302 Royal Society. 



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

 stration is given of the properties of the author's formulae 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 

 formulae 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 formulae 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 

 independent of one another, instead of standing in the relation of 

 primitive and derivative, as in Sturm's theorem ; this leads to the 

 theory of interpositions, of which it is shown that the Sturmian 

 theorem may be treated (not so much as a particular case) as an 

 easy corollary. In this part, the author obtains an entirely new 

 rule for determining, in an infinite variety of ways, a superior and 

 inferior limit to the real roots of any algebraical equation, whether 

 numerical or literal. 



The fourth section is also divided into two parts. In the first 

 part, the index of interposition for two functions of the same degree 

 is shown to be determinable by means of the quadratic form, pre- 

 viously termed the Bezoutiant ; and as a corollary, it follows that 

 the number of real roots of an equation of the degree m depends in a 

 direct manner on the number of positive roots in another equation 

 of the degree m — 1, all of whose roots are real, and the coefficients 

 of which are quadratic combinations of the coefficients of the given 

 equation. 



In the second part of this section, the Bezoutiant is considered 

 under a purely morphological point of view. It is shown to be 

 a combinantive invariant of the two given functions (each treated 

 as homogeneous functions of two variables), remaining unaltered 

 when any linear combination of the two given functions is substi- 

 tuted for the functions themselves, and also when any linear substi- 

 tutions are impressed upon the variables of the given functions, pro- 

 vided that certain corresponding substitutions are impressed upon 

 the variables of the Bezoutiant. The family of forms to which the 

 Bezoutiant belongs is ascertained, and a method given for finding 

 the constituent forms of this family (one less in number than the 

 number of odd integers not exceeding in magnitude the degree of 

 either of the given functions which, throughout this section, are 

 supposed to be of equal dimensions in x), of which all other forms of 

 the family will be numerico-linear functions. The numerical coeffi- 

 cients connecting the Bezoutiant with this constituent group, are 

 calculated for the cases corresponding to any index from 1 to 6 in- 

 clusive. Finally, the author remarks upon the different directions 

 in which the subject matter of the ideas involved in Sturm's 

 justly celebrated theorem admits of being expanded, and of which 

 the most promising is, in his opinion, that which leads through the 



