Osculatory Interpolation by Central Differences; with an application to Life Table Construction

Abstract

The new method described in Mr. King's recent paper, “On the Construction of Mortality Tables from Census Returns and Records of Deaths”, marks such a great advance on that employed in the construction of the official English Life Table, that it occurred to me that it might be worth examining whether the numerical application could not be further simplified by the use of central differences. Everett's formula (J.I.A., xxxv, 452) lends itself admirably to the construction of tables by subdivision of intervals. It involves only even central differences of each of the two middle terms of the series between which the interpolation has to be made, and, as was pointed out by the author in communicating his formula to the Journal, “each sum of three terms does double duty, serving both for “the preceding and the succeeding interval. In an extended “computation, the number of ‘sums of three terms’ to be “calculated is accordingly practically identical with the number “of intervals, and the labour of calculation is only about half “what it appears to be on the face of the formula.”The problem before us is that of fitting between consecutive pairs of a series of points a series of partial interpolation curves, which shall have the same slope and curvature at their points of junction. In what follows, the central difference notation is that introduced by Dr. W. F. Sheppard in his paper on “Central Difference Formulæ” (Proceedings of the London Mathematical Society, xxxi, 449).