264 
A SECOND MEMOIR UPON QU AN TICS. 
[141 
(0) 
(1) 
(2) 
(3) 
(4) 
(5) 
and subtracting from each coefficient the one which immediately precedes it, we have 
the table : 
1 
0 
1 
1 
2 
1 
2 
1 
2 
0 
1 
1 
0 
1 
1 
2 
2 
2 
2 
3 
2 
2 
1 
1 
(0) 
(1) 
(2) 
(3) 
(4) 
(5) 
We thus obtain the following irreducible covariants, viz.: 
Of the degree 1 ; a single covariant of the order 5, this is the quintic itself. 
Of the degree 2 ; two covariants, viz. one of the order 6, and one of the order 2. 
Of the degree 3; three covariants, viz. one of the order 9, one of the order 5, and 
one of the order 3. 
Of the degree 4 ; three covariants, viz. one of the order 6, one of the order 4, and 
one of the order 0 (an invariant). 
Of the degree 5; three covariants, viz. one of the order 7, one of the order 3, and 
one of the order 1 (a linear covariant). 
These covariants are connected by a single syzygy of the degree 5 and of the 
order 11; in fact, the table shows that there are only two asyzygetic co variants of 
this degree and order; but we may, with the above-mentioned irreducible covariants 
of the degrees, 1, 2, 3 and 4, form three covariants of the degree 5 and the order 
11; there is therefore a syzygy of this degree and order.