E 1 
XIII. Addition to Memoir on the Transformation of Elliptic Functions. 
By A. Cayley, Sadlerian Professor of Mathematics in the University of Cambridge. 
Received February 0, — Read March 7, 1878. 
I have recently succeeded in completing a theory considered in my c Memoir on the 
Transformation of Elliptic Functions,’ Phil. Trans., vol. 1G4 (1874), pp. 397-45(5 — 
that of the septic transformation, n=7. We have here 
1 — y 1 —x / a — /3x + yx 2 — SP \ 2 
1 + y 1 + x \a + /9.x 2 -t- 7 X 3 + ST ) ’ 
a solution of 
Mdy dx 
\/l — ?/ 2 . l—lfilf fl — -v? . 1 — 
12/9 
where — =M-|- — ; and the ratios a : /3 : y : 8, and the wv-modi dar equation are 
determined by the equations 
14 9 , 9 ^ 9 , 
u L a=vo, 
u G (2ay + 2af3+F) = v*(y~ + 2yS+2f3b), 
y 2 -f" 2 /3y -J- 2 aS -p 2/3$ = v z w 3 ( 2 ay -f- 2 /3y + 2 aS -f- /3 ~ ) , 
S° + 2y8= r 3 ?d°(a~-t-2a/3) ; 
or, what is the same thing, writing «.= ] , the first equation may be replaced by 
§=— , and then, a, S having these values, the last three equations determine /3, y and 
the modular equation. If instead of /3 we introduce M, by means of the relation 
1 .1 _ / 1 v y \ 
1 + 2/3, that is 2/3= — — 1, then the last equation gives 2y=w 3 'r 3 (— — — j ; and 
a, A, y, § having these values, we have the residual two equations 
A’(2ay-j- 2 <x/3 -fi- /3 ' ) = v~(y 2j r 2yS-j-2/3S), 
y ' + 2/3 y -{- 2aS + /3S = v~U'(2ay + 2 /3y -f- 2 aS -{- /3~) , 
viz., each of these is a quadric equation in — ; hence eliminating — , we have the 
modular equation ; and also (linearly) the value of and thence the values of 
a, /3, y, S in terms of u, v. 
