This is the fourth, and, let it be hoped, the last, of a series of papers on mortality tables in their various aspects which the Institute has honoured me by accepting. The investigation had its origin many years ago in efforts to improve upon the then methods of constructing mortality tables from Census Returns; and, when the enquiry seemed to be nearing its conclusion, the fact became evident that the formulas which had presented themselves could be applied effectively in summation form to the graduation of tables derived from other sources. There was, however, a preliminary objection to be overcome. Summation Formulas of Graduation were under a cloud. Dr. Sprague had condemned them in unmeasured terms, and Mr. Todhunter, in the head-note to his paper (J.I.A., xxxii, 378), had accepted Dr. Sprague's authoritative judgment. Hence my first paper of the series, “On the Error introduced into Mortality Tables by Summation Formulas of Graduation”, (J.I.A., xli, 54), wherein it was attempted, and, I think, successfully, to show that that error is generally so minute as, with ordinary care, to be safely negligible in practice.

The presentation in November 1905, of three papers on “methods of valuation with allowance for selection” (J.I.A., vol. xl, pp. 1, 15, 42) might well be considered to have left room for no further word on the subject. Any attempt to produce a new method merely to achieve an equal degree of accuracy would undoubtedly be presumptuous, for Mr. King's method affords absolute accuracy, Mr. Diver follows with a microscopic error of £9 on a reserve of more than £3,000,000, and Mr. Ackland closes the series with a larger but still relatively insignificant error of £1,183 on a total reserve of upwards of £13,500,000. It is also shown that in both Messrs. Ackland's and Diver's methods, the error may be expected to be always less than ·01 per-cent.

The practical application of all three methods involves, however, a departure from the familiar system of classifying whole-life assurances according to attained ages, with the loss of all the advantages attaching to that system. This will be viewed by many with regret, and was especially referred to by Mr. Lidstone and Mr. R. P. Hardy in the subsequent discussion. Their remarks, together with Mr. Ackland's cordial expression of goodwill to “other workers in this field” led me to consider the subject from the standpoint of a grouping according to attained ages without any sub-grouping according to duration.