III. General Observations
7. Derivation of Formulae.—The advancing-difference formula (1) may be written, in the symbolical notation of finite differences,
uθ = (1 + Δ)θ u0 = Eθ u0
(13);
and it is an extension of the theorem that if n is a positive integer
| un = u0 + nΔu0 + | n (n − 1) | Δ2 u0 + ... |
| 2! |
(14),
the series being continued until the terms vanish. The formula (14) is identically true: the formula (13) or (1) is only formally true, but its applicability to concrete cases is due to the fact that the series in (1), when taken for a definite number of terms, differs from the true value of uθ by a “remainder” which in most cases is very small when this definite number of terms is properly chosen.
Everett’s formula (9), and the central-difference formula obtained by substituting from (4) in (2), are modifications of a standard formula
| uθ = u0 + θδu1/2 + | θ (θ − 1) | δ2 u0 + | (θ + 1) θ (θ − 1) | δ3 u1/2 + | (θ + 1) θ (θ − 1) (θ − 2) | δ4 u0 + ... |
| 2! | 3! | 4! |