Mr. J. J. Sylvester on the New Rule of Limits. 211 



lated sign + or — as a sequence of zero changes. Thus, for 



instance, H h + + H r- + H h -, may be treated 



as made up of the variation sequences 



+ -; -+; +i + ; + -+; +3 + - + -; -*; 



2ndly. I observe that if Xj, X^ ... X f be all linear functions 

 of x, and the signs of the coefficients of x in these functions con- 

 stitute a single unbroken series of variations, the denominator of 

 the continued fraction 



1 1 1 J_ 



x 1+ x 2 + x 3 + ••• X. 



(reduced to the form of an ordinary algebraical fraction) will have 

 all its roots real. 



3rdly. Suppose, for greater simplicity, that <f>x is of one degree 

 in x lower than fx, and that by the ordinary process of common 

 measure we obtain 



^x_ i i i_ j_ 



fx " X T + X 2 + X 3 + X n ' 



where X p Xg, Xg, . . . X n are all of them linear functions of x. 



Let JjLjj X 2 , . . . X w be divided into distinct and unblending 

 sequences, 



X L X 2 ...X f ; X f+1 Xi +2 ..tXi/; Xi/ +1 ,..Xf//; X w+1 X( t -) +2 ...X w ; 



so that in each sequence the signs of the coefficients of x present 

 a single unbroken series of variations, which by virtue of obser- 

 vation (1), may be considered to be always capable of being done, 

 and let 



<h%___J. 1 1_ J_ 



f x x ~~ X x + X 2 + X 3 + ' * *Xi 



6# l i i 



f$X X <+1 + Xf + 2 X 



($)x _ 1 J^ 



_ (/)^-x, )+1 + ••• xj 



then, according to observation (2), the equations 



/r*=0 />=<) . . . (f)x=0 



have each of them all their roots real ; and the observation now 

 to be made is, that the highest of the highest roots and the 



* The rule is, that the given series of signs is to be separated into di- 

 stinct sequences of variations, so that the final term of one sequence and 

 the initial term of the next shall form a continuation, i. e. we must have 

 variation sequences connected together by continuations at their joinings. 



P2 



