216 Prof. Sylvester on the New Rule for the Separation 



Postscript, 



It often happens that the pursuit of the beautiful and appro- 

 priate, or, as it may be otherwise expressed, the endeavour after 

 the perfect, is rewarded with a new insight into the true. So it 

 is in the present instance ; for the substitution of the H for the 

 G series, devised solely for the purpose of giving greater clear- 

 ness to the enunciation of a known theorem, leads to a sup- 

 plemental theorem which combines with and lends additional 

 completeness and harmony to the original one. 



At present the theory stands thus : a superior limit to the 

 number of real roots between two limits a and b is afforded by 

 counting, as x ascends from the one to the other, the loss of 

 changes or gain of permanences (these two numbers are identical) 

 in the /or Fourierian progression, and also by counting the loss 

 of double changes, or gain of double permanences, in the /and 

 H progressions combined : these two are distinct. We have 

 thus the choice of three superior limits. 1 shall show that a 

 fourth independent one is afforded by considering the loss of 

 changes or gain of permanences in the single H progression, and 

 combining it with such loss or gain in the single /progression. 



We have 



H r #=/ r a?.G r a?, 

 where 



y r being essentially positive for all values of 7. 



It has been proved (see Mr. Pur-kiss' s paper above referred 

 to) that, when G r #=0, 



7r+l Jr+1& 



€ being an infinitesimal. 



Now suppose li r x=0. This may happen in two distinct 

 ways, namely, either when G r # = 0, or wheji/ r # = 0. 



1. Let G r #=0, then 



Hence 



s H '* =/ '* i G * w - 



H r (*+«) = -5-.^!.G r+I « 



7r + l Vr+lff/ 



