In the following essay we shall attempt the solution of a problem which has hitherto remained unsolved, though it is one of the most important problems connected with the science of life contingencies.

[Dr. Wittstein then explains the notation he uses; but we have substituted for it the usual notation, with which all our readers will be familiar; and he states that the Table B, on page 167, may serve as a specimen of a Mortality Table.]

Mortality Tables are based upon observations. We do not now propose to discuss the mode of collecting these observations. Upon this subject the author's treatise on Mathematical Statistics (Hanover, 1867), may be consulted. By whatever method the observations are made, the data resulting from them are always put in the form given in columns 1–3 of Table A, page 166. This table contains the data which Brune published in the year 1847 from the experience of the Prussian Widows' Fund, and from which he constructed his Mortality Table for Males (Table B); and we select this table in illustration of our subject, simply because it is readily accessible.

There are few matters in connection with actuarial science of which it can be said with truth that professional knowledge has gained little during the past two decades—a period which has witnessed a remarkable expansion in both the theoretical and practical branches of the subject of life contingencies. Questions relating to extra risks, and the premiums which should be charged for them, scarcely appear, however, to have occupied the attention of actuaries to a degree at all commensurate with their commercial importance. Within the space of time alluded to above the practical side of the subject has received consideration at the hands of Mr. A. H. Bailey (J.I.A. xv, 77), and Mr. J. Meikle (J.I.A. xix, 268) ; while lines of inquiry and investigation bearing upon points of theory have been pursued by Mr. W. M. Makeham (J.I.A. xiv, 159 and 242), and Mr. J. R. Macfadyen (J.I.A. xvii, 77). Mr. A. G. Mackenzie has also contributed to our available stock of information on the subject in his Note on War Mortality, read before the Actuarial Society of Edinburgh in 1872. I venture to submit, however that there still remains much to be done to bring this branch of the main subject to the same state of enlightenment and completeness that other parts of the science have now attained. The following paper is an attempt to throw light upon a point which has hitherto escaped scientific treatment; and I entertain the hope that, in some cases of official routine, the methods proposed will be found of service.

Some interesting statistics, based on the returns of 77 offices, and showing the sources from which profit is derived, are given by Mr. Archibald Hewat, in a paper read before the Actuarial Society of Edinburgh, entitled, Bonuses—how Earned and how Distributed, an abridged reprint of which appears in the Institute Journal, vol. xxii, pp. 286–292. Therein it is stated that £2,285,000 of hard cash is on the average annually divided among the policyholders by way of bonuses, such sum being made up from

These statistics afford a practical demonstration of the important bearing which interest—or, perhaps, more appropriately speaking, the surplus interest,—has on the profits earned. We see that no less than 45·50 per-cent of the profits of each year, or, if the profit on investments be included, that fully one-half of the surplus proceeds from this source; the remainder being principally contributed by the loading.

Twelve years have now elapst since the Married Women's Property Act, 1870, gave legislative sanction to the principle that a married man may effect a policy for the benefit of his wife and children, in such a way that the policy will be effectually settled upon them, notwithstanding his subsequent insolvency. I believe that the facilities thus afforded by the Legislature have been greatly appreciated by some classes of the community,—both by the lower middle classes who usually make no settlement upon their marriage, and by professional and mercantile men who have desired to supplement the provisions they made upon marriage, and to settle a portion of their accumulated savings so as to be secured to their families in the event of unforeseen reverses overtaking them.

This principle has since been extended to Scotland. In England the provision was included in an Act dealing generally with the rights and liabilities of married women; but in Scotland the law relating to married women is fundamentally different from that in England, and it was therefore thought better that the provision should be made by a short Act dealing only with this matter, which Act is printed in the Journal for July 1881.

It has now been considered desirable to consolidate and amend the law relating to the property of married women, and for this purpose to repeal the Married Women's Property Act, 1870; and the question of Settlement Life Policies therefore came up for reconsideration. The result has been that the new Married Women's Property Act, which came into operation on 1 January last, re-enacts with some material improvements the clause on the subject in the old Act.

The combination of circumstances, which for generations past has impelled large numbers of our countrymen to establish homes in distant lands, has wrought vast changes on the surface of the globe. The progress and welfare of Englishmen call, from every continent, for our interest and sympathy; and, while there is, probably, no portion of “Great Britain” which usually attracts more lively interest than Australasia, special attention is just now directed to that important member of the British Empire. The richness of its soil, its varying and salubrious climate, embracing those degrees of temperature most conducive to health, and, above all, its immense possibilities for the future, combine to fascinate the mind, and inspire a wish for further knowledge.

A comparison of the population returns of Australia with those of the United Kingdom, presents a contrast as striking as can be found in any department of social science. The area of the continent,—which is estimated to be somewhat under three million square miles,—contained at the end of the year 1882 about 2,296,000 inhabitants. The average number of persons to the square mile in each colony was as follows:—

In the course of an investigation that was recently being made into the rate of mortality prevailing among a body of assured lives subject to extra risks, it occurred to me that it would be advisable. if possible, to procure some standard of comparison in addition to the ordinary mortality tables at present in use, it being apparent that the rate of mortality in question would depart too markedly from the existing tables for them to serve as any criterion. Several modes of forming hypothetical tables exhibiting features similar to those characterizing the actual observations—namely, an increased rate of mortality, especially heavy at the earlier ages—suggested themselves to me; but the difficulty of fixing the degree in which they should differ from the table on which they were to be based, and the arbitrary nature of the assumptions it was necessary to make, were obstacles I was unable to remove in a satisfactory manner. Finally, influenced, among other considerations, by the practice in vogue with Life Assurance companies of the present day, I selected the basis explained in the following article, and deduced a table of mortality from the HM 3 per-cent whole-life pure-premium increased at each age by the net addition of 01.

At a recent meeting of the Institute, Mr. Sutton read an instructive paper on the method used by Milne in the construction of the Carlisle Table of Mortality, and the paper gave rise to an interesting discussion. I was surprised to find that it was the general opinion of members that Milne's description of his method is vague, and still more so to discover that more than one able investigator had expended much pains and ingenuity, unsuccessfully in my view, to unravel the mystery. Years ago I had studied Milne's explanations, and they seemed to me perfectly clear; and I had more or less tested his figures, and proved that they worked out correctly. So much was I satisfied, that it never occurred to me that the matter required further elucidation. But the remarks of Mr. Sutton in his paper, and of Mr. Manly in the-subsequent discussion, show that Milne has been very much misunderstood; or else that he has not only failed to explain his method, but that he has explained a method which he did not in the slightest degree adhere to, and that he thus has almost wilfully misled the student. Under these circumstances I may, perhaps, be pardoned if I give with some detail the results of my own researches,—which, since Mr. Sutton's paper was read, have been carried out to greater fulness—especially as I think that thereby may be vindicated the character of Milne for both lucidity and accuracy.

Before proceeding to a description of the formulas which form the main subject of this paper, it will be worth while to notice some points in connection with the celebrated “Summation Formula” of Maclaurin, the application of which to life contingencies we owe to Mr. Woolhouse. As Mr. Sutton has shown (J.I.A. xv, p. 307), the formula in question may be very simply established by symbolical methods. Thus, putting D for , we have

If we transpose the quantities ∑Ux and ∫uxdx, we get an expression for the value of the integral in terms of the sum of the function; and inserting the limits of integration, we obtain

Assuming, as in life calculations, that everything vanishes at the upper limit, we have

This important formula, which in practice is very powerful, is usually, at first, rapidly convergent, but ultimately becomes divergent. It has been shown, however, that the convergent portion will always give approximate numerical results. The main reason for reproducing the above demonstration, which will be familiar to most readers, is to show how, by a similar method, we may arrive at another summation formula, slightly more convergent than (1).

Questions as to the rate of mortality amongst assured lives, and enquiries based on re-arrangements of the mortality experience of life assurance companies collected by the Institute of Actuaries, have so frequently been discussed that I entertain some doubts as to whether the subject has not already been worn threadbare. Knowing, however, the kindness and consideration with which the Council receive contributions from junior members of the Institute, I venture to submit the results of some calculations I have lately made with a view of testing the probable effect of lapses and surrenders on the rate of mortality—a branch of the general question of mortality of assured lives which has not hitherto been directly dealt with.

The subject of this paper was some years ago suggested to me by a careful perusal of Mr. Sprague's paper (J.I.A. xv, 328), entitled “On the Rate of Mortality prevailing among “Assured Lives as influenced by the length of time for which they “have been assured”; but, although I had frequently turned the matter over in my mind, it was only recently I saw my way to give practical effect to my ideas.