It is demonstrated, with particular reference to the OM, OM(5) and A1924–29 Tables, that over the range age 20 to age 65 of a normal mortality table, μx can, with a close degree of approximation to the tabular values, be expressed in the form

leading to the following analogous expressions:—

Generally, all functions involving mortality within the stipulated range can be expressed in a similar form as purely mathematical and easily evaluated functions of the initial age and interest in combination.

It is claimed that the evaluation of such temporary functions at any rate of interest is thereby greatly facilitated.

The retrospective and prospective values of an endowment assurance policy in terms of the formula are investigated, and check methods of valuation, continuous in process and independent of the working principles and practice of the principal method, are put forward.

Numerical results and examples of the procedure are given in the paper.

The two main objects of the paper are :—

(1) To ascertain what modifications in Makeham's Law are necessary in order that the principle of Uniform Seniority may be applied to the A1924–29 ult. table.

(2) To ascertain whether some approximate method is readily available for the calculation of joint life annuity values according to the A1924–29 ult. table.

As regards (1) developments of the suggestions put forward in the discussion following Mr. C. L. Stoodley's paper to the Faculty in February 1934 are examined, and the conclusion is reached that, while the results are somewhat disappointing so far as the A1924-29 table is concerned, some of the formulae brought out may yet prove of practical use when applied to a mortality table more free from heterogeneous matter.

In the second part of the paper the results of applying to the A1924–29 ult. table several standard approximate methods are first examined, and it is shown that such methods are liable to produce systematic errors when applied to this table. Two new methods of approximation are then developed, which are shown to yield much more satisfactory results, and tables are reproduced to facilitate the practical application of these methods.

The paper is intended as an elementary introduction and companion to the paper on “Orthogonal Polynomials,” by G. J. Lidstone, J.I.A., vol. briv., p. 128, and the paper on the “Sum and Integral of the Product of Two Functions,” by A. W. Joseph, J.I.A., vol. lxiv., p. 329; and also to Dr. Aitken's paper on the “Graduation of Data by the Orthogonal Polynomials of Least Squares,” Proc. Roy. Soc. Edin., vol. liii., p. 54.

Following Dr. Aitken Σux is defined for the immediate purpose to be u0+…+ux−1.The scheme of successive summations is set out in the form of a difference diagram and is extended to negative arguments. The special point to which attention is drawn is the existence of a wedge of zeros between the sums for positive arguments and those for negative arguments.

The rest of the paper is for the greater part a study of the table of binomial coefficients for positive and for negative arguments. The Tchebychef polynomials are simple functions of the binomial coefficients, and after a description of a particular example and of its properties general methods are given of forming the polynomials by means of tables of differences. These tables furnish examples of simple, differences, of divided differences, of adjusted differences, and of a system of special adjusted differences which gives a very easy scheme for the formation of the Tchebychef polynomials.