144] 
A THIRD MEMOIR UPON QUANTICS. 
323 
The discriminant is 
which is equal to 
No. 55. 
a, h, g 
h, b, f 
ih f c 
abc — a/ 2 — bg 2 — ch 2 + 2fgh. 
The reciprocant is 
which is equal to 
No. 56. 
& V> ? 
£ a, A, 0 
77, h, b, f 
t g, f, C 
(bc-f 2 , ca-g 2 , ab-h 2 , gh-af, hf-bg, fg - ch%%, v , 
The discriminant is, it will be noticed, the same function as the Hessian. The reci 
procant is the evectant of the discriminant. The covariants are the quadric itself and 
the discriminant; the reciprocant is the only contravariant. 
Next, for a ternary cubic, we have the following Tables : 
Covariant and other Tables, Nos. 57 to 70 (a ternary cubic). 
No. 57. 
The cubic is U = 
(u, b, c, f, g, h, z, j, k, y, z) 3 , 
which means— 
ax 2 + by 3 + cz 3 + 3fy 2 z + 3gz 2 x + 3 hx 2 y + Siyz 2 + 3jzx 2 + 3 kxy 2 + Qlxyz. 
No. 58. 
The first derived functions (omitting the factor 3) are 
(a, 
k, 
g, 1, 
j> 
h l^x, 
y> z , 
(h, 
b, 
*. f 
l, 
k §x, 
z ) 
(j, 
f, 
c, i, 
g> 
l \x, 
lh z ) 
41—2