327 



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 whch 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 ^vays, 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 tM^o 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 foUov/s 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 w^hose 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 

 elusive. 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 

 theory of interpositions. Several of the theorems in this memoir 

 have been previously published by the author, but they are here given 

 along with a great deal of new matter in a connected form, and with 

 the demonstrations annexed, for the first time. 



4. "On the frequent occurrence of Indigo in Human Urine, 

 and on its Chemical and Physiological Relations," - By Arthur Hill 

 Hassall, M.D., Physician to the Royal Free Hospital, &c. Com- 

 municated by Professor Sharpey, F.R.S. Received June 9, 1853. 



The author was led to the investigations laid before the Society 

 in the above communication, by the following circumstances : — 



Some three or four years since, when examining urinary deposits 



