410 
MR. SYLVESTER ON A THEORY OF THE CONJUGATE 
by the number of roots of / and p conjointly, which appear in any term of such 
coefficient). Now in the succeeding articles I revert to the Bezoutic system of the 
first section, and beginning with the supposition of m and n being equal, I demon- 
strate that the most general form of a conjunctive of any degree in a: wiU be a 
linear function of the Bezoutics, from which it is easy to deduce that the simplified 
residues of any given degree in x are the conjunctives whose weight in respect of the 
roots is a mimmum ; so that all conjunctives having that weight must be identical (to 
a numerical factor pr^s), and any integral form of less weight apparently representing 
a conjunctive must be nugatory, every term vanishing identically. These results are 
then extended to the case of two functions of unlike degrees. The conclusion is, that 
the weight of the forms assumed in (16.) and ( 17 -) being equal to the minimum weight, 
they must (unless they were to vanish, which is easily disproved) represent the 
simplified residues, or which is the same thing, the Bezoutian secondaries.^ 
We thus obtain for each simplified residue a number of essentially distinct forms 
of representation, but all of which must be identical to a numerical factor p)'es, a 
result which leads to remarkable algebraical theorems. 
The number of these different formulae depends upon the degree of the residue; 
there being only one for the last or constant residue, two for the last but one, three 
for the last but two, and so on. The formulae continue to have a meaning when their 
degree in ^ exceeds that of / or <p ; but then, as although always representing con- 
junctives, they no longer represent residues, this identity no longer continues to sub- 
sist. In articles (22.), (23.), (24.), (25.), I enter into some developments connected 
with the general formulae in question: these, it may be observed, are all expressed 
by means of fractions containing in the numerator and denominator products of 
differences; the differences in the numerator products being taken between groups^ 
of roots of / and groups of roots of <P; and in the denominator between roots of/ 
inter se and roots of (p inter se. A great enlargement is thus opened out to the 
ordinary theory of partial fractions. 
In art. (26.) I find the numerical ratios between the different formulae which 
represent (to a numerical factor jore^) the same simplified residue, and in arts. (27.) 
and (28.) I determine the relations of algebraical sign of these formulae to the sim- 
plified residues or Bezoutian secondaries. In art. (29.) I determine the syzygetic 
multipliers corresponding to any given residue in terms of the faetors and roots of 
the given functions; but the expressions for these, which are closely analogous to 
those for the residues, cease to be polymorphic. They are obtained separately from 
the syzygetic equation, and it is worthy of notice, that to obtain the one we use the 
first of the polymorphic expressions for the residue, and to obtain the other the 
opposite extremity of the polymorphic scale. In the subsequent articles of this 
section, by aid of certain general properties of continued fractions, I establish 
a theorem of reeiprocity between the series of residues and either series of syzygetic 
multipliers. 
