1883.] of Elliptic Functions. 21 



Then, if — = V6, 



k = (^3 - V2) (2 - V3), 

 a case already considered. 



But if Z = V3' 



& = (V3 + V2) (2 - V3) ; 

 and if x = sn iTv, y = sn (-KT+ 3/ZsT') u ; 



then 



1 ^- 1- 



sn 2 2&)/ v sn 2 4&)/ V sn* 6<y 

 y = (1 +j\/o) # 



00 



(1 -&Vsn 2 2o) (1 - &Vsn 2 4a>) (1 - fcVsn 2 6ft)) ' 

 where co = ^ (K - SiK'). 



In the preceding transformations the factor | has been intro- 

 duced in the numbers a and b of the complex multipliers a + bi\/n; 

 it is interesting to see how other factors like ^, \,... may come in. 



For instance, if in the expression of y = en \ (1 + i V5) u in 

 terms of x = en it, we put 



then m = ^ (1 — i */5) v, 



and the same expression will give #=cni(l — i*Jo)v in terms of 

 y = en ?; by the solution of a cubic equation. 



Again, if z = en \ (1 + i a/5) v, then 

 /i #? i flu 

 y = t J(-ic)J[ ^V £, a = cni(/v-z7r), 



a , = cn^(Z+«X'), 



connecting a? = en ^ (1 — i a/5) » or x = en m, 



and « = en £ (1 + i \/5) v or z = en £ (2 — i a/5) m ; 



