I 
526 
A FOURTH MEMOIR UPON QUANTICS. 
[155 
90. If the two qualities are the differential coefficients or first derived functions 
(with respect to the facients) of a single quantic of the order m, then we have in 
like manner the Bezoutoidal Emanant of the single quantic ; this is a function of the 
order (m—2) in each set of facients, and the coefficients whereof are quadric functions 
of the coefficients of the single quantic. Thus the Bezoutoidal émanant of the quartic 
is 
(a, b, c, d, e\x, y) 4 
( 3 (ac — b 2 ), 
i 3 (ad — be), 
| ae — bd, 
3 (ad —be) , ae — bd \x, y) 2 (X, Yf 
ae + 8 bd — 9c 2 , 3 (be — cd) 
3 (be — cd) , 3 (ce — d 2 ) 
and of course the determinant formed with the matrix which enters into the expression 
of the Bezoutoidal Emanant, is the discriminant of the single quantic. 
91. Professor Sylvester forms with the matrix of the Bezoutic emanant and a set 
of m facients (u, v, ...) an in-ary quadric function, which he terms the Bezoutiant. 
Thus the Bezoutiant of the before-mentioned two cubics is 
( 3 (ab'— a'b), 3 (ac' — a'c) , ad' — a'd lfw, v, w) 2 ; 
i 3 (ac' — a'c), ad' — a'd + 9 (be' — b'c), 3 (bd' — b'd) 1 
ad' — a'd, Sbd' — b'd , 3 (cd'—c'd) 
and in like manner with the Bezoutoidal emanant of the single quantic of the order m 
and a set of (m — 1) new facients (u, v,...), an (m — l)ary quadric function, which he 
terms the Bezoutoid. Thus the Bezoutoid of the before-mentioned quartic is 
( 3(ac — b 2 ), 3 (ad —be) , ae — bd $ii, v, w) 2 . 
3 (ad — be), ae + 8bd — 9c 2 , 3 (be — cd) 
ae—bd, 3 (be — cd) , 3 (ce — d 2 ) 
To him also is due the important theorem, that the Bezoutiant is an invariant of 
the two quantics of the order m and of the adjoint quantic (u, v, ..-\y, — x) m ~ l , being in 
fact a linear function with mere numerical coefficients of the invariants called Cobe- 
zoutiants, and in like manner that the Bezoutoid is an invariant of the single quantic 
of the order m and of the adjoint quantic (u, v,...\y, —x) m ~ 2 , being a linear function 
with mere numerical coefficients of the invariants called Cobezoutoids. 
The modes of generation of a covariant are infinite in number, and it is to be 
anticipated that, as new theories arise, there will be frequent occasion to consider new 
processes of derivation, and to single out and to define and give names to new co 
variants. But I have now, I think, established the greater part by far of the definitions 
which are for the present necessary.