692] 
333 
692. 
ADDITION TO THE MEMOIR ON THE TRANSFORMATION OF 
ELLIPTIC FUNCTIONS. 
[From the Philosophical Transactions of the Royal Society of London, vol. clxix. 
Part II. (1878), pp. 419—424. Received February 6,—Read March 7, 1878.] 
I have recently succeeded in completing a theory considered in my “Memoir on 
the Transformation of Elliptic Functions,” Phil. Trans., vol. clxiv. (1874), pp. 397—456, 
[578],—that of the septic transformation, n — 7. We have here 
1 — 2/ 1 — (CL — fix + 7Æ 2 — &r 3 \ 2 
1 +y 1 + & \ a + /3# + 7& 2 + 
a solution of 
Mdy _ dx 
Vl — y 2 . 1 — v 8 y 2 Vl — x 2 .1 — u 8 x 2 
where i = 1 + — : and the ratios a. 
M a ’ 
determined by the equations 
: (3 : 7 : 8, and the m>-modular equation are 
u li a 2 = v 2 8 2 , 
u 6 (2«7 + 2a/3 4- /3 2 ) = v 2 (f + 2yS 4- 2/3S), 
7 2 4- 2/3y + 2 a8 + 2/S 8 = v*v? (2ay + 2/3y + 2aS 4- /3 2 ), 
8 2 4 27S = v 2 u 10 (a 2 4 2a/3) ; 
or, what is the same thing, writing a = 1, the first equation may be replaced by 
8 = —, and then, a, 8 having these values, the last three equations determine /3, 7 
and the modular equation. If instead of /3 we introduce M, by means of the relation