I propose in this paper to deal with a question which has an important bearing upon those calculations as to deathrates, which are now so widely circulated and so generally felt to possess interest.

It has occurred to the Registrar-General, and, as I believe, to other inquirers, to remark that the mortality happening in London is diminished, “because domestic servants, shopwomen and milliners, “who have come from the country, retire when health fails them “to their native air.”

The first question that must be considered in connection with this subject is,—When is a Life Insurance Company insolvent? This question has recently acquired greater practical importance in consequence of the passing of the “Life Assurance Companies Act, 1870,” by which it is for the first time in effect enacted that an insolvent Life Office may be wound up, altho it has not committed any act of bankruptcy. Under the old law, even if such a Company were notoriously insolvent, it may be said that practically there was no means of putting a stop to its operations until it failed to pay an accrued claim. This has now been altered, and a Company that is proved to be insolvent can be wound up. The 21st section of the above Act provides that the Court of Chancery “may order “the winding up of any Company in accordance with the Companies “Act, 1862, on the application of one or more policyholders or “shareholders, upon its being proved to the satisfaction of the “Court that the Company is insolvent, and in determining whether “or not the Company is insolvent the Court shall take into account “its contingent or prospective liability under policies and annuity “and other existing contracts.”

The formula for the law of mortality which I am about to explain and apply to a practical case, is not a merely empirical formula. but is based on a presumed property of the causes of death. The value of the formula is however a question which is in a great measure independent of the correctness of my hypothesis; for it is well known that even a false hypothesis may be of great service. While I shall certainly be very much interested to learn the opinion entertained by actuaries as to the theoretical part of the subject, yet the practical application of the formula is the point to which I attach most importance, and to which I would by preference invite criticism.

The past session of Parliament has witnessed the passing of an Act for the regulation of Life Assurance Companies in the United Kingdom, which, while introducing great changes in the law, still stops very far short of the system of legislation which has been for several years in operation in a few of the United States of America, and which is warmly approved of and urgently recommended for adoption by some persons in this country. The present may therefore be considered a fitting time for reviewing what has been done and considering whether any further legislation is desirable, and if any, of what nature it should be.

The question,—how insolvent Life Assurance Companies are to be dealt with,—which all who take an interest in assurance affairs have for some considerable time been aware could not much longer be evaded, has during the last three years assumed a pressing importance, and although the subject has lately been discussed in several pamphlets and essays by members of this Institute and others it can hardly be considered, exhausted. One of our most distinguished public men, himself one of the ablest lawyers of the day, after a searching and diligent enquiry has come to the conclusion that no better way can be found to settle the affairs of the “Albert,” than by distributing among its more than 20,000 assured a sum which, on an average, may be expected to yield about one year's premium for each. It is surely worth consideration whether it is desirable that this decision should be drawn into a precedent, and besides, the subject may interest the members of this Institute for its own sake, because it involves questions concerning estimates of liabilities with regard to which actuaries are by no means agreed, and at which lawyers are in the habit of looking from a different point of view altogether. Certainly we may learn something from the legal profession, possibly they may pick up a little from us, towards the elucidation of a somewhat intricate matter, and the solution of a by no means easy problem.

The various Pension Funds which have been established in the Presidencies of India in the Military, Medical, and Civil Services of the late Honourable East India Company have afforded the means of bringing together a great variety of interesting facts relating to the rates of mortality and marriage amongst Europeans in India. They were originally founded as mutual charitable schemes, giving assistance or pensions to the widows or children of such Members of the services as had not left sufficient pecuniary means for the support of their families or for putting them forward in life. The Company saw fit to encourage these laudable designs by allowing high rates of interest on their invested Funds, and making favourable arrangements in the exchanges for payments to be made in England.

In a paper read before this Institute many years ago, by Mr. S. Brown, occur the following judicious remarks relative to Statistics of Mortality and Sickness:—

“It has frequently been remarked” says Mr. Brown, “that “however varied and uncertain may be the occurrence of the events “to which the life of a single individual is exposed, the average “return in a large mass of persons is so regular as to be predicted “with confidence within very small limits of error. On the “uniform happening of these events under similar circumstances, “or the discovery of the laws by which they are governed, the “actuary depends for the application of the theories of probabilities “to the many useful purposes for which they may be rendered “available. The principal difficulty which he has to encounter is “the indefinite character of the event itself. In endeavouring to “ascertain the law of mortality at any period of time, or in any “particular country, the event is certain and definite whenever it “does occur. The attention is only required to the difference of “the circumstances which cause it—the age of the party, the “nature of the disease, or the locality within certain boundaries of “which a greater or less intensity of disease may prevail. But a “much greater difficulty arises when the event itself which is under “observation is of an uncertain character; such for instance as the “average duration or effects of sickness in a large number of cases.

The honour I have received by the invitation of the President and members for organising the Fourth Session of the International Statistical Congress, to be held daring the week commencing the 16th July instant, which the state of my health prevents me having the satisfaction to attend, induces me to offer to the notice of the distinguished members some observations which, I think, may be considered of some importance in connexion with the objects of the Congress; having for many years paid attention to the subject of vital statistics, and to the mode of rendering the information derivable from its pursuit interesting, beneficially and scientifically, to the public, who may gain consoling pecuniary advantage from it, and to the student, who may enjoy the contemplation of the paths of science; because the services which have been obtained by the public from Assurance Societies, and from the proper management of Friendly Societies, have been obtained by the attentive and philanthropic study of the subject, and the mode of pursuing that study depends on statistical inquiries, and, I may say, on deep mathematical reflection,—a reflection which, whilst it may and has been of important service to society at large, may offer many new views in philosophical and mathematical branches of enquiry even quite unconnected with statistics or its objects.

The accompanying illustration gives a top view of the instrument. The following description is taken from the Engineer, 20th May, 1870.

It is constructed chiefly of a brass plate A A furnished with eight slots as shown; directly under these slots are mounted eight drums, each having nine elongated cog-teeth of successively decreasing length; over each drum, and between it and the slot, is mounted a square shaft on which slides a pinion wheel, so as to catch any number of teeth on the drum. Each of these pinion wheels is moved by a button a, of which there is one in each slot, the figures at the sides of the slots showing the proper position of each button a for any work to be performed by the instrument, so that not the least trouble is encountered in arriving at the result. The cogged drums gear by bevel wheels with a long horizontal shaft which is also in gear with the vertical shaft moved by the handle b, by which the instrument is worked. B B is a moveable brass plate, which can turn and slide on a round bar hinge at the back; in this plate there are sixteen holes, c, under each of which is a moveable disk numbered from 0 to 9, and arranged so that any one figure of each disk may be brought under its corresponding hole c.

The development of the nth differential coefficient of a function in terms of the finite differences is involved in the expansion of the multinomial . A general expression for the coefficient of each power, to the seventh inclusive, is given in De Morgan's Diff. and Int. Calculus, p. 264,—but the extremely intricate character of such general expression renders it desirable to obtain a more convenient practical method of computing these coefficients to any extent that may be required.

When our mutual life companies began their business in 1843, they had noAmerican table of mortality to guide them in determining the premiums ofinsurance that ought to be charged at the different periods of life. Therewere no American statistics public or private, good, bad, or indifferent, towhich they could refer, except the mortuary reports of cities, and thesewere so imperfect and unreliable as to be utterly useless, except toencourage the opinion that the chances of long life were about the same hereas in the countries from which our people had emigrated. It was known thatthe numbers of the dying, as reported by our city registers, were below thereal deaths; that the ages were full of errors; that the boundaries of themortuary limits were constantly changing; that residents of the city wereoften buried in the country, and sometimes country people were interred intown; that the population was fluctuating; that the immigration from therural districts and from foreign countries was large and irregular; that thecensus of the population, whether taken by the United States, or by thestates, or by the cities themselves, was full of errors; that the ages ofthe living, both among males and females, were wrongly reported, sometimesintentionally, but always carelessly and thoughtlessly; and that theseerrors in the numbers and ages of the people and of the deaths were sonumerous that no confidence could be placed in the ratio of the living andthe dying at any particular age, while this ratio at all ages is anindispensable element in determining the proper premiums to be charged inany of the contracts made by our life companies.

Dr. T. N. Thiele makes the following remarks in reply to those we prefixed to his adjusted table of mortality given in the last number of this Journal:—

I must consider the constant difference of about 0·1 after the age of 50 between my adjustment and the numbers of living, which follow from the observations, but are not directly observed, and therefore not independent of each other, as the probabilities of living are, as an agreement completely satisfactory. The apparent disagreement I consider to be the result of one observation (52–53), where the observed probability of life disagrees strongly with the numbers of the other adjacent ages, and is not compensated by the following numbers disagreeing sufficiently in the opposite direction. Whence in comparing the calculated probabilities with the observed no systematical disagreement appears in these ages, as the following table shows; but if the numbers of living are compared, every other adjustment must show a disagreement of the same kind at the age of 50 as mine.

In every branch of business, even in the very rudest stage of barter, men have to count. A skin is exchanged for so many cowries, a rifle for so many skins. Do what we will, go where we will, the necessity for counting meets us. Yet, though numbers be essential to all our operations, we do not easily form an idea of large numbers. Thus in no language do the separate names for numbers reach to twenty. In one, I believe, the numeration goes as far as fifteen; in the northern languages of Europe it reaches to twelve, but in most languages it stops short at ten.

I have recently tad occasion for a special purpose to apply the formula to a case which, I think, presents points of sufficient interest to justify me in laying it before the readers of the Journal of the Institute of Actuaries.

The chief difficulties encountered in the construction of the following Table have arisen, first, from the extreme paucity of the data for adult ages,—the whole number of deaths observed above the age of 14 being 27 only,—and secondly, from the want (hitherto) of a simple and satisfactory method of adapting the law of mortality prevailing in adult life to the observations for infancy and childhood. I proceed to explain the mode in which these difficulties have (as I venture to think) been overcome.

While life insurance premiums, irrespective of the additions for the cost of management, are so calculated, that assuming a given mortality table and a given rate of interest, the receipts of the Insurance Office improved at compound interest will exactly suffice to pay the claims that occur according to the table of mortality; yet the Office suffers a loss in those cases where death occurs soon, and on the contrary makes a profit in the case of the persons who live long. In the grant of life annuities, the contrary is the case. It can be calculated exactly how long an insured person must live in order that the Office may make neither profit nor loss on his insurance; and in the same way it can be calculated what gain or loss the Office experiences when the life assured dies after a given number of years, n. If all these gains and losses are added together, the losses being considered as negative gains, the sum must be = 0. This, however, is far from being the case in practice. The datums of the mortality table, altho' they may be deduced from observations extending over a long series of years, are to be considered as mean values, which are strictly applicable only when there is a very great number of deaths; and we ought to expect that with a small number of insurances, and still more in single instances, considerable deviations from the mean will take place.