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fHpÿf- • Tïnhrrr^i 
155] 
A FOURTH MEMOIR UPON QUANTICS. 
525 
then we have what are termed the Cobezoutoids of the single quantic, viz. the cobe- 
zoutoid is an invariant of the single quantic of the order m, and of an adjoint quantic 
of the order (m — 2) ; and treating the coefficients of the adjoint quantic as facients, 
the cobezoutoid is an (m — l)ary quadric, the coefficients of which are quadric functions 
of the coefficients of the given quantic. The number of the cobezoutoids is \(m — 1) 
or \(m — 2), according as m is odd or even. 
88. Consider any two quantics of the same order, 
(a, ...$#, y) m , (a', y) m , 
and introducing the new facients (X, Y), form the quotient of determinants, 
(a, ...\x , y ) m , (a, y Y 
(a,...£X, Y) m , (a',...][X, Yy 
which is obviously an integral function of the order (m — 1) in each set of facients 
separately, and lineo-linear in the coefficients of the two quantics; for instance, if the 
two quantics are 
(a, b, c, d\x, yf, 
(a', b', c, d'\x, yY, 
the quotient in question may be written 
( 3 (ab' — a'b), 3 (ac — a'c) ad' — a'd \x, yY (X, Y)*. 
3 {ac' — a'c), ad'— a'd + 9 (be'— b'c), 3 (bd'—b'd) 
ad' — a'd, 3 (bd' — b'd) , 3 (cd' — c'd) 
The function so obtained may be termed the Bezoutic Emanant of the two quantics. 
89. The notion of such function was in fact suggested to me by Bezout’s abbre 
viated process of elimination, viz. the two quantics of the order m being put equal to 
zero, the process leads to (m - 1) equations each of the order (m-1): these equations 
are nothing else than the equations obtained by equating to zero the coefficients of 
the different terms of the series (X, F) m_1 in the Bezoutic emanant, and the result 
of the elimination is consequently obtained by equating to zero the determinant 
formed with the matrix which enters into the expression of the Bezoutic emanant. 
In other words, this determinant is ¡the Resultant of the two quantics. Thus the resultant 
of the last-mentioned two cubics is the determinant 
3 (ab' — a'b), 3 (ac'—a'c) , ad' —a'd 
'3 (ac' — a'c), ad' — a'd + 9 (be' — b'c}, 3 (bd' - b'd) 
ad' — a'd, 3 bd' — b'd 3 (cd' — c'dj) 
mmmms