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;