MR. A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 
J2I 
In order to render the covariants 0 definite as well numerically as in regard to 
sign, vve may suppose that the covariant is in its least terms {i. e. we may reject 
numerical factors common to all the terms), and we may make the leading term 
positive. The leading term with the proper numerical coefficient, if different from 
unitv and with the proper power of x, or the order of the function annexed, will, when 
the covariants of a quantic are tabulated, be sufficient to indicate, without any 
ambiguity whatever, the particular covariant referred to. I subjoin a table of the 
covariants of a quadric, a cubic and a quartic, and of the covariants of the degrees 
], 2, 3, 4 and 5 respectively of a quintic, and also two other invariants of a quintic. 
Co variant Tables (Nos. 1 to 26). 
No. 1. No. 2. 
( +1 « 
+ 2 b 
+ 1 c 
yY 
+ 1 ac 
-1 P 
The tables Nos. 1 and 2 are the covariants of a binary quadric. No. 1 is the qua- 
dric itself; No. 2 is the quadrinvariant, which is also the discriminant. 
No. 3. 
No. 4. 
+ 1 
-f- 3 i 
3 c 
+ 1 d 
3/)'- 
+ \ ac 
+ 1 ad 
+ \ bd 
-1 P 
— 1 be 
-1 r 
No. 5. 
No. 6. 
-f 1 a"d 
-|- 3 dhd 
— 3 acd 
— 1 ad^ 
— 3 abc 
— 6 ac^ 
-f6 P d 
+ 3 bed 
-1-2 P 
+ 3 Pc 
-3 bP 
—2 c^ 
yY’ 
+ 1 «"c?" 
— 6 ahcd 
+ 4 ac^ 
-t-4 Pd 
-3 
The tables Nos. 3, 4, 5 and 6 are the covariants of a binary cubic. No. 3 is the 
cubic itself ; No. 4 is the quadricovariant, or Hessian ; No. 5 is the cubicovariant ; 
No. 6 is the invariant, or discriminant. And if we write 
No. 3 = U, 
No. 4=:H, 
No. 5 = 0, 
No. 6=V, 
O"— VU"+4H*=0. 
then identically, 
