1883.] of. Elliptic Functions. 11 



These transformations shew that it is not possible to express 

 en (1 + i*J2) u in terms of en u, or dn (1 + i \J2) u in terms of dn u, 

 by a rational transformation. 



Again, if -^ = 2, 



then A; = 3 - 2 V2 ; 



and if x = sn u, y = sn (1 + 2i) u, 



(i ^_Vi l-\ 



V sq 2 2co) V sn"' W 

 then y = (1 + 2.) « ——-——-— --,— , 



where w = i (K — iK') ; 



leading to the equations 



i__^_V/i + 



1 — y __1 — kx I sn o> \ / sn 3a> 

 1+2/ 1 + &# I x 1 \ i x 



sn a) sn 3w ' 



l—ky_l — x/ sn 2w \ / sn 4w 

 1 + % 1 + a; I 1 . a? 



+ — 5-/ M- 



sn2co / sn 4o> 



so that en (1 + 2i) u has a factor dn u, and da (1 -f 2i) u a factor 

 en u. 



K' 



Also, if -=f = V6, 



then fc=(V3-V2)(2-V3); 



and if # = sn ?/-, y = sn (1 + i \/6) w, 



then 

 2/ = (1 4- i *J6) x 



l-^-)[l--^T-lfl- 



sn 2 2a> J V sn 2 4co A sn 2 6 



>to 



(1 - &V sn 2 2g>) (1 - lix z sn 2 4<y) (1 - ¥x l sn 2 6o>) ' 

 where w = \{K — iK') ; 



