7 PROCEEDINGS 
THE ROYAL IRISH ACADEMY. 
1846-7. No. 66. 
May 10th, 1847. 
REV. PUMPAREY LLOYD, D.D., President, in the 
Chair. 
Edward Barnes, Esq., and Henry Freke, Esq., were 
elected Members of the Academy. 
The Rev. Charles Graves read the following note on the 
development of a function in factorials of the variable upon 
which it depends. 
The process of integration for factorials being simpler than 
that for powers, in the inverse calculus of finite differences, 
we sometimes have occasion to resolve a proposed function of | 
_ @ into a series of the form 
Ay + A, x + Aga (w—1) +434 (x—1) (w—2) + &e.; 
and we may readily determine the coefficients Ap, Aj, Ao, Az» 
&e., by making x successively equal to 0, 1, 2, 3, &e. In 
‘this way Sir John Herschel, in his Collection of Examples of 
the Applications of the Calculus of finite Differences, has 
_ solved the more general problem of developing a function F (x) 
5 in a series of factorial terms of the form 
ho + A (@—fi) + 42 (@—Si) (@ — A) + Be. 
_ F (x) being any function whatever of a, and fi, fi, &c., parti- 
"VOL. III. be Si 
