ries is ... [Sigma]u_(n-1), [Sigma]u_n,
[Sigma]u_(n+1).... The suffixes are chosen so that we may have
[Delta][Sigma]un = un, whatever n may be; and therefore (S 4)
[Sigma]un may be regarded as being the sum of the terms of the series
up to and including un-1. Thus if we write [Sigma]u_(n-1) = C + un-2,
where C is any constant, we shall have

[Sigma]u_n = [Sigma]u_(n-1) + [Delta][Sigma]u_(n-1) = C + u_(n-2) + u_(n-1),
[Sigma]u_(n+1) = C + u_(n-2) + u_(n-1) + u_n,

and so on. This is true whatever C may be, so that the knowledge of
... u_n-1, u_n, ... gives us no knowledge of the exact value of
[Sigma]u_n; in other words, C is an arbitrary constant, the value of
which must be supposed to be the same throughout any operations in
which we are concerned with values of [Sigma]_u corresponding to
different suffixes.

There is another symbol E, used in conjunction with u to denote the
next term in the series. Thus Eun means u_(n+1), so that Eun = u_n +
[Delta]u_n.

10. Corresponding to the advancing-difference notation there is a
_receding-difference_ notation, in which u_(n+1) - u_n is regarded as
a difference of u_(n+1), and may be denoted by [Delta]'u_(n+1), and
similarly u_(n+1) - 2u_n + u_(n-1) may be denoted by [Delta]'^2u_(n+1).
This notation is only required for certain special purposes, and the
usage is not settled (S 19 (ii.)).

11. The _central-difference_ notation depends on treating u_(n+1) -
2u_n -u_(n-1) as the second difference of un, and therefore as
corresponding to the value x_n; but there is no settled system of
notation. The following seems to be the most convenient. Since un is a
function of x_n, and the second difference u_(n+2) - 2u_(n+1) + u_n is
a function of x_(n+1), the first difference u_(n+1) - u_n must be
regarded as a function of x_(n+1/2), i.e. of 1/2{x_n + x_(n+1)}. We
therefore write u_(n+1) - u_n = [delta]u_(n+1/2), and each difference in
the table in S 9 will have the same suffix as the value of x in the
same horizontal line; or, if the difference is of an odd order, its
suffix will be the means of those of the two nearest values of x. This
is shown in the table below.

In this notation, instead of using the symbol E, we use a symbol [mu]
to denote the mean of two consecutive values of u, or of two
consecutive differences of the same order, the suffixes being assigned
on the same principle as in the case