^{page 377 note} * In view of the various developments of the method, the name ‘n-ages’ has ceased to be appropriate. ‘n-variates’ would be a better name, but, as Elderton indicated (J.I.A. Vol. LXIV, p. 309) and as will further appear from comments in this paper, the method is closely akin to quadrature. Accordingly the suggestion is made that the name be changed to ‘weighted quadrature’. [But see p. 398 where ‘n-point method’ is suggested.–Eds. J.I.A.]

^{page 380 note} * It may assist students to note that each formula represents an n-term ‘pocket’ representation of the full distribution u_xy , all the moments of this ‘pocket’ distribution up to the order used in the basis of the formula being identical with those of the full distribution.

page 381 note * Prior to the developments in this paper, R. E. Beard reached formula(11) as an ‘n-ages’ solution of the problem of the group valuation of joint and last survivor annuities on two lives, recently discussed by the Faculty (T.F.A. Vol.XVII, p.39), and obtained close results for various test distributions, but his consideration of the two-variable aspect of the ‘n-ages’ method in the practical field was interrupted by the transfer of his activities outside the life assurance sphere.

^{page 389 note} * Many quadrature formulae can be obtained as special cases of a corresponding ‘n-ages’ formula; for example, Jones's fourth-order formula for Σu_xf(x) applied to (the frequencies u_x being treated as constant, so that σ₂ = ⅓ and β₂ = 1.8) produces the quadrature formula correct to the fifth order. See also Whittaker and Robinson, The Calculus of Observations, p. 163, where the special case of Jones's fourth-order formula, when u_x is the normal curve, is given as due to A. Berger.

It seems that Tchebycheff, following up a special case by Bronwin, was the originator of the single-variable ‘n-ages’ method, reaching a general solution for an n-term equally weighted formula, correct up to the nth order, including the remainder term (see The Calculus of Finite Differences, by L. M. Milne-Thompson. p. 177). When the distribution u_x is rectangular, Tchebycheff's quadrature formulae result and the simple quadrature formulae referred to by Elderton in J.S.S. Vol. II, No. 2, are special approximate cases. The fifth-order Tchebycheff quadrature formula given by Whittaker and Robinson (p. 159) is easily obtained by postulating a symmetrical distribution and reaching an ‘n-ages’ formula in the form of x ₃ = 0, x ₅ = –x ₁, x ₄ = –x ₂, by solving the equations and . The quadrature formula results when the moments of the rectangular distribution are inserted. J. E. Kerrich's note Approximate Integration, J.I.A. Vol. LXIV, p. 545, is also of interest in this connexion.