1. The development of this paper involves the consideration of certain arithmetical, mechanical, and photo-electrical technicalities, but the ultimate aim is a very simple one and it may therefore be advisable to state it at once. The ultimate aim is to persuade the whole civilized world to abandon decimal numeration and to use octonal numeration in its place; to discontinue counting in tens and to count in eights instead.

It is over 20 years since the surrender values and paid-up policy amounts of ordinary policies were made a subject for discussion in this Hall. Industrial surrender values have been recently discussed. This paper is therefore primarily concerned with ordinary policies. In practice a company can grant whatever scales of surrender values or paid-up policies it may choose, provided it does not depart from any written, verbal or implied obligations to its policy-holders. There are, however, certain principles of equity to which consideration should be given, and it is along these lines that the problem is investigated.

This is a new subject, and it is with some diffidence that I bring it before the Institute for discussion. We are treading in unfamiliar ways and the pitfalls are many. Aviation hazard is, however, a problem that is daily assuming more and more importance in our business, and it is one that we shall have to face.

(1) The purpose of this paper is to describe the results of further investigations into the possibilities of trigonometrical representation of actuarial functions. The authors’ previous paper (J.I.A. Vol. LIX, p. 256) was concerned mainly with the use of Fourier’s Series, and the simplicity of that series in practice invites its use in any work of this nature. To a large extent Fourier’s Series, or a modification thereof, is employed in the present work, but the authors do not insist that the final formulae suitable for use in practice should be Fourier’s Series.