to the summation of certain extensive classes of Series. 81 
posing a^ itself to vanish for every value of x greater than n ; 
2dlj, By supposing F (1 + A)0^ to do so; and, Bdly, By sup- 
posing the former of these functions to vanish for some value of 
a?, as, for example, the alternate odd or even, while the latter 
vanishes for the others, (as the even or odd), or any other such 
disposition. The first hypothesis it is not necessary to dwell on, 
as it gives a rational integral function of and therefore the 
value of (p (f) is resolvable into several series, whose sums may 
be separately ascertained by (4), which is a particular case of 
this. The second will, on a short examination, appear to be 
contradictory to the previous supposition that F (f) is develop- 
able in positive powers of t ; and there remains only the third 
method of verifying our equation of condition. 
(10.) Suppose, therefore, we first assume 
= F(1 + A)0^" = 0; [iv>n]-, 
the former gives f (^) = any even function of ^ (such as cos 
sec &c.) provided only it be developable in powers of The 
latter gives F (f) = (any odd function of ^) -|- R (0 where H (^) 
represents a rational integral function of ^ of dimensions, that 
is 
F («')=: 4 . (eO—’f («“') + R(0 
or 
F (0=:4(0-4'(J)+R(iog0 
where 4^ (t) denotes an arbitrary function of t. In this case, 
then, we have 
F (1 + A) 0 { 4 (1 + A) - 4 ( 
+ R(logl+A) 0“ 
The former part of this expression, as I have elsewhere demon- 
strated, vanishes, whatever be the form of 4" or the value of Xy 
and we therefore have 
F (1 + A) 0"'^= R (log 1 -h A) 0"^ 
and therefore 
<p (0) = Uq. R (log 1 -j- A) 0® + a^. R (log 1 + A)0 -j- &c. 
But if we take 
^ 
