155] A FOURTH MEMOIR UPON QUANTICS. 515 
U the faeients (%, y) are replaced by the first derived functions (d y U, —d x U) of the 
quantic, the result will contain U as a factor, and the other factor will be also a 
covariant. 
Without defining more precisely, we may say that the function obtained by replacing 
as above the faeients of a covariant or contravariant by the first derived functions of a 
contravariant or covariant is a Transmutant of the first-mentioned covariant or contra 
variant. 
64. Imagine any two quantics of the same order, for instance, the two qualities 
U=(a, b,...Jx, y 
V = {d, V, y ...) w , 
then any quantic such as XU + yV may be termed an Intermediate of the two quantics; 
and a со variant of A,U + yV, if in such covariant we treat A, у as faeients, will be a 
quantic of the form 
(A, B, ... B\ Aft A, yj\ 
where the coefficients {A, B, ... B\ A') will be covariants of the quantics U, V, viz. A 
will be a covariant of the quantic U alone ; A s will be the same covariant of the quantic 
V alone, and the other coefficients (which in reference to A, J.' may be termed the 
Connectives) will be covariants of the two quantics; and any coefficient may be obtained 
from the one which precedes it by operating on such preceding coefficient with the 
combinantive operator 
d'à a + b djj + ..., 
or from the one which succeeds it by operating on such succeeding coefficient with the 
combinantive operator 
ад a ' + Ъду 4-..., 
the result being divided by a numerical coefficient which is greater by unity than 
the index of у or (as the case may be) A in the term corresponding to the coefficient 
operated upon. It may be added, that any invariant in regard to the faeients (A, y) 
of the quantic 
{A, B, ... B\ A'$A, y) n 
is not only a covariant, but it is also a combinant of the two quantics U, V. 
As an example, suppose the quantics U, V are the quadrics 
(a, b, c$x, yf and (a', b', c'Qæ, y)‘\ 
then the quadrinvariant of 
ATJ + yV is (Au + ya)(Ac + yc'') — (Ab + yb'^ 2 , 
which is equal to 
(iac — b 2 , ac'— 2bb'+ ca', dc' — b >2 ^X, yf, 
and ad — 2bb' + ca' is the connective of the two discriminants ac — b 2 and a'c' — b4 
65—2