to the summation of certain extensive classes of Series. 
and to infinity simply, 
^ (1 -f a) 
which reduced to a form adapted to numerical computation, 
becomes 
— + 
(6.) Not, however, to multiply instances, we shall noAv go on 
to the more extensive series, whose general term is 
A^^a + hicy'. (a’ + h’xf'. {a” -^h" x)^" 
of which the series, 
2® S? -f S’lf 4^ + &c. 
is a particular instance : but not to encumber ourselves with un- ^ 
necessary symbols, the method being the same for any num- 
ber, we will confine our analysis to two factors in each term. 
If we denote by S the sum of the series, and observe that 
(a’ + K xf = (1 + A) 0 “'- 
we have 
S = A„ (1 + A)“0 ” X (1+ A)‘‘'o“' + (1 + A)“ + '^ 0” 
X(l + A)“'+^'0’‘' + &C. 
In order to separate the symbols of operation from those of 
quantity in this case without confusion, we must resort to the 
system of accentuation I have explained in the paper in Philo- 
sophical Transactions 1816, above alluded to ; thus, applying 
an accent to the latter A, and a corresponding accent to the 0 
with which it is to be connected, we mark them as it were for 
each other, and a temporary separation by way of abbreviation 
or transformation will not be any source of confusion. We 
have then 
S= I A,, (1 + A)“ (1 + + A, (1 + A)“ + 
(l+A')“' + ^' + &c. I ©’■O'”' 
- (1 + A)“ (1 + ^Y■ F I (1 d- A)* (i + A')'*' 1 0“ 0'"'- 
