ARTICLE XL 



Expansion of F {x + h). By Pike Powers, of the University of Virginia. 



Read April 2, 184L 



A FUNCTION may be regarded as the general expression of a series of num- 

 bers which vary according to some given law; the place or number of the 

 term being denoted by x, and its value by Fx. The series may always be 

 represented wholly or in part by a curve whose ordinates correspond to the 

 different values of Fx, and its abscissas to those of x. 



In any function certain values may always be assigned to x, between which 

 the difference of any two consecutive values of i^2; will not be infinitely greater 

 than the difference of the corresponding values of x. 



The only functions which the writer can conceive of as not subject to the 

 preceding remark are, 1. Such as undergo, incessantly, abrupt changes from 

 increase to decrease, or the reverse : 2. Those which, while they vary in the 

 same sense through a finite interval, yet undergo always an infinite change for 

 a finite chano^e in x. It seems obvious, however, that if such functions can be 

 analytically expressed, they cannot admit of Taylor's theorem. (See note.) 



Supposing X to be confined within the limits referred to above, we have 



F (x -\- h) — Fx ^ f i.\ /,% 



^ jI^ = ^{x.h)- (1), 



^ {x .h) reducing to a finite quantity ^ [x .0) when h — 0. 

 In like manner we may write 



ti^l^Izili£ijy = ^' (:, . A), 



it 



or ^{x .h) = ^{x .0) +11. ^' {x . h), 



where h ip' {x . h) ~ when ^ = 0. « 



Now if 4)' {x . h) should be finite when ^ = 0, we should have 

 <?)' {x.h) = ^' {x.O) + h. <?)" {x . h); 



