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. Letu represent any function ofx . Letx +h ,x +2h ,x +3h , &c. be successive values ofx , and1 u ,2 u ,3 u &c. corresponding successive values ofu . Let Δn u represent the first term of then th order of differences of the quantitiesu ,1 u ,2 u &c. And let also Sn u represent the first term of a series of quantities, of which the first term of then th order of differences isu . Then (e representing the series 1 + 1 + 1/1.2 + 1/1.2.3 +, &c.)