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.