In the course of an investigation that was recently being made into the rate of mortality prevailing among a body of assured lives subject to extra risks, it occurred to me that it would be advisable. if possible, to procure some standard of comparison in addition to the ordinary mortality tables at present in use, it being apparent that the rate of mortality in question would depart too markedly from the existing tables for them to serve as any criterion. Several modes of forming hypothetical tables exhibiting features similar to those characterizing the actual observations—namely, an increased rate of mortality, especially heavy at the earlier ages—suggested themselves to me; but the difficulty of fixing the degree in which they should differ from the table on which they were to be based, and the arbitrary nature of the assumptions it was necessary to make, were obstacles I was unable to remove in a satisfactory manner. Finally, influenced, among other considerations, by the practice in vogue with Life Assurance companies of the present day, I selected the basis explained in the following article, and deduced a table of mortality from the HM 3 per-cent whole-life pure-premium increased at each age by the net addition of 01.

During the last session of the Institute of Actuaries, I had the honour to submit a paper, the purpose of which was to devise a method of adjustment as even and correct as that of Mr. Woolhouse, and more easy in application. This was to be done by combining two summations of numbers increasing by third differences; the summations being so chosen as to introduce small factors only into the formula, and to admit of its being worked in columnar shape. (Journal, vol. xxiii, page 335.)

Thereupon, Mr. G. F. Hardy showed that Mr. Woolhouse's formula can be used in columnar shape; and Mr. Ackland, improving on Mr. Hardy's arrangement, contributed a table in which an adjustment of the HM decrement was so calculated. (Journal, xxiii, 354.)

This important and unexpected result of the reading of my paper rendered it of little value, except that it supplied a short formula which may be convenient for use where a rough graduation only is required. Any degree of smoothness can be obtained by repeating the process; but this is tedious, and I would now furnish a formula derived in the same manner, and yielding a satisfactory adjustment by a single operation.