[ 214 J 



XXXIII. On an improved form of Statement of the New Rule for 

 the Separation of the Roots of an Algebraical Equation, with a 

 Postscript containing a New Theorem. By Professor Sylves- 

 ter, F.R.S.* 



MY new rule (of which the demonstration will be found in a 

 paper by the late lamented Mr. Purkiss in the last Num- 

 ber of the ' Cambridge, Oxford, and DublinMathematical Mes- 

 senger ') for separating the roots of an algebraical equation, I 

 mean the rule which bears to Newton's rule generalized the same 

 relation as Fourier's to Descartes's, is susceptible of a certain slight 

 improvement as regards the mode of statement, which appears to 

 me deserving of notice. 



If we suppose fx = to be the equation in the theorem as 

 originally stated, I have employed the double progression 



JX, J^X, Jq 



Gx, G Y x, 



G, 



vx n X f 



where f r x means ( j- J fx, and G?x means (/^) 2 —y r f r - \X .f r+ x x, 



y r being a known function of r involving an arbitrary parameter, 

 confined between limits of which one is dependent on n. 



In applying the theorem, it becomes necessary to count the 

 number of compound successions for which, on writing a given 

 value a for x, f r .f r +i and G r .G,. + 1 are both simultaneously 

 positive, and also the number of the same for which f r .f r+l , 

 and G r .G,. +1 are simultaneouly negative and positive respec- 



tively, the succession 



fr 



a. 



fr+l 



G 



r+l 



in the first case constituting 



what I have called a double permanence, and in the other case 

 a variation-permanence. This latter is of course to be distin- 

 guished from a permanence- variation, which corresponds to the 

 supposition of/ r ,/ r+1 bearing like, and G r , G r+1 unlike signs — 

 there being in fact four kinds of succession, viz. double per- 

 manences, variation-permanences, permanence-variations, and 

 double variations. 



If the enunciation of the theorem can be made to refer to 

 double variations and double permanences exclusively, it is evi- 

 dent that something will have been gained in point of simplicity 

 of statement t; and this can easily be effected in the manner 

 following. 



* Communicated by the Author. 



f Moreover, so stated the theorem becomes more closely analogous to 



