to the summation qf certain extensive classes of Series. 
f AO"" > 1 
{/(l). 0* + f (1). -r +/" (!)• Tl- + } X 
Now, if we observe that the symbol operated on (0^) is the same 
in every term, and that any expression, such as f{X A) 0^ is 
only an abbreviated mode of denoting that, when developed in 
powers of A, each power so produced is to be applied imme- 
diately before the power of 0 annexed, with its proper co-effi- 
cient, we shall have for the abbreviated expression of the above 
co-efficient, A r= 
E { /<’> <'»■ 1+^" <')T8 + } »■ 
=^44^ (A) 
(2.) — Thus, provided we have the means of estimating the value 
ofy‘(l -j- A) 0*, we are furnished with that of and vice versa, 
if by any method the value of A^ become known, the value of 
^*(1 -f A) 0^ is given by the equation, 
f(l + A)0-=:1.2....xA^ (B) 
Now, here it must be remarked, that whatever be the form off 
whether algebraic or transcendental, the value of this expression 
is necessarily limited to a finite number of terms, by reason of 
' the property of these differences, which gives a”^ 0^ =0 for 
every value of m greater than x ; so that the series of which 
^(1 -P a) 0 is an abbreviation, necessarily breaks off after a? -f- 1, 
terms though apparently continued to infinity. 
(3.) The following values of (1 -}- a) 0^ for different forms 
of the function will be useful to us hereafter. 
(1 + A) 0 := whatever be the value of m (C) 
1 ^QX 
-t 9 
1 -P A 
0 
2-pa 
0 
Qx- 
=(-iy 
C)2X 
— 1 
X 
B 
(D) 
(E) 
where Bg^^ ^ is the number of Bernouilli ; 
/ 1 (1 + Af I 0"=M*X/(1 + A) 0" . (F) 
which are immediate consequences of the equation B, if the par- 
ticular form of^ be substituted for the general. 
