XVI. Two general propositions in the method of differences

1. Though so many ingenious writers have demonstrated, and, in various respects, extended the celebrated formulas of La Grange, for ∆n ϕ(x ), Ʃn ϕ (x ) no one appears to have entertained the idea, that these, and the more general cases, in which the quantities under the functional sign have their differences variable, might be included in one simple form. Mr. Prony* is, I believe, the only mathematician who has given a form of any regularity to ∆n ϕ(x ), when the difference ofx is variable; but he does not seem to have been aware of the capability of the method he was employing; and instead of embracing, as he might have done, all cases in one simple expression, he has proposed a formula which has neither any particular elegance in itself, nor any apparent relation to that which, in the simpler case, had been given by La Grange.