[Delta]u_(n+1) -[Delta]u_n denoted by [Delta][Delta]u_n, or, more
briefly, [Delta]^2u_n; and so on. Hence the table of differences in S
2, with the corresponding values of x and of u placed opposite each
other in the ordinary manner of mathematical tables, becomes

+---------+---------+----------------+-----------------+-----------------+----------------------+
| x | u | 1st Diff. | 2nd Diff. | 3rd Diff. | 4th Diff. |
+---------+---------+----------------+-----------------+-----------------+----------------------+
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| | | | | | |
| x_(n-2) | u_(n-2) | | [Delta]^2u_(n-3)| | [Delta]^4u_(n-4) ... |
| | | [Delta]u_(n-2) | | [Delta]^3u_(n-3)| |
| x_(n-1) | u_(n-1) | | [Delta]^2u_(n-2)| | [Delta]^4u_(n-3) ... |
| | | [Delta]u_(n-1) | | [Delta]^3u_(n-2)| |
| xn | u_n | | [Delta]^2u_(n-1)| | [Delta]^4u_(n-2) ... |
| | | [Delta]u_n | | [Delta]^3u_(n-1)| |
| x_(n+1) | u_(n+1) | | [Delta]^2u_n | | [Delta]^4u_(n-1) ... |
| | | [Delta]u_(n+1) | | [Delta]^3u_n | |
| x_(n+2) | u_(n+2) | | [Delta]^2u_(n+1)| | [Delta]^4u_n ... |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
+---------+---------+----------------+-----------------+-----------------+----------------------+

The terms of the series of which ... u_(n-1), u_n, u_(n+1), ... are
the first differences are denoted by [Sigma]u, with proper suffixes,
so that this se