ON THE HISTORICAL DEVELOPMENT OF ABELUN FUNCTIONS. 279 



Then by Abel's theorem we have in algebraic form the complete in- 

 tegrals of the system of equations. 



n j X (y(,)) 



(i=l,2,. . .y). 



(40) Hermite then takes for the inverse functions the quantities 

 a;,, X2, . . . Xj defined by the y equations 



IT'ij^ x(yo,) 



{i=l, 2, . . . y), 

 and writes 



Xi=\ (Mj, W2. • • • «,)> 



(i=l, 2, . . .y). 



It follows ' without difficulty that the y functions 



{i=l, 2, . . . y), 



are the roots of an algebraic equation of the degree y, whose coefficients 

 are rational functions of the different functions 



\i (Ui, M2, . . «,), \ {Vi, V2, . . . v^), 



{i=l, 2, . . . y). 



In the third section of this memoir, Hermite discusses the periodic 

 properties of these functions, and determines their periods. 



The theorem relative to the addition of the arguments leads to the 

 expression of the inverse functions in all their generality in terms of the 

 simplest particular functions, in which we may suppose successively that 

 only one argument varies, the others being constant, zero for example {cf. 

 art. 38). 



As an illustration, we saw in art. 24 that functions connected with 

 the hyperelliptic integrals of the second order arise, in which appear three 

 arguments, n (u, v, lu), say ; and from Abel's theorem it follows that 



n {u + li' + u", V + v' + v", 10 + w' -\- iv")= n (m, V, w) + n (u', v', w') 

 + n (m", v", w'') + alg. and logarith. function. 



Now writing 



u=zO,v = 0; u' = 0,io' = 0; v" =0, iv" = 0, 



we have 



n {u", v', w) = n (0, 0, to) = n (o, v', o) + n (u", o, o) 



4- alg. and logarith. function. 



At the end of this memoir are found certain theorems relative to the 

 transformation of elliptic integrals (cf. Hermite, ' Cours a la Faculte des 

 Sciences de Paris,' 4*"® ed., 1891) ; from these theorems formulae are 

 deduced which set forth in a beautiful manner many problems of trans- 



" See also Hermite (' Sur la division des fonctions Ab61iennes,' Memoires des 

 Savants etrangers, 1848, p. 672) ; and Eichelot (Crelle, bd. xxix. p. 281 ; and 

 Liouville's Journ. 1843, p. 505). 



