Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., late sadlerian professor of pure mathematics in the University of Cambridge (Vol. 11)

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.
	        
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