NOTICES AND ABSTRACTS 



MISCELLANEOUS COMMUNICATIONS TO THE SECTIONS. 



MATHEMATICS AND PHYSICS. 



Mathematics. 



On the Expressions for the Quotients which appear in the application of 

 Sturm's Method to the discovert/ of the Meal Roots of an Equation. Ey 

 J. J. Sylvestkr, M.A., F.R.S., F.R.A.S. 



Many years ago I published expressions for the residues which appear in the 

 application of the process of common measure to fx and f'x, and which constitute 

 Sturm's auxiliaiy functions. These expressions are complete functions of the factors 

 of/x and of differences of the roots of /«, and are therefore in effect functions of the 

 factors exclusively, since the difference between any two roots may be expressed as 

 the difference between two corresponding factors. Having found that in the prac- 

 tical applications of Sturm's theorem the quotients may be employed with advan- 

 tage to replace the use of the residues, I have been led to consider their constitution ; 

 and having succeeded in expressing these quotients (which are of course linear func- 

 tions of x) under a similar form to that of the residues, i. e. as complete functions of 

 the factors and differences of the roots of/ar, I have pleasure in submitting the result 

 to the notice of the Mathematical Section of the British Association. 



Let hih^hg A« be the n roots offx. 



Let Ciabc.l) in general denote the squared product of the differences of 

 a, b, c. . . .1. 



Let Zi denote in general Sf (^ As,... . h^.), where di, 62,... di indicate any com- 

 bination of t out of the n quantities a,h, c,.. , I, with the convention that Zq=\, 



Zj=n ; and let (i) denote ^ {l-(-(— 1)'}, being zero when i is odd, and unity when 

 t is even ; then I find that the tth quotient Q, may be written under the form 



Q,=;pJ . {x-h,) + ,^,{x-h,) + &c +.Vl{x-h^), 



where in general 



Zy2 72 <72 



» e 



"7 va '72 rji 



x^i;(he,he^....hei_,)x(h^-he,XK-^e,) (A.-Aj.,,;}. 



1853. 1 



