. Annual report of the Board of Regents of the Smithsonian Institution. Smithsonian Institution; Smithsonian Institution. Archives; Discoveries in science. 322 METHODS OF INTEHPOLATION. APPENDIX I. IMPROVED ADJUSTMENT FORMULAS. We have seen that in (IG) and similar formulas used for making pre- paratory adjustments by the second method, thelocal weight of the middle term can be increased or diminished if desired, and that, when the for- mula includes more than five terras, the weights of other terms besides the middle one can also be made to vary. We have employed this pro- lierty in assigning
. Annual report of the Board of Regents of the Smithsonian Institution. Smithsonian Institution; Smithsonian Institution. Archives; Discoveries in science. 322 METHODS OF INTEHPOLATION. APPENDIX I. IMPROVED ADJUSTMENT FORMULAS. We have seen that in (IG) and similar formulas used for making pre- paratory adjustments by the second method, thelocal weight of the middle term can be increased or diminished if desired, and that, when the for- mula includes more than five terras, the weights of other terms besides the middle one can also be made to vary. We have employed this pro- lierty in assigning to the several terms, weights increasing in arithmeti- cal progression, from the extreme terms to the middle one, as in formula (20). But further investigation has shown that this arrangement of the weights, although it gives formulas which are very simple and easy of application, is not the best one in theory. To determine what the best arrangement is, we must consider that when one of these formulas is ap- plied at any part of a series, all those terms which are not included by the formula have the weight zero; that as the adjustment progresses, when a term is first included by the formula its weight is negative, it then becomes positive, attains its maximum when the term occupies the mid- dle position, then diminishes till it becomes negative again, and finally resumes the weight zero when the term is no longer included by the for- mula. To make this transition as unbroken and contiiuious as possible, it is evident that if we regard the weights as ordinates to a curve, the form of this curve should be as shown in the annexed figure, for a formula including seven terms whose positions 1, 2, 3, . . 7, are laid ofl* equidistantly on the axis of X. The curve is symmetrical with respect to the middle ordinate or axis of Y, and is tangent to the axis of X at the points 0 and 8, which are the positions of the two nearest terms not included by the formula. Such a curve has four poi
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