MR. A. CAYLEY’S THIRD MEMOIR UPON QUANTICS. 639 
No. 53. 
The operators which reduce a covariant to 
zero 
are — 
( K 
b, 
2/IB„ 
B.)— ^B^ 
(%. 
/, 
Ba, 
B^)— ^B^ 
( a, 
2h, 
^ I 
Sa, 
B/)-3/B, 
( g^ 
2/, 
C I 
Ba, 
B/)-^B, 
( a, 
A, 
2^ I 
B/, 
Be) — ^B, 
(2h, 
b, 
Ba, 
BJ-^B^. 
No. 54. 
The evector is 
n, l)\ 
No. 55. 
The discriminant is 
a, h, g 
h, h, f 
g, L c 
which is equal to 
ahc—af'^ — hg^—ch^-\-^fgh. 
No. 56. 
The reciprocant is 
I, n, ^ 
I, h, g 
>1, h, h, f 
g, 
whieh is equal to 
{bc—f\ ca-g\ ah — h\ gh—af, hf-hg, fg-ch\l, ri, 
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). 
The cubic is U= 
which means — 
No. 57. 
(a, b, c, f, g, h, i, j, k, IJ^x, y, zf, 
ax^ -b by^ -f -j- ^fy'^z -f- 3g z^x -\-^hx^y iyz"^ + 2>jzx^ -f- 3 kxy'^ 6 Lxyz. 
