The appended notes, based upon the population statistics of England and Wales, fall naturally, as their title suggests, into two sections. The first consists of an attempt to identify and to measure, so far as may be possible, the more important errors introduced into the successive records through the misstatements of age to which the general population has been addicted ever since its returns were collected, and may be considered a necessary preliminary to the second, which reviews briefly the mortality experienced by that population, noting certain features in the changes which have occurred in the past and making some suggestions with regard to its possible course in the future.

In the Presidential Address delivered to the Institute in December 1918, Mr. Geoffrey Marks attempted a comprehensive description of the actuary, and indicated some directions in which his usefulness might be enlarged. It was this portion of the address which attracted the particular attention of the present author, chiefly because at the time the address was delivered he was engaged by a number of large engineering firms upon a statistical problem which had arisen in connection with certain repairing contracts involving payments totalling over one million pounds, a problem incapable of correct solution without the use of knowledge possessed exclusively by the actuary, and presenting, indeed, an almost exact analogy to a valuation for the purpose of amalgamation of a number of Sickness Insurance Societies.

The object of the present paper is to examine a formula deduced by Mr. Duncan C. Fraser in his “Notes on a Stationary Insurance Fund and on a Uniformly Progressive Insurance Fund” (J.I.A., vol. lvii, pp. 231, et seq.); and to investigate whether, and if so, to what extent, the formula can be put to practical use. In considering a uniformly progressive life assurance fund, where the numbers of new entrants in successive years are continually increasing at the rate of (1 + i'), Mr. Fraser derives the expression