1882.] Multiplication of Elliptic Functions. 275 



The form of the transformation may be inferred from the con- 

 siderations that 



(1) when 1 — y = 0, 



I (1 + i *Jn) a = 2mK+ 2im'K' (m + m even), 



(in + hri Jn) (i — i \Jn) ir 

 a = - - -k, 



P 



m + ran „ m —m . „, 

 = - - A + — - iA', 

 P V 



— 4»i'A"+ (m — ni) to, 

 = 4un K + 2sw, 

 and therefore x = en 2sco ; 



(2) when 1+^ = 0, 



i (1 + i */n) a = 2mK + 2im'K' (in + m odd), 



a — 4f»iK+ (in — in) w, 

 = 4tm'K+(2s+ l)w, 

 and x = en (2s + 1) w ; 



(3) when y + ic= 0, 



I (1 + i a/>0 a = (2m + 1) K + (2m' + 1 ) ill' (in + m even), 

 a = (4sm + 2) K + (m — in) &>, 



= (4>m' + 2)K+2sa>, 

 x — — en 2so) ; 



(4) when y — ic = 0, 



I (1 + i V«) a = (2m + 1) K+ (2m' + 1) iK' (m + m odd), 

 a = (4»i' + 2) K+ (2s + 1) &>, 

 # = — en (2s + 1) ft). 

 Therefore 



1 - y = PI4 (en 2s&) - x) -r- D, 



1 + y = QI1 {en (2* + 1) ft) - a?) -f- A 

 y + ic = ATI (en 2s&> -f a?) -f- A 



y-ic= SU {en (2s + 1) ft) + a-] - D ; 



and P, Q, R, S are determined from the simultaneous values x = 1 

 y = 1 and u; = 0, ?/ = \J(— ic). 



