16 Mr. J. J. Sylvester on a remarkable Modification 



Call the roots of fa a v a^ . . . a* ; we know that/.* is 



2{(*-a 2 )(tf-a 3 ) •••(*-«»)}> 

 and I am able to state that /, . x is (to a constant factor prh) 

 equal to >tkte vjeuo. v*q a* 



2{(K fla, . . .0{(*-«2)(*-«8) • • • (*-^j}tt£ sw 

 f(fl 2 , fl s , . . . fl») denoting the squares of the products of the dif- 

 ferences between the (n— 1) quantities « 9 , 03, . . . a n . Accord- 

 ingly it will be seen that whenever x is indefinitely near, whether 

 on the side of excess or defect, to a real root off(x), fx and/j(^) 

 will have the same sign ; which serves to show, upon an inde- 

 pendent and specific algebraical ground, why the two series of 



fx f x 



residues corresponding to J j- and ^— are as (by a deduction 



J x J x 

 from a general principle they have been previously shown to be) 

 rhizoristically equivalent. 

 7 New Square, Lincoln's Inn, 

 May 31, 1853. 



Observation. 



In comparing the relative merits of the old and new methods 

 of substitution for the purposes of Sturm's theorem, the effect 

 of the introduction of positive multipliers into the dividends in 

 order to keep all the numerical quantities integral ought not to 

 be disregarded. If we call the quotients corresponding to this 

 modification of the dividends Q v Q^, Q 3 , Q 4 , &c, and the factors 

 thus introduced m l} m 2 , m^ m 4 , &c, the true quotients will be 

 Q m m m ; m 



m, w?o ^ m,.Wo a ' m*.m d ' 



and it will be found that we may employ as our rhizoristic index 

 either the number of continuations of sign in the series 



(the law of formation of the successive terms u Q , u lf m 2 , &c. being 



Wt + l = Qt + l .Mi — Wi + i .M1-1), 



or the number of positive signs in the series 

 ^1 O — —2 



Qj 



the law of formation of the successive terms v v % t> 3 , &c. being 



Qmi 



Vi-l 



There may therefore, in fact, be in each case (m— 1) more mul- 

 tiplications than have been taken account of in the text above. 



