The question of the reserve which a life assurance society should hold, in order to meet the claims under the policies issued by it as they mature, is one of such vital importance that I am led to hope that, notwithstanding the many and valuable papers on the subject which have from time to time appeared in the Journal of the Institute of Actuaries, the following reflections may not be without interest, even though they may not throw much fresh light on the subject. I may premise by stating that the conclusions arrived at, and hereafter referred to, are the result of an investigation undertaken as a matter of curiosity, and carried on in the midst of laborious and urgent official duty; consequently the scheme propounded may appear somewhat crude and ill-defined, and doubtless lacks the completeness which a more mature consideration of the point taken up would possibly have developed. Further, the results given here were arrived at before Mr. Sprague's paper “On the Construction of Select Mortality Tables” (Journal of the Institute, vol. xxi, p. 229) appeared in print; otherwise I should probably not have undertaken the investigation.

What I have aimed at is the discovery of some method of arriving at the requisite reserves for new policies, which shall be somewhat further removed from the realm of pure theory than some others which have been suggested; a method which can be readily applied in practice; and, what is of more importance, one which can be made intelligible to the ordinary man of business.

When a series ov quantitys proceed by a regular law, ther is no difficulty in interpolating between each ajacent two ov them any number ov terms with any desired degree ov accuracy. The methods to be adopted for this purpos hav been described in varios original and reprinted papers containd in the Jurnl ov the Institute ov Actuarys : (see, for instance, Mr. Woolhouse's paper On Interpolation, Summation, etc., xi, 61 , 301, and xii, 136; Mr. Gray's paper On the Construction ov Tabls by the Method ov Differences, xiii, 61, 149, 293, and xiv, 307; and the translations ov M. Maurice's Essay on Interpolation, xiv, 1; and ov Briggs's explanation ov his Method ov Quintisection, xiv, 73). We somtimes, however, hav to interpolate between quantitys which do not accuratly folio any law. For instance, we may hav calculated premiums for every quinquennial age, and wish to obtain the premiums for the intermediat ages by interpolation. In this case, it wil somtimes happen that the ordinary formulas ov interpolation do not giv satisfactory results, unles we take a very large number ov differences, and then the amount ov labor is offen more than the result is worth. In my paper On the Value ov Anuitys payabl half-yearly, quarterly, etc., I indicated briefly a method ov interpolation which I tho't miht be employd with advantage in such cases (xiii, 322), and it is now my intention to work out the formulas resulting from that method, and giv a practical ilustration ov their use.