of a Series in the inverse Method of finite Differences . 121 
Let the equation (2) be multiplied by L-, and the 72th fluent 
taken, and we have 
S' : u — — fl." ux n 4- — - — fl. M 1 ux n 1 . . . 7- fl. ux mi 
h n T b n-i r h 
2 
nf-h + 0 -i/i See. 
X 1 X 
Also from the above value of %, it is easy to see that the ex- 
4 - h \ - » 1 f h 
pansionof e — 1 
hr 
A h 
gives 
— 1 
in the seventh volume of their Transactions, furnish a general method of reducing 
any function of x to a series ascending by the powers of x, and that either by assign- 
ing at once the coefficient of x n , or by equations of relation between the coefficientSo 
By the converse of that method we are enabled, either from the general coefficient, or 
from the equations of relation, to arrive at the primitive function. The converse, 
therefore, applies to the summation of series, to the investigation of the general term 
of a recurring series, and to several other important purposes. It is evident, that it 
applies to finding the general term of a recurring series, because from the given scale of 
relation, the primitive function can be deduced, and from the primitive function, the 
general coefficient may be determined. The same method extends to the reduction 
of any function of x, y, z, See. and therefore the converse to finding the general term 
of double, triple, & c. recurring series. 
I had not considered the converse of the method of reduction of analytical functions 
afforded by my theorems for finding fluxions per saltum, till I had seen M.Arbogast’s 
ingenious work, entitled “ Du Calcul des Derivations.” As those theorems furnish 
every thing that is given in the former part of his treatise, and likewise admit of more 
extensive application ; so also the converse of them serve for deducing, with greater 
facility, every thing respecting recurring series. Sec. contained in the same Treatise. 
The important uses to be derived from finding fluxions per saltum in the reduction 
of analytical functions, and from the converse, induced me to draw up a particular 
work on that subject. Its publication has hitherto been delayed by my unwilling- 
ness to offer a fluxional notation different from either that of Newton or Leibnitz, 
each of which is very inconvenient as far as regards the application of the theorems for 
finding fluxions per' saltum. 
MDCCCVII. 
R 
