1. Before dealing with the question of mortality in Australia it will perhaps be well to indicate briefly what is meant by “Australia,” since the name is used with different significance in different cases. At an early stage of Australian history the name was restricted to the island continent itself, and what is now known as the Commonwealth of Australia was generally spoken of as Australia and Tasmania, in somewhat the same way as the homeland pair is spoken of as Great Britain and Ireland. With the advent of Federation, however, this mode of reference became less usual and has now practically disappeared.

In Canada and in the United States the writing of disability coverage in connection with life insurance policies is a matter of great importance, and the demand on the part of the insuring public for this additional coverage in a life contract is a marked feature of underwriting conditions in those countries.

The object of these Notes is to obtain for the valuation of pension funds a method in which commutation functions operating over groups of ages are calculated directly from the decremental rates applicable to the groups. By this means the labour involved in the construction of a full service table and commutation functions at each age can be avoided. Valuations have been effected by various methods, and it is shown that the results brought out by the short methods are as reliable as those obtained by using functions at individual ages. In proceeding to the particular short method advocated, other possible lines of attack have been suggested, and it is not disputed that formulae based on similar assumptions, but different from those in the Notes, might give equally satisfactory results.

In a note to his paper on "the Approximate Valuation of Endowment Assurances" (J.I.A. Vol.XLVIII, p. 32) Mr Elderton suggested that it might be possible to approximate to the true net premium reserves by accumulating the premium at interest only without having any direct regard to mortality, and I have made an attempt in this paper to arrive at a suitable method of approximation.

This paper is an account of some experiments which were made with the data relating to the O[M] and allied tables chiefly with a view to ascertaining if the graduation of the tables based on such data could have been shortened or simplified. The possibility of some means of condensing the tables was also borne in mind.

1. The ideas submitted in the present paper have their origin in an investigation carried out by the author with a view to determining whether it was possible to estimate the mortality likely to be experienced by married men in England and Wales in the course of the next few years. The English law relating to registration of deaths does not require that the marital condition of a deceased man shall be recorded when death is registered, in this respect differing from the law in Scotland and in various other countries. Thus no statistics could be obtained of married men’s deaths in England and Wales, and the question had to be approached by indirect methods.