256 
MR. A. CAYLEY’S INTRODUCTORY MEMOIR UPON QUANTICS. 
m in the coefficients. Hence to a covariant of the degree 0 in the coefficients, of a 
quantic of the order to, there corresponds a covariant of the degree to in the coeffi- 
cients, of a quantic of the order 0 ; the two covariants in question being each of 
them of the same order p. And it is proper to notice, that if we had commenced 
with the covariant of the quantic V, a reverse process would have led to the covariant 
of the quantic U. We may, therefore, say that the covariants of a given order and 
of the degree 0 in the coefficients, of a quantic of the order to, correspond each to 
each with the covariants of the same order and of the degree to in the coefficients, of a 
quantic of the order 0; and in particular the invariants of the degree 0 of a quantic of 
the order to, correspond each to each with the invariants of the degree to of a quantic 
of the order 0. This is the law of reciprocity demonstrated by M. Hermite, by a 
method which (I am inclined to think) is substantially identical with that here made 
use of, although presented in a very different form : the discovery of the law, con- 
sidered as a law relating to the number of invariants, is due to Mr. Sylvester. The 
precise meaning of the law, in the last-mentioned point of view, requires some expla- 
nation. Suppose that we know all the really independent invariants of a quantic of 
the order in, the law gives the number of invariants of the degree m of a quantic of 
the order 0 (it is convenient to assume 0>m), viz. of the invariants of the degree in 
question, which are linearly independent, or asyzygetic, i. e. such that there do not 
exist any merely numerical multiples of these invariants having the sum zero, but the 
invariants in question may and in general will be connected inter se and with the 
other invariants of the quantic to which they belong by non-linear equations ; and 
in particular the system of invariants of the degree m will comprise all the invariants 
of that degree (if any) which are rational and integral functions of the invariants of 
lower degrees. The like observations apply to the system of covariants of a given 
order and of the degree to in the coefficients, of a quantic of the order 0. 
21. The number of the really independent covariants of a quantic ( , y) m is pre- 
cisely equal to the order m of the quantic, i. e. any covariant is a function (generally 
an irrational function only expressible as the root of an equation) of any m indepen- 
dent covariants, and in like manner the number of really independent invariants is 
ni — 2 ; we may, if we please, take to— 2 really independent invariants as part of the 
system of the to independent covariants ; the quantic itself may be taken as one of 
the other two covariants, and any other covariant as the other of the two covariants ; 
we may therefore say that every covariant is a function (generally an irrational 
function only expressible as the root of an equation) of to— 2 invariants, of the 
quantic itself and of a given covariant. 
22. Consider any covariant of the quantic 
(a, b..b\ a'Jx, y) m , 
and let this be of the order and of the degree 0 in the coefficients. It is very 
easily shown that m0—^ is necessarily even. In particular in the case of an invariant 
