284 EEPORT— 1897. 



where X is a rational integral function of the fifth or sixth degree ' in x. 

 By means of the quotients of two such ^-functions Goepel and Rosenhain 

 showed how to represent the functions (p{'n,v) and yl{ii,i-) (art. 37), and 

 thus completely solved the problem of representing the inverse functions 

 of the hyperelliptic integrals of the first order. 



The zeros and periodic properties of these two ^-functions, the relations 

 between the squares of such functions and of the constants that enter 

 these relations, the number of independent relations, Goepel's biquadratic 

 relation, the connection between these functions and Rummer's sixteen- 

 nodal quartic surface, and similar questions, are found in Harkness and 

 Morley's ' A Treatise on the Theory of Functions,' p. .341 et seq. 



(46) Goepel remarked that his investigations could be extended to any 

 number of variables ; but in this connection Jacobi showed that there is 

 a troublesome paradox ( Werke,' bd. ii. p. 521 ; and Weierstrass, 'Werke,' 

 bd. i. p. 142) ; since, when there are more than two variables, tlie 

 generalised 6- function contains more essential constants than the hyper- 

 elliptic functions with like number of variables. 



(47) We must mention next papers by Hermite, ' Sur la theorie de la 

 transformation des fonctions Abeliennes ' ('Compt. Rend.' xl. pp. 249, 

 303, 365, 427, 485, 536, 704, and 784). 



Besides the sum and the quotient of x and y (of art. 37), which we 

 saw could be expressed througli fractions whose numerator and denomi- 

 nator are functions of the argument u and v, and have unique and finite 

 values for all finite real and imaginary values of these arguments, Goepel 

 and Rosenhain gave in an analogous form the analytical expression of 

 thirteen other functions of m and v, which depend algebraically but in an 

 irrational manner upon the first two. 



Hermite- designates by f^{u,v), f.^ {u,v), . . .f\:Xu,v) this complete 

 system of fifteen functions which appear in the study of the integrals 



dx I y dx 



xdx , ( ydy 



n/?>0*^) I,, N/'/'(yy 



when (.t) denotes a polynomial of fifth or sixth degree in x, and which 

 are analogous to the functions sn it, en u, and dn tt, of the elliptic inte- 



' Cayley (' Memoir on the Single and Double 0-fuDCtion,' P7(27. Trans. 1880, pp. 

 897-1002) treats the whole theory in a manner analogous to that emplo}-ed by Goepel. 

 In this paper special attention is paid to the relations among the squares of the 

 functions and to the derivation of the biquadratic relation among four of the 

 functions, which is the same as Kummer's sixteen-nodal quartic surface. See also 

 Cayley (Crelle, bd. Isxxiii. pp. 210 and 23.5 ; and Forsyth (Biographical Notice on 

 Arthur Cayley, 'Obituary Notices' of the Proc. Boyal Society, vol. Iviii.), and 

 Cayley's Math. Fa'pe.rg, vol. viii. p. ix, where other references are given. Other papers 

 on the same subject by Cayley are found in Crelle, bd. Ix.xxv. Ixxxvii. and Isxxviii. 

 Prof. Forsyth, ' Memoir on the Tbeta-function, particularly those of two Variables ' 

 {PJiil. Tratis. 1882, vol. clxxiii. p. 783) follows more closely Kosenhain's paper, and 

 extends it in many directions. Cf. also Konigsberger (Crelle, bd. Ixiv. p. 17; bd. 

 Ixxxi. p. 193; and especially bd. Ixxxvii. p. 173, where the problem of transforma- 

 tion is discussed fully), and also Math. Ann. bd. xv. p. 174. 



'^ Hermite, Sur la tliiorie de la traiisforniaf ion des fonctions Aleliennes (Comj>t. 

 Rendus, t. xl. pp. 249, 303, 365, 427, 485, 536, 70i, 784). 



(i.)-i 



