ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS. 281 

 Hermite next proves that 



vi"'=yt~l 7n"=n— 1 m'=n— 1 tn=Ti — 1 



(B.) 5 ^ ^ :s /[n'* + ,v^+« ■ 



m"'=0 m"=0 »n'=0 m=0 >- ^ ' 



X J M + , V + -)1 T?"' ?'"' r"'" S"'"' 



= !y A + BA(A(mm, 7i7;)) + CA(A|(7m, ««)) + DA(A(nM, nv)) A(.\,('jm, ?tu)) 

 where A, B, C, D are rational functions of \(7iu, nv) and K^(nu, nv). 



The first member of this equation may be denoted by f {u, v), and 

 may be expressed rationally in terms of A (u, v) and Xj (u, v), since, owing 

 to the fundamental properties of the functions X and Xj, this is true of 

 each of the terms constituting ^ (->(, r). 



It is easily proved that 



, / iv-r'i n/ — 1 + k'i., + k"t3 ^/ — 1 + K"'ii 

 \ n 



Ki,'\/ — 1 + K'i'2 + K"i's\/ — 



1+k"VA 



11 

 =2^"1~'' '>'~'" ^"" 9{^h ■^')) 



whatever be the values of the integers k, k, k" and k". 



The «th power of <!> is a rational function in X (;/,, i'), X, (;/, v), which 

 does not change when for these quantities are substituted any two other 

 of the simultaneous roots of the proposed equations. It follows, there- 

 fore, from the theory of the symmetric functions of the roots of a system 

 of equations in several unknown quantities, that this function may be 

 rationally determined in the coefficients of the equation (A.) ; and since 

 any rational function of two roots A (X («?/, nv)), A (Xj (iiii, nv')) may be 

 put under the form 



A+ BA (\ {nu, nv)) + C A (A i (?m, nv)) + DA (X (nu, nv)) A (X , (nu, oiv)), 

 the theorem is proved. 



(42) Hermite says, in continuance of the above discussion, it seems, 

 that the preceding considerations may be extended to the hyperelliptic 

 integrals in general. 



For let ^x)=s/x{l—x){\-Xi-^x) . . . (l.-X22„,i.x-), 

 d,(x)=a, + /3^x + y,x-'+ . . . +7,X, 



and write 



.*, / \ P d,(x)dx 



Ui='-i{x,) + <i>i{x^) + <blx.i)+ . . . +*i(a;„), 

 and Xi=\^{u^, w„ u^, . . . u„), 



(i=--0, 1, 2, . . . n). 



Then -{x^)=d,{x,)^-':i + 6,{x^)^^ + l),{x,)^^+ . . . +0„{x,)p, 



(i=0, 1, 2, . . . n), 



where the partial derivatives may be rationally expressed in terms of the 

 functions A. Since W is of the wth degree, it appears that the roots of the 

 equation of the wth degree 



Q = %{x)P+ti,{x)P+e,{x)l^+ ■ • • +«n(-^)?— 



are the n functions a^o, Xi, . . . a!„. 



