348
ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC.
[770
24 E" = $(8 2 )E.
30 Ë = —-^Jac (K, U, A).
31 Ë' = -\(8)E.
32 E" = -^(8 2 )E.
25 M = ^■Jac(C r , y i r , A).
26 M' = -(8)M.
33 M = - i [Jac] (P, F, A).
34 M'= i(8)M.
In explanation of the notations, observe that
U = ax 3 + by 3 + cz 3 + 6lxyz,
H = l 2 (ax 3 + by 3 + cz 3 ) — (abc + 21 3 ) xyz.
Hence, writing
QH = a'x 3 +b'y 3 + c'z 3 + 61'xyz,
we have
a', b', c', V = 6al 2 , 6bl 2 , 6cl 2 , — (abc + 21 s ).
And this being so, we write
X, Y, Z = ax 2 + 2 lyz, by 2 + 2 Izx, cz 2 + 2 Ixy,
a, b, c, f, g, h = ax, by, cz, lx, ly, Iz,
for £ of the first differential coefficients, and ^ of the second differential coefficients
of U ; and in like manner
X', Y', Z' = ax 2 + 21'yz, b'y 2 + 21'zx, c'z 2 + 2Vxy,
a', b', c, fi, g', h' = a'x, b'y, c'z, l'x, I'y, I'z,
for ^ of the first differential coefficients, and £ of the second differential coefficients
of 6H.
Jac is written to denote the Jacobian, viz. :
Jac ( Z7, H, ^) =
3 X U, d y U, d z ll
3 X H, d y H, 3 Z H
d x % 3 y % d z V
and in like manner [Jac] to denote the Jacobian, when the differentiations are in
regard to (£, y, £) instead of (x, y, z): 8 is the symbol of the 3-process, or sub
stitution of the coefficients (a', b', c', l') in place of (a, b, c, l); in fact,
3 — alb a + b 3& + c'd c + I'di :
3, 3 2 , &c., each operate directly on a function of (a, b, c, l), the (a', b', cl, V) of the
symbol 3 being in the first instance regarded as constants, and being replaced ultimately
by their values; for instance,
8abc = a'bc + ab'c 4- abc', 8 2 abc = 2 (ab'c' + abc' + a'b'c), 8 3 abc = Qa'b'c.