Page:Edinburgh Review Volume 59.djvu/299

1834. Adding each number to the number on its left, and repeating 24, we get the following as the second terms of the several series:—

And, in the same manner, the third and succeeding terms as follows: — No. T D' D2 D^ D^ 1 1 15 50 60 24 2. 16 65 110 84 24 3 81 175 194 108 24 4 256 369 302 132 24 3 625 671 434 156 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 succession of terms, and so on.

By this principle it appears, that all tables in which each series of differences continually increases, may be produced by the operation of addition alone; provided the first terms of the table, and of each series of differences, be given in the first instance. But it sometimes happens, that while the table continually increases, one or more serieses of differences may continually diminish. In this case, the series of differences are found by subtracting each term of the series, not from that which follows, but from that which precedes it; and consequently, in the re-production of the several serieses, when their first terms are given, it will be necessary in some cases to obtain them by addition, and in others by subtraction. It is possible, however, still to perform all the operations by addition alone: this is effected in performing the