638 
MR. A. CAYLEY’S THIRD MEMOIR UPON QUANTICS. 
the covariant operator 
may, if not with perfect accuracy, yet without risk of ambiguity, be termed the ^cec^or, 
and a covariant obtained by operating with it upon an invariant oi- covariant, or 
rather such covariant considered as so generated, may in like manner be termed an 
Eivectant. 
61. Imagine two or more quantics of the same order, 
(fl, h, ..fe?/)’" 
(a, VT 
we may have covariants such that for the coefficients of each pair of quantics the 
covariant is reduced to zero by the operators 
Such covariants are called Comhinants, and they possess the property of being inva- 
riantive, quoad the system, i. e. the covariant remains unaltered to a factor pr^s, 
when each quantic is replaced by a linear function of all the quantics. This extremely 
important theory is due to Mr. Sylvester. 
Proceeding now to the theory of ternary quadrics and cubics, — 
First for a ternary quadric, we have the following tables : — 
Covariant and other Tables, Nos. 51 to 56 (a ternary quadric). 
No. 5 1 . 
The quadric is represented by 
{a, b, c,f,g, h\x,y, zf, 
whi(di means — 
cz‘^ + + '2gzx -{-2hxy. 
No. 52. 
The first derived functions (omitting the factor 2) are — 
(a, h, gXx, y, z) 
{h, h, /Xj7, y, z) 
ig, f> cXx, y, z). 
