sion of series may be reproduced in the other direction by addition. But let us suppose that the first number of the original table, and of each of the series of differences, including the last, be given: all the numbers of each of the series may thence be obtained by the mere process of addition. The second term of the original table will be obtained by adding to the first the first term of the first difference series; in like manner, the second term of the first difference series will be obtained by adding to the first term, the first term of the third difference series, and so on. The second terms of all the serieses being thus obtained, the third terms may be obtained by a like process of addition; and so the series may be continued. These observations will perhaps be rendered more clearly intelligible when illustrated by a numerical example. The following is the commencement of a series of the fourth powers of the natural numbers:—
| No. | Table. | |
| 1 | . . . | 1 |
| 2 | . . . | 16 |
| 3 | . . . | 81 |
| 4 | . . . | 256 |
| 5 | . . . | 625 |
| 6 | . . . | 1296 |
| 7 | . . . | 2401 |
| 8 | . . . | 4096 |
| 9 | . . . | 6561 |
| 10 | . . . | 10,000 |
| 11 | . . . | 14,641 |
| 12 | . . . | 20,736 |
| 13 | . . . | 28,561 |
By subtracting each number from the succeeding one in this series, we obtain the following series of first differences:
| 15 |
| 65 |
| 175 |
| 369 |
| 671 |
| 1105 |
| 1695 |
| 2465 |
| 3439 |
| 4641 |
| 6095 |
| 7825 |
In like manner, subtracting each term of this series from the succeeding one, we obtain the following series of second differences:—
