1883.] of Elliptic Functions. 15 



o/ , o\ i-l-2j2c-c-ic n I — -. a /kA 



* — 1 + z^Sc — c — %c 



From (4) and (5) we have a/3 = — ic, which satisfies (2); 

 then by (3) and (5) 



(a + /3) 2 = - 2j=Tc(l-ic), 



o / , o\ i-1-2 J2c -c-ic ( I — - N2 



2 (a + /3) = - - — — ? = r (1 - V- icf; 



% - 1 + 2 72c - c - *c 



therefore we must have 



(i— 1 — 2 J 2c- c-icY ^ . ~ / — -, 2 , — - ~ 

 U — 1 + 2 v 2c — c — icj 



or 



(i - 1 - 2 72c - c - ic) 2 [1 - ic - (1 - 72c] 2 



= - 4 (1 - i) J¥c (1 - w) («-l + 2 V2c- c- tc) 2 . 



The right-hand side of this equation 



= - 2* + 8 J 2c + 32ic - 40c J 2c - 68i'c 2 



+ 40c 2 J2c + 32ic* - 8c 3 JYc - 2ic\ 

 and the left-hand side 



= 8 (i + 1) sJ2c - 64ic + 40 (i -Y)cj2c 



+ 40 {i + 1) c 2 J 2c - 64ic 3 + 8 (i - 1) c 3 72c, 

 therefore equating we get 



2% + 8* x/2c - 9Qic + 40i'c 72c + 68ic 2 



+ 40tc 2 J¥c - 9Qic 3 + 8ic 3 J 2c + 2ic* = 0, 



or 1 + 4 J2c - 48c + 20c 72c + 34c 2 



+ 20c 2 JYc - 48c 3 + 4c 3 J¥c + c 4 = 0. 



Substituting the above values of c and J 2c, the left-hand side of 

 this equation becomes 



= 1-12-873 + 875 + 4/15-1344-76873 



+ 576 J 5 + 336 JlE 



+ c {- 60 - 40 73 + 40 7-5 + 20 Jl5 + 952 + 544 73 



- 408 75 - 238 715} 



