' 
266 
[210 
210. 
ON THE CUBIC TRANSFORMATION OF AN ELLIPTIC FUNCTION. 
[From the Philosophical Magazine, vol. xv. (1858), pp. 363—364.] 
Let 
z = 
(<ab', c', d'\x, l) 3 
(a, b, c, d$x, l) 3 
be any cubic fraction whatever of x, then it is always possible to find quartic functions 
of z, x respectively, such that 
dz _ dx 
V(a, b, c, d, o§z, l) 4 *J{A, B, G, D, E^x, l) 4 
This depends upon the following theorem, viz. putting for shortness, 
U = (a, b, c, d $#, y) 3 , 
U' = (a', b', c', d’Jx, y) 3 , 
and representing by the notation 
disct. (aU' — a'U, bU'-b'U, cU'-c'U, dU'-d'U); 
or more shortly by 
disct. (a U' — a' U,...), 
the discriminant in regard to the facients (A,, /¿) of the cubic function 
(aU'-aU, bU'-b'U, cU-c'U, dU' - d'UJ\, /x) 3 ; 
or what is the same thing, the cubic function 
(a, b, c, d /¿) 3 . (a', b', o', d , \x, y) 3 
-« b', c', d'$\, pi) 3 . {a, b, c, djx, y) 3 ; 
and by J (U, U') the functional determinant, or Jacobian, of the two cubics U, U'; 
the theorem is that the discriminant contains as a factor the square of the Jacobian, 
or that we have 
disct. (aU' — a'IT, ...) = {J(U, U')}\{A, B, G, D, E\x, y) 4 .