541 
156 | A FIFTH MEMOIR UPON QUANTICS. 
that is, as the Jacobian of the cubic and Hessian; and under the form 
h (9 a , d b , d c , d d \y, - x) 3 □, 
that is, as the evectant of the discriminant. 
The discriminant, taken negatively, may be written under the form 
+ 4 (ac — b 2 ) (bd — c 2 ) — (ad — be) 2 , 
that is, as the discriminant of the Hessian. 
117. We have 
(a, b, c, (Tfybx 2 + 2cxy + dy 2 , — ax 2 — 2bxy — cy 2 ) 3 = U<£>, 
which expresses that a transmutant of the cubic is the product of the cubic and the 
cubicovariant. The equation 
{(3«, 3 6 , d c , defey, -x) 3 } 2 U =2 U 2 
expresses that the second evectant of the discriminant is the square of the cubic. 
The equation 
d 2 , 
— 3 cd , 
— 3bd + 6c 2 , 
— 3 be + 2 ad 
- 27 D 2 
— 3 cd , 
— 3c 3 +12 bd, 
— 3ad — 6bc , 
— Sac + 6b 2 
— 3bd + 6c 2 , 
— 3ad — 6bc , 
— 3 b 2 + 12ac, 
— Sab 
— 3 be — YLad, 
— 3ac + 6b 2 , 
3 ab , 
a 2 
expresses that the determinant formed with the second differential coefficients of the 
discriminant gives the square of the discriminant. 
The covariants of the intermediate aU + /3<b are as follows, viz. 
118. For the Hessian, we have 
H(<xU + №)= (1, 0, -□£«, /3) 2 H 
= (a 2 -/3 2 D )H; 
for the cubicovariant, 
and for the discriminant, 
0> («Z7H- /SO>) = (0, □, 0, -D 2 5«, /3) 3 U 
+ (1, 0, 0\a, /¡Y* 
= (a 2 - / S 2 D)(aT> + y SDH); 
Ù(aU+0®)= (1, 0, - 2D, 0, D 2 -%a, /3)^T> 
= (a 2 — /3 2 □ ) 2 □, 
where on the left-hand sides I have, for greater distinctness, written H, &c. to denote 
the functional operation of taking the Hessian, &c. of the operand all + /3<1>.