23] 
ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
133 
and thus fax is an inverse function, the complete functions of which are \ fa, i fa : and 
fa, fa are connected with co, v by the equations 
O) —— — (0.03 -f- gfa), 
fa = 1 (a'co + g'v), 
p 
a, g, fa, g' being any integers subject to the conditions that a, g' are odd and fa, g 
even ; also 
ag' — fag = p, 
pg' — vfa — Vp, 
pg —va= Ip, 
l, l' being any integers whatever. In fact, to prove this, we have . only to consider 
the general form of a factor in the numerator of fax. Omitting a constant factor, 
this is 
(1 mo> + nv + 2rd) < ^ ’ 
and it is to be shown that we can always satisfy the equation 
or the equations 
ma> + nv + 2 r6 = m fa + fa fa, 
pm + 2 vp = vfa & + fa fa, 
pn + 2rv — vfag + fag'; 
and also that to each set of values of m, n, r, there is a unique set of values of 
vfa, fa, and vice versa. This is done in the paper referred to. Moreover, with the 
suppositions just made as to the numbers cl, g' being odd and fa, g even, it is obvious 
that vfa is odd or even, according as m is, and fa according as n is, which shows 
that we can likewise satisfy 
m + ho)+n-\-%v + 2r6 = vfa + ^ fa + fa + \ v'; 
and thus the denominator of </>,# is also reducible to the required form. 
Now proceeding to the immediate object of this paper, cl, g, fa, g', and consequently 
w!, fa are to a certain extent indeterminate. Let A, B, A', B' be a particular set of 
values of a, g, fa, g', and 0, P the corresponding values of fa, fa. We have evidently 
A, B' odd and A', B even. Also 
AB'-A'B= p, 
fiB' — vA' = L'p, 
fiB — vA — Lp, 
0 (A co + B v), 
P 
U=-(A'a> + B'v). 
p y