Interpolation by means of Given Differences (Central-Difference Notation)

Abstract

Formulae of interpolation in terms of given central differences might be regarded as falling into two groups, A and B. In group A, the simplest cases are those in which each given difference is one of the two which in the difference table lie nearest to the preceding given difference; the differences are all natural differences (i.e., are not mean differences), and are all expressed in the centraldifference notation. Any such formula can be a central-difference formula for a certain range of the variable: but that is a matter with which we are only incidentally concerned. What I have to do is to examine the formula as determined by the series of differences given. I have then to see how the formula is affected when an ordinary difference is replaced by a mean difference. This brings us to group B, which comprises two formulae only: the Newton-Stirling formula, which expresses the required quantity in terms of a tabulated value and its central differences; and the Newton-Bessel formula, which expresses it in terms of the mean of two tabulated values and the central differences of this mean.