I believe there was a general concensus of opinion that the discussion which followed the reading of my paper at the sessional meeting of the Institute on 17 December 1888 (J.I.A., xxvii, 362) as an eminently practical one, and that another evening might usefully be devoted to a further discussion of many of the questions which were then raised.

With the object, therefore, of starting the discussion again, I propose to examine, in a general way, some of the views which were then expressed; and to better carry out that object, I shall consider the subject under four heads. (1) Whether, in the formula , Px, should be the pure premium according to the mortality table and rate of interest used in the valuation, or some other value. (2) Whether d should be taken at the rate of interest used in the valuation, or at some other rate. (3) To what extent special refinements can be introduced, and how far they may be neglected in practice. (4) The superiority of ray tables for practical use over other methods.

I believe that the first publisht paper on the subject of premiums for insurance against issue was that of Mr. Archibald Day, which was read before the Institute on 31 January 1859 (J.I.A., viii, 127). In this paper he gave from the Census Statistics the numbers of bachelors and widowers living in 1851, and the numbers of marriages of bachelors and widowers in the same year; and from these he deduced the probabilities of a bachelor and of a widower of any age marrying in a year, Then by means of the last-mentioned probabilities, combined with the probabilities of life given by the English Table No. 1, he calculated the value, at 3 per-cent interest, of £1 payable at the end of the year in which a husband shall contract a second marriage. The formula he used for this purpose may be written npH(1−npw)ϕH+n−1 × vn, where H and W represent the respective ages of husband and wife, and ϕ is the probability of a widower marrying in a year. It may be remarked in passing, that this formula is only approximately correct, inasmuch as there is a contingency that is not provided for in it, namely, that the husband may be alive and his wife dead, but that he may already have married a second time.

As the marriage rate among widowers depends not only on their age, but also upon the length of time for which they have been widowers, it was necessary to form the combined marriage and mortality table for widowers on principles analogous to those adopted by Mr. Sprague in the construction of his Select Life Tables, where the rate of mortality among assured lives for each age at entry was, after the lapse of five or a less number of years, joined on to the HM(5) rate of mortality. The first step, therefore, was to form a combined marriage and mortality table for chronic widowers—that is to say, those among whom the marriage rate has reached its ultimate magnitude; in other words, to find (wl)x, the number of widowers alive and still unmarried at any age, and number marrying, (wm)x, and dying, (wd)x, in the following year.