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



lowest of the lowest roots of these equations furnish respectively 

 a superior and inferior limit to the roots of/#=0*. 



N.B. The single root of any one or more of these which may 

 be of the first degree in x is to be treated, in applying the pre- 

 ceding observation, as being at the same time the highest and 

 the lowest root of such equation or equations. 



4thly and lastly. The problem of assigning limits to the roots 

 of/#=0 reduces itself to that of finding limits to 



f 1 x=0f 9 x=0...{f)x=0; 



for the greatest and least of these collectively will evidently, a 

 fortiori, by virtue of the preceding observation, be limits to the 

 roots of/o7=0. Of any such of these as are linear, the root or 

 roots themselves may be treated as known ; leaving these out of 

 consideration, the functional part of any other of them, such as 

 fx, is the denominator of a continued fraction of the form 



1 1 1 1 



(a l x + b l )+ (a4c + b 2 )+ (fl 3 #4-6 3 ) + (ax + b.)' 



in which a v a 2 , a S} . . . a, present a single sequence of variations 

 of sign, and the limits to the roots of f x x — may be found as 

 follows. 



Form the two systems of equations (in which /i v /j, 2 , . . . fi^i 

 are numerical quantities having all the same algebraical sign, but 

 are otherwise arbitrary and independent), 



* This theorem may be more concisely stated as follows : — " If U with 

 any subscript be understood to mean a linear function of x in which the 

 sign of the coefficient of % is constant, then the finite roots of the equation 



i i i j i_ __i__ J_ i i l 



U x - U 2 - U,-"'UH- Uh-i- Ui+2-'"U,v+ U(i )+ i- U(0+i** #M IE 



lie between the greatest and least finite roots of the equations 



1 1 J_ = oc 



U 2 - TV 3 ,M U, 



_J L_...± = ao 



Ut+i— Ut+2— U,' 



_J_ _i_J...JL=ao.» 



U (t -)- U (0 + i- UX 



The theorem under this form suggests a much more general one relating 

 to para- symmetrical determinants, ». e. determinants partly normal and 

 partly gauche, which will be given hereafter; one example among the many 

 confirming the importance of the view first stated in this Magazine by the 

 author of this paper, whereby continued fractions are incorporated with the 

 doctrine of determinants. 



= CD 



