286 



Developing the symbol of operation (a) we shall now find, as 

 an extension of Taylor's theorem, that 



,, /u , v 1 d6(x) 1 1 d 1 dd,(x) . 



Here, as in Taylor's theorem, the first term is <j> (x) : and 

 each of the rest is deduced from the one which precedes it by 

 a uniform process. \p being given, the form of/'(ar), or of its 

 primitive f (x), must be determined by the equation (b). This, 

 as it stands, is a functional equation, but it may be reduced to 

 an equation in finite differences, of the first order, and of a de- 

 gree which depends on the nature of the function \p. For, if 

 we make x =/" 1 (y), it becomes 



*{/->)} -/Ky+-i). 



When, on the other hand, we desire to ascertain the power 

 of any proposed symbol of the form (a), we must first inte- 

 grate f\x), and then invert the function /(#), in order to de- 

 termine the form of ip. Upon the possibility of effecting these 

 two operations depends the success of this attempt to interpret 

 the symbol. Pursuing this method, we obtain interesting re- 

 sults, of which the following are examples : — 



d_ 



1. The effect of e dx is to change x into ex. 



2. The effect of e d * is to change a; into {x 1 ' n + l-n) 



log m [ x + 



3. The effect of e m-iidx j s ^ change x into mx + n. 



A 



4. The effect of e ^ is to change x into afi. 



It is worthy of notice, that the general solution of the 

 equation (b) would lead to important results in the theory of 

 functional equations. For we shall have 



