272 Mr A. G. Greenhill, On the Complex [Nov. 27, 



in terms of sn u, by means of a transformation of the pth order, 

 where p = a 2 + nb 2 , and a and b are integers. 



Denoting sn (a + ib s/n) u by y, and sn u by x, and putting 



cK + iK' 

 = &>, -where c 2 +rc is a multiple of p, then Jacobi states 



P 

 that 



, . , JTP- 1 ~ sir 2 Ysa> 

 y = (a+ib f Jn)xU^ ———-. 



It is necessary however to introduce the restriction that a is 

 an odd number, and b an even number : and it is easily seen that 

 this restriction is required in order that sn (a + ib *Jn) u may be 

 expressed entirely in terms of sn u. 



In the other two cases to be taken into account, namely, a 

 even and b odd, and a odd and b odd ; it is easily seen that if a 

 is even and b is odd, then sn (a + ib *Jn) u must have a factor en u; 

 and that if a is odd and b is odd, then sn (a + ib */n) u must have 

 a factor dn it. 



If however we work throughout with the function en, then 

 these restrictions disappear, as well as for the function dn. 



We shall therefore, henceforth, seek to determine en (a + ib <Jn) u 

 in terms of en u. This is equivalent to taking Abel's canonical 

 form of the elliptic integral 



dx 



V{(1 - *?) (1 + eV)} ' 

 and then e = tan 6, where 6 is the modular angle ; or 



dx 



V{(l-^)(c 2 + * 2 )}' 



and then c = cot 6 ; instead of taking Jacobi's canonical form 



dx 



Vl(l-aO(l~£V;}' 



where k = sin 6. 



Then Abel's function fat, is equivalent to Jacobi's function 

 cn(K-u). (Abel, CEuvres Completes,]). 265.) 



First, suppose -p = \Jo ; then as we know, the modular angle 

 is 15°, and c = cot^ = 1 -=2+ v/3. 



