[ 300 ] 

 XL VI. Proceedings of Learned Societies. 



ROYAL SOCIETY. 



[Continued from p. 231.] 



June 16, " /~*\N a Theory of the conjugate relations of two rational 



1853. V-r integral functions, comprising an application to the 

 Theory of Sturm's Functions, and that of the greatest Algebraical 

 Common Measure." By J. J. Sylvester, Esq., M.A., F.R.S., Bar- 

 rister at Law. 



The memoir consists of four sections. In the first section, the 

 theory of the residues obtained by applying the process of the common 

 measure to two algebraical functions is discussed. It is shown that 

 a certain superfluous or allotrious 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— 1 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 shown 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 



