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 and 8, 



which are the positions of the two nearest terms not included by the 

 formula. Such a curve has four points of inflexion, so that if it is of 

 algebraic form, it must be of a degree not lower than the sixth. As- 

 suming, then, that the series of Aveights from to 8 inclusive is of the 

 sixth order, and that it has maxima at the points and 8, these two 

 conditions will sufliceto determine the two arbitrary numbers 1c and h' 

 in the foruuila — 



^{S-Jc){u,+nc)-5{u,+ u,)] 



which holds good, as has been shown, for any seven consecutive terms 

 in a series of the third or any lower order. Since the nine weights — 



0, _5, (8-7>-), (13-f4A-), (13+4 A- -f A-'), (13-4-4 A-), (8-A;), -5, 



