ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 321 



Expanding this in terms of v', and equating the coefficients of v', we have 

 at once 



•P., </>3, fo, (V, ^)-^^^^^ = "^0, 3 0'iXO, 0) 03, , {V, w) 0,, , (V, lU) 

 and similarly, 



0., 03. 0^. (^. «')— ^l^T^ = V-o. , <PXl K 3 0'» «') 0a. 3 (^. *^) 



- 00. 2 0'1"2 03, 2 (^' "") 02, 2 i^y W)' 



(Z.02^*'jLi<') 



02. 00, 0\. ('^^ «') ^"'^f'"''"^ = 03. 3 9X1 03. 3 (^ ''') 0, . 3 (". "0 



-03, 2 0'i"2 03, 2 (^' '") 01, 2 (^'» "-')» 

 f^.02,_J,(!''Jf) 

 02. 00, 0'o. (^ '") "^""il'"^^ = 03. 3 0'r3 03, 3 (''- '^) 0.. 3 (^. ^^) 



-03,2 0'ir2 03,2(^.'<')0i,2(^'^«'); 



and substituting in these equations the expressions we have obtained in the 

 last section, we have equations of the form 



dv dii d 



u 



ds/_x^ ^ ^ d\^x^x^ ^, ds/x^x^ 

 dw du ' du' 



where A, B, A', B' are certain constants ; and we have two similar expres- 

 sions for 



d\^(l-xXl-^\) ^ dx^jl-xX^-^,) ^ 

 dv du 



whence we have 



du=adu+bdiv, du'=a'dv + b'dw, 



by properly choosing 



or and a', h and h' ; 



and therefore, finally, 



adv + bdxu = — . ax -\ .—— cix , 



1—u^x, , 1 — u'x. , 



Hence our formulae give us the solution of the hyperelliptic differential 

 equations. 



1873. T 



