EXPANSION OF F (a; -f h). 159 



NOTE. 



The validity of the reasoning used in the foregoing demonstration, to show the existence of F' x 

 in a finite form in all cases where x is confined within certain limits, will perhaps appear more 

 evident from the following remarks. 



Postulate 1. There are no functions which, throughout their whole range of values, change in- 

 cessantly from increase to decrease as x varies, and that by quantities infinitely greater than the 

 change in x. It is scarcely possible to give the graphic representation of such functions, much 

 less their analytical expression. A line continually returning upon itself thus, ^//^» or a spiral 

 whose coils are compressed into almost absolute contact, would be an approximative expression of 

 them. We conclude, then, that in any function Fx, values a and a -\- n h may he assigned to x, 

 differing by a finite quantity n h, and such that from Fa to F {a + n h), Fx shall constantly in- 

 crease or constantly decrease. 



Postulate 2. There are no functions which, while they undergo a constant increase or decrease 

 through finite intervals of value, yet always receive an infinite change for a finite change in x. 

 And here we again appeal to observation, and the apparent impossibility of exhibiting such func- 

 tions in either a geometric or algebraic form. 



Tlieorem 1. Now suppose that . '■' ' v ~ — L approaches infinity as h approaches 0, for all 



h 



values of x. Then the following ratios, 



F{z-\-h) — Fx F{x+'2h.) — F{x+h.) F {x -[- 2,h) — F {x -\- ^h) . . . F (x + nh) — F [x+ (n — l) K] 

 h h h h 



will all be infinitely great when h is infinitely small. 



Let n be taken so great that n h shall be finite, and let x be such that Fx constantly increases or 



constantly decreases from Fx to ^(a; + nh). The numerators of the preceding ratios will be 



all of the same sign ; their sum is obviously F {x -\- n h) — Fx ; and if P denote the least of these 



numerators, n P <^F {x -{- nh) — Fx. But 



P nP , F(x + nh) — Fx 



— = — r = CD ; hence — ^^ — j^ = oo. 



h n h nh 



But this result is impossible by postulate 2. Hence in any function Fx there must be some values 



of x, such that 



F {x ■\- K) Fx 



— ^^ r~ = a finite quantity F' x, when A = 0. 



From which we readily derive 



F {x + h) = Fx + h . F' X + h . R, 

 R vanishing with h. 



As to the method used by Lagrange to determine the limits of the expansion oi F {x -f h), it 

 may not be amiss to observe that when the existence of the differential coefficients in a finite form 

 is admitted, this method furnishes in all cases an exact and simple mode of exhibiting the true 

 value oi F {x -j- h). This fact has been most singularly overlooked. 



