114 
MR. A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 
and the asyzygetic invariants are of the form and the number of the asyzygetic 
invariants of the degree m is equal to the coefficient of in 1 -f-(l— 
Conversely, if these formulae 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 
and the coefficients are given by the table : 
1 
1 
2 
3 
5 
6 
8 
9 
11 
11 
12 
11 
14 
16 
18 
19 
20 
( 0 ) 
( 1 ) 
( 2 ) 
( 3 ) 
( 4 ) 
( 5 ) 
and subtracting from each coefficient the one which immediately precedes it, we have 
the table ; 
1 
0 
1 
0 
1 
0 
1 
0 
1 
1 
2 
1 
2 
1 
2 
0 
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). 
