) (f^ , in which the path of integration xy is the conjugate of the xy path xy, point for point. When we place this restriction on the path x'y', formula (33) reduces to (36) 2/^+iog J^^Mi ^ qS Ql'+naM^y+ «.(^^/] xy (^12/2 + ^2^2)" ^n e^] the paths of integration now being such that m = 0. Since the o- functions are covariants ^) of the sextic /'(^j, ^2)? we may perform our computations on the canonical form (7), or, as we may also write it, to which the original sextic has been reduced by a linear trans- formation of determinant 1. § 7. The Odd a -Functions. To every decomposition of the sextic into the product of a linear and a quintic factor corresponds an odd o-function. If, for example f = m- w^jj/, where m = mJ^z^—cLZ^, n = nJ^2^-{- az^, and ^, X have the same meaning as in (6) , the corresponding function we shall denote by ci^(w^, iv^ , and , in accordance with Klein ^), it may be defined by the equation where ^ __ ;^ ,,{zdz) ^„ _ r ^.(^^^) r z,{zdz) r y ^f{^v ^2) -4 \lf^v ^2) On the other hand, let Oi(w), <:ij^u), aj^u) be the three even a- functions belonging to the invariants *) g^, g^, and g^{u), ci^{u), Oj(w) those belonging to g^, g^, where 1) 1. c, pag. 449. 2) 1. c, pag. 439, 8) 1. c, pag. 449. ,^ 4) Cf. § 2 of the preceding. — 28 — Then, in accordance with Klein's definitions (1. c. pg. 455), and in view of formulae ^) (15), we write (u) = fe y^x-'^^g.)(->]-p'7),)(ir]-T'iri,)+\/4a„Y],(y3 - «\)(6-p-g(g-f ^J \\ ^1 (36)/ 2Vi2.(^)i2,(7,) e CI ^ ,,(w) = V4ao^.(e.-«^aY],-p\)(7j,-Y^^+V/4^o-ri.(T],-a\)(S,-13^$,)6-Y'y .^^g^ while o,(m), 02(«/) are obtained directly from o,(m), a,(w) by inter- changing a and p, and 03(w), ^.(w) are similarly obtained by inter- changing a and y. The odd elliptic o- functions are (37) I tX«)(B.(il) I * = «««•'< It may be remarked, in passing, that the sign ambiguities in the above expressions need not be taken into account, since the functions in which they occur will hereafter be used only in com- binations from which these ambiguities virtually disappear. In order to express a„(tfi. w^ in terms of the elliptic o- functions, we observe, in the first place, that oK^i, w'g) is the product of an exponential e^xy ^ and an algebraic part whose denominator is ^\lf\x)f{y) and whose numerator is rational in ^ii^vVvVv Further, if each of the four products (i^{u)o^{u), <32(m)o,(u)j 1) If we denote the left hand of (36) by ai(M), and let 5^(u), a,,(tt) be ob- tained from Ci{u) by the permutations (ap), (a-jf) respectively, then, since cfjj(«) = 1 — I e;iM'-f . . . , by applying to this last equation, and also to (15), the permutations (ap), (ay) and comparing results, we find X = 1, p. = 2, v = 3. — 29 ^ "sW^aC^)* o(w)a(M) be multiplied by the exponential c J which will be denoted briefly by s{u,u), the resulting quantities will each consist of the product of an algebraic part together with the exponential factor (38) e(u,u).e ^^i ^]. By means of (35) the exponential (38) can be replaced by 2 \Jx^x,y,y, ^ Q^y When we make the substitution the four products e(M, u) o^(m) o^Wj ^^^' > will each consist of the exponential factor , and an algebraic factor, rational in x^,x.^,y^,y^ except for ^f{x)f{y) which occurs in the denominator. We see , on inspection , that o^(«(;„ w^) cannot be expressed in terms of any one of the above four products alone. The next simplest expression being a linear combination, we proceed to determine whether constants A,B,C,D can be found such that (39) ol{ti\,w^) = e(M,M)[^o,(M)aXM) + Ba,{u)(i,{u) + Ca^uXiu) + 2>o(w)o(w)J When we make the substitutions indicated above, divide out the common factors, and clear of fractions, we have the relation / i^iV, + ^22/,) (^1^2 - ^22/i)' • <{^, - a^ J (2/1 - a2/2) ,= 2JaJ^,^./^^a^a;D(t/:FP^2/D(2/^T'2/D+M2(2/!-a^2/D(^:-r^D(^:-TX^)] (40)<;+25a,[a;,^,(a;:-p^^^)(l/^-aVD(2/:-YVD+2/x2/2(2/^P^^2)(^!-aX^)(^:-TX)] + 20aJ;r,a;,(a;^Y';rD(y^r2/D(y:-«VD+2/.y.(y^YVD(^;-r^D(^!-a'^D] + 4{A^B + CXx,y, + x,y,) \JMf'(y) + ^Di^y.-Kyiy which must be an identity for all values of x^, x^^ y^, y^. •— 30 — In this equation we make the two specializations x^ = a. a;, = Y 1) y, = p '• 2) y, = a ^2 = 2/2 = 1 ^2 = 2/2 = 1 and obtain the conditions ^ Aa,^{f-a.*)-Ba,a{^'-f) + B = Aa.-da'- p^) - Ca, a (p«- f) -B = 0. If, further, we make w^^w^ = u^u = in (39), we have the additional condition A+B+G = 0. These three equations of condition give for the ratios of the constants A, By C, D the values By putting iCj = — a, j^^ = p, a;^ = 2/2 = 1, D is found to have the value , D = — | »»^a . If, now, these values be substituted in (39), the relation is found to be, in fact, an identity. Accordingly we have (41) oj. (m;„ w^ = B b(u,u)[<5^{u)o^{u) - ap^Xiu) + a,a(P' - fHu)o{u)] , where B= *^" 2aor-T') ' It only remains to express the coefficients of (41) in terms of the simultaneous invariants of m, n, (}>, x- For this purpose let m ^ m, = m[ = m^z^-\-m^e^ n = n^ = n', = w, e^ ■\-n^z^. Then, in the symbolic notation, we have ' (42) (wm) = 8w„n(,a - 31 - from which we obtain _g ^ i(^^) (t{>n)(x«')((l;X) Also, from (12) and (42), remembering the notation of (15), § 8. The Even a -Functions. The even a -functions are defined by the equation xy where ^3, (jjg are two cubies such that 93 ^^ = f. In the present special case the ten functions divide themselves into two classes ac- cording to the way in which the linear factors of f are distributed in ipj and ^^. The linear factors of f may be denoted by m, n, p, q, r, 5, where mn = (p, pq = (jj, rs = /. The two classes of 0- functions are, then, I- ?») '^a of the form The corresponding o-functions may be designated by the notation There are six such functions. ^' ?3} ^3 of the fofm 'I's (^1, ^2) = w(^^, ^j • q (^1, ^2) • s (^x) ^2) • The corresponding a-functions will be denoted by O/mpM^v ^2)- They are four in number. OF THl UNIVERSITY % OF — 32 "<''-^ Class I, The six functions of this class must he expressible in the form (43) ^)^p\ i^v ^i) = s («j «) [-4 Oi (w) <3, (m) + -B Oj (m) o, (m) 1x2/ + Co3{tt)a,(«) + 2)o(u)o(u)]. For, by the addition, in equation (41), of a properly chosen set of half periods to w^, w^, and the corresponding half periods to u, u, the equation may be made to take the form of (43). This process would, in fact, lead us directly to the values of A, B, C, D, but, in the absence of sufficiently explicit formulae for effecting such a transformation, it is easier to determine these values by the method of indeterminate coefficients. Substituting the explicit expressions for the o - functions dividing out common factors, and clearing of fractions, we have mi + Ba, [x^x, {x]-^X) (y!-a VD if-fyl) + y.y. iy'-^yl) K-^'^l) (^'--(X)] + Ca, [x,x, {xl-fxD (yl-^Y,) {y]-«.Y,) +y,y, {y\-fyl) {x\-^X) (•*!-«'a:J)J + {A + B-^C){x,y, + x,y,) \IMM + JD{xlyl-xlyy. Making the four specializations. Ix^ = a 1) k = P 'X^ = —Cf. I^. =^3 = 1 X — Y (^, = 2/2 = 1 u = u = we obtain the following conditions, Aa, p (T'-a*) - Ba, a (p'-r') + D = Aa,^{f-a') + Ba,a{^'-t') + D = J5a„ Y (a'-p^) + Ca, p (T'-a') + i) = A + B + C = 1, ' from which me find B = 0, A == C = h ^ = -i«oP(T-a') = -^{pq)F. Class II, The four functions of this class are expressible in a form similar to (48). By the same method as in the prece- — 33 — ding, we derive the equation ^1^2 + ^2^/1 (45). 2 L«o(^i-a-^2) (^ -P^s) (^•1-1^2) {y^+ ci.y,) {y,+ ^y,) {y^+ yyj +a^{y~cLy,) (y-^ (2/1-T2/2) (^i+a-^^) (^i+P^a) (^1+ T-^i,)+2V^/'(a;)/"(^)] = Aa,[x,x, (x^clX) if-K) iyl-fyl)+y^y. {yl-<^'yl)i^l-?'^l){^\-f<)] + Ba, [x,x, {xi-^-'xi) (2/:-aVD {y\-fyl)+y.y. {y\-m) (^^-aVj (x^-T^^D] + c\ [^,^, (a^!-TX') (2/:-I3'>:) { y:-^'yl)+ y.yM-fyl){^\-^'K) {<-<^'^\)] + {A + B + C){x,y, + x,y,)\lflx)f(y)+Dix\yl-xlyl)\ By making tlie following specializations, ix^ = a Ix^ = a. Ix^ = p yi = P 2) L^ = Y 3) 3/^ = Y 4) ^2 = 2/2 = 1 la^2 = 2/2 = 1 1^2 = 2/2 = 1 we obtain the conditions, M? if- a") - -Sao* (P - t') + ^ = ^«oT (« - P') - ^«o« (P - f ) - D = A + B + C = 1 from which we find _ a(P^-f) (a-P)(P-T)(Y-a) jg _ P(f-a") |t(;j^ = tt;^ = 'w = M =0 C = (a-P)(P-T)(T-a) t(«-P^) (a-PJ(P-T)(T-a) D = 0. These constants are readily expressed in the invariant form when we observe that «^oa(P-f) = ifnn)E «oP(T-«^) = ipg[)F «oT(a-P') = M^ «o(«-P)(P-T)(T-a) = «o«(P-f)+«oP(T-a^)+«^oT(a^-P^) = {mn)E+{pq)F+{rs)G. The results of the present and the preceding section we will summarize, for convenience, in the following theorem. 3 - u - Theorem VI, Let tp = mn, ^ = pq^ -^ = rs he three qita- dratic forms in the variables z^^ ^„ whose skew invariant R is zero, and let m, n; p, q; r, s be their linear factors. Further, let J.,, A^; A[, A[ he constants such that cp, ^, )^ are each linearly expressible m terms of {A^z^ + A2Z^f,(A[z^ + A'^z^Y. If, besides, we put r {A^z, + A,z,){zdz) ^ w,= r V94'X ' " Jy V/(p. - 36 — written as the product of its linear factors as follows, f{,,,,,) = (^a)(^p)(^Y)K)^13')(n'). Then, ommiting the arguments of the o- functions for the sake of brevity , we will denote by o the function corresponding to the decomposition of the sextic into the linear factor (^a) and the remaining quintic, and by oi ,^, ,1 the function corresponding to The Weber equation, in its general form, will be a relation between the functions p = a^,, q = ^\, r = a^, s = o^(^,P,^,). From the relation^) we eliminate the even functions by means of the formulae '^) (pT)a'(pY^') = (pT)5 + {aa')(Tp')(TT')2-K)(PP')(PTV (46) {(Ta)^\ppY) = (Ya)5-(Pa'){Tp')(T7')P + (M{ap')(aT')r and obtain immediately the relation sought, viz., \l'mPm)s + (aa')(Tp'r(n')2- K) W(Pt')^J (47) + V(Ta) g [(Ya) s - (^a') (tP') (ttO P + (?«') («?') {ay') ^] + V{aP) r [(ap) s - (ya') (a^') (ay') g + ha') (PP') (Pt') 3] = 0. If we compare this with Kummer's equation^) 1) Burkharidt — Systematik der hyperelliptischen Functionen, Math, An- nalen, Bd. 35, pg. 242 (1890). 2) 1. c. pg. 241. 3) Kummer — Ueber die Fiachen vierten Grades, mit sechzehn singularen Punkten. Monatsberichte der Berl, Akad., 1864. — 36 SJpiH + V + 8s + W(a'i> + 'i'r + S's) + \Jr (a> + ^"q + h"s) = it is found that the constants satisfy the required conditions a'Y + a"p - Py = a'Y + P"T-aT = 0. Referring to the reduction formulae (§§ 6 — 7), we observe that a new set of variables a?,, a;,, x^^ x^ may be introduced by means of a linear transformation such that X, = a (u)a (u) ^8 = <5^(w)a^(w) x^ = (m) a (m) . The resulting equation is that of the "Wave Surface '). B. Borchardt's Equation. Borchardt's equation ex- presses the relation between the four functions of a Gropel's quadruple, such as, for example, aPY\ /ap'YX /a p t'\ /a p Y\ a'pYJ' U'PtV' Wt/' WPtJ' Starting with the first equation of (46), we derive two others by permuting : 1", a with p , and a' with p' ; 2"^, a with y , and a' with 7'. Solving these three equations with respect to 0', we obtain (48) \m in') KM M («P') + (P't) (t'«) (^'P)] o« = (P't') (t^) («?) <'^(^,^,J,) (PY0(Ta)(aP0a'(^,^'^,)-(P'T)(7'*P)c»'(^,^,;[)+(pT)(T and from this, by permuting a with a', p with p', and 7 with y', we have (48') (-(P'7)(T'a')(a'P)a'(^,P'.^,)- (PT')(Ta')(a'P')o'(^|:[') + (P'T')(T«')(a'?)<'.(^|'j') 1) Study — Spharische Trigonometrie , orthogonale Substitutionen und elliptische Functionen, pg. 225. — 37 - Burkhardt's formula (94), (1. c. pag. 242), gives us Pt7 Va'P'T (49) Squaring each member and eliminating al, o^ , by means of (48) and (48'), we obtain, after a little reduction, tbe equation sought, viz.. In the special case in which a, a'; p, P'; Y) Y' ^^^^ pairs in an involution, the 15 Gopel quadruples composed of even functions divide themselves into three classes. Class I. One quadruple, „(<'?i\ „('='P't\ J^^A J'^^'A Class II. Six quadruples of the type /a^yX /otp'yN (aa'^\ Wp'\ HarT'J'Ha'pW'HTT'PrHTY'pJ' - 38 - Class III. Eight quadruples of the type /a,3Y\ m'a\ /aa'T\ M'^\ °Vp'Tr lnvj'Hi3?YJ' 'wrr I. In the equation of Class I (see formula (49)) one of th? terms disappears, since (PT')MW) + (?'T)(A)(a'p) = 0, as we may immediately see where we take the sextic i^ the canonical form of § 6. Equation (49) then easily reduces to where < - \/(P'T)(T«)(a?0(?T'){TV)Wa'(^,^'^,) = ^^^ta'pT') x\ = V(?T')(T'«)(--P)(i3'Y)F')W) -'{^J^?^) = ^a^ta'^'T) It is to be observed that the sign of one of the radicals may be chosen arbitrarily. The signs of the others are there deter- mined by the conditions \ = -^-(y'P«P') A3 = -A,.(YaYp) A, = -A,.(p'aYa') where , for brevity , (7'p a p') is used to denote the anharmonic The Kummer surface, in this case, becomes a doubly - covered quadric surface. n. Using the quadruple given for illustration un^r Class 11, the equation connecting these functions may be obtained at once from (49) by permuting a' and y'- If in this we make the sub- stitution — 39 — ^. = v'G3T)(Ta)(aP)(i3'T')(TV)(a'P') a (^, J,^^,) = ^K ^Qi^] X., = v/(P'T)(Ya)(a.3')(P7')(Y'a')(a'P) o(^,^'^,) = VA; a (^,^^';^,) ^, = (/(P'a') (a'a)(ap') (S)(Y7)Wr ^ (^'^^,') = "^^ 'i^^''^^) the resulting equation takes the form x\-\-xl + xl + xl + Xx^x^x^x^ + {1 (ic^ic* + ;r|ii7^) The roots of unity involved in the above radicals we may choose arbitrarily, subject only to the conditions When we make the linear transformation ^1 = 2/1 + 2/2 ^2 = 2/3 + 2/* ^z = 2/1-2/2 ^4 = 2/8-2/4 the equation assumes the simpler form 2/t + 2/2 + 2/3 + 2/4 + ?^'(2/Jy^ + 2/8 2/D + P-'(2/i2/I + 2/2 2/D + v'(2/>: + 2/l2/D = 0. This reduces to the well known equation of the Wave Surface — = when we make the substitutions DC ^ ca ^ ah a» — ^^2/2 .,^ _ ca«/| _ aby\ ^ r" = ic'^ + «/^ + ^». — 40 — The equations of Class III represent a surface more properly associated with the reduction problem for a transformation of the fourth degree , and will not he considered further in this connection. It is what Humbert calls an Elliptic Kummer Surface of index 4*). 1) G. Humbert — Sur les surfaces de Kummer elliptiques. Americaa Journal, Vol. XVI, pg. 221. University of Chicago, August 1895. ^Vita. I, John Irwin Hutchinson, was born at Bangor, Me., on the 12*^^ of April, 1867. I completed my preparatory coarse at the Edward Little High School, Auburn, in 1885, and in 1889 was graduated from Bates College , Lewiston , with the degree, Bachelor of Arts. During the two years 1890—92, I was Scholar, and Fellow in Mathematics at Clark University, where I attended lectures by Bolza, Perott, Story, Taber, and White. The two following years I was Fellow in Mathematics at the Uni- versity of Chicago, attending lectures by Bolza, Hale, Laves, Maschke, Moore, and Young. Also for a considerable portion of the time during one year I was occupied in laboratory ex- periments and astronomical observations at the Kenwood Astro- Physical Observatory of the University of Chicago. To the above professors and lecturers, and in particular to the Professors Bolza, Hale, and Moore I gratefully acknowledge my indebtedness. ("- Of THL \ UNIVERSITY ) ?---"t'J^^^ 0^25 CENTS I'D 21-100..12. 43 (8796s) mm