the second terms of the several series:—
| T | D 1 | D 2 | D 3 | D 4 |
|---|---|---|---|---|
| 16 | 65 | 110 | 84 | 24 |
And, in the same manner, the third and succeeding terms as follows:—
| No. | T | D 1 | D 2 | D 3 | D 4 |
|---|---|---|---|---|---|
| 1 | 1 | 15 | 50 | 60 | 24 |
| 2 | 16 | 65 | 110 | 84 | 24 |
| 3 | 81 | 175 | 194 | 108 | 24 |
| 4 | 256 | 369 | 302 | 132 | 24 |
| 5 | 625 | 671 | 434 | 156 | 24 |
| 6 | 1296 | 1105 | 590 | 180 | 24 |
| 7 | 2401 | 1695 | 770 | 204 | 24 |
| 8 | 4096 | 2465 | 974 | 228 | 24 |
| 9 | 6561 | 3439 | 1202 | 252 | 24 |
| 10 | 10000 | 4641 | 1454 | 276 | |
| 11 | 14641 | 6095 | 1730 | ||
| 12 | 20736 | 7825 | |||
| 13 | 28561 |
There are numerous tables in which, as already stated, to whatever order of differences we may proceed, we should not obtain a series of rigorously constant differences; but we should always obtain a certain number of differences which to a given number of decimal places would remain constant for a long succession of terms. It is plain that such a table might be calculated by addition in the same manner as those which have a difference rigorously and continuously constant; and if at every point where the last difference requires an increase, that increase be given to it, the same principle of addition may again be applied for a like