160 EXPANSION OF F {x + Jl). 



Cauchy apparently, and De Morgan confessedly, have made Lagrange's method the basis of 

 their demonstrations of Taylor's theorem. We will now exhibit the method of Lagrange, after 

 premising that the equation 



F [x + h) = Fx + h . F' X + h . R 

 will readily establish the following well known theorem. 



Theorem 2. When F' x is positive, Fx and x vary in the same sense, and when negative, in 

 an opposite sense; consequently, if F {Qi) = 0, and F' (0) be not infinite, Fx will be positive or 

 negative at the same time with x, when F' x is positive. 



Let us suppose that for x = a, and x = b, and for all intermediate values, F' x, F" x , . . . i^'") x 

 are all finite and continuous, and let us replace a; by a + h, h admitting all values from to 6 — a. 

 Now let A and B be the greatest and least values of F' {a -f h) : then 

 Jl — F' {a + h)^ {), and F' {a + h) — B y 0. 

 Hence the primitives of these expressions taken with regard to h, and so as to vanish with h, will 

 likewise be positive. Theorem (2). 



A h — F [a + h) + Fa *:> 0, and F {a + Ji) — Bh — Fa^ 0. 

 Next let A' and B' be the greatest and least values of F" {a + li) : then 



£' _ F" {a + h) > 0, and F" (a + h) — ^' > 0. 

 By taking the primitives as before, we have 



A'h — F' [a + h) + F'ay 0, and F' [a + h) — B'h — F'ayO; 

 and by taking them again, we have 



A' - — F {a + h) + h . F' a + Fay 0, and F{a + h) — B' -^- — h . F' a — Fay 0, 

 or 



F{a + h)<^Fa + h. F'a +--^ A', znd F {a + h)y Fa + h. F' a + ^B'. 



1.2 l . i 



By continuing this process, we shall finally get 



F{a + h)<Fa + h.F'a+ | ./"'«+ _^ ./""a + __^1__ . ^W 



F{a + h)y Fa + h . F' a + ^^. F" a + ^-^ . F'" a + . . . . ^J'^^_^ ■ ^("' 



./S^"^ and 5(") representing the greatest and least values of i^(") [a + h). 



Hence if F^ [a + h) be continuous, and h =s b — a, there will be some value Z'" (a -{• e h) 

 intermediate between A (") and B (") such that 



F{a + h) = Fa + h . F'a + ^ . F" a + , /" . F- {a + eh) , 



where e <^ 1. This is precisely the expression obtained by Cauchy. It is general, since a is 



any value of x subj ect to the conditions stated, and it gives always the exact value of F {x + h) 



/i" 



when X and h are such that i^" ix-\-Q h) shall be finite, since :; — t-—z will always finally 



\ . i , 6 . .n 



converge. 



