Abstract
SIR GEORGE DARWIN has directed attention (Mess, of Math., 1877; Phil. Trans., A, 1891; “Collected Works,” vol. i., p. 319) to the problem of interpolating values of a function at points each half-way between two consecutive points of an equidistant set, e.g. for determining probable half-hourly values when the hourly ones are found from observations. Let q′, Q′, Q, q be four points (Fig. 1) with equidistant ordinates u′, U′, U, u. It is required to find P where the graph through these four points cuts the ordinate half-way between Q and Q′. By taking the origin on the half-way ordinate and writing the function as we find that if we neglect terms beyond x3, then A rule for determining the point P is accordingly:—join QQ, qq and let them cut the central ordinate in V, v respectively, then P lies in vV produced, and PV = Vv. This rule, although theoretically identical, is simpler in form than that discovered by Sir George Darwin, and seems to be safer, especially near a point of inflexion. It may. be worth noticing that in the special case where QQ and qq are parallel, the cubic reduces to a parabola, and the rule for finding P is involved in. the relation PV: Pv = QV2: qv2 = 1: 9. At the beginning and end of the series the rule breaks down, but it can be adapted by assuming the parabolic form for the first and last arcs. In the latter case q is indeterminate, and qv must be drawn parallel to QQ (Fig. 2).
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WHIPPLE, F. Graphical Interpolation. Nature 77, 103 (1907). https://doi.org/10.1038/077103a0
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DOI: https://doi.org/10.1038/077103a0
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