1883.] of Elliptic Functions. 17 



Again from the original equation, we have 

 . „_ fo~l) s/*~c{ (ic - x*)* - 2s» J^ic~(l-ic)-(ic-x*) (1- J^k)*x] 



y D 



therefore ic + y = 



(icj2c- 2c - 2ic + jYc) { (ie-a?) a -2x' i l J^ic{l - ic) } - ( J 2 i+icj-2c + 2c + 2ic)(ic - a»)(l - J^ ic'f 2 x 



D 



This will be in the form 



ic + y 



D 



where Q = ic J 2c -2c- 2ic + J2c, 

 provided we have 



7 2c + ic JYc + 2c + 2ic _ i-1 -2j2c-c-ic 

 J2c + ic J2c -2c- 2ic i-l + 2 J2c- c - ic 



This gives J^c + icjjte = i-l-c-ic 



2c+2ic _ 2x /2c" ' 



1 + ic _ i — 1 — c — ic 

 1+7" ^2~ ' 



-2(l+ic) = Q.+i){(i-l)- c (i + l)} 



= - 2 - 2ic 

 an identity. 



Therefore ic + y takes the required form. 

 Thus l + y- f < g t«r<P + < 



therefore 1+y = P («+^(g+^ 



tC + y Q {QL-CB)*(fi-xf ' 



where - = *'~ 1 +2 ^ ~ c ~ *j 



Q icj2c-2c-2ic + j2c 



i-l + 2j2~c-(l+i)c 

 JTc [ic + 1 - J¥c (1 + ?') 



VOL. V. PT. I. 



