240 ON THE INSUFFICIENCY OF TAYLOR's THEOREM, 



Denote by a the value of z corresponding to that part of the portion where 

 the alteration of one of the terms P or Q is not exceeded in slowness by the 

 values of P or Q either there or at any part of the portion, and assume 



^h = F(a + h)^Fa, 

 it is then clear that 



F (x + h) — F h 

 F(a + h) — Fa 



can never become infinite between the required limits, supposing, as is always 



done, that h does not exceed the small values employed in estimating the rates 



of increase above mentioned. 



We thus prove that ^ h is a factor of the function 



{F(x + h) — Fx} 



. a + £ 



and, consequently, that we may write 



F (x + h) = F X + Q. |. h 

 But it will follow from this equation that whatever function Q is of x and h, 

 it must also, between the required limits, have a continuity of the class which 

 we have assumed for F (x + h). As will be seen by considering that a finite 



and definite increase of F (x + h) will require a finite and definite increase in 



1 

 Q. ^ (h) or R, and that as ^ (h) has been so assumed as to increase in this man- 

 ner, Q must do so likewise. 



1 2 



By this reasoning we therefore establish that all the functions ^h, eh, &c., 

 are continuous functions of the first order, and may be taken not to contain x ; 

 whence it appears that the development (8) is just; that it constitutes a con- 



n 



verging series when h is sufficiently small; and that Q is of less magnitude 

 than any of the terms which precede it, and may be made as small a portion 

 of the whole as we please by taking h within proper limits. 

 Having thus established the equation 



{F(x + h) = Fx + ^h. F(x) +|h. F (x) + &c R} 



X = X . . X 



for all functions that have between x and x a continuity of the first order, and 



excluding from such continuity all serpentine forms that have infinitely small 

 periods,* we proceed to investigate some of the simplest of the series 



^ (h), I (h), I (h), &c. 



* The rates in such forms would not become definite as r was decreased. 



