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.

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.

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.