| x. | u. | 1st Diff. | 2nd Diff. | 3rd Diff. | 4th Diff. |
| · | · | · | · | · | · |
| · | · | · | · | · | · |
| · | · | · | · | · | · |
| · | · | (−1, 0) | · | (−2, −1, 0, 1) | · |
| x0 | u0 | (−1, 0, 1) | (−2, −1, 0, 1, 2) | ||
| (0, 1) | (−1, 0, 1, 2) | ||||
| x1 | u1 | (0, 1, 2) | (−1, 0, 1, 2, 3) | ||
| (1, 2) | (0, 1, 2, 3) | ||||
| x2 | u2 | (1, 2, 3) | (0, 1, 2, 3, 4) | ||
| · | · | (2, 3) | · | (1, 2, 3, 4) | · |
| · | · | · | · | · | · |
| · | · | · | · | · | · |
| · | · | · | · | · | · |
The formulae (1) and (15) might then be written
| u = u0 + | x − x0 | (0, 1) + | x − x0 | · | x − x1 | (0, 1, 2) + | x − x0 | · | x − x1 | · | x − x2 | (0, 1, 2, 3) + ... |
| h | h | 2h | h | 2h | 3h |
(19),
| u = u0 | x − x0 | (0, 1) + | x − x0 | · | x − x1 | (−1, 0, 1) + | x − x0 | · | x − x1 | · | x − x−1 | (-1, 0, 1, 2) + ... |
| h | h | 2h | h | 2h | 3h |
(20).
The general principle on which these formulae are constructed, and which may be used to construct other formulae, is that (i.) we start with any tabulated value of u, (ii.) we pass to the successive differences by steps, each of which may be either downwards or upwards, and (iii.) the new suffix which is introduced at each step determines the new factor (involving x) for use in the next term. For any particular value of x, however, all formulae which end with the same difference of the rth order give the same result, provided tabular differences are used. If, for instance, we go only to first differences, we have
| u0 + | x − x0 | (0, 1) = u1 + | x − x1 | (0, 1) |
| h | h |
identically.