262 REPORT— 1897. 



shall obtain the set of roots x^ by equating E to zero ; and denote by 

 U any algebraic function of x^, x^, . ■ . x^. 

 Then 



Z\/{x,)j / F„F„...F,_ A-(g>L/(^,)j2; J / F.,F„...F,_A +^- 



J \ ^2) •^3' • • • ^r J \ ^li ^Z^ . . . X^ I 



The summation on the right-hand side is taken over all the roots of E^O, 

 which are assumed as the upper limits of the integrals ; while on the 

 right-hand side the summation is over all the roots F,=0, F2=0, . . . 

 Fr_i=0, considered as r — 1, simultaneous equations giving ccj, X3, . . . x^ 

 in terms of rcj. 



In connection with this paper we note a paper by Professor Cayley, 

 * A Memoir on the Abelian and Theta Functions ' (' American Journ. 

 Math.' vol. V. p. 137, and vol. vii. p. 101). 



The first chapter treats of Abel's theorem ; the second, a proof of 

 Abel's theorem. The connection between the lines of thought presented 

 in this paper and those of Professor Forsyth are particularly interesting. 

 In the further developments of Professor Cayley's paper, which is founded 

 upon Clebsch and Gordan's ' Treatise,' some geometrical results are brought 

 into prominence. The theory is illustrated by examples in regard to the 

 cubic, the nodal quartic and the general quartic respectively. 



The general case where the fixed curve is any curve whatever has been 

 solved with great generality by Nother, ' ' Zur Reduction Algebraischer 

 DifFerentialausdriicke auf der Normalform,' and ' TJeber die Algebraischen 

 DifFerentialausdriicke ' ('Sitzungsber. der Phys. Med. Soc. zu Erlangen,' 

 Dec. 10, 1883, and Jan. 14, 1884). Other addition-theorems, especially 

 for the hyperelliptic functions, are given in art. 32. 



(21) Periodic Functions of Several Variables. — In art. 10 the periodic 

 properties of functions of one variable were considered, and it has been 

 seen that Abel's theorem embraces the integrals of all algebraic functions. 

 Considering the inverse of these transcendental integrals, Jacobi dis- 

 covered the existence of the 2^s'>^'iodic functions of several variables, and 

 thus revealed the real significance and hitherto hidden properties of such 

 functions. 



Some of Jacobi's investigations ^ relative to hyperelliptic transcen- 

 dents are next given, since they may be used to illustrate Abel's theorem 

 for the more general integrals, and set forth the properties of the inverse 

 functions that are comprised in this theorem. 



If X denotes a rational integral function of the fourth degree in Xy 

 then by Euler's theorem (art. 5) transcendents of the form 



f ^ ^=U{x) 



enjoy the singular property that if 



n(.T,) + n(x2) =]!(«), 

 then a may be found algebraically in terms of rrj and x^. Owing to 



' See also Nother, Math. Ami. bd. ii. p. 314, bd. ix. p. 17; Brill and Nother, 

 Math. Ann. bd. vii. p. 269, and continuation in bd. vii. ; Klein-Fricke, Mliptische 

 Modulfunctionen, bd. 1. 1890, p. 533; Baker, Cambridge Phil. Trans, xv. ; Math. 

 Ann. bd. xlv. p. 133, &c. 



- Jacobi, Considerationes Generates de Transcendentihus Ahelianis, Crelle, bd. ix. 

 p. 394, 1832 ; Werhe, bd. ii. p. 7. 



