20 



Mr Greenhill, Complex Multiplication [Oct. 29, 



= -1. i ~ 1+ J> +iC \ic+a?) (1- J- icfj2c {(a/3 + a 2 / 



D 2 



(a+/3)<V} 



= - 2--- T> 2 (tc + O (1 - J - zc) 2 J2c (a 2 - x 2 ) (/3 2 - x 2 ) 



2 (i-l)(l-ic) 



i> 2 



therefore 



(ic + x'){l-J- icf J 2c (a 2 - x 2 ) (/3 2 - x 2 ) ; 



i dy 



(1 - ic) J 2c (a 2 - x 2 ) (/3 2 - tf 2 ) (ic + x 2 ) = 1 



D* 2(i-l){l-J-icf' dx' 



Substituting in (A), we have 



(l-yw+c 2 ) 



dy 



or 



(1 - x 2 ) (a? + c 2 ) 4 (i - 1) 2 (1 - 7 - ic) 4 " \dx) ' 



dyV (l-^ 2 )(^ 2 + c 2 ) = _ 4(i-l) 2 (l- N /^7c"y 

 dx) (l-tf)(tf + c?) 



2i (i - 1) (1 - ic - 2 V- ic) 



p. 





+ 2V2c — c — ic -J 



2i(l-ic-2V-7c)T 

 _l-(*+l) V2c + *c J 



• 2{o-y2o + »(l-V2o)} T 



. 1 - V2c + 1 (c - V2c) J 



' 2(31 + 18^ -14^5 -8 s /i5+»(4 + 2 N /3-2 < y5-^ 15)}T 

 _ 4 +i^/3 -2JJ-JU +1(31+18^/3 -MjHsJlE) . 



~l + i Vl5 



<fy 



±£(l + iVl5) . d# 



tlieref0rG V(l-2/ 2 )V + c 2 ) V(1-^(^+c 2 ) J 

 which proves the second part. 



As another example of the case when -^ = *Jn, where n is a 

 composite number, suppose n = 6. 



