Let represent any permutation of the first n natural numbers. If we take the first number, a1, and put it last, we get a new permutation; and if we perform this operation (n−1) times, we get the (n−1) permutations

This is the problem discussed in my paper bearing the not very happy title “On the different non-linear arrangements of eight men on a chess-board”, which was read to the Edinburgh Mathematical Society on 14th March 1890, and is printed in its Transactions, Vol. VIII, p. 30. At that time I was not aware that the problem had been discussed by any previous writer, and I treated it as an entirely new one. I have since learnt that a good deal has been written about it, and I propose on the present occasion to give briefly the history of the problem, and the results which have been arrived at; also to communicate some new results which I have obtained.

Let f(t), ø(t), be two functions of t, such that they both vanish with t, that is, f(0) = 0, ø(0) = 0; and put

If we put θ = π/2 in the well known equation

we get eiπ/2 = i; and raising both sides to the power i,

In the general equation of a curve that passes through the origin,

u1 ≡ a1x + b1y = 0 is the equation to the tangent at the origin; for, y q g when x and y are very small, we may neglect all the terms in comparison with those involving the first powers of x and y. If neither a1 nor b1 vanishes, we may, without loss of generality, write the equation in the form

The subject of lost policies is one that is constantly coming before Life Insurance Companies in different ways, and it is one regarding which it seems very desirable that they should all act on the same principles. At present, however, I believe there is a good deal of diversity of practice; and I think it may therefore be useful if I submit to my professional brethren the conclusions to which I have been led by my business experience and careful consideration of the various points involved.

Gentlemen,—This Society was privileged during a recent Session to listen to an eloquent and highly original address by Professor Chrystal:— ‘On some fundamental Principles in the Theory of Probability.’ The value of such addresses, dealing, as Professor Chrystal's did, with questions on which wide differences of opinion exist, is not to be measured simply by the amount of information they contain ; they serve a much more useful purpose, by leading those who hear them, or afterwards read them, to think out for themselves the various questions discussed. Speaking for myself, I have to thank the Professor, not only for a pleasant evening spent in listening to him, but for causing me to consider carefully the fundamental principles of the theory with which, as Actuaries, we are all supposed to be so much concerned. I have found the subject a most interesting one, and I propose to lay before you this evening, some of the reflections that have occurred to me, and the conclusions at which I have arrived, after carefully studying what has been written on the subject by a number of authors.

In the course of investigations that I have lately been making into the transformation of permutations, I have found it convenient to employ several new symbols of operation, which combine with each other according to laws that differ materially from the ordinary algebraical laws; and it is my object in this paper to explain those laws, and give a few examples of the manner in which various propositions relating to permutations may be demonstrated by means of my symbols.

In the course of an investigation I have lately made, into the rate of mortality that has prevailed among the female lives insured in the Scottish Equitable Life Assurance Society, I have tried various methods of arranging the facts furnisht by the books of the office; and I think it may be useful to persons engaged in similar investigations, if I describe these methods, and state the conclusions at which I have arrived as to the best course of procedure.

I believe that the first publisht paper on the subject of premiums for insurance against issue was that of Mr. Archibald Day, which was read before the Institute on 31 January 1859 (J.I.A., viii, 127). In this paper he gave from the Census Statistics the numbers of bachelors and widowers living in 1851, and the numbers of marriages of bachelors and widowers in the same year; and from these he deduced the probabilities of a bachelor and of a widower of any age marrying in a year, Then by means of the last-mentioned probabilities, combined with the probabilities of life given by the English Table No. 1, he calculated the value, at 3 per-cent interest, of £1 payable at the end of the year in which a husband shall contract a second marriage. The formula he used for this purpose may be written n p H(1−n p w)ϕH+n−1 × v n , where H and W represent the respective ages of husband and wife, and ϕ is the probability of a widower marrying in a year. It may be remarked in passing, that this formula is only approximately correct, inasmuch as there is a contingency that is not provided for in it, namely, that the husband may be alive and his wife dead, but that he may already have married a second time.