140 Mr. J. J. Sylvester o» \w«wA 



Hence it follows immediately that /(A,)=0 cannot have imagi- 

 nary roots ; for, if possible, \et'\,=p + q \/ — I, and write 

 a+p = a' b+p = b' c+p = (/ X+pz=\', 



f{X) becomes 



a' + X' 7 /8 



7 b' + X' a. 



or say 0(X'), and the equation 4>{\') x<p{—X')=0 will be of the 

 form X'6-F.\'4+G'A,'2-H'=0, where ¥', G', H' are all essen- 

 tially positive. Hence, by Descartes' rule, no value of X.'^ can be 

 negative, i. e. {\ — pY cannot be of the form — q'^; that is to 

 say, it is impossible for any of the roots o{f{X) = to be impos- 

 sible, or, as was to be demonstrated, all the roots are real. 



I may take this occasion to remark, that by whatever linear 

 substitutions, orthogonal or otherwise, a given polynomial be 

 reduced to the form SAj^, the number of positive and negative 

 coefficients is invariable : this is easily proved. If now we pro- 

 ceed to reduce the form (expressed under the umbral notation) 

 (ffia^j + a^2 + • • • + *n • *»)^ *^ *^^ ^°^^ 



by first driving out the mixed terms in which x^ enters, then 

 those in which ,v^ enters, and so forth until eventually only a;^ of 

 the original variables is left, it may readily be shown that 



. flj Gc^ • ' • On , '^1 d^ ' • • dn—l 



"' ~ a^ a^. . .(In Oi a^- • ' «n-i ' 

 It follows, therefore, that in whatever order we arrange the 

 umbrae a^ a^. . . a^, the number of variations and of continua- 

 tions of sign in the series 



' «i a^a^ ' ' ' Ci a^. . . On 

 will be invariable, and in fact will be the same as the number 

 of positive and negative roots in the generating function in X 

 above treated of, i. e. since all the roots are real, will be the same 

 as the number of variations and continuations in the series formed 

 by the coefficients of the several powers of X, i. e. 



V, Zt , 2ii ... 



The first part of this theorem admits of an easy dii-ect demon- 

 stration ; for by my theory of compound determinants, given in 



