o/ Sturm's Theorem, 



455 



and the reasoning is of course general, which establishes the 

 theorem in question. 



It seems only proper and natural that I should not leave un- 

 stated here the signaletic properties of the series of numerators 



fx 

 to the convergents to -p- expanded under the form of a con- 

 jx 



tinned fraction. 



Let the number of changes of sign in the denominator series 

 for any given value {a) of x be called D(«), and for the nume- 

 rator series N(«). Then N(«) — N(6) may be equal to, or at most 

 can only differ by a positive or negative unit from D(a)— D(6), 

 The relation between these differences depends on the nature of 

 the interval between the greater of the two limits («) and (6), 

 and the root of f{so) next less that limit, and of the interval 

 between the less of the two limits {a) and (6), and the root <difx 

 next greater than such limit. If a root oi f{x) is contained in 

 each such interval, 



N(«)-N(6)=D{a)-D(^>) + l; 



if a root of f{x) is contained within one interval, but no root 

 within the other, 



NW-N(6)=D(«)-D(6); 



if no root of f{x) is contained within either interval, 



N(fl)-N(6)=D(«)-D(6)-1. ,,, 



I may conclude with noticing that the determinative form 6f 

 exhibiting the successive convergents to an improper continued 

 fraction affords an instantaneous demonstration of the equation 

 which connects any two consecutive such convergents as 



KT ^""^ % ""''• Nt.Dc.i~Nc-i.De=l. 



For if we construct the matrix, which for greater simplicity I 

 limit to five lines and columns. 



(M) 



and represent umbrally as 



ezj «g «3 a^ «5 



