MR. A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 
113 
and the coefficients are given by the table. 
1 
1 
1 
1 
1 
1 
2 
2 
3 
1 
1 
2 
3 
4 
4 
5 
1 
1 
2 
3 
5 
5 
7 
7 
8 
1 
] 
2 
3 
5 
6 
8 
9 
11 
11 
12 
1 
1 
2 
3 
5 
6 
9 
10 
13 
14 
16 
16 
18 
(0) 
( 1 ) 
( 2 ) 
(3) 
(4) 
(5) 
( 6 ) 
And subtracting from each coefficient the coefficient immediately preceding it, we 
have the table — 
1 
1 
0 
0 
1 
0 
1 
0 
1 
1 
0 
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 
( 0 ) 
( 1 ) 
( 2 ) 
(3) 
(4) 
(5) 
(6) 
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 I ; a co variant 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 O ; but that the irreducible covariants are 
connected by an equation of the degree 6, viz. there is a linear equation or syzygy 
between O®, PH®, PJFPU, and J®U®; 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^PPFT, or else of the form 
and the number of the asyzygetic covariants of the degree m is equal to 
the coefficient of in or what is the same thing, in 
1 — 
(1 — a?)(l— 
