fferences_ of the original series; and so on. The successive
differences are also called _differences of the first, second, ...
order_. The differences of successive orders are most conveniently
arranged in successive columns of a table thus:--

+-----+----------+-----------+-----------------+----------------------+
|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 | | | | |
+-----+----------+-----------+-----------------+----------------------+

_Algebra of Differences and Sums._

[Illustration: FIG. 1.]

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 S
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 (