+-----+----------+-----------+-----------------+----------------------+ |Term.| 1st Diff.| 2nd Diff. | 3rd Diff. | 4th Diff. | +-----+----------+-----------+-----------------+----------------------+ | | | | | | | a | | | | | | | b - a | | | | | b | | c - 2b +a | | | | | c - b | | d - 3c + 3b - a | | | c | | d - 2c +b | | e - 4d + 6c - 4b + a | | | d - c | | e - 3d + 3c - b | | | d | | e - 2d +c | | | | | e - d | | | | | e | | | | | +-----+----------+-----------+-----------------+----------------------+
3. The formal relations between the terms of the series and the differences may be seen by comparing the arrangements (A) and (B) in fig. 1. In (A) the various terms and differences are the same as in § 2, but placed differently. In (B) we take a new series of terms [alpha], [beta], [gamma], [delta], commencing with the same term [alpha], and take the successive sums of pairs of terms, instead of the successive differences, but place them to the left instead of to the right. It will be seen, in the first place, that the successive terms in (A), reading downwards to the right, and the successive terms in (B), reading downwards to the left, consist each of a series of terms whose coefficients follow the binomial law; i.e. the coefficients in b - a, c - 2b + a, d - 3c + 3b - a, ... and in [alpha] + [beta], [alpha] + 2[beta] + [gamma], [alpha] + 3[beta] + 3[gamma] + [delta], ... are respectively the same as in y - x, (y - x)², (y - x)³, ... and in x + y, (x + y)², (x + y)³,.... In the second place, it will be seen that the relations between the various terms in (A) are identical with the relations between the similarly placed terms in (B); e.g. [beta] + [gamma] is the difference of [alpha] + 2[beta] + [gamma] and [alpha] + [beta], just as c - b is the difference of c and b: and d - c is the sum of c - b and d - 2c + b, just as [beta] + 2[gamma] + [delta] is the sum of [beta] + [gamma] and [gamma] + [delta]. Hence if we take [beta], [gamma], [delta], ... of (B) as being the same as b - a, c - 2b + a, d -3c + 3b - a, ... of (A), all corresponding terms in the two diagrams will be the same.
Thus we obtain the two principal formulae connecting terms and differences. If we provisionally describe b - a, c - 2b + a, ... as the first, second, ... differences of the particular term a (§ 7), then (i.) the nth difference of a is