22 
PROFESSOR CAYLEY’S NINTH MEMOIR ON QUANTICS. 
powers, the required number is =coeff. a e x 4 in A(x). The formula, in fact, indicates 
that the covariants are made up of (ax' 3 , a 2 x 2 , a 3 x 3 , « 4 ), the cubic itself, the Hessian, the 
cubicovariant, and the discriminant, these being connected by a syzygy (a 6 x 6 ) of the 
degree 6 and order 6. Compare Second Memoir, No. 50, according to which the 
number of covariants of degree 0 is 
= coeff. a 9 in 
1 — a 6 
( T- c)(l-« 2 )(l-« 3 )(l— a 4 ) ' 
335. For the quartic (a, b, c, d, e\x, y) 4 the number of asyzygetic covariants a e x M is 
1 —x 
=coeff. a e x e ~^ in 
or transforming as before, this is 
=coeff. a e x ti in 
the function is here 
T — a) (1 — ax)(\ — ax 1 ) (1 — ax 3 )( 1 — ax 4 ) ’ 
l-x - 2 
where 
Ato- 
ll — ax 4 ) (1 — ax 2 ) (1 — a)(l — «« _2 )(1 — ax~ 4 ) 
1 — a 6 x 12 
(1 — ax 4 ) (1 — cdx 4 ) (1 — a 2 ) (1 —a 3 )(l — a 3 x 6 ) ’ 
and the second term containing only negative powers, the required number is 
= coeff. a 0 x 11 in A(x). The formula indicates that the covariants are made up of 
(ax 4 , a 2 x 4 , a 2 , a 3 , a 3 x 6 ), the quartic itself, the Hessian, the quadrinvariant, the cubin- 
variant, and the cubicovariant, these being connected by a syzygy ( a 6 x 12 ) of the degree 6 
and order 12. Compare Second Memoir, No. 51, according to which the number of 
covariants of degree $ is 
= coeff. a e in 
1 — a 6 
(l-«)(l-a 2 ) 2 (l-« 3 ) 2 ' 
336. For the quin tic (a, b, c, d, e, f\x, yf the number of asyzygetic covariants 
a 9 x IJ - is 
= coeff. a B x 9 ~^ in 
1 —x 
(1 — «)(1 — ax)[\ — ax 2 ) (l — «,z' 3 )(l — ax 4 )( 1 — ax 5 ) ’ 
or transforming as before, this is 
= coeff. cdx' 1 ' in 
1 —x~ 2 
(1 — «^ 6 )(1 —ax 3 ) (1 —ax) (1 — ax~') (1 —ax~ 3 )(l —ax~ 5 )‘ 
The developed expression is 
1 
+ ax 5 
Aa 2 (x 10 -j-x 6 Ax 2 ) 
1 
+ ax~ 5 
-j-a 2 (x~ 10 -{-x~ 6 +x~ 2 ) 
