MR. A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 
119 
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 in 
The first factor of the numerator is the irreducible factor of 
and the second factor of the numerator is the irreducible factor of 
l-x\ =( 1 - x 4 ) h -( 1 - 
substituting these values, the number is 
1 
■X^) ; 
coefficient x^ in 
(1 — a;)(l — a:^)(l —x^)’ 
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 I is the 
cubic itself U, the other covariants are the covariants already spoken of and repre- 
sented by the letters H, O and V respectively (H is of the degree 2 and the order 3, 
0 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 
coefficient x® in 
1 —x + x"^ 
(1 — a?)^(l —x’^) (1 — 
the numerator of which is the irreducible factor of 1— .r®, i. e. it is equal to 
(1— a?®)(l— Making this substitution, the number is 
coefficient x® in 
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 U, the other irre- 
ducible covariants are those already spoken of and represented by I, H, J, 0 respect- 
ively (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, O is of the degree 3 
and the order 6). 
52. For a quintic, the number of irreducible covariants of the degree 6 is 
CO0lT« OC in ' " ' / 1 o\ 2 / 1 4w~i fiVTi 8\ - , 
the numerator of which is 
(l-l-.x')^(l — x-\-2x'‘-]-x^-{-2x^-\-3x^-\-x^-\-5x‘^-\-x^-\-3x^-\-2x^°-{-x^^-{-2x ^^ — ; 
the first factor is (1— < 2 :’) ^{l—x^Y, the second factor is 
