ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 345 



In precisely the same manner we shall find, continuing the process, 

 Al(«i + 2mKj + 2rKj + 2stI, . . . . ) 



oy ya py Py, 



from which we may infer the truth of the theorem. 

 Section 8. — Now assume 





;„= i(K„.ii;,+K„,2v,+ .... +K,,,.i^„) ; 





whence we obtain equations of the form 



Vj = n-(G,, 1^1 + 62,1^2+ .... +G„,iM„). 



t-^=:7r(Gi_ 21*1 +62,2^*2+ • • • • +G-„,2Wn)j 



V„ = 7r(Gi,„nj + Cr2,nWa+ +G„,„M„) ; 



from these equations we have manifestly 



Then from the first of equations (2), section 6, we have 

 2.(G..,o'K,A.,-G,,A,cK.,) = 0, 

 ^AK o2 A'. o'K, . - J. o2c,G,, ,,K,, ,,} = 0, 



or 2^{ K^_ c21c'(Cf „', c'^v, c' ~ ^v', e'Gfy, c') } 



+ S^{K,, ,S,,G,, , J,, , - J,, ,2,G,, ,,K,, ,, } = 0. 

 But S,{K^, eS,rG,, ,,J,,, ,, - J,, ,2,,G^, ,,K^, ,,} 



(B) 



