141] 
A SECOND MEMOIR UPON QUANTICS. 
253 
Now every rational factor of a binomial 1— x m is the irreducible factor of 1 — x m ', 
where in' is equal to or a submultiple of m. Hence in order that the series a 1} a. 2 , cz 3 ,... 
may terminate, (f>x must be made up of factors each of which is the irreducible factor 
of a binomial 1 — x m , or if cf>x be itself irreducible, then <fi.v must be the irreducible 
factor of a binomial 1 — x m . Conversely, if (fix be not of the form in question, the 
series a 1} a 2 , a 3 , &c. will go on ad infinitum, and it is easy to see that there is no point 
in the series such that the terms beyond that point are all of them negative, i.e. there 
will be irreducible covariants or invariants of indefinitely high degrees; and the number 
of covariants or invariants will be infinite. The number of invariants is first infinite in 
the case of a quantic of the seventh order, or septimic; the number of covariants is first 
infinite in the case of a quantic of the fifth order, or quintic. [As is now well known, 
these conclusions are incorrect, the number of irreducible covariants or invariants is 
in every case finite.] 
29. Resuming the theory of binary qualities, I consider the quantic 
(a, a'fi^x, y) m . 
Here writing 
{yd x } = ad b + 2bd c ... + mb'd a ', =X, 
{xdy} = mbd a + (m — 1) cd b ...+ dd b ', = Y, 
any function which is reduced to zero by each of the operations X - yd x , Y—xd y is a 
covariant of the quantic. But a covariant will always be considered as a rational 
and integral function separately homogeneous in regard to the facients (x, y) and to 
the coefficients (a, b,...b\ a). And the words order and degree will be taken to refer 
to the facients and to the coefficients respectively. 
I commence by proving the theorem enunciated, No. 23. It follows at once from 
the definition, that the covariant is reduced to zero by the operation 
which is equivalent to 
Now 
X — yd x . Y — xdy — Y— xdy . X — yd x , 
X . Y — Y. X + yd y — xd x . 
X. Y = XY + X (Y) 
Y. X = YX + Y (X), 
where XY and YX are equivalent operations, and 
X (Y)= 1 mad a + 2 (m — 1) bd b ...+ mlb'd^, 
Y (X) = mlbd b ...+ 2 (m - 1) b'd b ' + 1 mad a ', 
whence 
X (Y) — Y(X) = mdd a + ( m — 2)bd b ...-(m — 2) b'd b - mdd a ', = k suppose, 
and the covariant is therefore reduced to zero by the operation 
k Y ydy — xd x . 
Now as regards a term a a b^...b s ^a' a .x i fi, we have 
k = ma. + (m — 2) /8..., — (m — 2) /3' — mot 
ydy - xd x =j - i ;