693] 
A TENTH MEMOIR ON QUANTICS. 
341 
The Numerical and Real Generating Functions. Art. Nos. 366 to 374, 
and Table No. 96. 
366. I have, in my Ninth Memoir (1871) [462], given what may be called the 
Numerical Generating Function (N.G.F.) of the covariants of a quartic ; this was 
. . _ 1 — a 6 « 12 
1 — ax 4, . 1 — a 2 x 4 .1 — a 2 .1 — a 3 .1 — d s x s ’ 
the meaning being that the number of asyzygetic covariants a 9 x> J -, of the degree 9 
in the coefficients and order g in the variables, or say of the deg-order 9. g, is equal 
to the coefficient of a e x<*■ in the development of this function. And I remarked that 
the formula indicated that the covariants were made up of (ax 4 , a¥, a 2 , a 3 , a 3 af), the 
quartic itself, the Hessian, the quadrinvariant, the cubinvariant, and the cubicovariant, 
these being connected by a syzygy a 6 x 12 of the degree 6 and order 12. Calling these 
covariants a, b, c, d, e, so that these italic small letters stand for covariants, 
Deg-order. 
1.4 a, 
2.0 b, 
2.4 c, 
3.0 d, 
3.6 e, 
then it is natural to consider what may be called the Real Generating Function 
(R.G.F.): this is 
1 — e 2 
1 — a.l — b.l — c.l—d.l — e » 
the development of this contains, as it is easy to see, only terms of the form a a b ? crd 8 
and a a №crd s e, each with the coefficient +1, so that the number of terms of a given 
deg-order 9. g is equal to the coefficient of a e x^ in the first-mentioned function : and 
these terms of a given deg-order represent the asyzygetic covariants of that deg-order: 
any other covariant of the same deg-order is expressible as a linear function of them. 
For instance, deg-order 6.12, the terms of the R.G.F. are a 3 d, a 2 bc, c 3 : there is one 
more term e 2 of the same deg-order ; hence e 2 must be a linear function of these : 
and in fact 
e 2 = - a 3 d + a 2 bc — 4c 3 , 
viz. this is the equation 
= — U 3 J + U 2 IH - 4H 3 .