ON DEFINITE INTEGRATION. 465 
which will apply, equally well, to the summation 
of all kinds of series. And in consequence of 
this, they have by various methods been enabled 
to give a solution to this problem, only in some 
particular classes of series. 
One of the most celebrated of these methods, 
which was, I believe, first given by Newton, is 
that, which is derived from the calculus of Finite 
Differences; and, is called the Differential me- 
thod. Its success may be said to depend upon 
the circumstance of one and every succeeding 
order of differences entirely vanishing. There- 
fore, all series, whose general term is a rational 
Algebraical Function, may then be summed. 
Another method is, by means of a formula, 
which was first given by the celebrated Euler, 
and subsequently investigated by the late Rev. 
Robert Murphy, in his valuable Treatise on the 
Theory of Algebraical Equations. Mr. Murphy 
derived his formula from the expansion of the 
well-known expression —~—, in powers of h z 
p Blz_y ? p = ics 
by means of the decomposition of rational frac- 
tions. This method reduces the difficulties of the 
summation in question, to the determination of 
