408 
MR. SYLVESTER ON A THEORY OF THE CONJUGATE 
to be of these lowest dimensions ; accordingly there must enter into each of them a 
certain unnecessary factor, which, however, as it cannot be properly called irrelevant, 
I distinguish by the name of the Allotrious Factor. The successive readues, when 
divested of these allotrious factors, I term the Simpliaed Residues, and in article (3.) 
and (4.) I express the allotrious factors of each residue in terms of the leading coeffi- 
cients of the preceding simplified residues of/ and p, In article (o.) I proceed to 
determine by a direct method these simplified residues in terms of the coefficients o 
/ and <p. Beginning with the case where/ and (p are of the same dimensions (m) m x, 
I observe that we may deduce from / and 9 m linearly independent functions of j: 
each of the degree (m-l) in all of them syzygetic functions of/ and (p (vanishing 
when these two simultaneously vanish), and with coefficients which are made up of 
terms, each of which is the product of one coefficient of / and one coefficient of 0. 
These, in fact, are the very same (m) functions as are employed in the method which 
goes by the name of Bezout’s abridged method to obtain the resultant to (i. e. the 
result of the elimination of x performed upon) / and (p. As these derived functions are 
of frequent occurrence, I find it necessary to give them a name, and I term them the (w) 
Bezoutics or Bezoutian Primaries ; from these (m) primaries m Bezoutian secondaries 
may be deduced by eliminating linearly between them in the order in which they are 
generated,— -first, the highest power of x between two, then the two highest powers of 
X between three, and finally, all the powers of ^ between them all: along with the 
system thus formed it is necessary to include the first Bezoutian primary, and to con- 
sider it accordingly as being also the first Bezoutian secondary ; the last Bezoutian 
secondary is a constant identical with the Resultant of/ and (p. When the ?n times m 
coefficients of the Bezoutian primaries are conceived as separated from the powers of -r 
and arranged in a square, I term such square the Bezoutic square. This square, 
as shown in art. (7.), is symmetrical above one of its diagonals, and corresponds 
therefore (as every symmetrical matrix must do) to a homogeneous quadratic function 
of (m) variables of which it expresses the determinant. This quadratic function, 
which plays a great part in the last section and in the theory of real roots, I term the 
Bezoutiant ; it may be regarded as a species of generating function. Returning to 
the Bezoutic system, I prove that the Bezoutian secondaries are identical in form 
with the successive simplified residues. In art. (6.) I extend these results to the case 
of/ and 9 being of different dimensions in x. In art. (7.) I give a mechanical ru e 
for the construction of the Bezoutic square. In art. (8.) I show how the theory of 
f{x) and (p{x), where the latter is of an inferior degree to / may be brought under 
the operation of the rule applicable to two functions of the same degree at the 
expense of the introduction of a known and very simple factor, which in fact will be 
a constant power of the leading coefficient in/(^). In art. ( 9 .) I give another method 
of obtaining directly the simplified residues in all cases. In art. (10.) I present tbe 
process of successive division under its most general aspect. In arts. (11.) and (12.) 
I demonstrate the identity of the algebraical sign of the Bezoutian secondaries wit 1 
