141] 
A SECOND MEMOIR UPON QUANTICS. 
271 
leading terms of © x and © 2 do not either of them enter into © 3 ; and there is but 
one such covariant, © 3 . And so on, until we arrive at a covariant the leading term 
of which is higher than the leading terms of the other covariants, and which does 
not contain the leading terms of the other covariants. We have thus a series of 
covariants ©j, © 2 , © 3 , &c. containing the proper number of terms, and which covariants 
may be taken to represent the asyzygetic covariants of the degree and order in question. 
In order to render the covariants © definite as well numerically as in regard to 
sign, we 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 
unity) 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 
1, 2, 3, 4 and 5 respectively of a quintic, and also two other invariants of a quintic. 
[Except for the quantic itself, the algebraical sum of the numerical coefficients 
in any column is = 0, viz. the sum of the coefficients with the sign + is equal to 
that of the coefficients with the sign —, and I have as a numerical verification 
inserted at the foot of each column this sum with the sign + ]. 
Covariant Tables (Nos. 1 to 26). 
No. 1. No. 2. 
a + 1 
b + 2 
c + l 
5 x , yf- 
ac +■ 1 
b 2 - 1 
± l 
The tables Nos. 1 and 2 are the covariants of a binary quadric. No. 1 is the 
quadric itself; No. 2 is the quadrinvariant, which is also the discriminant. 
No. 3. 
No. 4. 
a + 1 
6 + 3 
c + 3 
d + 1 
y) 3 - 
ac + 1 
ad + 1 
bd+ 1 
b 2 - 1 
be - 1 
c 2 - 1 
± 1 
± l 
± 1 
\x, yf 
No. 5. No. 6. 
d-d 2 
+ 
1 
abed 
— 
6 
ac 2 
+ 
4 
b 3 d 
+ 
4 
b 2 e 2 
- 
3 
± y 
a~d 
+ 1 
abd + 3 
acd — 3 
ad? — 1 
abc 
- 3 
ac 2 — 6 
b 2 d + 6 
bed + 3 
b 3 
+ 2 
b' 2 c + 3 
be 2 -3 
c 3 — 2 
±3 ±6 ±6 ± 3 
The tables Nos. 3, 4, 5 and 6 are the co variants 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 iNo. 3=U, No. 4 = H, 
No. 5 = No. 6 = V,