20 
PEOEESSOE CAYLEY’S NINTH HEMOIB ON QUANTICS. 
but a 2 -f-C The l 2 identical relations are here relations between composite covariants, 
and the effect (if any such relation could subsist) would, it appears, be to increase a . 2 ; 
between syzygies such identical relations do actually exist, and the effect is contrariwise 
to diminish the a ; we may, for instance, for the degree s have irreducible covariants less 
irreducible syzygies = a s — l s . 
331. Assume for a moment that, for a given value of s, u s is positive; but for the 
term l s it would of course follow that there was for the degree in question a certain 
number of irreducible covariants ; and it was in this manner that I was led to infer that 
the number of the covariants of a quintic was infinite — viz. the transformed expression 
for the number of asyzygetie covariants is 
=coeff. a 6 in (l-a^l-a 8 )- 3 (1 -a l2 )~ 6 (1 
a product which does not terminate, and as to which it is also assumed that the series 
of negative indices does not terminate. 
332. The principle is the same, but the discussion as to the number of the irreducible 
covariants becomes more precise, if we attend to the covariants as involving not only the 
coefficients (a, b, . . .) but also the variables (x, y) ; we have then to consider the covari- 
ants of the form («, b, . . .) e (x, yf, or, say, of the form a n x IM (degree & and order y,), and the 
number of the asyzygetie covariants of this form is given as the coefficient of a e x ]L in 
a given function of (a, x), (I write a instead of the z of my Second Memoir in the 
formulae which contain x and z) : by taking account of the composite covariants and 
syzygies, we successively determine, from the given number of asyz) r getic covariants for 
each value of 0 and y>, the number of the irreducible covariants for the same values of 
b and [jj. This is, in fact, done for the quintic in my Eighth Memoir up to the covariants 
and syzygies of the degree 6. But before resuming the discussion for the quintic, I 
will consider the preceding cases of the quadric, the cubic, and the quartic. 
Article Nos. 333 to 336 . — New formula? for the number of Asyzygetie Covariants. 
333. For the quadric (a, b, cfx, y) 2 , the number of asyzygetie covariants a e x' ± 
= coeff. a e x e ** in 
1 — X 
( 1 — «) (1 — ■ ax) (1 — «a’ 2 )’ 
(see Second Memoir, No. 35, observing that q is there =6— \g,, and that the subtraction 
of successive coefficients is effected by means of the factor \—x in the numerator. See 
also Eighth Memoir, No. 251, where a like form is used for the quintic). "Writing ax 2 for 
«, and "2 for x, this is 
1 - 
- coeff. in - 
(l — <7x 2 )(l — 
