518
A FOURTH MEMOIR UPON QUANTICS.
[155
70. The lineo-linear invariant ab' — a'b of two linear functions may be considered as
giving the lineo-linear covariant d x U. d y V — d y U. d x V of the two quantics U and V,
and in like manner the lineo-linear invariant ac' — 2bb' + ca! may be considered as giving
the lineo-linear covariant d x 2 U . d y 2 V— 2d x d y U . d x d y V+ d y 2 U. d x 2 V of the quantics U, V.
And generally, any invariant whatever of a quantic or quantics of a given order or orders
leads to a covariant of a quantic or quantics of any higher order or orders: viz. the
coefficients of the original quantic or quantics are to be replaced by the derived functions
of the quantic or quantics of a higher order or orders.
71. The same thing may be seen by means of the theory of Emanants. In fact,
consider any emanants whatever of a quantic or quantics; then, attending only to the
facients of emanation, the emanants will constitute a system of quantics the coefficients
of which are derived functions of the given quantic or quantics; the invariants of the
system of emanants will be functions of the derived functions of the given quantic or
quantics, and they will be covariants of such quantic or quantics; and we thus pass
from the invariants of a quantic or quantics to the covariants of a quantic or quantics
of a higher order or orders.
72. It may be observed also, that in the case where a tantipartite invariant, when
the several quantics are put equal to each other, does not become equal to zero, we may
pass back from the invariant of the single quantic to the tantipartite invariant of the
system ; thus the lineo-linear invariant ad — 2bb' + ca' of two quadrics leads to the quadrin-
variant ac —b 2 of a single quantic; and conversely, from the quadrinvariant ac — b 2 of a
single quadric, we obtain by an obvious process of derivation the expression ad — 2bb' + ca'
of the lineo-linear invariant of two quadrics This is in fact included in the more general
theory explained, No. 64.
73. Reverting now to binary quantics, two quantics of the same order, even or odd,
have a lineo-linear invariant. Thus the two quadrics
(a, b, cjsc, y) 2 , (a, b', c'Jcc, y) 2
have (it has been seen) the lineo-linear invariant
ad — 2bb' + ca!;
and in like manner the two cubics
{a, b, c, d!§x, y) 3 , (a', b', d, d'\x, yf
have the lineo-linear invariant
ad! — 3 bd + 3 cb' — da',
which examples are sufficient to show the law.
74. The lineo-linear invariant of two quantics of the same odd order is a combinant,
but this is not the case with the lineo-linear invariant of two quantics of the same even
order. Thus the last-mentioned invariant is reduced to zero by each of the operations
ad a ' + bdy + cd c > + ddj;