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 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.

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.

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.

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.

Sir,—In one of the first papers read before the Institute of Actuaries, Mr Jellicoe pointed out “the method of procedure which should be observed in determining the value of life contingencies, where the risk was of an isolated character, as distinguished from cases in which the usual considerations of average presented themselves.” The paper was not, I believe, printed for the use of the members, bnt a short account of it is given in the Post Magazine of the 10th March, 1849. This was followed by one from Mr. Hardy, “On the values of annuities which are to pay certain given rates of interest on the purchase money during the whole term of their continuance, and to replace their original values, on their expiration, at certain other given rates”—where for the first time also is given, in an intelligible form, an expression for the value of an annuity certain under such conditions; in accordance with this a set of tables was constructed, and appended to the paper read before the Institute 25th November, 1850.

Sir,—The letter of your correspondent “Verus,” in the last Number of the Magazine, induces me to trouble you again upon this subject; especially as, if the views there expressed are entertained by any number of your readers, it is evident that so much difference of opinion exists on this question as to render further discussion desirable.