236 ON THE INSUFFICIENCY OF TAYLOR's THEOREM, 



or, in other words, the factors, and especially the prime factors, into which any- 

 given function may be divided. 



That any function may be made an " apparent " factor of any other function 

 is evident, since we have only to write 



F (x) = f X. ?^ = f X ^ (x) 



in order to make f x act the apparent part of a factor of F x. The existence of 

 such an error, however, if made unintentionally, would become manifest on 

 substituting for x the value a that caused f x to disappear, since such a substi- 

 tution would render (|)x infinite, whilst, on the other hand, the substitution of 

 a value b that caused f x to become infinite would cause ^ x to disappear. 



When F X is an algebraic function consisting of the sums of powers of x, 

 and which, therefore, can be actually produced by multiplying together factors 

 of the same kind, we have the most exact idea of the nature of these last, but 

 the same is not the case when F x and its factor are functions of different 



kinds ; it might even be shown that 



{Fxl 



X = a . . . b 



{Fx} 



X =: C . . . d 



or F X with X varying from a to b, and F x with x varying from c to d, may 

 be distinct functions, having distinct systems of factors; and thus, in speaking 



of an equivalence 



Fx = P X Q 

 as expressing the more correct proposition 



{Fx = PxQ} ' 



X = a . . . b 



we omit a factor R, which acts in these expansions by products, the part of a 

 remainder in the expansions by sums, and which has the property of being 

 unity between the values x = a . . . b, and of being variable, and, perhaps, 

 even infinite for other values. 



These remarks will be sufiicient to show that F x vanishing when x = a, 

 f X may be considered as a factor of {F x} * provided that we have 



a — e . . . a -{- i 



fa = 



-^ — not infinite; 

 I a 



* J is a quantity which may be infinitely small, or, in general, as small as may be required in 

 the proposition wherein it occurs. 



