18 
PEOFESSOE CAYLEY’S NINTH MEMO IE ON QU ANTICS. 
exhibit certain new formulae which appear to me to place the theory in its true light. 
I remark that although I have in my Second Memoir considered the question of finding 
the number of irreducible covariants of a given degree & in the coefficients but of any 
order whatever in the variables, the better course is to separate these according to their 
order in the variables, and so consider the question of finding the number of the irre- 
ducible covariants of a given degree Q in the coefficients, and of a given order in the 
variables. (This is, of course, what has to be done for the enumeration of the irreducible 
covariants of a given quantic ; and what is done completely for the quadric, the cubic, 
and the quartic, and for the quintic up to the degree 6 in my Eighth Memoir, Phil. 
Trans. 1867.) The new formulae exhibit this separation ; thus (Second Memoir, No. 49), 
writing a instead of x, we have for the quadric the expression 7 showing 
° (1— a)(l — a) 
that we have irreducible covariants of the degrees 1 and 2 respectively, viz. the quadric 
itself and the discriminant: the new expression is J- showing that the cova- 
( l — ax 1 ) ( 1 — « ) 
riants in question are of the actual forms («, . .fx, y ) 2 and (a, . .) 2 respectively. Simi- 
larly for the cubic, instead of the expression No. 55, — — — we have 
J (l — «) (1 — « ) (1 — ft 3 )(l— «) 
^ qG'JS ... • • 
5— 5-5-7 q-*— — tt, exhibiting the irreducible covariants of the forms 
(1 — aar)(l — a l x l )(\ — <rar)(l — <r) ° 
(a, . .fx, y) 3 , (a, . f(x, ?/) 2 , ( a . .) 3 („r, yf, and (a , . .) 4 , connected by a syzygy of the form 
(a, . ,)%x, yf; and the like for quantics of a higher order. 
In the present Ninth Memoir I give the last-mentioned formulae ; I carry on the theory 
of the quintic, extending the Table No. 82 of the Eighth Memoir up to the degree 8, 
calculating all the syzygies, and thus establishing the interconnexions in virtue of which 
it appears that there are really no irreducible covariants of the forms ( a , . .) 8 (a, y) u , and 
(a, . . 8 3(a, yf°. I reproduce in part Gordan’s theory so far as it applies to the quintic, 
and I give the expressions of such of the 23 covariants as are not given in my former 
memoirs ; these last were calculated for me by Mr. W. Barrett Davis, by the aid of a 
grant from the Donation Fund at the disposal of the Royal Society. The paragraphs of 
the present memoir are numbered consecutively with those of the former memoirs on 
Quantics. 
Article Nos. 328 to 332 . — Reproduction of my original Theory as to the Number of the 
Irreducible Covariants. 
328. I reproduce to some extent the considerations by which, in my Second Memoir 
on Quantics, I endeavoured to obtain the number of the irreducible covariants of a 
given binary quantic («, b , . . .jur, y) n . 
Considering in the first instance the covariants as functions of the coefficients (a, b, c . .), 
without regarding the variables (x, y), and attending only to the following properties — 
1°, a covariant is a rational and integral homogeneous function of the coefficients; 
