of the Roots of an Algebraical Equation, 21 7 



2. Let/ r # = 0, then 



also 



G,._ I ,r=(/ r _ 1 a:) 2 ; G r ,r= — 7 r ,/ r _, a? ./ r+ i#; G r+1 a?=(/,. +1 a:) 2 . 



Thus H r _i#, H r (a? + e), H r+1 # are conformable in signs to 



fr-l, —fr-lX.€, fr+\X (a) 



in this case, and in the case preceding to 



ffritff, H r+l x.€, U r+l x (ff) 



The above cases have reference to any intermediate H beco- 

 ming zero ; the final H is (foe) 3 , and the last but one is 



(A) s - Vl (/% ./'*./*); 



and accordingly when fx =0, H,#, H (x + e) become of the same 

 signs as 



/'*j e /^ (y). 



By combining the results (a), (/3), (7), and denoting by v the 

 number of real roots included between (a) and (b), it is easy to 

 infer the equation 



v=P-2<£ + 2?7, 



where P is the number of permanences gained in passing up x 

 from a to b in the H progression, </> is the collective number of 

 times that any intermediate G vanishes at a moment when the 

 preceding and subsequent H's have like signs, and t) is the col- 

 lective number of times that any intermediate f vanishes at a 

 moment when the two adjacent/ 1 ' s have like signs. But if p is 

 the number of permanences gained from the / (Fourier's series) 

 by passing up x from a to b, we have v=p—2ri, where rj repre- 

 sents the same quantity as above. 

 Hence 



2v=P+;>-2<£; 



and accordingly there emerges a new superior limit to v, viz. 



- P + » 



, an unlooked for and striking conclusion. 



Thus, ex. gr., if jo = P + 2, v cannot be greater than P + l, and 

 therefore not greater than P, because it must differ from p or P 

 (whose sum is necessarily even) by an even number*. To make 

 the preceding demonstration absolutely rigorous, it would be 

 necessary to consider the singular cases when several consecutive 



* And so in general, when p — P is positive and not divisible by 4(p), 



the superior limit given by Fourier's theorem may be replaced by "'*" — 1. 



