developement of exponential functions , &c. 35 
the co-efficient of z*—P in the product &S (i),we shall have 
wanted, the principal part of the work consists in calculating 
the first n terms of the successive products, which, (being 
derived from one another) except n is considerable, is attended 
with very little trouble. 
The remarkable form of our equation (2) enables us to ex- 
hibit a variety of properties of the functions comprehended 
under the expression A n o x , some of the principal of which I 
shall now proceed to notice. 
Suppose//') == a 0 + 0, . t -j- a 2 t* + &c. 
Then, as we have shown. 
from which we find 
/(l + A)o* = i.2 x.a x . (14) 
If then the developement of f (s') be given, we are enabled to 
assign the value off (1 + A )o* in functions of x, and the con- 
verse. It is scarcely necessary, however, to remark, that the 
extent of these equations is not limited to cases in which the 
actual developement off (1 -f- A) in powers of A is practicable , 
or in which the form of/ is known, or even dependent on 
analytical relations. > 
Let us suppose a function F(f), and any two others//) 
and f ! /), so related that 
and, as every value of n 1 S j ~ l , from x—n up to x= 00 is 
Let also 
F (0 =/(*)•/'(*) 
F /) = A o + Aj . t + A 2 1 * + &c. 
F 2 
