10 Mr Qreenhill, Complex Multiplication [Oct. 29, 



For instance in the general transformation of an odd order, if 

 x=cnu, y = cn(J^, >J 



1 



1 - y = (i - x ) 



OS 



1 + 



cn2s<w 

 x 



T 



en 2sco J 



%C \ ICJ 



1 + 



IC \ IC/ 



1- 

 1 + 



V 



en (2s — 1) &>J 

 x 



en (2s — 1) o>_ 



When n is an even number however, the sn function must be 

 employed ; for instance, if 



K! 



vjr = a/2, then k = *J2 — 1 ; 



and if a; = sn u, y = sn (1 + * \/2) w ; 



/-i • /on sn a 2o) 



y = (1 + * V2) * ., — Ta-2 — ?o— > 

 * v y l-&Vsn 2 2&> ' 



where to = ^{K — %K') ; 



leading to the equations 



rp 2 



1 _ N 



1 — y \-\-kx I sn &> \ 



1 + 2/ 1 - Aa: I j 



+ 



snor 



Also 

 so that 



and 



1—Jcy_l —x /1+ &# sn 6) 

 1+&?/ 1 +% \1 — kccsxico 



snco = J-i, 



sn 2 2&> = 



V2_-j 

 V2-1' 



^ 1 + * V2 ^-(V2-W2-*>* 



_ (1 + z V2) a; - (V2 - 1) ^ 3 



1 - (V2 - 1) (V2 - i) a? ' 



