and Simultaneous Sign-successions. % 



469 



+ + + + + 



+ + 



+ 



+ + + 



— + 



— 



+ + + 







+ 



+ - + - + 



+ - 



— 



+ 



+ + 



+ 



+ + + 



- + 



+ 



+ - + + - 







+ 



+ + —.+ - 



+ - 



— 



abcdlmnp 

 badcmlpn 

 cdabnplm 

 dcbapnml 

 Imnpabcd 

 mlpnbadc 

 nplmcdab 

 pnmldcba 



The lettered matrix forms (as in the preceding cases) a " conjugate 

 system [in Cauchy's sense] of regular substitutions." The right- 

 hand matrix, interpreting + and — to mean plus and minus 

 units, is a direct and inverse orthogonal matrix corresponding to 

 8 represented as 2 . 2 . 2 ; the lines produced by the term-to- term 

 multiplication of the three matrices gives the quantities A, B, C, 

 D, L, M, N, P, which satisfy the equation 



2A 2 =(2« 2 )2(« 2 ), 



and the term-to-term product of the two matrices actually above 

 written is an orthogonal matrix of the 8th order. 



9. I now pass to another and more important illustration of 

 such matrices, which presents itself in the application of New- 

 ton's rule (or my extension of it) for finding a superior limit to 

 the number of real roots in an algebraical equation. That rule 

 deals with permanencies and variations of sign in two series of 

 quantities. It will be more simple to consider the two simulta- 

 neous successions of signs obtained by multiplying together the 

 signs of the consecutive terms in the series 



G, G, 





•/. 



G„. 



We obtain in this way two series of n signs each, written respec- 

 tively over one another ; and the quantities with which the 

 theory is concerned are the numbers, say it and (j>, of compound 



signs and which occur in these simultaneous progres- 

 sions : the/ series and G series both consist of functions of a?; 

 the increase of it and the decrease of <£, when x ascends from 

 one given value a to another b, each of them gives a superior 

 limit to the number of real roots in fx contained between a 

 and b. 



It is of course obvious that it corresponds to the number of 

 double permanencies, and (/> to that of variation permanencies in 



