514 
A FOURTH MEMOIR UPON QUANTICS. 
[155 
and an adjoint linear form, the operative quantic 
(a, b, ...$0*, 9„, ...) m , 
or more generally the operative quantic obtained by replacing in any covariant of the 
given quantic the facients (x, y, ...) by the symbols of differentiation (9$. 9,,...) (which 
operative quantic is, so to speak, a contravariant operator), may be termed the Pro 
vector ; and the Provector operating upon any contravariant gives rise to a contra 
variant, which may of course be an invariant. Any such contravariant, or rather such 
contravariant considered as so generated, may be termed a Provectant; and in like 
manner the operative quantic obtained by replacing in any contravariant of the given 
quantic the facients (£, 77, ...) by the symbols of differentiation (d x , d y ,...) (which operative 
quantic is a covariant operator), is termed the Contraprovector; and the contraprovector 
operating upon any covariant gives rise to a covariant, which may of course be an 
invariant. Any such covariant, or rather such covariant considered as so generated, 
may be termed a Contraprovectant. 
In the case of a binary quantic, 
(a, b, ...fa, y) m , 
the two theorems coalesce together, and we may say that the operative quantic 
(a, b, ...fay, — d x ) m , 
or more generally the operative quantic obtained by replacing in any covariant of the 
given quantic the facients (x, y) by the symbols of differentiation (d y ,—d x ) (which is 
in this case a covariant operator), may be termed the Provector. And the Provector 
operating on any covariant gives a covariant (which as before may be an invariant), 
and which considered as so generated may be termed the Provectant. 
63. But there is another allied theory. If in the quantic itself or in any covariant 
we replace the facients (x, y,...) by the first derived functions (9%P, 9 V P, ...) of any con 
travariant P of the quantic, we have a new function which will be a contravariant of 
the quantic. In particular, if in the quantic itself we replace the facients (x, y, ...) by 
the first derived functions (9¿P, d v P, ...) of the Reciprocant, then the result will contain 
as a factor the Reciprocant, and the other factor will be also a contravariant. And 
similarly, if in any contravariant we replace the facients (£, 77,...) by the first derived 
functions (d x W, 9 y W,...) of any covariant W (which may be the quantic itself) of the 
quantic U, we have a new function which will be a covariant of the quantic. And in 
particular if in the Reciprocant we replace the facients (£, 77, ...) by the first derived 
functions (9 X U, d y U, ...) of the quantic, the result will contain ii as a factor, and the 
other factor will be also a covariant. In the case of a binary quantic (a, b, ...fas, y) m 
the two theorems coalesce and we have the following theorem, viz. if in the quantic 
U or in any covariant the facients (x, y) are replaced by the first derived functions 
(d y W, — d x W) of a covariant W, the result will be a co variant; and if in the quantic