141] 
A SECOND MEMOIR UPON QUANTICS. 
263 
and subtracting from each coefficient the coefficient immediately preceding it we have 
the table: ’ 
1 
(0) 
(1) 
(2) 
(3) 
(4) 
(5) 
(6) 
1 
0 
0 
1 
0 
1 
0 
1 
1 
o 
1 
1 
1 
0 
1 
1 
0 
1 
1 
2 
0 
2 
0 
1 
1 
0 
1 
1 
2 
1 
2 
1 
2 
0 
1 
1 0 
1 
1 
2 
1 
3 
1 
3 
1 
2 
0 
2 
the examination of which will show that we have for the quartic the following 
irreducible covariants, viz. the quartic itself U; an invariant of the degree 2, which I 
represent by /; a covariant of the order 4 and of the degree 2, which I represent by H; 
an invariant of the degree 3, which I represent by J; and a covariant of the order 6 
and the degree 3, which I represent by ; but that the irreducible covariants are 
connected by an equation of the degree 6, viz. there is a linear equation or syzygy 
between <h 2 , PH 3 , PJH 2 U, IPHU 2 and J 3 U 3 ; this is in fact the complete system of 
the irreducible covariants of the quartic: the only irreducible invariants are the 
invariants I, J. 
38. The asyzygetic covariants are of the form U p PH r J s , or else of the form 
U p I q H r J s< $>, and the number of the asyzygetic covariants of the degree m is equal to 
the coefficient of x m in (1 + of) 4- (1 — x) (1 — ¿c 3 ) 2 (l — oc 3 ), or what is the same thing, in 
1 - x 6 
(1 -x)(l -,r 2 ) 2 (l -a?f' 
and the asyzygetic invariants are of the form I p J q , and the number of the asyzygetic 
invariants of the degree m is equal to the coefficient of x m in 14- (1 — x 2 ) (1 — x 3 ). 
Conversely, if these formulas were established, the preceding results as to the form 
of the system of the irreducible covariants or of the irreducible invariants, would 
follow. 
39. In the case of a quintic, the function to be developed is 
1 
(T - z) (1 - xz) (1 - <x?z) (1 - Pz) (1 - x A z) (1 - Pz) ’ 
and the coefficients are given by the table: