1883.] of Elliptic Functions. 5 



These transformations shew that it is not possible to express 

 sni (1 + i\Jn) u in terms of saw, or dn| (1 + ijn)u in terms of duu, 

 by a rational transformation, when n = 3 (mod. 4). 



For the remaining series of odd values of n, namely where n is 

 of the form 4^ + 1, or 



n = 1 (mod. 4), 



it is not possible to express cn|- (1 -\-i^/n)u rationally in terms of 

 cm*, although it is possible to express en (1 + isjn)u rationally in 

 terms of enw by a transformation of the n + 1 th order. 



I have however received a letter from Mr G. H. Stuart, 

 formerly Fellow of Emmanuel College, in which he points out that 

 when n = 5, that is, when 



K' 



~E = ^°' 



., en u 



cn 6(o en eo 



a transformation, so to speak, of the order 1 + \. 



This theorem can be immediately generalized; for if 



x = cn u, y = cn | (1 + i *Jn) u ; 

 where -^ = *Jn, 



and n is of the form 4p + 1, or 



n = 1 (mod. 4); 



k' 

 and, if c = -v 



and 



K-iK' 



2p + l ' 





- /(- in , cn(2 r 1)m > 



cn(zs — 1) <» 

 a transformation of the order p + -|, equivalent to 



i_ 2 ,s =(1 ^ c)(1 _«) H ( 1 __y^i>, 



