Mr Oreenhill, Complex Multiplication [Oct. 29, 



3. Complex Multiplication of Elliptic Functions. By A. G. 

 Greenhill, M.A. 



In a previous communication to the Cambridge Philosophical 

 Society on Nov. 27, 1882, 1 had the honour of shewing that it was 

 possible to express 



in terms of x — cnu, 



by a rational transformation of the p th order, 



where 



K 



= \/n, 



when n is of the form 4<p — 1, or 



n = 3- (mod. 4). 



h' K — iK' 



Denoting -^ by c and by co, the transformation is 



rC p 



of the form 



2 



1 — y .. . N l — ^ tt 



1+4- 5 = 1 



1" 



cn2s&) 



or 



2C 



1 + 2/ // • \ *C TT 



^ = V(- ^c) 11 



V t x 



1 - -f- 1 + — s=i 



%C IC 



1 + 

 1- 



X 



cn2seo 



T 2 



en (2s — 1)g> 

 + cn(2s-l)ft)J 



if p is odd : and of the form 

 .= -V(-*c) 



r p-2 r 



or 





1 + ;/ 



1 +a? _ a; 



1 + — s=l 



1- 



I 2 



cn2s« 



1- 



y 



V(-w)II - 



if p is even. 



1 + 



1- 



1 + 



x 



h 



cn2so> 



en (2g — l)<p 



fl? 



en (2s— l)w 



