[574 
575] 
103 
writing 
575. 
ON A SPECIAL QUAETIC TEANSFOEMATION OF AN ELLIPTIC 
FUNCTION. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xii. (1873), 
pp. 266—269.] 
ant without the term in 
It is remarked by Jacobi that a transformation of the order n'n" may lead to a 
on of ¿4 b y and 
modular equation 
A' ri K' 
A “ n" K’ 
and in particular when n' = n", or the order is square, then the equation may be 
A' K' 
— = -g. ; viz. that instead of a transformation we may have a multiplication. A quartic 
transformation of the kind in question may be obtained as follows: writing 
4(0©)' . . 
(<9©) 4 ; and similar ‘y 
, ., . - 1 . 4(6»©)' 
s£ that of d in 
X = (a, b, c, d, effx, l) 4 = a (x — a) (x — ¡3) (x — 7) (x — 8), 
H the Hessian, <I> the cubi-covariant, I and J the two invariants, then there is a 
well known quartic transformation 
2 H 
Z= ~X' 
ng powers of 6 contains 
leading to 
dz 2 V(— 2) dx 
nic term; hence differ- 
V(£) * 
and the expressions in 
where Z = 0 3 - Iz + 2 J. In fact we have 
3 6 71 - 1 in terms of the 
that is, 
jsting. 
V(Z) = ^=^V(X), 
Murphy.