n6 Dr. Brinkley's Investigation of the general Term 
In the case of n — 1 and h — l the formula for S u is of the 
form ^ fh ux -f- (3u -f- y 4- -f -f- & c - an d exhibits the sum 
of a series of which u any function of x is the general term. 
For which purpose it was first given by Euler in the Vlth 
Vol. Com. Petropol. and afterwards demonstrated in the VUIth 
Vol. of the same work. It has been differently investigated 
since by several authors.^ But Laplace appears to have been 
the first who gave a general term for the coefficients. Euler 
seems to have sought it in vain, for he says “ Ipsa series 
“ coefficientum «, (3, y See. ita est comparata ut vix credam 
“ pro ea terminum generalem posse exhiberi.” 
As preparatory to the main object of this enquiry it has 
been thought proper to give investigations of the above two 
theorems, which will probably be found as simple as any that 
have appeared. It has also been found necessary, for avoid- 
ing very complex formula, to adopt a peculiar notation, 
which requires some explanation. 
Notation. 
The first term of the «th order of differences of the series 
o m , i m , 2 OT , &c. is denoted by - - - - A" o m 
X n 
is denoted by ------ £" 
1.2. 3 . .11 J 
n 
n 
• — - — is denoted by------x 
1.2.3 ..n J 
— — - is denoted by - - - - A” o'” 
which omission of the denominators cannot produce any in- 
convenience, because the indices sufficiently point them out 
whether those indices refer to powers, fluxions, or differences. 
* Mac l a u r 1 n’s Fluxions, Vol. II. p. 672, &c. Wasi ng’s Med. Analyt. p. 581, 
&c. 
