ON THE THEOREM OF THE FINITE NUMBER 
two or more monomials is not a monomial, and we are thus in no wise concerned 
with identities such as 
(£ ~ 7) 0 ~ «) - 0 ~ 7) 0 ~ /3) + (a - /3) O - 7) = 0, 
or 
( a - 8) (/3- 7) - (/3 - 8) (a - 7) + (7 - 8) (a - /3) = 0 ; 
notwithstanding these syzygies respectively, 
(/3 - 7) 0 - a), (a - 7) 0 - /3), and (a - /3) <> - 7) 
are regarded as independent covariants of the cubic, and 
(a-8) (£-7), (/3—8) (a - 7), and (y-8)(a-/3), 
as independent invariants of the quartic. 
It is only when a monomial covariant is equal to a power or product of simple 
or other powers of lower monomial covariants that it is regarded as a function of 
these lower monomial covariants and therefore as not irreducible. Thus 
(a - /3) (a - 7) (/3 - 8) (7 - 8) = (a - /3) (7 - 8). (a - 7) (/3 - 8), 
is a reducible monomial covariant, expressible in terms of the lower irreducible 
monomial covariants 
(a —/3) (7 — 8) and (a-7)(/3-8). 
The theorem of the finite number of the irreducible monomial covariants (as just 
explained) of the root-quantic is a question of the same kind as, but entirely distinct 
from, that of the finite number of the covariants of the quantic in the ordinary form ; 
and there are thus the two questions; (zl), that of the finite number of the irreducible 
monomial covariants of the root-quantic; and (C), that of the finite number of the 
irreducible covariants of the ordinary quantic. 
But we can pass from (^1) to (C) by means of a lemma (B), which I have not 
proved, but which seems highly probable, and which I enunciate as follows: (B) The 
infinite system of terms X, rational and integral functions of a finite set of letters 
(a, b, c, ...) which remain unaltered by all the substitutions of a certain group 
G{a, b, c, ...) of substitutions upon these letters, includes always a finite set of terms P 
such that every term X whatever is a rational and integral function of these terms P. 
In explanation of this lemma, observe that, if G (a, b, c,...) denotes the entire 
group of substitutions upon these letters, so that the functions which remain unaltered 
by the substitutions of the group are in fact the symmetrical functions of (a, b, c,...), 
then the theorem is “ The infinite system of rational and integral symmetrical 
functions of (a, b, c,...) includes always a finite set of terms P such that every such 
rational and integral symmetrical function is a rational and integral function of the 
terms P, viz. the terms P are here the several symmetrical functions