The basis of this Paper is the new method of graduation introduced by Professor E. T. Whittaker, and expressed in the equation

where δ indicates central differencing, so that δ6u′x=Δ6u′x−3;u′=the graduated and u the ungraduated values of a series of observations, and ∈ is a measure of the importance assigned in any particular case to adherence to the original data as against perfect smoothness in the graduated curve.

Certain terminal conditions attach to this equation, and it is suggested that if no attempt be made to carry the method to the extremes of the series of data, these terminal conditions may be disregarded.

By methods described in the Paper, different summation formulae of graduation are derived from the equation, giving ε various values. The results of test graduations of the a(f) and a(m) ultimate tables by certain of these formulae are shown, and compared with the results under the exact solutions obtained by Dr. A. C. Aitken. A graduation is also made of the Female Government Annuitants 1900-1920 Experience, and the results are compared with these produced by the methods suggested in Dr. Buchanan's recent paper (T.F.A., x., p. 289), and in the discussion thereon.

It is suggested that ε will prove of practical value as an aid to the selection and classification of graduation formulae, and that summation formulae obtained by the methods described have thus a practical advantage as well as one of theory.

The Paper is a review of the theory of selection as developed by British actuaries.

The fundamental ideas underlying the theory were present in the minds of actuaries from a very early date. Apparently the earliest allusion to the subject was that of Joshua Milne in his treatise on Annuities and Assurances; but its importance was clearly recognised by Arthur Morgan in his Introduction to the Experience of the Equitable Society, and by Thomas Galloway in the Preface to the Experience of the Amicable Society.

It is desirable that investigations into the mortality experienced by assured lives and more especially by annuitants should be carried out by continuous methods which involve a minimum of time and labour in order that the nature and extent of changes in the mortality from time to time may be observed. The Census Method* meets these requirements, and this method is examined “with the view of establishing how far it can in “itself or with practicable modifications cope with certain questions “arising (1) out of the rapid increase in the rate of mortality during the “period in which selection is in operation and (2) from the disturbance “caused by the heaping-up of business towards the end of each financial “year.” No suitable experience on which to test the method was available, and it was decided to construct hypothetical data in which known rates of mortality prevail in order that some measure of the probable numerical magnitude of the errors in practice might be obtained.

The Paper is an attempt to suggest a method of estimating the mortality of the future from the records of the past.

In Part I., after a brief discussion of the existing methods of calculating rates of mortality, a new method of employing past results called the “cumulative process” is suggested. The chief distinction between the new method and those already in use lies in the fact that an attempt is made to employ existing knowledge of mortality rates when examining new data. It is claimed for the cumulative process that it is suited to a steadily increasing accumulation of knowledge such as should be the result of systematic mortality studies.