326 



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

 other, and with the simplified residues. The formulas 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 numiber 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 syzygeticaiiy 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 formulse 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 v/hich 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 formulse 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 formulse 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 b}^ 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- 

 cated to the author by M. Hermite ; and an instantaneous demon- 

 stration is given of the properties of the author's form^ulse for Sturm's 

 auxiliary functions in determining the real roots of an equation by a 

 method quite irrespective of the theory of the common measure, and 

 depending upon a certain extremely simple but unobserved law of 

 quadratic forms, which he terms the law of inertia. In place of these 

 formulse it is shown that others greatly more general, and possessing 

 the same properties as regards the determination of the real roots, 

 may be substituted ; the known formulse are, however, the most 

 simple that can be employed. The author then proceeds to inquire 

 as to the nature of the indications afforded by the signs of a series 

 of successive simplified residues, taken between any two functions 



