132 
[23 
23. 
ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
[From the Philosophical Magazine and Journal of Science, vol. xxvn. (1845), pp. 424—427.] 
In a former paper [19] 1 gave a proof of Jacobis theorem, which I suggested [21] 
would lead to the resolution of the very important problem of finding the relation 
between the complete functions. This is in fact effected by the formulae there given, 
but there is an apparent indeterminateness in them, the cause of which it is necessary 
to explain, and which I shall now show to be inherent in the problem. For the sake 
of supplying an omission, for the detection of which I am indebted to Mr Bronwin, 
I will first recapitulate the steps of the demonstration. 
If I«, £ v (*) be the complete functions corresponding to 0, x, then this function 
is expressible in the form 
<px = adl f 1 4- 
1 4 
mw + nu/ v ra +1 w + n+^v' 
Let p be any prime number, p, v integers not divisible by p, and 
0 = 
gw + vv 
P 
The function 
0 (x 4- 26) <p (x + 4$) <b (x + 2 (p — 1) 6) 
fax = <f)X 
<f)2d 
040 
4>(-2( P -i)e) 
is always reducible to the form 
a?II 1 4- 
+n 9+s 
m! + -I w + n' +1 v ' 
Analogous to the K, K'i of M. Jacobi.