ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 337 



We shall now indicate the method by which these formulse are to be 

 proved ; it will bo sufficient if we put n=3, which will guide at once to the 

 investigation for (n) greater than 3. Let 



(a;-aJ(x-a^X'^-a,)(a;-py(x-qy(x-ry~ 



(x—a,)(x-a,Xa;-aJ(a; - a,)(c„ + c^x+c^f^ 



and put in this equation 

 which also necessitates 

 and the equation becomes 



(a'-aJ(^-«3)(.^'-a,)(.^'-«o)C'^'-«J('-l-•-«J-('■«^-«6)(«o + <'l•'»-'+''2•-^'')' 

 = {x-x\){x-x'Xx-x\){x-x;){x-xX^-x;) (4) 



Putting in this equation successively x^x^, x=x.^, x^=x^, we have 



2 VEotT 



c Jf-C^X^-\-C^X^= — . L, 



° ' ■* ^3 — «6 



J , v/rZ" 



whence 



<'i = 





_jv/R^ x^ + x^ \/nx^ _ --^'^l + •^', 



o-'^-a, ■ ix'^-x,)(x^-x.^ x^-a, ' (x^-x^){x^-xj 



c - '^^ 1 a/R^ 1 



'^ ■" a7^-a„ ■ (.1?,- d?0(^-3 - ^',) ^2 - «G ' (^1 - •''2)(^'3 - -^'2) 



. -/R^ 1 



•-^'3-«6'(^l-''^'3)(^2-^'3) 



Substitute these values in equation (4), and we have, putting at the 

 same time x=^a^, 



1872. 2 A 



