William Fleetwood Sheppard

“ Completion of Table of Differences.—If no values of u outside the range within which we have to interpolate are given, the series of differences will be incomplete at both ends. It may be continued in each direction by treating as constant the extreme difference of the highest order involved; and central-difference formulae can then be employed uniformly throughout the whole range. ”

“ For interpolation between u1 and u2 we have, with the same succession of values of θ,
u1+θ = u1 + θV1, V2 + 1−θV1
(12) . The values of 1−θV1 in (12) are exactly the same as those of θV1 in (11) , but in the reverse order. The process is therefore that (i.) we find the successive values of u0 + θΔu0, &c., i.e. we construct a table, with the required intervals of x, as if we had only to take first differences into account; (ii.) we construct, in a parallel column, a table giving the values of θV1, &c. ”

“ The methods of interpolation are usually the same in the two groups of cases, but special considerations have to be taken into account in the second group. The line of demarcation of the two groups is not absolutely fixed; the tables used by actuaries, for instance, which are of great importance in practical life, are based on statistical observations, but the tables formed directly from the observations have been “smoothed” so as to obtain series which correspond in form to the series of values of mathematical functions. ”

“ This theorem (see Infinitesimal Calculus) gives the formula
(2) ,
where, c1, c2, c3, . . . are the values for x = x0 of the first, second, third, . . . differential coefficients of u with regard to x. The values of c1, c2, . . . can occasionally be calculated from the analytical expressions for the differential coefficients of u; but more generally they have to be calculated from the tabulated differences. For this purpose central-difference formulae are the best. If we write ”

“ To find u for an intermediate value of x we assume the principle of proportional parts, i.e. we assume that the difference in u is proportional to the difference in x. Thus for x = 4.342945 the difference in u is .945 of 1000 = 945, so that u is .6376898 + .0000945 = .6377843. For x = 4.34294482 the difference in u would be 944.82, so that the value of u would apparently be .6376898 + .000094482 = .637784282. This, however, would be incorrect. It must be remembered that the values of u are only given “correct to seven places of decimals,” i.e. ”