Royal Society* 301 



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 terms 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 weight 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 

 consequently identical (save as to a numerical factor) with one an- 

 other, and with the simplified residues. The formulae thus obtained 

 for the simplified residues deserve particular attention on their own 

 account, being double sums of terms, any single series of which is 

 made up of fractions whose denominators are the products of the 

 differences between a certain number of the roots of each one of the 

 functions and a certain other number of the same combined in every 

 possible manner, thus containing a vast extension of the ordinary 

 theory of partial fractions. The author subsequently determines 

 under a similar form, the value of each of the multipliers which con- 

 nects the given functions syzygetically with the simplified residues, 

 and establishes a general theorem of reciprocity, by aid of certain 

 general properties of continued fractions, between the series of resi- 

 dues and either series of syzygetic multipliers. 



The third section is divided into two parts. The first part is de- 

 voted to a determination of the values of the preceding formulae in 

 the case to which Sturm's theorem refers, where one of the given 

 functions is the first differential derivative of the other ; when this is 

 the case the roots and factors of the second function are functions of 

 those of the first, and it will be found that one of the polymorphic 

 representations for the residue of any given degree will consist of 

 terms, each of which is convertible into an integral function of the 

 roots and factors of the given primitive function ; in this way are 

 obtained the author's well-known formulas for Sturm's auxiliary 

 functions. In like manner, the multiplier which affects the deri- 

 vative function in the syzygy between the primitive, the derivative 

 and any simplified residue, may also be expressed immediately as a sum 

 of integral functions of the roots and factors of the primitive, com- 

 plementary in some sort to the formulae for the residues. The for- 

 mula for the remaining syzygetic multiplier, (that which attaches to 

 the primitive itself,) cannot be obtained directly by a similar method, 

 but it is deduced by aid of the syzygetic equation itself, all the 

 other of the five terms of which are known, or have been previously 

 determined. The process of obtaining this last-named multiplier is 

 one of great peculiarity and interest, and results in a form far more 

 complex than that for the residues or for the other syzygetic multiplier. 

 In the second part of the third section are contained some curious 

 and valuable expressions for the residues and multipliers, communi- 



