770] 
ON THE 34 CONCOMITANTS OF THE TERNARY CUBIC. 
349 
In several of the formulas, instead of 8 or 8 2 , the symbol used is (3) or (8 2 ); 
in these cases, the function operated upon contains the factor {abc + 81 3 ) or (abc + 81 3 ) 2 , 
and is of the form (abc + 81 3 ) {a TJ + b V+ c W) or (abc + 81 s ) 2 {a 2 U+ abV + &c.) : the 
meaning is, that the 8 or 8 2 is supposed to operate through the {abc + 81 3 ) a, or 
{abc + 81 3 ) 2 a 2 , &c., as if this were a constant, upon the TJ, V, &c., only; thus: 
{8).{abc +8l 3 ){aU+bV+cW) is used to denote {abc + 8l 3 ) {a8U + 6SF+ cSTF). As to 
this, observe that, operating with 8 instead of {8), there would be the additional 
terms U8 {abc + 81 3 ) a + &c.; we have in this case 
8 {abc + 81 3 ) a, = a {2a'bc 4- ab'c + abc + 24l 2 V) + 8l 3 a', 
= 24a 2 bcl 2 — 24<al 2 {abc + 21 3 ) + 48al 5 , = 0 ; 
or the rejected terms in fact vanish. For (8 2 ) . {abc + 8l 3 ){aTJ+ bV + cW), operating 
with 8 2 , we should have, in like manner, terms U8 2 {abc + 8l 3 ) a, &c.; here 
8 2 {abc + 81 3 ) a = a' 2 bc + 2aba'c' + 2aca'b' + orb’d + 24*l 2 aT + 24all' 2 , 
which is found to be = — 24a {abc + 81 3 ) {— abcl + l 4 ), that is, = — 24$ {abc + 81 3 ) a; and 
the terms in question are thus = — 24$ {abc + 8l 3 ) {aTJ + bV + cIF), viz. 
{abc + 81 3 ) {aU + bV + cW) 
being a covariant, this is also a covariant; that is, in using (S 2 ) instead of 8 2 , we 
in fact reject certain co variant terms; or say, for instance, 8 2 E being a covariant, 
then (S 2 ) E is also a covariant, but a different covariant. The calculation with {8) 
or {8 2 ) is more simple than it would have been with 8 or S 2 . See post, the calcula 
tions of K, K’, &e. 
I give for each of the 26 covariants a calculation showing how at least a single 
term of the final result is arrived at, and, in the several cases for which there is 
a power of abc + 81 3 as a factor, showing how this factor presents itself. 
Calculations for the 26 Covariants. 
13. ¥ = 3 (be' + b'c - 2ff', ..., gh' + g'h - afi - a'f, .. .$X, F, Z^X’, F, F) + TU 2 , 
= 3 {{be + b’c) yz - 2ll’x 2 , ..., 2U'yz-{aV + al)x 2 , ...\ax 2 + 2lyz, ...\a'x? + 21'yz, ...) 
+ T (a¥ +...). 
The whole coefficient of x 6 is 
- m’aa + Ta 2 , = 36a 2 l 3 {abc + 21 3 ) + Ta 2 , 
viz. the coefficient of a 2 x 6 is 
= 367 3 {abc + 21 3 ) + a 2 b 2 c 2 - 20abcl 3 - 81 6 
= a 2 b 2 c 2 + 16a6cZ 3 + 64£ 6 
= {abc + 81 3 ) 2 . 
X, 
X’, 
irv 
14. 
n = I ^Jac(i7, H, T r ), 
F, 
F', 
№ 
z, 
F,