MR. A. CAYLEY’S THIRD MEMOIR UPON QUANTICS. 
637 
termed the Tactmvariant 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 
(*Xx,p, 
and let the facients x,p, .. be replaced by Xa^+iO-X, .. the resulting function 
may, it is clear, be considered as a quantic with the facients {X, (jij) and of the form 
(X, Y,...) 
(A, 
MX, Y,..) 
L 
The coefficients of this quantic are termed Emananis, viz. excluding the first coeffi- 
cient, which is the quantic itself (but which might be termed the 0-th emanant) ; the 
other coefficients are the first, second, and last or ultimate emanants. The ultimate 
emanant is, it is clear, nothing else than the quantic itself, with (X, Y, ..) instead of 
{x, y, ...) for facients : the penultimate emanant is, in like manner, obtained from the 
first emanant by interchanging {x,y,..) with (X, Y, ...), and similarly for the other 
emanants. The facients (X, Y, ..) may be termed the facients of emanation, or simply 
the new facients. The theory of emanation might be presented in a more general 
form by employing two or more sets of emanating facients ; we might, for example, 
write i^X-f-i'X', Xy-^-fY-^vY ', ... for x,y , ..., but it is not necessary to dwell upon 
this at present. 
The invariants, in respect to the new facients of any emanant or emanants of a 
quantic {i. e. the invariants of the emanant or emanants, considered as a function or 
functions of the new facients), are, it is easy to see, covariants of the original quantic, 
and it is in many cases convenient to define a covariant in this manner ; thus the 
Hessian is the discriminant of the second or quadric emanant of the quantic. 
60. If we consider a quantic 
{a, h, ..Xx,y, ..y, 
and an adjoint linear form, the operative quantic 
(which is, so to speak, a contravariant operator) is termed the Evector. The proper- 
ties of the evector have been considered in the introductory memoir, and it has been 
in effect shown that the evector operating upon an invariant, or more generally upon 
a contravariant, gives rise to a contravariant. Any such contravariant, or rather 
such contravariant considered as so generated, is termed an Evectant. In the case of 
a binary quantic, 
{a, b, ..X-r, yf, 
4 P 2 
