C »4 3 
VI. An Investigation of the general Term of an important Series 
in the inverse Method of finite Differences. By the Rev. John 
Brinkley, D. D. F. R.S. and Andrews Professor of Astro- 
nomy in the University of Dublin. Communicated by the 
Astronomer Royal. 
Read February 2 6, 1807. 
1 he theorems relative to finite differences, given by M. La- 
grange in the Berlin Memoirs for 1772, have much engaged 
the attention of mathematicians. M. Laplace has been par- 
ticularly successful in his investigations respecting them ; yet 
an important difficulty remained, to endeavour to surmount 
which is the principal object of this Paper. The theorems 
alluded to may be thus stated. 
Let u represent any function of x. Let x -f- h, x -f- 2/2, 
responding successive values of u. Let A n u represent the 
first term of the 22th order of differences of the quantities 
u r u &c. And let also S"z2 represent the first term of a series 
i 2 
of quantities, of which the first term of the 72th order of dif- 
ferences is u. Then (0 representing the series 1 -f- 1 + ~ 4 * 
— +,&c.) 
j.2.3 * 5 ' 
—h —h —n 
1. A” 72 =(tT —1)" 2. S n u = (e* — 1) provided that 
— b 2 3 - 
in the expansion of (e* ■ — 1 )”, &c. be substituted for 
