468 ON THE SUMMATION OF SERIES, 
Subtract equation (1) from equation (2) and 
there will remain 
F(+1) -F@=ffowtneyt 2 anoles le (3) 
The solution of equation (3) depends upon the 
integration of Finite Differences; and by using 
the notation which is adopted in that science, 
equation (3) becomes 
A F(z) foc tit nt, and by integrating 
we have F(@)==". fie (e+h+1t Rttydn Ae octets (4) 
It is, then, the object of this paper to exhibit a 
solution of the equation (3) in terms of the differ- 
ential coefficients of the xth term of the series. 
If we, now, apply Taylor’s theorem* to the 
* IT shall here avail myself of the opportunity, to notice 
more particularly, the mode of treatment of the celebrated 
theorems of Maclaurin, Taylor, Lagrange, and Laplace, by 
most writers on the Differential and Integral Calculus. 
The object of Maclaurin’s and Taylor’s theorems is, to 
develope, if possible, the functions f(a) and f(#+h). The 
first in ascending powers of 2, and the second in ascending 
powers of Ah. 
These two developements, when they are effected by the 
