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.