PROFESSOR CAYLEY’S TENTH MEMOIR ON QUANTICS. 
605 
The Numerical and Real Generating Functions . — Article Nos. 366 to 374, and 
Table No. 96. 
366. I have in my Ninth Memoir (1871) given what may be called the Numerical 
Generating Function (N.G.F.) of the covariants of a quartic ; this was 
. . 1 —a 6 * 12 
X ' 1 —ax*. l—a 2 x 4 . 1 — a 3 . 1 —a 3 . l—a 3 x 6 ’ 
the meaning being that the number of asyzygetic covariants a ff xf, of the degree 0 in the 
coefficients and order /x in the variables, or say of the deg-order O.g is equal to the 
coefficient of a e x' J - in the development of this function. And I remarked that the 
formula indicated that the covariants were made up of (ax 4 *, err 1 ', or, a 3 , c« 3 x 6 ) the 
quartic itself, the Hessian, the quadrin variant, the cubinvariant, and the cubico variant, 
these being connected by a syzygy «°r 13 of the degree 6 and order 12. Calling these 
covariants a, h, c, d, e, so that these italic small letters stand for covariants, 
Deg-order. 
1.4 a 
2.0 h 
2.4 c 
3.0 d 
3.6 e 
then it is natural to consider what may be called the Ileal Generating Function 
(Pv.G.F.) : this is 
1 — e 2 
1 -a . 1-b . 1-c . 1-d .1-e’ 
the development of this contains, as it is easy to see, only terms of the form a a b p c y d s and 
a a Fc y d s e, each with the coefficient +1, so that the number of terms of a given deg- 
order O.ji is equal to the coefficient of a e af 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 odd, a~bc, c 3 : there is one 
more term e 2 of the same deg-order ; hence e 3 must be a linear function of these : and 
in fact e 3 = — a^d-\-a 2 bc — 4c 3 , viz., this is the equation — U 3 J + U 3 IH — 4H 3 . 
