This subject, as it left the hands of Simpson, Halley, Dodson, &c., is fully discussed in the appendices of Francis Baily's work on interest and annuities. These, with Milne's note on the subject (vol. ii., p. 680), Jones's notice of the problem, and the various modes of approximation announced in the pages of this Journal, give all I have found written about the matter in modern times. Referring to the citations in Baily for the history of the question subsequent to 1680, I give what I have found in previous time; and I add a few new formulæ for consideration. The problem is, I doubt not, of infrequent occurrence. It never occurred to myself to want more than a look at the tables would furnish; but it may be suspected that, as the requirements of business become closer, inverse problems will acquire more importance than they now possess.

No greater encouragement can probably be given to the practice of life assurance than the making its contracts as secure as the nature of them will admit, and the rendering as light as possible the consequences of any breach of them. How strongly this is felt by those more immediately interested in the extension of the system may be seen in the constant efforts, more or less legitimate, made by them in this direction. Thus, from more than one quarter the public has the offer of “indisputable” policies, and, from others, “unchallengeable” ones; and Companies are not wanting which liberally enough intimate that they hold, for a considerable time, at the disposal of the assured, the value of the assurance which he has accidentally or intentionally altogether abandoned.

Since the last article on the settlement of fire-claims under average policies appeared (April, 1857), a very important change has been adopted. The third clause of the Liverpool floating policies, known under the title of the “independent liability clause,” has been introduced, by the common consent of nearly all the Insurance Companies, and made part of the conditions of all London mercantile floating policies. The alteration is, beyond all doubt, a very considerable one in its practical effect. It will change entirely the mutual relation of the Companies in very many cases of joint insurances, and, if very great care be not taken at the outset, introduce no small share of doubt and difficulty in the relation of the Offices with the assured. It may, therefore, be worth while to offer a few remarks at the present moment on the change actually effected by the recent alteration of the average clause, and then make some practical suggestions as to the regulation of claims under the altered form of policy.

Most writers on the subject of life annuities have had occasion to lament the paucity of tables available for the performance of calculations involving two or more lives. The late Mr. David Jones has done much to supply this deficiency by the publication of complete sets of tables for two lives at various rates of interest; but, beyond this, it is extremely improbable that, under the present system, any considerable progress will be made, owing to the multiplicity of the different combinations when three or more lives are concerned, and the consequent magnitude of the task involved in the construction of complete sets of tables for such cases.

In the former paper, it was laid down, as the basis of any improvements I might be able to suggest as to life contingency calculation, that, in place of premises of a specific nature leading to special conclusions, we should rather, from the very commencement, recognise the variable nature of the results we might have to encounter, and accordingly construct a system of limits rather than one of particular values.

It is commonly known that in calculating an annuity on three joint lives, two of the lives are made to count as one life of the same annuity value; and that the result is approximately true. Twenty years ago (Phil. Mag., Nov., 1839), I showed that if the law of mortality explained by Mr. Gompertz were accurately true for the whole of life, this substitution of one life instead of two would also be accurately true. I now repeat the same proposition in a more general form.

Some three or four years ago, I submitted to the members of the Institute a few observations on the valuation of property held for life and in reversion, and on the due apportionment of it when questions arise between the life tenant and the owner of the fee. I was induced to bring this subject under the notice of the members from observing the great difference of opinion prevailing in regard to it generally, and especially in the discussions on the subject of church leases and of other property similarly circumstanced—in which discussions no sort of agreement appears to have been come to as to the true principles upon which the value of the interests of the several parties concerned should be estimated. In the paper in question, I endeavoured to point out some of these principles; and my object now is to enlarge somewhat on the arguments therein laid down, to show that the question at issue almost always resolves itself into one whether a property is to be bought or sold, to call attention to the vast difference in value which arises under the two conditions, and to point out the imperative character of the causes, from which this difference originates; finally, to urge the necessity of great care and circumspection in dealing with the questions submitted to us, so that all risk of confounding one of the conditions above referred to with the other may be avoided.

In the course of an actuary's practice, the question not uncommonly arises—What is the value of a life interest, accompanied by a policy of assurance on the same life ? The most obvious course for solving this question is to value the life interest and the policy separately—the former by the well-known formula . With regard to the policy, it is not so clear how it should be valued. If valued on the same principles as the life interest, the value of a policy for £1 would be 1 − (p + d) (1 + A); but this formula is quite inapplicable, for it gives a negative value for many years to a policy, and almost always a much smaller value than the surrender value allowed by the Office. There is, therefore, apparently no choice but to take the value of the policy at the latter amount.

The steady and gradual diminution in the rate of interest that took place from towards the end of the sixteenth to the middle of the eighteenth century, was generally attributed, during the period, to the great additions made, after the discovery of America, to the amount of gold and silver circulating in Europe.

The principal circumstances affecting the value of individual live, have reference to sex, age, personal peculiarities, habits, occupation, residence, family history, previous ailments, and present health. These are brought before the medical referee, in the printed forms of examination and inquiry adopted by Life Assurance Societies, in a series of questions calculated to elicit the information required. In the present treatise they will be considered under several heads.

The extension of life assurance in the United States, and, especially, the more or less successful attempts of English Companies to compete with the American Companies, render an inquiry into the mortality of American assured lives of the greatest interest to the members of this Institute. The question assumes the greater importance, because, from various reasons, the population statistics of the United States have hitherto been, and will for a long period probably continue to be, in a very imperfect state, notwithstanding the talents and skilful labours of the able men to whom the collection of the data in the different censuses has been entrusted, and the readiness of the people to afford the information desired. The marvellous rapidity with which the population has there increased; the vast extent of the country showing such a diversity of soil and climate; the conversion of the country into town districts, or of wild insalubrious localities into lands reclaimed and rendered healthy by the industry of a dense population—are all so many obstacles to prevent our obtaining at present any Government statistics that could be relied on for the discussion of this grave question.

Among the numerous classes of risks now undertaken by Insurance Companies, those against the contingency of leaving issue are beginning, from their increasing numbers, to assume a position of greater importance than heretofore. Nothing has hitherto, to my knowledge, been written upon the subject, and the premiums for insurances of this nature have been left to the judgment of the actuary as each individual case has come before him, there being no recognized and accurate formula for the calculation of the risk. Under these circumstances, and as a considerable amount of material exists from which approximate results may be deduced, I venture to submit to the Institute a few considerations upon the subject, but confined principally to the probabilities of marriage so far as they affect the calculation of such premiums.

Dr. Farr. in the 12th Report of the Registrar-General, has given the value of annuities and the expectation of life, as deduced from the male English Life Table.

Observing a very uniform difference between the results thus obtained and those from the Carlisle Table of Mortality, it appeared to me probable that a simple hypothesis might be constructed which would produce a close approximation to the expectation of life between certain ages, but somewhat less favourable than the hypothesis. I proposed in 1857 for the expectation of life according to the Carlisle Table of Mortality.

The idea of periodic or discontinuous functions was primitively introduced by Euler, and has since been the subject of extended investigation by M. Fourier, who has made some new and important applications of it in. his mathematical theory of heat.

The history of the United Provinces, and of Holland especially, from the close of the Spanish rule down to the establishment of the modern monarchy of the Netherlands, is distinguished for its manifestation of a permanent struggle between different opposite principles. Liberty and authority, municipal principle and state principle, republic and monarchy, the spirit of federal isolation and that of centralization, appear to give battle to each other upon a territory itself with difficulty defended from the waves of the ocean by the watchful industry of its inhabitants.

The recommendations were—“That a work should be published, in which should be given the history, and an analytical and comparative table, of the different systems of weights, measures, and coins of the different nations of the earth, to be translated and published, under the supervision of the branch societies, in all the languages of the nations represented in the Association. 2. That, to ensure the accuracy of such a work, the branch societies should furnish information as to the monies, weights, and measures prevalent in each country, with their values in terms of the Metrical System. 3. That each branch society should use every means, especially by aid of the press, to enlighten public opinion on the subject, and to prepare for the meeting of an official International Congress for discussion thereon. 4. That, in the meantime, the branch societies should make every effort to procure that, in all statistical tables and documents, public and private, the different quantities and values should be accompanied by their reduction into the monies, weights, and measures of the Metrical System, so that all nations may have a common medium of comparison. And 5. That unity in the fineness of the coin, unity in the standard of value, and unity in weights and measures of all kinds, should be pursued, in order to facilitate the adoption of a uniform system.”

The extent to which the interchange of risks is made by Assurance Companies at the present day, renders it of no small importance that a clear understanding should exist as to the terms upon which these contracts are to be carried out. There was a time, we believe, when a proposition from one Company to effect an assurance with another was looked upon with coldness, if not with suspicion; and the fact that such proposals were often made when the proposing Office declined to assume any portion of the risk itself, would seem, very naturally, to give rise to such a sentiment. This practice, however, we believe, is now quite on the decline; and with well-conducted Companies it is pretty clearly understood that these transactions must be in every respect perfectly bonâ fide, and that the arrangement of them must be entirely just and equal, as between all the parties concerned. As the wealth of the country increases, the cases are becoming more frequent in which the sums to be assured upon individual lives are larger than any single Company would care to sustain the liability of; and as propositions of this nature are being at intervals made to all, each becomes sensible of the advantages of the system of reassurance, and of the necessity of its being based upon principles of perfect fairness and equality.

The following problem was suggested by certain remarks in Mr. Buckle's introductory chapter to his History of Civilisation. In that chapter Mr. Buckle remarks upon the striking uniformity usually exhibited by classes of phenomena—such as not only births, deaths, marriages, but such things as those apparently depending upon mere caprice—such as the number of letters annually sent without directions through the Post-office, or those which appear likely to depend on the most capricious and irregular causes, such as murders and suicides; and he proceeds to draw certain moral conclusions from the fact of this uniformity, namely, the existence of certain moral laws, by which a section of the community, definite in number, is always impelled to such acts. But before entering upon a discussion of the legitimacy of such results, there appears to me always a previous question to be solved, namely, whether the observed degree of uniformity is remarkable or not. For supposing the observed uniformity to be not more than that which might be expected from events, the occurrence of which to individuals was conceived of as perfectly fortuitous, the whole argument would resolve itself into a pure metaphysical question, from which it would be hopeless to expect any practical issue.

As the forms above referred to are now numerous, and in constant requisition, we have thought that, by collecting some of them together and putting them into print, we might be affording certain facilities for, and leading the way to, more uniformity of practice in this particular than has hitherto been attainable. We, accordingly, append such specimens as we have been able to get together, and shall be obliged to any of our readers who will afford us the means of adding to their number, or who will suggest any improvements in their construction. Most of those now given have been in use for many years amongst some of the older Companies. It is quite possible that amongst those of more recent date better forms are rife, and we shall be very glad if the publication of these prove the means of bringing such to light.—Ed. A. M.