120 
[19 
19. 
INVESTIGATION OF THE TRANSFORMATION OF CERTAIN 
ELLIPTIC FUNCTIONS. 
[From the Philosophical Magazine, vol. xxv. (1844), pp. 352—354 ] 
The function sinam u ((fm for shortness) may be expressed in the form 
(f)U — u II ( 1 + j 
2mK + 2 m'K'i 
o™' ir',- ) ' ^ ( 1 + i 
2mK + (2 m' + 1) K'i 
•(1) 
where to, m! receive any integer, positive or negative, values whatever, omitting only 
the combination to = 0, m! — 0 in the numerator (Abel, Œuvres, t. I. p. 212, [Ed. 2, 
p. 343] but with modifications to adapt it to Jacobi’s notation ; also the positive and 
negative values of to, to' are not collected together as in Abel’s formulae). We deduce 
from this 
cp0 
= n 
v * 2mK + 2m'K'i + 6) 
2toA + (2to' + 1) K'i + 
Suppose now K = aH + a'H'i, K'i = hH + h'H'i, a, h, a', h' integers, and ah' — a'b a 
positive number v. Also let 6 =fH +f'H'i; f f integers such that a/'—af, bf—hf v, 
have not all three any common factor. Consider the expression 
_ <f>u <f>(u + 2a>) ... cf) (u + 2 (v — 1) ty) 
(/> (2<w) ... <fi (2 (v — 1) to) 
from which 
(3), 
V = «11 
2 mK + 2to K i + 2 rdj 
where r extends from 0 to v — 1 inclusively, the single combination to = 0, m =0, r = 0 
being omitted in the numerator. We may write 
mK + m'K'i + r9 = pH + p'H'i,