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:-- 50 110 194 302 434 590 770 974 1202 1454 1730 Proceeding with this series in the same way, we obtain the following series of third differences:-- 60 84 108 132 156 180 204 228 252 276 Proceeding in the same way with these, we obtain the following for the series of fourth differences:-- 24 24 24 24 24 24 24 24 24 It appears, therefore, that in this case the series of fourth differences consists of a constant repetition of the number 24. Now, a slight consideration of the succession of arithmetical operations by which we have obtained this result, will show, that by reversing the process, we could obtain the table of fourth powers by the mere process of addition. Beginning with the first numbers in each successive series of differences, and designating the table and the successive differences by the letters T, D^1 D^2 D^3 D^4, we have then the following to begin with:-- T D^1 D^2 D^3 D^4 1 15 50 60 24 Adding each number to the number on its left, and repeating 24, we get the following as 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:--