112 MR. A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 
of the degree ], viz. a covariant of the order 3 ; this is the cubic itself, which I repre- 
sent by U. The line (2) shows that there are two asyzygetic covariants of the 
degree 2, viz. one of the order 6, this is merely U^, and one of the order 2, this I 
represent by H. The line (3) shows that there are three asyzygetic covariants of the 
degree 3, viz. one of the order 9, this is U^; one of the order 5, this is UH, and one of 
the order 3, this I represent by <I>. The line (4) shows that there are five asyzygetic 
covariants of the degree 4, viz. one of the order 12, this is U^; one of the order 8, 
this is UTd ; one of the order 6, this is ; and one of the order 0, i. e. an invariant, 
this I represent by v- The line (5) shows that there are six asyzygetic covariants of 
the degree 5, viz. one of the order 15, this is U® ; one of the order 11, this is U^H ; 
one of the order 9, this is TPO; one of the order 7, this is UH^ ; one of the order 5, 
this is HO ; and one of the order 3, this is VU. The line (6) shows that there are 8 
asyzygetic covariants of the degree 6, viz. one of the order 18, this is U®; one of the 
order 14, this is U^H ; one of the order 12, this is U^O; one of the order 10, this is 
U^IP ; one of the order 8, this is UHO ; two of the order 6 {i. e. the three covariants 
H^, and VU^ are not asyzygetic, but are connected by a single linear equation or 
syzygy), and one of the order 2, this is VH. We are thus led to the irreducible 
covariants U, H, O, V connected by a linear equation or syzygy between H®, O* and 
VU^ and this is in fact the complete system of irreducible covariants ; V is therefore 
the only invariant. 
36. The asyzygetic covariants are of the form or else of the form 
; and since U, H, V are of the degrees 1, 2, 4 respectively, and is of the 
degree 3, the number of asyzygetic covariants of the degree m of the first form is 
equal to the coefficient of in 1 -r-(l— :c)(l — a?^)(l— and the number of the 
asyzygetic covariants of the degree m of the second form is equal to the coefficient 
of in — ^)(1 — .r^)(l — x*). Hence the total number of asyzygetic covariants 
is equal to the coefficient of a?™ in (1 -=-(1 — a:)(l — .r^)(l — x*), or what is the 
same thing, in 
1 
and conversely, if this expression for the number of the asyzygetic covariants of 
the degree m were established independently, it would follow that the irreducible 
invariants were four in number, and of the degrees 1, 2, 3, 4 respectively, but con- 
nected by an equation of the degree 6. As regards the invariants, every invariant is 
of the form i. e. the number of asyzygetic invariants of the degree m is equal to 
the coefficient of x'" in conversely, from this expression it would follow that 
there was a single irreducible invariant of the degree 4. 
37- In the case of a quartic, the function to be developed is 
1 
(1 — 2’)(l —xz)[\ —x‘^z){l —0(^z){\ —x'^z') 
