636 
MR. A. CAYLEY’S THIRD MEMOIR UPON QUANTICS. 
Resuming' now the geneial subject,^ 
54. The simplest covariant of a system of quantics of the form 
{*Xx,y, ..y 
(where the number of quantics is equal to the number of the facients of each quantic) 
is the functional determinant or Jacobian, viz. the determinant formed with the 
differential coefficients or derived functions of the quantics with respect to the several 
facients. 
55. In the particular case in which the quantics are the differential coefficients or 
derived functions of a single quantic, we have a corresponding covariant of the single 
quantic, which covariant is termed the Hessian ; in other words, the Hessian is the 
determinant formed with the second differential coefficients or derived functions of 
the quantic with respect to the several facients. 
56. The expression, an adjoint linear form, is used to denote a linear function 
\x-\-7iy-\--; oi‘ in the notation of quantics (|, ri...'fx,y, ..), having the same facients as 
the quantic or quantics to which it belongs, and with indeterminate coefficients 
(I, ;j..). The invariants of a quantic or quantics, and of an adjoint linear form, may 
be considered as quantics having (|, ??...) for facients, and of which the coefficients are 
of course functions of the coefficients of the given quantic or quantics. An inva- 
riant 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 contrava- 
riants ; for any contravariant is at once turned into an invariant by writing {y, —x) 
for (I, v)- 
57 . If we imagine, as before, a system of quantics of the form 
i*Xx,y , ..)“, 
where the number of quantics is equal to the number of the facients in each quantic, 
the function of tlie coefficients, which, equalled 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 witli 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 
