It is impossible in a short space to cover the whole ground suggested by the title of this paper. It is therefore proposed briefly to give an indication of the growth and scope of casualty insurance business in the United States of America, and then to examine the methods adopted there for calculating reserves, paying special attention to reserves in connection with workmen's compensation insurance.

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 two previous papers (vol. li, p. 77 and p. 211) an account has been given, of the published work of Newton on the subject of Interpolation by means of formulas of Finite Differences, and references have been made to the important letter which Newton sent on 24 October 1676, to Oldenburg, the Secretary of the Royal Society, with the intention that it should be communicated to Leibnitz. The second paper closed with a quotation from this letter, in which Newton mentions “a method [for the “construction of tables by interpolation] which I had almost “decided to describe here for the use of computers.” It has been my good fortune to discover that Newton had prepared an account of his method for inclusion in the letter and that the draft is preserved in the University Library at Cambridge among the Portsmouth Collection of Books and Papers written by or belonging to Sir Isaac Newton. By the freely granted permission of the Syndicate of the Library, I am able to give a reproduction and transcription of the manuscript, and I have added a translation.

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

The mortality of Europeans in West Africa has on several occasions been the subject of discussion in this Institute or of note in the pages of the Journal. In 1886 Dr. T. B. Sprague read a paper on the mortality of Europeans in the Congo, based on the statistics from Stanley's work on the Pounding of the Free State: In 1892 Dr. T. G. Lyon contributed a paper on mortality in various regions of the world from data gathered from Government Reports of the Colonial Office. In 1897 Dr. A. E. Sprague read a paper on the experience of 971 Belgian Government officials on the Congo and 178 employees of a Dutch Trading Company. In the same year Mr. J. E. Hart gave the experience of Colonial Office officials on the Gold Coast and Sierra Leone. Lastly, in 1911, Mr. Hart furnished a copy of the Report on Vital Statistics of European Government officials in the employ of the Colonial Office for the previous year.