1^8 Mr Herschel on the application of a new mode of Analysis 
(7.)'For example, the series Vt t ^ 2? 3” f f has fbr 
its sum to infinity 
(1 a) (1 -|- a')^ t . 
4_ A) (1 4- A)' 
which is reduced to calculable terms by developeinent in powcrt 
of A and A', rejecting all those of the former beyond m, and of 
the latter beyond m Its value to sc terms may in like manner 
be readily expressed. 
Thus in the case of the series 1.2 — 2.3 + 3.4 — &c. we find 
J for the value of the above expression, the terms multiplied by 
A. A' being the only ones to be retained in the developement ; and 
this is the same result with that afforded by the usual process, 
though it is obvious that, to arrive by that process, or any gene- 
ral expression, (for arbitrary values of m and would be next 
to impracticable. 
(8.) In the Philosophical Transactions for 1814, 1 pointed out, 
(I believe for the first time), the possibility of expressing the 
sum of the series 
_i &c 
a; 4- 1 ^ ^2 x-t- 1 
by means of the transcendent ^ and the numbers of Bernouilli 
alone. The following demonstration of this truth will afford a 
remarkable instance of the use of the numbers comprised in the 
form 0” , in what concerns the transformation of series. In- 
deed many reasons conspire to recommend the introduction of 
these numbers into analysis, in the place of those of Bernouilli. 
Simplicity of expression, and facility of transformation, are al- 
ways to be aimed at, and would be attained in the highest de- 
gree by this substitution. We ought, too, to add, that the lat- 
ter numbers are expressible with ease and neatness, in terms of 
the former, an advantage by no means reciprocal. 
The series in question being denoted by S, Euler has proved 
( v 2jt4* 1 
X I the co-efficient of in the develope- 
ment of sec t 
Now sec t 
— 2 I {Sp ‘ + (-^)‘*^ ' I andjapplyi 
inff our 
