| uθ = u0 + θΔu0 − | θ (1 − θ) | Δ2 u0 + | θ (1 − θ) (2 − θ) | Δ3 u0 − ... |
| 2! | 3! |
(1).
Tables of the values of the coefficients of Δ2u0 and Δ3u0 to three places of decimals for various values of θ from 0 to 1 are given in the ordinary collections of mathematical tables; but the formula is not really convenient if we have to go beyond Δ2u0, or if Δ2u0 itself contains more than two significant figures.
To apply the formula to Example 3 for x = 6.277, we have θ = .77, so that uθ = .79239 + (.77 of 695) − (.089 of −11) = .79239 + 535 15 + 0 98 = .79775.
Here, as elsewhere, we use two extra figures in the intermediate calculations, for the purpose of adjusting the final figure in the ultimate result.
3. Taylor’s Theorem.—Where differences beyond the second are involved, Taylor’s Theorem is useful. This theorem (see [Infinitesimal Calculus]) gives the formula
| uθ = u0 + c1θ + c2 | θ2 | + | θ3 | + ... |
| 2! | 3! |
(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
| μδu0 | = ½ (Δu0 + Δu−1) |
| δ2u0 | = Δ2u−1 |
| μδ3u0 | = ½ (Δ3u−1 + Δ3u−2) |
| &c. | |