In a paper which I read before the Institute of Actuaries in April 1887 (J.I.A., xxvi, 357) I gave some tables which seemed to me likely to be of use, in the case of certain life offices, for the purpose of forming estimates of the amounts of bonuses arising on ordinary whole-term with-profit policies. The offices contemplated were mutual societies, making their periodical valuations every five years by the HM Table, at 3 per-cent interest. Life offices differ so much in regard to rates of premium charged and rate and incidence of expenses, and also in regard to the rate of interest earned by their funds, that it appeared to me it would be desirable to construct tables in such a form that the bonuses, which would fit the circumstances of any particular office, might, without difficulty, be calculated from them, rather than to calculate bonuses for a hypothetical office. An illustration was given, it is true, but this was subsidiary to the main purpose of the paper, and was furnished with the object of showing how the tables might be applied.

Referring again to Art. 4 of Gompertz's treatise, let us denote by a0dx the actual probability of dying in the infinitely small time dx, at the initial age of the mortality table, and by axdx the abstract chance of death (in the time dx) at the end of x years; that is to say, the chance considered “independently of” (as Gompertz expresses it) or as “abstracted from” the deterioration resulting from increased age, then, according to Gompertz's law, the actual probability of dying, in the time dx, at the latter age will be axqxdx, in which expression q only is an arbitrary constant denoting the rate of deterioration. If, in axqxdx, we put x=0, it becomes a0dx, which coincides with the expression first assumed. Hence, Gompertz's law, which supposes that “the “vital force or recuperative power loses equal proportions in equal “times”, is expressed, in its general form, by the equation μxdx=axqxdx, or, which is the same thing, by μx=axqx.