ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 347 



Hence J'^,^ c' = 2 • ^'''. <=' + ^y^d^y'. c*^-. c)^'-. '' 



— 2 y', o''^^y^y'. v^ y, c'» 



This formula may be written thus, by merely changing the letters : 



= i Gy, c'^'y, + \2,(G„, ,J,, ,E:'„, ,K'.,. ,) ; 



therefore 2,(J;, ,K;,, ,, - j;, ,,KV. J = ^ S,(G,, ,K;, , - G^, ^.K',, ,) 



= |^(G..eK',,,-G,,,,KV,,). But 2,(J;,K;,-JV,,K;, J = 0; 

 hence 2,G,, ,K;, , = 2,G,, ,K;, , (4) 



Section 9. — Let us now put 



E(M^M^tt3 ) = 2 2 w^ w^ e^_ ,,, then 



y, c 



+ 22A2^a)^w,G^,/,,^. 

 Now 2,G^, f.<^v=2.G,, ^(miK^, i+^^zl^a, 2+ • • • • ) = '''V = 



hence 2^2,2^w^ti,G^^ /,_ ^ = 2,2,2^ w„i(^G,_ /,^ ^ = S.e^tt^ ; 



also 22,S,2^G,^ ^ J,_ ^w,w„ = 22e,w, 



E(«, + 2.^,. . . .) = E(«,. . . .) + 2%^eX%+w^), 



