274 



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



from the modular equation; and if 



cc = sn a, y = en \ (1 + i V7) a, 

 l_# = P(l-a;) {ic-x) + D, 



l+y=QWic-z) 2 +D, 



y + ic = R (1 + x) (ic + x) -h-D, 

 y — ic = S {\Jic + xf -f- D i 



a transformation of y in terms of x of the second order. From 

 the simultaneous values x = 0, y = \/(- ic), corresponding to a = K, 

 it follows that 



1 — y _ ji 1 — x ic — x 



y + i c V c ' 1 + x ' i c + x ' 



i + y _ /i /v*c — «y 



y — ic V c ' \\A'c + a;/ 



Conversely these two equations may be proved to be consistent, 

 and to lead to the differential relation 



dy 



— \{\+i V7) dx 



VKWMr + c 2 )} V{(i-0(* 2 + c 2 )}' 



where c = 8 + 3 \/7. 



I am indebted to Captain P. A. Mac Mahon, R.A., for the dis- 

 covery of a misprint in Cayley's Elliptic Functions, which retarded 

 the verification ; and also to Mr Pilkington, Fellow of Pembroke Col- 

 lege, for the algebraical verification of this theorem. The misprint 



1 — i Jc 



is on p. 74, and consists in putting en h (K+iK')= — - K / y , or 



— sf(—ic), instead of V(- * c )> as it should be. 



Generally, if -^ = *Jn, where n is of the form 4tp — 1 ; or 

 = 3 (mod. 4); then if 



and 



and 



a transformation of the order p. 



