262 
A SECOND MEMOIR UPON QU AN TICS. 
[141 
order 14, this is Z7 4 üf ; one of the order 12, this is U 3 <& ; one of the order 10, this is 
U 2 H 2 ; one of the order 8, this is UH<& ; two of the order 6 (i.e. the three covariants 
H 3 , d? 2 and V U 2 are not asyzygetic, but are connected by a single linear equation or 
syzygy), and one of the order 2 V this is VH. We are thus led to the irreducible 
covariants U, H, d?, V connected by a linear equation or syzygy between H 3 , d> 2 and 
V U 2 , and this is in fact the complete system of irreducible covariants ; V is therefore 
the only invariant. 
36. The asyzygetic covariants are of the form U p H q V r , or else of the form 
[7AiT?V r d> ; and since U, H, V are of the degrees 1, 2, 4 respectively, and d? is of the 
degree 3, the number of asyzygetic covariants of the degree m of the first form is 
equal to the coefficient of x m in 1 4- (1 — x) (1 — ¿c 2 ) (1 — xf), and the number of the 
asyzygetic covariants of the degree m of the second form is equal to the coefficient 
of x m in x 3 4- (1 — x) (1 — x 2 ) (1 - x 4 ). Hence the total number of asyzygetic covariants is 
equal to the coefficient of x m in (1 + x 3 ) 4- (1 — x) (1 — x 2 ) (1 — x A ), or what is the same 
thing, in 
1 — x 3 
(1 — x) (1 — X 2 ) (1 — X 3 ) (1 — X 4 ) ’ 
and conversely, if this expression for the number of the asyzygetic covariants of the 
degree m were established independently, it would follow that the irreducible invariants 
were four in number, and of the degrees 1, 2, 3, 4 respectively, but connected by 
an equation of the degree 6. As regards the invariants, every invariant is of the 
form V p , i.e. the number of asyzygetic invariants of the degree m is equal to the 
coefficient of x m in ^ ., and conversely, from this expression it would follow that 
JL 0u 
there was a single irreducible invariant of the degree 4. 
37. In the case of a quartic, the function to be developed is: 
1 
(1 — z) (1 — xz) (1 — x 2 z) (1 — x?z) (1 — x*z) ’ 
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 
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)