Questions of combination are so connected with an actuary's business, that no apology is necessary for offering solutions to the notice of the readers of this Journal.

When two systems, A and B, of combinations, each having its own individual cases, are so related that each case of A admits of being converted into a case of B, and into one only, by a rule which will not convert any other case of A into that same case of B, it may be said that the system A is convertibly within B. It is obvious that if A be convertibly within B, and B convertibly within A, the numbers of cases in the systems A and B are equal.

The series proposed for solution in the following paper is—

The law of this series is manifest. Each term is the product of two factorials—the first consisting of p, and the latter of q, factors; and in each successive term, the factors of the first factorial are each diminished by one. and those of the latter increased by one.

The prevailing system of life contingency calculation is one not of variable but of invariable quantities. At the very threshold the admission of two such important assumptions is asked for, as that the rate of mortality is always invariable at the same age, whether old or young, and that the rate of interest is equally invariable for all periods, whether long or short. Upon these admissions of invariability a system is formed for assessing the relative values of different cases, thereby necessarily in every instance indicating an invariable answer; and with such indications the system rests content. Whether such assessments, however logically fair in connection with agreed postulates of invariability, are themselves eventually justified by the same invariability of actual result as was à priori assumed, has not hitherto been commonly brought within the general scope of the actuary’s studies. Directly, however, he is called on to take upon himself the practical responsibility of upholding this theory of invariability, he is somewhat surprised to find that, good as the mere logic of his studies may have been, it is by no means an easy task to connect such logic with the nature of the events he may see passing around him.

1. A Short time back I transmitted to the Philosophical Society of Cambridge some remarks upon the construction of tables of annuities: my object in that paper was to show how the probabilities upon which annuities depend should be deduced from tables of mortality, and I gave in illustration some tables of annuities calculated from observations of the mortality at Chester, by Dr. Haygarth, which appear to have been made with very great care. I have since compared these tables with a great many others, and I now present the result of this comparison.

The method of valuation once so universal—viz., that by means of tables of annuities involving the rates of premium charged—is now, I believe, admitted on all hands to be erroneous, and is, so far as I can learn, pretty generally abandoned. I have on more than one occasion endeavoured to demonstrate the fallaciousness of that method, and it is therefore needless again to draw attention to the peculiar consequences resulting from it; more especially as the great majority of the Companies now estimate their liabilities, as most actuaries agree that they should do, with the aid of tables based upon rates of interest and mortality approximating as closely as possible to those which observation and experience have shown to be the actually prevailing ones.

1. The object of the following investigation is to show how the probabilities of an individual living any given number of years are to be deduced from any table of mortality. All writers (with the exception of Laplace) have considered the probability of an individual dying at any age to be the number of deaths at that age recorded in the table, divided by the sum of the deaths recorded at all ages. This would be the case if the observations on which the table is founded were infinite; but the supposition differs the more widely from the truth the less extended are the observations, and cannot, I think, be admitted where the recorded deaths do not altogether exceed a few thousand, as is the case in the tables used in England. The number of deaths on which the Northampton Tables are founded is 4,689 (Price, vol. i. p. 357). The tables of Halley are founded upon the deaths which took place at Breslau, in Silesia, during five years, and which amounted to 5,869.

It is now somewhat more than twelve months ago that the decimal coinage question was brought under the notice of this Institute, and that a resolution was thereupon unanimously come to approving of the plan recommended by the Committee of the House of Commons. Since that decision, public attention has been repeatedly called to the subject; and although various plans have been proposed and discussed, founded on a different basis from the one thus recommended, it must, I think, be conceded, that none of them have made any serious impression on the public mind, but that, on the contrary, the one proposed by the Committee has made some if not considerable progress in the course of the sifting which the question generally has undergone.

The Institute of Actuaries is devoted to the consideration of all questions relating to assurance; and as life assurance occupies a prominent place in the daily occupations of many of its members, it is not surprising that papers on almost every conceivable topic in connection therewith have been read before it. There are, however, various other kinds of assurance well worthy of inquiry, and among these may be classed fire, marine, fidelity, and, last but not least, when we consider the number of persons affected by it, sickness assurance. It is true this last kind of assurance is almost confined to the working classes: they have no wealthy boards of managers; they have no man of science on their staff to direct their proceedings; and they are conducted by individuals who often possess no qualifications for such occupations. For all this, we must admit that every Friendly Society is a mutual Assurance Company; and, compelled by necessity, these Societies have taken up the most difficult and complicated branch of assurance. Not many actuaries have been much occupied with the subject, for it is a branch which does not sufficiently reward those who grapple with its difficulties; we are therefore under the greater obligation to those who have attempted it. Up to the present time, however, I am afraid that the success has not equalled the endeavour.

It is not without some degree of pride, that we recall the fact that England has taken the lead in fostering and extending those social institutions which appear destined to carry out the beneficent design of procuring the greatest possible amount of happiness for the greatest possible number. She has not been afraid to encourage a spirit of self reliance in the mass of the people, and to allow her working classes to associate freely in the effort to equalize the uncertainties of life, so that those who may have a little better fortune than the average may assist those who have a little less. Now and then, indications may have been observed of fear amongst the ruling powers lest this free association should be used for political purposes, and doctrines dangerous to the Government or the good order of society be thereby enabled to circulate too easily amongst classes who may have real wrongs to redress or fancied rights to assert; but, in the long run, good sense has prevailed, and Government has fortunately perceived that Friendly Societies for mutual aid in sickness or want, and other associations for bettering the condition of the working classes, gave them a direct interest in the preservation of the public peace, and formed by their very principles the antidote to the fears they had excited.

In a Postscript to the introductory matter of the Assurance and Annuity Tables, compiled and published by Messrs. Smith, Orchard, and myself in April, 1851 (and which Postscript was written by me), it is stated that I had “devised a new table, which, while answering all, and more than all, the purposes of Tables V. and VI. [which show the single and annual premiums for a survivorship assurance of £1 for every combination of two ages], by means of a single value for each pair of ages, and thus occupying; little more space than one of them, would also be very much less laborious in its construction than the former of those tables. The new tables would, in fact, consist of series of columns supplementary to Mr. Jones's Commutation Tables for Two Lives, and possessing all the properties, mutatis mutandis, with respect to the formation of temporary and deferred, as well as present, whole life benefits and payments, that give to tables of the form of those mentioned their great value.” The form of the table to which reference is here made suggested itself to me while occupied in superintending the printing of the work which contained the foregoing announcement. Having, however, had just then, as I thought, quite enough to do with survivorship assurances for a time, I contented myself with making the announcement cited, and communicating my ideas in a general way to my coadjutors, Messrs. Smith and Orchard.

Let us suppose France and the United States to enter into a convention with us to issue pieces of gold of the same weight and fineness as our sovereign—that is, containing 113 grains of fine gold; the three countries would then have a common standard of gold money. Let us examine whether, in the face of the operations of commerce, the international quality of value of these coins could at all times be maintained. The temporary balance of trade is always fluctuating in favour of one or the other country; a permanent balance against any country is impossible.

After the inaugural address of M. Quetelet, the regulations for facilitating the business of the Congress were proposed and agreed to, and M. Ducpetiaux read a list of the papers and correspondence received.

The representatives of the Statistical Departments in the different Governments of Europe proceeded to give a summary view of the position occupied by statistics in their respective countries.

Among the indirect advantages to be derived by the public from the establishment of Life Assurance Companies, may be fairly cited that of fostering a system of elaborate calculation so closely allied to the fiscal requirements of the times, that, allowing for hyperbole, the nation's finance minister might in the present day be in some degree characterized as the nation's actuary. Whether the incidental possession of analogous qualifications to those expected in insurance managers will ever be indirectly exacted by the general community from its finance ministers, would perhaps be presumptuous in us as actuaries to further discuss; but certain it is that, apart from insurance and its ramifications, no other definite school has hitherto been presented in which financial questions, and especially those relating to contingent finance, have ever been so elaborately and usefully developed. The interest and annuity tables alone, of insurance writers, form indeed an addition to financial literature of considerable importance.

The above is the title to a pamphlet just published by Mr. Thomson. The subject is one of such importance, that we should have been glad had our space permitted to give it in extenso; as it is, we place before our readers the more prominent parts of it. Mr. Thomson says—

“It has been the practice in Scotland for a considerable period to fix the rate of interest on landed securities periodically, with reference to the views expressed on the subject by, or more correctly the decision of, a meeting held in Edinburgh three months before each money-term of Whitsunday and Martinmas.

The mortality, according to age, of the general population of England in all localities, was first satisfactorily determined for the period of seven years, ending with the year 1844. The observations and results were published by the Registrar-General in the year 1849; and in the following year I was allowed the opportunity of exhibiting in The Lancet (vol. i. 1850, pp. 297–330) the principal results in new combinations, with my conclusions therefrom. The knowledge thus obtained of the mortality of the general population (with distinction of town from country districts) is valuable for testing the correctness of independent observations made on the mortality of classes of the labouring population, such as are contained in “Friendly Societies,” whose mortality and sickness it is now proposed to discuss.

When I began to direct my particular attention to the balance-sheets of the Hamburg Marine Insurance Societies, I hoped that a minute examination would lead to conclusions of general interest, and enable us to judge of the risks and state of navigation in general. I hoped that the collection of data and the comparison of the different balance-sheets would not offer great difficulties, and that my examinations might embrace a long period. My hopes have not been fulfilled. I perceived very soon that the work would be very troublesome, and that it would be difficult to form a complete collection of balance-sheets. There were none at the Public Commercial Library, and it was only by the aid of private friends that I at length succeeded in forming a complete collection of the balance-sheets since 1847. Thus I was obliged to limit my comparisons to the years 1847-53; and I think this is the principal reason why the law which governs the events cannot be deduced, and why no general conclusions can be drawn.

The subject proposed for discussion this evening is one of considerable importance to all who engage in the pursuits of an actuary. The old practice of valuing life contingencies by a pet table of mortality and at one fixed rate of interest has long since been found to work unprofitably; and parties seeking investments in securities of this description have been compelled, for their own safety and advantage, to adjust the valuation of the actuary before determining the price to be given for the particular property under consideration. If such a state of things is unsatisfactory to the public, it must be equally so to the actuary to find his calculations used only as a partial guide, instead of being, as in fact they ought to be, the index by which a proposed purchaser may know the rate of interest and the other advantages to be derived by obtaining the security in question at the price fixed by the actuary. In the ordinary transactions of a Life Assurance Office, where the fluctuation in the rate of mortality is protected by the admission of large numbers, it is necessary also to protect its pecuniary interests by limiting the amount to be assured on each life. Even with this precaution, it not unfrequently happens that particular years have proved less favourable than might have been expected—not so much from an increase in the number of deaths, as from the average amount of the sums insured by the policies so lapsed being greater than the average of the whole number. When, however, the average of a number of years is taken, and the amount insured upon any one risk is properly limited, these fluctuations become less apparent and generally disappear.

Having for some years been an advocate for the abolition of the measures, weights, and coins, now used in Great Britain, and the adoption in their stead of the système métrique, employed throughout a great part of the Continent, I have had occasion to consider the various difficulties which present themselves in the prosecution of this object.

I have not found it easy to select a proper title for this paper; and I am very conscious of the difficulty of explaining my reasons for instituting the somewhat laborious experiments of which I am now to state the results. Perhaps, indeed, I ought to apologize for offering to the Institute of Actuaries a communication which some of its members, who are conversant with the leading treatises on the doctrine of probabilities, may know to have been anticipated and rendered unnecessary by the labours of men much better qualified than I can pretend to be, to do justice to so profound a subject. I have some reason, however, to believe that the experiments I am about to describe are new; inasmuch as, though I have consulted one or two works which are likely to have contained some reference to such experiments, had they been already made, and have questioned more than one eminent member of your Society upon the subject, I have not been able to learn that any such experiments are upon record.