VI. An investigation of the general term of an important series in the inverse method of finite differences

The theorems relative to finite differences, given by M. Lagrange in the Berlin Memoirs, for 1772, have much engaged the attention of mathematicians. M. Laplace has been particularly 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 + h , x +2 h , x +3 h , &c. be successive values of x , and 1 u , 2 u , 3 u &c. corresponding successive values of u . Let Δ n u represent the first term of the n th order of differences of the quantities u , 1 u , 2 u &c. And let also S n u represent the first term of a series of quantities, of which the first term of the n th order of differences is u . Then ( e representing the series 1 + 1 + 1/1.2 + 1/1.2.3 +, &c.)