325 



a certain superfluous or cdlotrious factor enters into each, the value 

 of which, in terms of the leading coefficients of the residues in their 

 simplified form, is determined ; and the simplified residues them- 

 selves are subsequently obtained from the given functions by a di- 

 rect method. 



In the case where the two functions are of the same degree (m) 

 in X, m functions of the degree m — 1 in x are formed, which, being 

 identical with those employed in the process which goes by the name 

 of Bezout's abridged method, the author terms the Bezoutics or 

 Bezoutic primaries. By linear elimination performed between these, 

 a second system of functions, whose degrees in x extend from 

 m—\ to 0, are formed, which he terms the Bezoutic secondaries: 

 these Bezoutic secondaries are proved to be identical with the sim- 

 plified residues. A similar theory is show^n to be applicable in the 

 general case of the functions being of unlike degrees. Other modes 

 of obtaining the simplified residues by a direct method are also given. 

 The coefficients of the primary system of Bezoutics form a square 

 symmetrical about one axis, to which (as to every symmetrical ma- 

 trix) a certain homogeneous quadratic function of {m) variables is 

 appurtenant. This quadratic function is termed the Bezoutiant, the 

 properties of which are discussed in the fourth section. 



Every residue is what may be termed a syzygetic function or con- 

 junctive of the two given functions ; these being respectively multi- 

 plied by certain appropriate rational integral functions, their sum 

 may be made to represent a residue. These multipliers are termed 

 the syzygetic multipliers ; and they form two series, one corre- 

 sponding to the successive numerators, the other to the successive 

 denominators of the convergents to the algebraical continued frac- 

 tion which expresses the ratio of the two given functions. The 

 residues are obviously a particular class of the conjunctives that can 

 be formed from the given functions ; every conjunctive has the pro- 

 perty of vanishing when the two functions to which it is appurtenant 

 vanish simultaneously ; and in general, for any given degree in x, an 

 infinite number of such conjunctives can be formed. 



In the second section, the author commences with obtaining in 

 terms of the roots and factors of the two given functions, a variety 

 of forms, all containing arbitrary forms of function in their several 

 terms, and representing a conjunctive of any degree not exceeding 

 the sum of the degrees of the two given functions in its most general 

 form. The author then reverts to the Bezoutic system of the first 

 section, and obtains the general solution for the conjunctive of any 

 given degree in x in terras of the coefficients of the given function ; 

 by aid of this general solution he demonstrates that the residues ob- 

 tained by the common measure process (divested of their allotrious 

 factors), are the conjunctives of the lowest iceight in the roots of the 

 given functions for their several degrees ; and obtains the value of 

 this weight. He then demonstrates that certain rational but frac- 

 tional forms ascribed to the arbitrary functions in the general ex- 

 pressions for a conjunctive in terms of the roots, will make these 

 expressions integral and of the minimum weight ; they will all be 



