ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS. 285 



grals. In a similar manner he denotes byF, (w,r), F2 (w.t;), , . . F^^{u,v) 

 the functions of a similar nature to which one would come in taking for 

 point of departure the equations 



(II. 



y + cx \^ y + fy 



\.^^ ^^ ' .2/0 ^^•^' 



where o, p, y, and are constants, and ^ {x) a polynomial of the fifth or 

 sixth degree in x. 



Hermite proposes as follows the problem of transformation : The poly- 

 nomial 4- (a;) in (I.) being given, determine the coefficients of i// (a;), and 

 the constants o, /3, y, and I in such a manner that the fifteen functions 

 F (it, v) be rationally expressed in terms of the fifteen functionsy(?/,r). 



By comparison of the linear relations that must exist among the periods 

 of the/(M, v) functions and the F (m, v) functions, and of the relations 

 that exist among the periods that belong to these respective functions, 

 many remarkable consequences are deduced. In this connection see a 

 letter of Eisenstein to Hermite ('Liouv. Journ.' xvii.) and also Eisen- 

 stein, 'Ueber dieVergleichung von solchen ternaren quadratischen Formen, 

 welche verschiedene Determinanten haben ' (' Sitzungsber. der Berlin. 

 Akad.' June 1852), and Hermite (Crelle, bd. xlvii. p. 343). From these 

 papers is seen the intimate relation that exists between the analytic theory 

 of transformation and the arithmetical theory of quadratic forms. 



In the execution of the transformation a system of sixteen ^-functions 

 is introduced, sixteen functions which may be algebraically expressed in 

 terms of any two of them. 



Four new functions, FTo, n,, rij, and II3, are introduced, which may be 

 expressed by an integral homogeneous function of degree k in four of the 



2 , 1 

 0-functions. The IT-functions contain linearly - constants. There 



are just enough conditions of the problem to determine these constants. 

 Further, the fl-functions are defined in terms of other 9-functions. From 

 this follow immediately relations among the quadruply-periodic quotients 

 which arise from the division of two 0-functions and those which arise 

 from the division of two fJ-functions. These last functions may be ret^arde J 

 as representing the more general periodic functions which orginate from 

 the hyperelliptic integrals of the first order, when the arguments x and y 

 have been replaced by others which depend linearly upon them in any 

 manner. 



Thus the proposed problem of transformation is solved.' On p. 704 

 of the ' Compt. Rendus,' t. xl., Hermite gives a method of division of the 

 H-functions, and compares them with Weierstrass' Al-functions of the fol- 

 lowing article. 



Liouville and Hermite made use of the periodic properties of the single 

 0-functions, and derived for the elliptic functions the results of addition, 

 multiplication, transformation, and division ; and Hermite by direct trans- 



' Cf. Brioschi {Cornet. Rend, xlvii. p. 310). 



