155]
A FOURTH MEMOIR UPON QUANTICS.
517
treatment of the theory of the invariants or covariants of any given degree whatever,
although the application of it becomes difficult when the degree exceeds 4. I remark,
in regard to this method, that it leads naturally, and in the first instance, to a special
class of the covariants of a system of quantics, viz. these covariants are linear functions
of the derived functions of any quantic of the system. (It is hardly necessary to remark
that the derived functions referred to are the derived functions of any order of the
quantic with regard to the facients.) Such covariants may be termed tantipartite
covariants; but when there are only two quantics, I use in general the term lineo-linear.
The tantipartite covariants, while the system remains general, are a special class of
covariants, but by particularizing the system we obtain all the covariants of the par
ticularized system. The ordinary case is when all the quantics of the system reduce
themselves to one and the same quantic, and the method then gives all the covariants
of such single quantic. And while the order of the quantic remains indefinite, the
method gives covariants (not invariants); but by particularizing the order of the quantic
in such manner that the derived functions become simply the coefficients of the quantic,
the covariants become invariants: the like applies of course to a system of two or more
quantics.
69. To take the simplest example, in seeking for the covariants of a single quantic
U, we in fact have to consider two quantics U, V. An expression such as 12 UV is a
lineo-linear covariant of the two quantics; its developed expression is
d x II . d y V — dyU. d x V,
which is the Jacobian. In the particular case of two linear functions (a, b^x, y) and
{a', b'\x, y), the lineo-linear covariant becomes the lineo-linear invariant ab' — a'b, which
is the Jacobian of the two linear functions.
In the example we cannot descend from the two quantics U, V to the single quantic
U (for putting V = U the covariant vanishes); but this is merely accidental, as appears
by considering a different lineo-linear covariant 12 2 f7F, the developed expression of
which is
d x *U . d y *V— 2d x d y U . d x d y V + 0 y *U. d x 2 V.
In the particular case of two quadrics (a, b, c^x, yf, (a', b' } c'^x, y) 2 , the lineo-linear
covariant becomes the lineo-linear invariant
ac' —2 bb' + ca'.
If we have F = U, then the lineo-linear covariant gives the quadricovariant
d m 'U .dy'TJ-WyUy
of the single quantic U (such quadricovariant is in fact the Hessian); and if in the last-
mentioned formula we put for U the quadric (a, b, c]£, x, yf, or what is the same thing,
if in the expression of the lineo-linear invariant ac' - 2bb' + ca', we put the two quadrics
equal to each other, we have the quadrinvariant
ac — b 2
of the single quadric.
