150 
operating on any function of 7 will change it into the corre- 
sponding function of 7 +p. 
The substitution of « for 7, and D for p, in (16), leads 
to a result which is of considerable value, viz., that 
eVfre) =f(a+ yD). 
If in this symbolical equation we suppose the subject to be 
unity, we shall have 
eV fre] = f (m+ YD) 1. (17) 
This is a remarkable extension of Taylor’s theorem, when 
stated in the symbolical form; and will be found useful in the 
interpretation of symbolical expressions which are met with 
in the solution of differential equations. In the development 
of the right-hand member of formule (15) and (16), the terms 
involving D may be all brought by means of the theorems 
(6) and (8) to the right or left hand at pleasure. The for- 
mule thus obtained will be found of considerable use. 
In the deduction and statement of theorems involving 7 
and p, we shall find it convenient to employ the symbols 
either of which denotes the operation of taking the derivée, in 
an algebraic sense, of any function of the symbol involved in it. 
d 
— must operate on 7 only where 
T 
According to this definition 7 
‘ ay. d 
it appears explicitly ; and so for —. 
dp 
Gnas : : : F 
Hence as inoperative on p, or any function of it, and is 
betes ad. : : 
commutative with p. So also a is commutative with 7, or 
any function of it. The two symbols £ and © are plainly 
7 dp 
commutative with one another; but they combine respectively 
