22 6 Mr. Woodhouse on the Integration 
shown, that several methods, apparently distinct and dissimilar, 
because expressed in different language, are fundamentally, and 
in principle, the same. 
The simplest mode, and the first that occurred to mathema- 
ticians, of finding the value of fdx [—7 — p-) was, to expand the 
differential expression into a series of terms ascending by the 
powers of e, and to take the integral of each term. This method, 
however, is very imperfect ; for, if e be nearly = 1, the series 
converges so slowly as to be unfit, or at least very incommodious, 
for arithmetical computation. It became necessary then to 
possess a series ascending by the powers of 1 — e 2 ; and such a 
series was first given by Euler, in his Opuscula, published at 
Berlin in 1750; and it must be manifest, that there can be no 
one single series, ascending by the powers of e , or by powers 
of the same function e , that can in all cases represent its value. 
I purpose to consider the several series that represent the value 
when e is small, 
when e is nearly = 1, or, when ^/(1—e') is small, 
when e is /_ yTi— e z ) and a -4=, 
v ' v 2 
when e is > y/( i— e‘) and > -T=, 
a/ 2 
when e and \/ (1 — e 2 ) are equal, or when each equals -1=. 
The series for the first and second cases, I shall deduce, because 
I wish to consider the subject in its fullest extent ; but those se- 
ries, when we regard practical commodiousness, are superseded by 
the methods by which the Jdx J (—7377) is to be found, in 
the third and fourth cases. Two methods then, are only requisite 
for finding the integral in all the values of e ; for the integral 
