(3),
so that, if (as in §§ 1 and 2) each difference is placed opposite the space between the two quantities of which it is the difference, the expressions δ2u0, δ4u0, ... denote the differences of even order in a horizontal line with u0, and μδu0, μδ3u0, ... denote the means of the differences of odd order immediately below and above this line, then (see [Differences, Calculus of]) the values of c1, c2, ... are given by
| c1 = μδu0 − 1⁄6μδ3u0 + 1⁄30μδ5u0 − 1⁄140μδ7u0 + ... c2 = δ2u0 − 1⁄12δ4u0 + 1⁄90δ6u0 − 1⁄560δ8u0 + ... c3 = μδ3u0 − 1⁄4μδ5u0 + 7⁄120μδ7u0 − ... c4 = δ4u0 − 1⁄6δ6u0 + 7⁄240δ8u0 − ... c5 = μδ5u0 − 1⁄3μδ7u0 + ... c6 = δ6u0 − 1⁄4δ8u0 + ... . . . . . . |
(4).
If a calculating machine is used, the formula (2) is most conveniently written
| uθ = u0 + P1θ P1 = c1 + 1⁄2P2θ P2 = c2 + 1⁄3P3θ . . . . . . |
(5).
Using θ as the multiplicand in each case, the successive expressions ... P3, P2, P1, uθ are easily calculated.
As an example, take u = tan x to five places of decimals, the values of x proceeding by a difference of 1°. It will be found that the following is part of the table:—
Example 4.—(u = tan x).