In the following pages I shall have frequent occasion to avail myself of a term which the progress of the analysis of life contingencies has rendered indispensable, but which is not found in any of the standard elementary works in that science. I think, therefore, that I cannot better commence this paper than by an attempt to give an explanation of the expression “force of mortality,” sufficiently ample to obviate any difficulties which might otherwise be experienced on this score.

Every mathematical table consists of a series of values of a function corresponding to successive values of the variable, which last series of values forms the argument of the table. Any such table may be constructed therefore, when the function to be tabulated—which may be called the characteristic function—is known, by the evaluation of that function in terms of the successive values of the argument. It is only, however, when the table to be constructed is of limited extent that this method of formation would be employed. If the table be extensive, and especially if the characteristic function be complex, this, which may be called the direct method, would become too laborious, each value when formed in this way also requiring separate verification. In these circumstances the Method of Differences becomes available for the end in view. This method dispenses with all reference to the characteristic function beyond what is necessary for the formation of a few values (which I call fundamental or primitive values), at stated intervals; and in applying it, each value being dependent on the preceding, verification is obtained by the periodical coincidence with those fundamental values of the corresponding terms in the series in course of being formed.

The present paper is a continuation of that in the Companion for 1840, on the application of Barrett's method of calculating Me contingencies. No tables were then published for contingencies involving two lives; but since that time Mr. Jones's tables have appeared, in his work on the subject in the Library of Useful Knowledge. It may therefore be worth while to collect and describe the formulæ relative to two lives which are most likely to be useful in practice; with some other matters, which may partly interest the actuary and partly the commencing; student.

In the last Number of this Journal I examined and compared the solutions of this question given by Milne, Baily, and Griffith Davies in their standard works. The first of these writers cannot be considered to have treated the question in a satisfactory manner. The two latter have investigated the problem more rigorously. They both proceed upon the supposition commonly adopted, viz. that the deaths occurring in each year of age are distributed uniformly over the year. They differ, however, in their views as to the interest to be allowed for fractions of a year, Davies throughout adopting Milne's view, by which the present value of £1 due in six months' time is , while Baily takes the value as . The latter, as I have already stated, appears to me the preferable view. Davies, however, has also given an investigation of the value of an annuity payable m times a year, on the supposition that interest is convertible into principal m times in the year; and his solution, as I have shown in the last Number of the Journal, is substantially the same as Baily's.

The mortality statistics of the Scottish Assurance Offices, whose experience it is proposed to develop before combining it with that of the English contributing Offices, have, after passing through various committees, and receiving the benefit of their consideration, eventually passed into my hands for final adjustment and development. Having accordingly considered the subject thus confided to my professional care, I venture to lay before the Institute a brief statement of the views which have occurred to me in regard to the processes which it is proposed to adopt for the arrangement of these data and transforming them into that shape which it is necessary they should assume before becoming the subject of comparison or of calculation.

The Calculus of Differences is a somewhat extensive subject, but it is not necessary now to go far into it, or to occupy ourselves with its application to other than the simplest class of functions, namely, rational, algebraical, and integer functions. The theorems we require might have been left to be deduced as occasion for them should arise; but it has been thought better— and the arrangement will certainly be more convenient for reference—to bring the greater portion of them together in this place.

A Table of mortality is designed to represent the number of lives which, according to the best deductions from past experience, may be expected to survive at the termination of each successive year of age, supposing these survivors to be derived exclusively from a certain number of persons originally taken either at birth or at a given age. As a practical index of the mathematical law of average mortality such a table may be taken as a trustworthy guide for the future, provided that the number of lives which enter into its formation be sufficiently large and the particulars respecting them be correctly registered; and, what is equally important, that the observations shall extend over a considerable number of years.

There is no country in which, from a combination of circumstances, a census, if accurately taken, is more likely to afford novel and interesting materials than the United States of America. The marvellous progress of the nation, which, in the last 70 years, has increased in numbers from less than 4 millions to nearly 31½ millions in 1860, is alone sufficient to suggest changes in population, in wealth, commerce, cultivation of land, means of transport and circulation, growth of towns, &c., which, if the facts were carefully examined, would throw light upon many obscure questions in social and political economy, and even by the disturbance of their ordinary laws modify and correct the theories which have been mooted.

Milne, in his Treatise on Annuities, chapter vii., has discussed this subject with much clearness and elegance; yet his method of investigation does not seem to be the best possible, nor do I think that his results are given in forms well adapted to numerical applications. It may therefore be not altogether useless to consider the subject in a somewhat different manner, and so to exhibit the formulas as best to meet the wants of the computer.

In former papers, I have investigated the value of an annuity on a single life, when payable half-yearly, quarterly, &c. It still remains to consider the values of annuities on the joint duration of two or more lives, or on the life of the last survivor of several lives, under similar circumstances. It will, however, be more useful to consider first the increase in the value of an annuity on a single life, when a proportionate part of the annuity is payable up to the day of death, instead of the annuity ceasing, as is generally assumed, with the payment that precedes the date of death.

It is my object in the present paper to discuss the principles which should guide the actuary in fixing the maximum amount of risk to be retained by a Life Insurance Company upon a single contingency.

Those principles are by no means limited to life insurance, but extend with suitable modifications to other kinds of insurance, as fire, marine, hailstorm, health, and accidental death insurance. They are also applicable to the risks of mercantile transactions, and in particular to the risks incurred by a bank in discounting its customers' bills. There can be little doubt that some recent failures of banks have been caused principally by the neglect of these principles; in other words, the manager has failed to put a proper limit on the risks undertaken, i.e., has trusted individuals to an imprudent extent, and the loss of heavy sums consequent on their failure has proved disastrous to the bank.

We have now to apply the principles laid down in the previous sections to the construction of tables, first of rational, and secondly of irrational, functions. It is only formations of the class last mentioned that are of real importance. A little previous attention to the class first mentioned, however, is desirable, as affording an opportunity of elucidating, under elementary conditions, the arrangement of the work which is found most convenient in practice. The present section will, therefore, be devoted to the formation of successive values of rational functions.

In the course of last year a very important document was published by the American Government, under the direction of the Secretary of the Interior, entitled “Statistics of the United States (including Mortality, Property, &c.) in 1860; compiled from the original Returns, and being the final Exhibit of the Eighth Census.” It contains an introduction by J. M. Edmunds, Commissioner of General Land Office, in charge of census, and a still fuller introduction to the mortality statistics for the year ending June 1st, 1860, by Dr. Edward Jarvis, a very able writer, who was a delegate to the Statistical Congress, when it was held in London, and is a Corresponding Member of the Statistical Society of London.

In considering the various expenses, or, as some writers term them, expenditures, that are the results of a general average act, the same test must be applied as in the case of sacrifices. Were they had recourse to as being, in the exercise of sound judgment and prudence, indispensably necessary for procuring the safety of all concerned in the adventure? Might the act from which they proceeded be the means of arresting any course of events which, if allowed to take their full scope, would terminate in the destruction of the vessel and the consequent loss of the freight and cargo? It is true human judgment is likely to err under these, as in other circumstances of like nature, and expenses may be incurred needlessly, for the ultimate consequences of an act cannot be always perceived; but if the end proposed will admit of no other course by which the vessel and the property in her may be preserved at the time, and if the measures resorted to are such as would not be employed in the ordinary prosecution of the voyage, there is reason for admitting such as a lawful subject of contribution. When common sense is exercised, and the circumstances are regarded comprehensively and clearly, the results are generally found to justify the means adopted in cases of danger needing prompt action.

In the Report before alluded to, this case is especially noticed as affording evidence of the concessions made by foreign countries to obtain uniformity. The practice abroad has hitherto conformed to the principle recognized universally, as to the propriety of admitting as general average the damage done by voluntary stranding which was attended by the preservation of the ship and cargo. They still retain, however, as a subject of contribution, any damage which may arise from running a ship ashore to avoid capture; but it is clear that the same objection cannot be raised as in the case of a vessel in danger of foundering; for she may be, under the former circumstances, perfectly seaworthy, and therefore there would not be the difficulty of determining the exact amount of damage done to the cargo or the ship.

In order to complete the consideration of this question as far as it relates to a single life, I will now give some numerical examples showing the magnitude of the corrections, or the additions to the value of annuities when payable half-yearly, as found from the formulæ of Mr. Baily and Mr. Woolhouse. Those formulæ will also be exhibited under some new aspects.