. |
+-------+-------+---------------------+----------------+----------------------+----------------------+

Similarly, by taking the means of consecutive values of u and also of
consecutive differences of even order, we should get a series of terms
and differences central to the intervals x_(n-2) to x_(n-1), x_(n-1)
to x_n, ....

The terms of the series of which the values of u are the first
differences are denoted by [sigma]u, with suffixes on the same
principle; the suffixes being chosen so that [delta][sigma]un shall be
equal to un. Thus, if

[sigma]u_(n-3/2) = C + u_(n-2),

then

[sigma]u_(n-1/2) = C + u_(n-2) + u_(n-1), [sigma]_(n+1/2)
= C + u_(n-2) + u_(n-1) + u_n, &c.,

and also

[mu][sigma]u_(n-1) = C + u_(n-2) + 1/2u_(n-1), [mu][sigma]u_n
= C + u_(n-2) + u_(n-1) + 1/2u_n, &c.,

C being an arbitrary constant which must remain the same throughout
any series of operations.

_Operators and Symbolic Methods._

12. There are two further stages in the use of the symbols [Delta],
[Sigma], [delta], [sigma], &c., which are not essential for elementary
treatment but lead to powerful methods of deduction.

(i.) Instead of treating [Delta]u as a function of x, so that
[Delta]u_n means ([Delta]u)_n, we may regard [Delta] as denoting an
_operation_ performed on u, and take [Delta]un as meaning [Delta].u_n.
This applies to the other symbols E, [delta], &c., whether taken
simply or in combination. Thus [Delta]Eu_n means that we first replace
un by un+1, and then replace this by u_(n+2) - u_(n+1).

(ii.) The operations [Delta], E, [delta], and [mu], whether performed
separately or in combination, or in combination also with numerical
multipliers and with the operation of differentiation denoted by D (:=
d/dx), follow the ordinary rules of algebra: e.g. [Delta](u_n + v_n) =
[Delta]u_n + [Delta]v_n, [Delta]Du_n = D[Delta]u_n, &c. Hence the
symbols can be separated from the functions on which the operations
are performed, and treated as if they were algebraical quantities. For
instance, we have

E.u_n = u_(n+1) = u_n + [Delta]u_n = 1.u_n + [Delta].u_n,

so that we may write E = 1 + [Delta], or [Delta] = E - 1. The first of
these is nothing more than a statement, in concise form, that if we
take two quantities, subtract the first from the second, and add the
result to the first, we get the second. This