On the Separation of the Roots of an Algebraical Equation. 215 



Let H r x = (f r x) 3 — y r . f r -\xfx .f+ix, 



so that 



Krx=f r xG r x, H r+1 #=/ r+ -ja?. G r+ i#; 



then, when/,.#, f r + x x have the same sign, the nature of the suc- 

 cession H,.#, H r +\X will evidently be the same as that of G r x, 

 G r+l x; but when f,x, f r +\X have unlike signs, the nature of the 

 succession H r x, H r+1 x will be contrary to that of G T x, G r+1 #. 



Accordingly when 



fr 



G r+1 



constitutes a double permanence, 



fr 



fr+1 



will also constitute a double permanence; but 



when 



fr 



G- 



fr+1 



G,. +1 



constitutes a variation-permanence 



fr 



H, 



fr+l 

 H r+ i 



will constitute a variation-variation, i. e. a double variation. 

 If, then, we take for our double progression 



{JXj J\X, JqX, . . . J n X, "^ 

 Ylx, Hja?, H 2 ,r, . . . H n x, J 



the rule, or rather the independent pair of rules referred to, will 

 take the following simplified form. 



Supposing a, b to be any two real quantities in ascending 

 order of magnitude, on substituting for x first a and then b, in 

 the simultaneous progressions above written, double permanences 

 (in passing from a to b) may be gained, but cannot be lost : 

 double variations may be lost, but cannot be gained. And the 

 number of real roots included between a and b will either be equal 

 or inferior to the number of double permanences so gained, and 

 also equal or inferior to the number of double variations so lost— 

 the difference, if there be any in either case, being some even 



number. The value of <y r is 



v + r— 1 

 v + r 



, where v is limited not 



to fall within the limits and — n. By ascertaining the 

 gain of double permanences and the loss of double variations 

 consequent on the replacement of a by b, we are furnished with 

 two independent superior limits to the number of real roots in- 

 cluded between a and b. 



Fourier's. It may not be unreasonable to imagine that a third progression 

 may remain to be invented such that the number of triple permanences 

 and triple variations of sign in the three combined may afford a new supe- 

 rior limit, and so on ad infinitum ; but this of course is at present a matter 

 of pure conjecture. 



Q2 



