869] 
505 
869. 
ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
[From the American Journal of Mathematics, voi. ix. (1887), pp. 193—224.] 
The algebraical theory of the Transformation of Elliptic Functions was established 
by Jacobi in a remarkably simple and elegant form, but it has not hitherto been 
developed with much completeness or success. The cases n = 3 and n = 5 are worked 
out very completely in the Fundamenta Nova (1829) ; viz. considering the equation 
Mdy _ dx 
VI-y 2 . 1 -xy ~ VI - x 2 .1 -Utf 5 
{k = u*, X = v 4 -, say this is the MkX- or Muv-form), Jacobi finds, in the two cases 
respectively, a modular equation between the fourth roots u, v, say the try-modular 
equation, and, as rational functions of u, v, the value of M and the values of the 
coefficients of the several powers of x in the numerator and denominator of the 
fraction which gives the value of y\ but there is no attempt at a like development 
of the general case. I shall have occasion to speak of other researches by Jacobi, 
Brioschi and myself; but I will first mention that my original idea in the present 
memoir was to develop the following mode of treatment of the theory : 
In place of the MkX-form, using the pa/3-form 
dy _ pdx 
Vl — 2 fiy 4 + y* V1 — 2 ax 2 + x 4, 
(I write for greater convenience 2a, 2/3 in place of the a of Jacobi and Brioschi and 
the fi of Brioschi), we can, by expanding each side in a series, integrating, and 
reverting the resulting series for y, obtain y in the form 
I 
C. XII. 
y = px( 1 +II 1 « 2 + n 2 # 4 + ...), 
64