^42 ON THE INSUFFICIENCY OF TAYLOR's THEOREM, 



shortened by having recourse to the differential calculus as establishing in the 

 expansion (9)* 



Thus it is shown in all our elementary works that 



d"F (x + h) = d°. F (x + h): 



X h 



and we may also show that when terms are arranged in the order of their mag- 

 nitude, their differential coefficients have the same arrangements — propositions 

 that will suffice for the investigation in question. 



To prove the latter of these propositions we remark — that if ^ x and i' x are 

 functions arranged by their magnitude 



4'X 



m='>i 



for every value of x that makes '4'X = 0; but 



d;; -t^x _ 4^ (x + h) — T^x _ 4' (h) — _ -i^-h 

 dj ^x ~ 4) (x + h) — ^ X ~ 4) (h) — ~ ^ 



I 



= 



x = 

 h = 



where the order of the successive substitutions x = 0, h = 0, is indicated by 

 their symbols below the right hand bracket. 



2 3 



To apply these propositions to the investigation of ^h, ^h, &c., denote the 



1 2 



latter by H, H, &c., and write 



W = F (x + h) = F + hd F + H.P + H P + &c. . . R; 

 when we shall have 



dW = dF+hd^F + HdP + &c dR 



X X X X X 



d W = d F + d H.P + d .H P + &c d R : 



h X h h h 



and as these terms are arranged in the order of their magnitudes, the equation 



h d ' F + &c. = d H P + &c. 



X h 



will have the first term on the right hand, of the same magnitude with that on 

 the left : whence it follows that by removing parts of these terms that do not 



* I owe this remark to Mr. Z. P. Powers, who supplied the proof that the differential coefii- 

 cients of terms arranged in the order of their magnitude, are also themselves arranged in the order 

 of their magnitude. 



