The paper is an attempt to bring out the exact relationship between what has been called the ‘modified osculatory process’ and the older formulae of osculatory interpolation. The conditions upon which the modified process is stated to be based are shown to be insufficient in themselves to produce a determinate formula even after the degree has been made as low as possible, and a general formula is derived which satisfies these conditions and of which the formulae hitherto employed are particular cases. The general formula contains, as might be expected, certain arbitrary constants which reflect the relaxation of certain of the conditions insisted on by the older formulae.

Illustrative graduations are performed by means of the general formulIllustrative graduations are performed by means of the general formula

and the results are compared with previous graduations of the same data by Jenkins's fifth difference formula (which may be obtained by putting b=0 in the general formula), viz.:—

This is the particular case of the general formula which has hitherto been used in practice.

The paper first discusses the chief tables at present in existence representing female assured mortality, and attention is drawn to the fact that a considerable period has elapsed since the publication of any experience of the mortality exhibited by such lives in this country. It is further pointed out that the Continuous Mortality Investigation now in progress makes no provision for such an enquiry, and this fact, together with the apparently considerable increase in proposals for assurances on female lives, gave rise to the present investigation, which is based on data representing the experience of eight Scottish offices over the period 1920-1930. The methods employed in the collection of the data having been discussed, a short explanation is given of the use made of Powers Machines in the investigation and the increased facilities for sorting and tabulating the particulars which they afforded.

The graduation of the experience formed one of the major problems of the investigation, and after considering the peculiarities of the ungraduated data, and the particular difficulties to which these gave rise, the method finally adopted is explained. Aggregate and ultimate tables on this basis are given, and the usual tests applied to the graduated rates to indicate to what extent smoothness and fidelity to data have been secured. Comparisons are made with other tables of mortality, and these are illustrated in both tabular and graphic form.