It is regretted that in these two places Jhr. G. M. M. Alting von Geusau was wrongly described as Chairman of the Association of Dutch Life Offices (Nederlandse Vereniging ter Bevordering van het Levensverzekeringwezen).

Although this paper deals with the application of mathematical formulae to mortality data, it is not concerned with ‘graduation’ if that word is held to imply the fitting of a particular curve to a particular experience with the object of satisfying statistical tests. Nor is the paper concerned with the development of any philosophical theory of mortality. The experiments which it describes were undertaken in the hope of finding a standard type of curve which would give a good over-all representation of adult mortality in general. If such a curve could be found, an examination of the variations in its parameters might contribute something to an understanding, not of the nature of mortality itself, but of the differences—more particularly the secular differences—between one mortality experience and another.

The original purpose of this paper—undertaken at request but not unwillingly—was to review the principles and practice of life-office valuations in the light of modern conditions. It was difficult, however, to deal satisfactorily with the principles of valuation in vacuo without reference to more fundamental principles. As a consequence the paper has become more ambitious in its scope than originally intended—and has threatened to run away with itself. The reader will perhaps be less disappointed if he is warned in advance that he is to be taken on a ramble through the actuarial countryside and that any interest lies in the journey rather than the destination.

The experiments described in this paper arose from some calculations made in connexion with the recent discussion on the mortality of life-office annuitants (J.I.A. lxxviii, 27) and are a natural development of the principles described in the paper submitted to the Centenary Assembly of the Institute (Proceedings, 11, 89). They fall into two groups. Those in the first, arising from an attempt to find an alternative method on which to base projections, have an affinity with those set out by Starke in his recent paper (J.I.A. lxxviii, 171), in that a formula for expressing mortality is derived essentially dependent on year of birth. There are, however, fundamental differences between the approaches, in that Starke is concerned mainly with an ad hoc expression for a derived function of qx without regard to the problem of computing functions, whereas the present experiments are concerned with (i) further developing actuarial technique based on the curve of deaths rather than on rates of mortality, and (ii) the calculation of actuarial functions without recourse to the usual methods of commutation columns. The second group of experiments is an extension of the gamma-function technique to the calculation of joint-life functions, including contingent assurances.

A quarter of a century has elapsed since the publication of complete mortality tables of life annuitants for office use. Statistics showing the results of the Continuous Mortality Investigation have been published from time to time, but, partly as a result of the war, the Joint Mortality Committee have not attempted to construct complete new tables of mortality and monetary functions until the present time.

The purpose of this paper is to provide a basis for a discussion on the treatment of proposals for life assurance on sub-standard lives. To assist in this discussion a body of mortality data in respect of the sub-standard lives of a particular office is presented. The data relate to ordinary life assurances issued in this country subject to a diminishing lien. In the light of these data certain suggestions are made regarding the relative financial importance of the extra mortality of impaired lives in modem conditions. Apart from this statistical material, there is little that is new or original in the paper, although it is hoped that the author's relatively simple outlook on the subject may be of general interest and of assistance to students. The paper is concerned with the actuarial and practical aspects of the subject rather than with the medical aspects.

When the employment of as a medium for first-difference interpolation was suggested in J.I.A. LXXII, 453, no mention was made of a corresponding medium for the amount of an annuity. It seemed that the matter might bear further investigation from this angle, and that it would be well to bring the allied continuous functions within the scope of the inquiry. These latter are simpler to deal with because they may be expressed in terms of only one variable (ηδ).

During the discussion on Beard's paper (J.I.A. lxxvii, 425) A. C. Edwards commented on the relatively heavy death-rate during the last twenty years among men in England and Wales who were aged 20–40 in 1916. He pointed out that, in the tables on the Trend of Mortality in England and Wales that appear in J.I.A. from time to time, there is a peak in the ratios of men's mortality to the 1930–32 standard that has moved from ages 45–49 in 1933 to 60–64 in 1944–48, and he put forward as a possible explanation that this age-group lost a number of its fittest members during the 1914–18 war and that the survivors who had been on active service have been suffering from delayed and cumulative after-effects. He added that the excess mortality at the top of the hump might be 10–15% and suggested that some marked changes in male annuitants' mortality in the 60's and 70's might be observed during the next few years.

Mr Elphinstone, in his recent paper to the Faculty of Actuaries, has reawakened interest in the method of graduation devised by Sir Edmund T. Whittaker some 32 years ago. Except for Professor A. C. Aitken who, in a paper( in the Proceedings of the Royal Society of Edinburgh and a note in the Transactions of the Faculty of Actuaries, gave a brilliant alternative solution of the equations on which Whittaker's method depends, English and Scottish writers have paid little attention to the method. Robert Henderson's paper published in 1924 stimulated American actuaries, and their investigations have continued up to the present time. The book by Kingsland Camp (reviewed J.I.A. lxxvii, 327) gives numerous practical developments of Robert Henderson's work. Students should not overlook a most valuable paper by Charles A. Spoerl which deals comprehensively with almost every aspect of the subject.