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Developing the symbol of operation (a) we shall now find, as 
an extension of Taylor’s theorem, that 
1 dp(a) 1 dil dg(a) 
f(a) dx ‘TaFS dz f (x) da 
Here, as in Taylor’s theorem, the first term is ¢ (x); and 
each of the rest is deduced from the one which precedes it by 
a uniform process. being given, the form of /'(«), or of its 
primitive f(a), must be iekeiaadl by the equation (4). This, 
as it stands, is a functional equation, but it may be reduced to 
an equation in finite differences, of the first order, and ofa de- 
gree which depends on the nature of the function y. For, if 
we make x = f"1(y), it becomes 
WF? (y)} =f y+). 
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 f (x), in order to de- 
termine the form of y. 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 :— 
+ &e. 
¢{¥(2)}= 
d 
1. The effect of e * is to change x into ex. 
1 
pid — 
2. The effect of e “ is to change z into {x!-"+1-n}'” 
d 
n 
log m{ +—}—. : 
m-1/dx is to change x into ma+n. 
3. The effect of e 
a 
4. The effect of e ** % is to change 2 into 2%. 
It is worthy of notice, that the general solution of the 
equation (4) would lead to important results in the theory of 
functional equations. For we shall have 
