12 Mr Greenhill, Complex Multiplication [Oct. 29, 



leading to the equations 



1 — y 1 + kx 



1 + y 1—kx, 



x 1 + 



and so on. 



If ra is a composite number rr, then the modular equation of 

 the rath order is obtained by combining the modular equations of 



K 



the orders r and /, and the modulus when -^ =*Jn is obtained 



by putting \ = Jc is the modular equation. 



But another root of the same modular equation will be the 

 modulus when 



K' _ y 



K ~V r' 



and by properly choosing r and r' it will be possible to represent 



any assigned value of -~ . 



The general problem of the "Complex Multiplication of Elliptic 



Functions " is then to express the sn, en, or dn of (aK+ biW) v in 



K' It' 



terms of the sn, en, or dn of Kv, where -^ = \/ > but ^ * s 



necessary that b should contain a factor r, so that the complex 

 multiplier always reduces to the form 



a + bi Jrr . 



. K' 



For in expressing the elliptic functions of i -^ u in terms of 



the elliptic functions of u, the first transformation of the order r 

 from the modulus k to a smaller modulus X, and then the second 

 transformation of the order r from A, to a larger modulus h' must 

 be employed ; so that if N, N' are the corresponding multipliers 

 of the transformations, 



N=K N' = *- 

 »'A' A' 



