of the Rule of Fourier. 7 



(a.) + - + 



{b.) + + + 



Let a -f h be one of the intervening roots ofy (x) = — the 

 least — and b — k the other : we shall assume h and k to be real. 

 From common algebraical principles we have 



/« [a + h) =/ 2 (a) +f, (a) h +/ 4 (a) £ +/ 5 (a) Jl 



yJn-2 



+ /w(a) 1.2.3... n-2 ; 



and the right-hand member of this is the second limiting po- 

 lynomial derived from 



these limiting polynomials being 



■A to * +/a W 1 +/4 (iJofK' /• ^ 2.3...;-l [2 ' ] 



A 2 Aw-2 



/»(«)+/ 8 ^)^+/4(«)^+— /»(«) 2 ,3.„ w , 2 • M 



The positive roots of the equations [1.] = 0, [2.] = 0, [3.] 

 = 0, when written in ascending order, are known to arrange 

 themselves as follows : — 



a x a 2 



b x b<z 



c l Ca •*• 



Consequently [1.] can suffer no change of sign as h proceeds 

 from h = up to h = c v the least positive root of [3.] = 0. 

 And a like conclusion has of course place when in [1.], [2.], 

 [3], — k is put for h*. 

 Now by hypothesis, 



/(« + h) =/(«) +f x (a) h +/ 8 (o)^ + . . .. = 



f{b-k) =f(b) -f x {b)k+f,{b) K --.... = o. 

 And by the conclusions just established, 



m*)* + 



and / 8 {b) - - 



* These inferences equally follow though a lt a. 2 , &c. be imaginary. 



