761] ON THE THEOREM OF THE FINITE NUMBER OF CO VARIANTS. 
273 
power of the first coefficient a is a rational function of the fundamental seminvariants. 
Thus, in the case of the cubic, we have the discriminant V, 
obtained from 
by the formula 
= a-<P — Gabcd + 4 ac 3 + 4 b 3 d — 3 b 2 c 2 , 
a, ac — b 2 , a 2 d — 3abc + 2b 3 , 
a' 2 V = (a 2 d — Sabc + 2 b 3 ) 2 + 4 (ac — b 2 ) 3 , 
and it is easily shown that this invariant V is the only new covariant thus obtainable, 
and that every other covariant is thus a rational and integral function of the 
irreducible covariants, the leading coefficients of which are 
a, ac — b 2 , a 2 d — Sabc + 2b 3 , 
and V. It appears a truism, and it might be thought that it would be, if not easy, 
at least practicable, to show for a quantic of any given finite order n, that we can 
in this manner, as rational functions of the n — 1 seminvariants, obtain only a finite 
number of new seminvariants, so that all the seminvariants would be expressible as 
rational and integral functions of a finite number of seminvariants; and, consequently, 
all the covariants be expressible as rational and integral functions of a finite number 
of irreducible covariants. But the large number, 23, of the covariants of the quintic 
is enough to show that the proof, even if it could be carried out, would involve 
algebraical operations of great complexity. 
The theory may be considered from a different point of view, in connexion with 
the root-form a(x- ay) (x- fry)..., or say (x - a) (x - /3) ... of the quantic; we have 
here what may be called the monomial form of covariant, viz. the general monomial 
form is 
(a — ft) m (a — y) n (ft — y) p ...(x — afi(x — ft) r ..., 
where in all the factors (whether a - /3 or x - a) which contain a, in all the factors 
which contain ft,..., and so for each root in succession, the sum of the indices has 
one and the same value, = 0 suppose. Thus, for the cubic 
(x -a)(x- ft) (x - y), 
we have the monomial covariants 
(a -ft) (a- y) (ft - y), 
(ft-y)(x-a), (a-y)(x-/3), (a - ft) (x - y), 
(x-a) (x- ft) (x-y); 
and so for the quartic 
(x -a)(x- ft) (x -y)(x- S), 
we have the monomial invariants 
(a-ft)(y-S), (a — y) (ft - S), (a — 8) (ft — y). 
Observe that the monomial form is considered as essential; a syzygetic function of 
C. XI. 35