466 ON THE SUMMATION OF SERIES, 
f(x) dx, and the successive derived functions of 
f(x), each of which is multiplied by a constant 
quantity, where f (x) is the general term of the 
series. The formula, as given by Mr. Murphy, 
does not, however, appear to me to be very well 
adapted to practice. 
This theorem of Euler, has been investigated 
also by Professor De Morgan, who has obtained 
the general form of it, by means of the calculus 
of Finite Differences, and, has determined the 
constant ete from the well-known expan- 
sion Werner sai in powers of a.—(See Diff. and 
ose Came pp- 265, 266.) 
It is, then, the object of the following investi- 
gation, to determine the solution of this propo- 
sition, by a method, which is, I believe, entizely 
new. Although, it has been justly remarked, by 
a mathematical writer of great reputation, that 
“it is dangerous for any one, at the present day, 
to claim anything as belonging to himself.” I 
wish it, therefore, to be understood, that the 
method here spoken of, is not to be found in any 
of the different works on the subject which I 
have yet had the opportunity of consulting. It 
