320 
A THIRD MEMOIR UPON QUANTICS. 
the quantic or quantics to which it belongs, and with indeterminate coefficients 
(£, 77The invariants of a quantic or quantics, and of an adjoint linear form, may 
be considered as quantics having (£, 77,...) for facients, and of which the coefficients 
are of course functions of the coefficients of the given quantic or quantics. An invariant 
of the class in question is termed a contravariant of the quantic or quantics. The 
idea of a contravariant is due to Mr Sylvester. 
In the theory of binary quantics, it is hardly necessary to consider the contra- 
variants ; for any contravariant is at once turned into an invariant by writing (y, — x) 
for (ff, t;). 
57. If we imagine, as before, a system of quantics of the form 
Ofe y> ••■)”*» 
where the number of quantics is equal to the number of the facients in each quantic, 
the function of the coefficients, which, equated to zero, expresses the result of the 
elimination of the facients from the equations obtained by putting each of the quantics 
equal to zero, is said to be the Resultant of the system of quantics. The resultant 
is an invariant of the system of quantics. 
And in the particular case in which the quantics are the differential coefficients, 
or derived functions of a single quantic with respect to the several facients, the 
resultant in question is termed the Discriminant of the single quantic; the discriminant 
is of course an invariant of the single quantic. 
58. Imagine two quantics, and form the equations which express that the differen 
tial coefficients, or derived functions of the one quantic with respect to the several 
facients, are proportional to those of the other quantic. Join to these the equations 
obtained by equating each of the quantics to zero ; we have a system of equations, 
one of which is contained in the others, and from which therefore the facients may 
be eliminated. The function which, equated to zero, expresses the result of the 
elimination is an invariant which (from its geometrical signification) might be termed 
the Tactinvariant of the two quantics, but I do not at present propose to consider 
this invariant except in the particular case where the system consists of a given 
quantic and of an adjoint linear form. In this case the tactinvariant is a contravariant 
of the given quantic, viz. the contravariant termed the Reciprocant. 
59. Consider now a quantic 
(*$>, y> 
and let the facients x, y,... be replaced by Xx + yX, Xy + yY, ... the resulting function 
may, it is clear, be considered as a quantic with the facients (X, y) and of the form 
'(*$«> y, •••)”* 1 
(*jx, 7,...)“ j 
(.JX, Y,...r