150 
operating on any function of 7 will change it into the corre- 
sponding function of 7+ wp. 
The substitution of w for 7, and D for p, in (16), leads 
to a result which is of considerable value, viz., that 
eVDfirevD =f (x+ YD). 
If in this symbolical equation we suppose the subject to be 
unity, we shall have 
eWPfre V1 =f(a+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- 
mulz 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 
d end d 
dir dp’ 
either of which denotes the operation of taking the deriveée, in 
an algebraic sense, of any function of the symbol involved in it. 
: ‘ eee 
According to this definition — must operate on 7 only where 
Tv 
d 
it appears explicitly ; and so for a 
p 
ss : : : , 
Hence i inoperative on p, or any function of it, and is 
Ett. = ad , ‘ ‘ 
commutative with p. So also dp is commutative with 7, or 
d, 
commutative with one another; but they combine respectively 
any function of it. The two symbols and are plainly 
