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