of the differences. Thus

[mu]u_(n+1/2) = 1/2{u_n + u_(n+1)}, [mu][delta]u_n = 1/2{[delta]u_(n-1/2)} + [delta]u_(n+1/2), &c.

If we take the means of the differences of odd order immediately above
and below the horizontal line through any value of x, these means,
with the differences of even order in that line, constitute the
_central differences_ of the corresponding value of u. Thus the table
of central differences is as follows, the values obtained as means
being placed in brackets to distinguish them from the actual
differences:--

+-------+-------+---------------------+----------------+----------------------+----------------------+
| x | u | 1st Diff. | 2nd Diff. | 3rd Diff. | 4th Diff. |
+-------+-------+---------------------+----------------+----------------------+----------------------+
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
|x_(n-2)|u_(n-2)| {[mu][delta]u_(n-2)}|[delta]^2u_(n-2)|{[mu][delta]^3u_(n-2)}| [delta]^4u_(n-2) ... |
| | | [delta]u_(n-3/2) | | [delta]^3u_(n-3/2) | |
|x_(n-1)|u_(n-1)| {[mu][delta]u_(n-1)}|[delta]^2u_(n-1)|{[mu][delta]^3u_(n-1)}| [delta]^4u_(n-1) ... |
| | | [delta]u_(n-1/2) | | [delta]^3u_(n-2 | |
|x_n |u_n | ([mu][delta]u_n) |[delta]^2u_n | ([mu][delta]^3u_n) | [delta]^4u_n ... |
| | | [delta]u_(n+1/2) | | [delta]^3u_(n+1/2) | |
|x_(n+1)|u_(n+1)| {[mu][delta]u_(n+1)}|[delta]^2u_(n+1)|{[mu][delta]^3u_(n+1)}| [delta]^4u_(n+1) ... |
| | | [delta]u_(n+3/2) | | [delta]^3u_(n+3/2) | |
|x_(n+2)|u_(n+2)| {[mu][delta]u_(n+2)}|[delta]^2u_(n+2)|{[mu][delta]^3u_(n+2)}| [delta]^4u_(n+2) ... |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| . | . | . | . | . |