Discussion
Everett's formula truncated after second differences is equivalent to ..................... truncated after third differences.
Answer: B
Everett's formula to second differences is equivalent to Bessel's formula to third differences.
Both are central-difference formulas for interpolating between \(x_{0}\) and \(x_{1}\). Everett's formula uses only even-order differences (\(y_{0}, y_{1}, \delta^{2}y_{0}, \delta^{2}y_{1}, \ldots\)), while Bessel's uses means of even differences and odd differences. Rewriting the odd differences as differences of even ones turns Bessel's formula up to third differences term for term into Everett's formula up to second differences. Both reduce to the same cubic through the four points \(x_{-1}, x_{0}, x_{1}, x_{2}\).
Stirling's formula (printed in option A as 'String's') is centred on a single tabulated point and is not equivalent. Lagrange's formula works with unequal spacing and is not a difference formula at all.
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