Journal . e difference of the two inclined diagonals. Applying this relation to the figure used for BesseFs formula,we can at once write Un = in),it, + (n + l)-A~iii>+{>i-i-2),Au.^, + 8ic.] \ . -(VI)— {n-1) iVo-in) 3^-H_,-{n +1), Put n^p, l—n = q; then, giving their values to the coefficients,the formula is easily expressed in the customary form +^-^^^A% (VIA) 10. The elaborate diagram of section 2 which has been usedfor purposes of demonstration can be replaced in practice by thesimple form given below, in which the difierences are left to besupplied mentally. The points to
Journal . e difference of the two inclined diagonals. Applying this relation to the figure used for BesseFs formula,we can at once write Un = in),it, + (n + l)-A~iii>+{>i-i-2),Au.^, + 8ic.] \ . -(VI)— {n-1) iVo-in) 3^-H_,-{n +1), Put n^p, l—n = q; then, giving their values to the coefficients,the formula is easily expressed in the customary form +^-^^^A% (VIA) 10. The elaborate diagram of section 2 which has been usedfor purposes of demonstration can be replaced in practice by thesimple form given below, in which the difierences are left to besupplied mentally. The points to be noticed are that, at anystage in a formula, the coefficient of the next term is on the rightof the difference last employed ; and that the coefficient may beused with either the difiereuce above it or the difference below. 1909.] Actuarial Xofef:. 241 Thus^ at every stage, tliere is a choice between two alternatives inproceeding to the next order of differences. K-l. 11, On Premiums pai/able by Instalments fur Whole-LifeAssurances. By S. T. Shoveltox, ^ , Felluiv ofMerton College, Oxford. J_HE problem dealt with in this paper is that of premiumspayable by instalments—m times a year—when those instalmentsfor the current year remaining unpaid at the time of death arededucted from the sum assured. Let it be assumed that the sumassured is payable at the end of that mth part of the rear inwhich the life fails, and let Uj denote the annual premiumspayable m times a year when unpaid instalments are to bededucted. Then the sum payable will be 1 m —1m n. 1- n. 1 n. according as the life fails in the tirst, second, . . rth, ... of them sections into which the year is divided. The value of this isA^-[l—IIj.] together with the value of an assurance of n. n, . according as death occurs in the lirst, 242 Actuarial Xotes. [April second, . . section of the year. The value of this assurance isreadily seen to be TT- f I ?> /• \ —-—
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