The general interpolation series, originated by Newton, has been studied mainly for its algebraic interest, only the special case of equidistant data being developed on the practical side. This is justified by the simplicity of this case, and by the numerous problems for which it suffices, but it may lead to undue simplification of data and to restrictions on experimental and computative methods. Thus tables of functions which are not in common use, or which are carried to many places, must often be limited to relatively few entries, and these might conceivably be not in arithmetical progression, with advantage both of easier tabulation and of more accurate interpolation. Data from experiment or statistics, again, are often fitted to a parabolic curve of arbitrarily chosen degree, and on rather inadequate grounds. The formation of a difference-table not only avoids the suppression of the original data, but supplies at a glance a useful analysis of them—indicating their consistency and regularity, showing with what accuracy a parabolic curve can represent them, and supplying its expression with minimum labour. For direct interpolation to new points Lagrange's formula, the usual alternative, fails in this respect and, when applied to unfamiliar data, is very apt to mislead. It is wasteful of labour and more liable to error, and cannot easily be extended to include fresh terms.
(Received June 28 1919)
(Revised June 11 1919)