141]
A SECOND MEMOIR UPON QUANTICS.
269
Considering next the covariants,—
49. For a quadric, the number of asyzygetic covariants of the degree 6 is
coefficient x 9 in 7-
(1 - x) (1 - x 2 ) ’
i. e. there are two irreducible covariants of the degrees 1 and 2 respectively; these
are of course the quadric itself and the invariant.
50. For a cubic, the number of the asyzygetic covariants of the degree 6 is
coefficient x e in + ^ ^ + x \ .
(1 - a; 2 ) 2 (1 - x*)
The first factor of the numerator is the irreducible factor of
1 — oc 2 , = (1 — x 2 ) -r- (1 — x),
and the second factor of the numerator is the irreducible factor of
1 — iri, = (1 — ¿ri) -r- (1 — x 2 );
substituting these values, the number is
coefficient x 9 in
1 — x 6
(1 -x) (1 -^)(1-^)(T - x 4 ) ’
i.e. there are 4 irreducible covariants of the degrees 1, 2, 3, 4 respectively; but these
are connected by an equation of the degree 6; the covariant of the degree 1 is the
cubic itself U, the other covariants are the covariants already spoken of and repre
sented by the letters H, <1> and V respectively (H is of the degree 2 and the order 3,
‘F of the degree 3 and the order 3, and V is of the degree 4 and the order 0,
i.e. it is an invariant).
51. For a quartic, the number of the asyzygetic covariants of the degree 6 is
1 — x + x 2
coefficient x 9 in
(1 — xf (1 — x 2 ) (1 — x?) ’
the numerator of which is the irreducible factor of 1 — x 6 , i.e. it is equal to
(1 — x?) (1 — x) -T- (1 — x 2 ) (1 — ic 3 ). Making this substitution, the number is
coefficient x 9 in
1 — x 6
(1 -*) (1 -xj (1 -O 2 ’
i.e. there are five irreducible covariants, one of the degree 1, two of the degree 2,
and two of the degree 3, but these are connected by an equation of the degree 6.
The irreducible covariant of the degree 1 is of course the quartic itself TJ, the other
irreducible covariants are those already spoken of and represented by I, H, J, <f>
respectively (I is of the degree 2 and the order 0, and J is of the degree 3 and
the order 0, i.e. I and J are invariants, H is of the degree 2 and the order 4,
is of the degree 3 and the order 6).
