518 MR. SYLVESTER ON THE FORMAL PROPERTIES OF THE BEZOUflANT 
evidently f(^, V) « covariant with / and f, and therefore 
(which is a mere truism) with the entire system /, So also is xy xy, and 
therefore the quotient of these two, is a covariant to the system. Hence, therefore, 
by virtue of a general theorem given in my Calculus of Forms, 
is a covariant to the system ; and again, therefore, 
d d\ - /d d 
is a covariant thereto. Now & is of (m- 1) dimOTsions in x, y and also of the same 
in x\ y. Consequently this latter form will contain only the quantities «i, m,, ... 
and the coefficients of / and ?>, so that the powers of x, 1/ ; j:', y' will not appear in it. 
’•^((5'’ ~(i?y 
d_ 
dif 
= Xn-liK r . WD + 2S1 _ 1 (A,, , . W, . ZL) , 
r and ^ being excluded in the latter sum from being made equal ; but this^ latter 
expression is the Bezoutiant to/, (p. Hence the Bezoutiant of / is an invariant to 
/ p, CL, L e. a covariant to the system / p, as was to be proved. The mode ot 
obtaining the covariant used in this and the preceding article, is very remarkable. 
I believe that the true suggestive view of the process for finding it, is to considei 
f{x, y) .<p{x', y')-fix', y').<p{x, y) 
as a concomitant capable of being expressed under the form of a function of ^ and u, 
CO standing for the universal covariant xy’—x'y ; S) is then to be considered, not pro- 
perly as a quotient, but rather as an invariant of the form a function of of the 
first degree, where ^ is treated as constant. 
Art. (64.). B is not an ordinary covariant of/ and p, it belongs to that special and 
most important family of invariants to a system to which I have given the name of 
Combinants*, viz. Invariants, which, besides the ordinary character of invariance, when 
linear substitutions are impressed upon the variables, possess the same cbaracter of 
invariance when linear substitutions are impressed upon the functions themselves 
containing the variables; combinants being, as it were, invariants to a system ot 
* For some remarks on the Classification of Combinants, see Cambridge and Dublin Mathematical Journal.. 
November, 1853. 
