and the Roots of Algebraical Equations. 175 



There are four changes of sign lost between 2 and 3 : 



1 



Puta> = 2 + 



u\ 



1 _ 6 + 1 + 50 - 62 - 



2 - 8 - 14 + 72 + 



104 + 170 



20 - 168 



-4-7 + 36 + 10- 

 + 2-4-22 + 28 + 



84 + 2 

 76 



- 2 - 11 + 14 + 38 - 

 + 2 + 0-22-16 



- 8 



0-11 



+ 2+4 



-8 + 22 

 -14 





. + 2- 7 - 

 + 2 + 8 



- 22 





+ 4 + 1 

 + 2 



+ 6 





The equation in u x is (say) 



FaOi) = 2^-8^ 22<-22i< 1 3 + u 1 2 + 6z< 1 + 1 = 0, 



F A '(0 = 12w 1 5 -40« 1 ' 1 + %W-£W + 2 Ul + 6, 



F A "( M i) =r,60w 1 4 -16(V + 264i< l 2 -132z<, + 2, 



F A // ^ 1 )=240u 1 3 -48(V + 528w l -132, 



F A IV Oi) = 720u 1 2 -960w, + 528, 



F/( Wl ) = 1440^-960, 



F/ I (w 1 )=1440. 



u x . 



F. 



F'. 



F". 



F". 



Fiv. 



Fv. 



Fvi. 



1 



+ 



+ 



+ 



+ 



+ 



+ 



+ 



Cf»/f F A (wi)^has no positive roots greater than unity; hence 



* . / there are no real roots o£ f(x) = between 2 and 3, and, as 



(w^ / there is no other interval in which positive roots can exist, 



v I we may pronounce that there are no positive roots of f{x) = 0. 



