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



By a repeated application of the process we can express 

 en {£(1 +in/n)]'a 

 in terms of en a by a transformation of the order p\ 



The case of n '= 11 presents peculiar difficulties ; we therefore 

 pass on to the case of -p = \/15, then i/(kk') = sin 18° (Joubert, 

 Comptes Rendus, t. 50); 

 and c = (7 + 4V3)(-t + V15), 



V(2c) = (2 + V3)(V5+V3). 



1 — y A' 1 — x ic — a; / \l ic — spy 



y + ic \c'l+x'ic-\-x \\/ic + x, 



1 -f- y li (a — x\~ f(3 — x 





If 



then 



y — ic \ c \a + xj \p + &v 



where a = en £ (if— £2T'), /9 = en f (A"— ?7i') ; and therefore a/3 = ic; 

 also 



cfo/ _ — ^ (1 + i a/15) dx 



These results have also been verified algebraically by Mr Pil- 

 ls ington. 



Returning to the case where -^ = a/d, then 



2MW5-2 = (^^" 



c = V5 + 2 + 2V(V5 + 2), 



, V5 + 1 /A/5 + 1 



(Abel, (Euvres. p. 382). 



It is not possible to express en ^ (1 + i a/5) a rationally in terms 

 of en a; but if x = en a, y = en (1 +* \/5) a, then ?/ can be expressed 

 in terms of a? by the sextic transformation 



i- y = i¥ i-a* ( cn 2 |(ir-a7r) -a: 2 | a 



1 + y x 2 + c* [en 2 £ (A" - iK') - x 2 ) ' 



or 



y + i c „ 2 f cn 2 i(A"+2t7r)-# 2 ] 2 

 y - ic °° (en 2 1 (2 A" + /if) - a?) ' 



