278 REPORT— 1897. 



exist simultaneously ; so that x^ and Xj are algebraic functions of the 

 four quantities x^", x.°, x^°\ and 0:2*'" ; that is, the functions of two 

 variables X(r«,'i;) and k-^{u,v) may be algebraically expressed in terms of the 

 four quantities of a single variable 



X(m,0), \{u,0\ 

 \{Q,v), \{0,v). 



Eisenstein (Crelle, bd. xxvii. p. 185) writes as follows regarding the 

 inverse functions : ' Great difficulties are found with the Abelian integrals 

 whose inverse functions have a triple or multiple periodicity. Under the 

 assumption that an integral with definite lower limit may take all possible 

 real and imaginary values for any given value of the variable, the Abelian 

 integral ceases to be a function of its variable. In order to meet these 

 difficulties, for example in the Abelian integrals of the first order, Jacobi 

 considered two such integrals connected by the relations (1) and (2) of 

 the preceding article. But if we grant that the function 11 (a;,) can have 

 all possible values for any given value of x^, the function n(a;.,^ may have 

 the same property for every given value of a;,, and so the sum u may for 

 a greater reason take all possible values for given values of x^ and x,^. The 

 same is true of ij ; so that it is not clear how we may speak in this wise 

 of a dependency between u, v, x^, and x.,.' Eisenstein then proposes, 

 in order to set forth the real nature of these functions after the analogue 

 of the elliptic functions (art. 11), to form the quotients of the quotients 

 of infinite triple products. 



Jacobi (' Werke,' bd. ii. p. 86) corrects Eisenstein's objections with the 

 remarks that Eisenstein did not understand the nature of the functions 

 }^(u,v), Xy{u,v), his mistake being that he did not sufficiently comprehend 

 the fundamental principle of the co-existence of the periods relative to the 

 two arguments m and v. He then asks if the quotients of quotients are 

 not simply quotients, and points out how Eisenstein has made some 

 fundamental mistakes in the theory of elliptic functions (Crelle, bd. xxvii. 

 pp. 185 and 285). 



(39) Hermite (Extrait d'une Lettre a M. Liouville, ' Comp. Rendus,' 

 t. xviii. ; 'Liouville's Journ.' t. ix. p. 353) introduces into the analysis of 

 the transcendents of any algebraic diSerentials the inverse functions of 

 several variables, after the example of that which had been done by Jacobi 

 for the hyperelliptic integrals of the first order. 



Using the notation of Abel and of Minding, he takes 



an irreducible algebraic equation, whose coefficients are rational and in- 

 tegral functions of x. The roots of this equation he denotes by y^, 1/2* 

 . . . y„. He further writes : 



f(^,y)=to+t\y+ • • • +t„-2y'"^ 



any rational integral function in x and y, in which the degrees of the 

 x's are subjected to certain restrictions. 



Finally, let y denote the number of arbitrary constants that are con- 

 tained in the function f{x,y) ; this function may then take y different 

 forms, which are represented by 



Designate by x^, X2, . . . x^. . . /j. variables where ^>y and by y(_, y^ 

 • • • y^.), irrational functions arbitrarily chosen among the n roots yi, y^. 



