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Published online by Cambridge University Press: 18 August 2016
The great value of quadrature formulæ for the calculation of survivorship and other benefits involving three or more lives has been already illustrated in the pages of the Journal. From certain remarks made by Mr. G. F. Hardy and Mr. George King in the discussion which followed the reading of Mr. King's paper (J.I.A., xxvi, pp. 297, 301), I inferred that it would be of interest to examine the errors involved in these formulæ, and to compare the results yielded by them and by other methods of approximation, such as the substitution of an equivalent life. Mr. Sheppard has shown (Proceedings of the London Mathematical Society, xxxii, p. 258) how most of the best-known quadrature formulæ can be readily derived from the Maclaurin Summation Theorem, with expressions for the errors in terms of differential coefficients. In a recent paper (Proceedings of the London Mathematical Society, xxxiv, p. 335), I have obtained similar formulas by direct integration from Everett's Interpolation Theorem, with expressions for the errors in terms of central differences.
page 385 note * The central difference notation adopted here is different from that employed in the paper quoted above. There I used two operators, σ and μ (the latter a new one due to Mr. Sheppard, cf. Proceedings of the London Mathematical Society, vol. xxx, pp. 428, 460), defined by the relations
This notation possesses certain advantages over that used above, and it would have been in some ways preferable to employ it again. But μ has to actuaries a well-defined meaning, and it would have been very inconvenient to use it in two senses in the same paper; the suffix notation has accordingly been adopted.
