... are _Bernoulli's numbers_.

(ii.) If we express differential coefficients in terms of advancing
differences, we get a theorem which is due to Laplace:--

_ x_n
1 /
- | udx = [mu][sigma](u_n - u0) - 1/12 ([Delta]u_n - [Delta]u0) + 1/24 ( [Delta]^2u_n - [Delta]^2u0)
h _/x0

- 19/720 ([Delta]^3u_n - [Delta]^3u_0) + 3/160 ([Delta]^4 u_n - [Delta]^4 u0) - ...

For practical calculations this may more conveniently be written

_ x_n
1 /
- | udx = [mu][sigma](u_n - u0) + 1/12 ([Delta]u0 - 1/2[Delta]^2u0 + 19/60 [Delta]^3u0 - ...)
h _/x0

+ 1/12 ([Delta]'u_n - 1/2[Delta]'^2u_n + 19/60 [Delta]'^3u_n - ...),

where accented differences denote that the values of u are read
backwards from un; i.e. [Delta]'un denotes u_n-1 - u_n, not (as in S
10) u_n - u_n-1.

(iii.) Expressed in terms of central differences this becomes

_ x_n
1 /
- | udx = [mu][sigma](u_n - u0) - 1/12 [mu][delta]u_n + 11/720 [mu][delta]^3u_n - ...
h _/x0
+ 1/12 [mu][delta]u0 - 11/720 [mu][delta]^3u0 + ...

/ 1 11 191 2497 \ / \
= [mu]([sigma] - -- [delta] + --- [delta]^3 - ----- [delta]^5 + ------- [delta]^7 - ...)(u_n - u0).
\ 12 720 60480 3628800 / \ /

(iv.) There are variants of these formulae, due to taking hum+1/2 as the
first approximation to the area of the curve between um and um+1; the
formulae involve the sum u_1/2 + u_3/2 + ... + u_n-1/2 := [sigma](u_n -
u0) (see MENSURATION).

20. The formulae in the last section can be obtained by symbolical
methods from the relation

_
1 / 1 1
- | udx = - D^1 u = --.u.
h _/ h hD

Thus for central differences, if we write [theta] := 1/2hD, we have [mu]
= cosh [theta], [delta] = 2 sinh [theta], [sigma] = [delta]^-1, and
the result in (iii.) corresponds to the formula

/ / 1 2 2.4 \
sinh [theta] = [theta] cosh [theta]/ (1 + - sinh^2[theta] - --- sinh^4[theta] + ----- sinh^6[theta] - ...).
/ \ 3 3.5 3.5.7 /

REFERENCES.--There is no recent Engl