PROCEEDINGS 



THE ROYAL IRISH ACADEMY. 



1846-7. No. 66. 



May 10th, 1847. 



REV. HUMPHREY 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 

 X into a series of the form 



Aq -4- Ai J? + ^■2X (x—l) 4- ^3x{x—\) (x—2) + &c. ; 



and we may readily determine the coeflBcients Aq, Ai, a^, A3, 

 &c., by making x successively equal to 0, 1, 2, 3, &c. In 

 this way Sir John Herschel, in his Collection oj^ Examples of 

 the Applications oj" the Calculus of finite Differences, has 

 solved the more general problem of developing a function f {x) 

 in a series of factorial terms of the form 



Ao + A, {x-f) + Ai(x-/) {x -/,) + &c. 

 F [x) being any function whatever oix, andyj,^, &c., parti- 



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