AN EIGHTH MEMOIR ON QUANTICS. 
182 
[405 
in terms of the other invariants), the coefficients (A, g, g, V) are expressed in terms 
of g, h, *, that is of A, B, G, viz. we have 
j" 72 V* 5 A =h(g— 1 6*) 2 - 9k (g + 16k) + (g - 16k) VZ, 
j 24 \/k 3 g = 9* 2 + 16hk — gh — Va, 
I 24 V£7/ = 9A; 2 + 16** — gh + VA, 
[ 72 V^A' = h (g - 16*) 2 - 9k (g + 16*) -{g- 16*) VZ ; 
these values of (A, g, g, A') could of course be at once expressed in terms of (J, K, L), 
but I have not thought it necessary to make the transformation. 
317. It has been already noticed that the linear covariant (G, = Pæ 4- Qy), was 
<IX, F) 3 , 
= V2 (V*, V*$Z, Y), 
it is to be added that the septic covariant (P'x 4- Q'y) is 
= VZ 3 (V*, - V*£Z, Y), 
and that the canonical forms of the cubicovariants fa (x, y), &c. are as follows: 
fa(X, Y) = VZ (g, 3 V*, 3 V*, g'\X, F) 3 , 
fa 2 (X, Y) = A(g, V*, - V*, - g\X, Y) s , 
[fa(X, Y)} = y/A*(g, - V*, - V*, Z$Z, F) 3 , 
{04 (Z, F)}= Z 2 (^, -3V*, 3 V*, -Z^Z, F) 3 , 
(Z, F) = VI 3 r (2 V* 3 - 3gk + gg 2 ), ^ 
3 ( V* 3 + gg V* — 2/i* ), 
j — 3 ( V* 3 4- gg V* — 2gk), 
[ — (2 V* 3 — 3gk 4- gg 2 ), j 
03 (Z, F) = VZ 3 (5/a, -V*, V*, 5^Z, F) 3 , 
0 4 (Z, F) = VZ 3 f (7 *JAg 4- 96 (2 V* 3 - Sgk + gg 2 )), 
— 3 (3 VZ V* — 96 ( Vk 3 4- gg V* — 2/x* )), 
j + 3 (3 A V* — 96 ( V* 3 4- gg V* — 2gk )), 
[ - (7 VZ,/ 4- 96 (2 V^ - 3//* + /V' 2 )) ; 
or, as the last formula may also be written, 
fa (Z, F) = VZ 3 r ((7/7 - 53* 4-110*) /a ~ 64Ag V*), 
| — 3 ((3/7 4- 151* — 90*) V* — 64A'/u. J )> 
j 4- 3 ((3/7 4- 151*. - 90*) V* - 64A// 2 ), 
[ — ((7/7 — 53* 4- 110*) g — 64A'g V*) 
<[Z, F) 3 
F) 3 .