The object of the following paper is to describe an investigation recently made, to determine, from observed facts, the risk incurred by persons making voyages to the Australian Colonies. This investigation was carried out under the superintendence of Mr. Sprague, at whose suggestion the following account of it was prepared.

Sir,—In compliance with your instructions, we have the honour to submit to you the following observations upon the Accounts and Statements submitted to the Board of Trade in pursuance of the Life Assurance Companies Act, 1870.

A “company” is defined for the purposes of the Act to mean “any person or persons, corporate or unincorporate, not being registered under the Acts relating to Friendly Societies, who issue, or are liable under policies of assurance upon human life within the United Kingdom, or who grant annuities upon human life within the United Kingdom.”

It may perhaps be thought by some persons, that the subject upon which I am to address you tonight, is one that requires no urging on your attention. It is admitted on all hands, I may be told, that a good actuary must necessarily go through a course of mathematical training, and nothing further need be said upon the point. But if an apology for the choice of my subject is required, I would remind you that a mathematician takes nothing for granted. Whereas many people will say that it is self-evident that “two and two make four”, and that this neither requires nor admits of proof, the mathematician knows that it is a proposition, which is not in the true sense of the word self-evident, but can be strictly proved. While we deal with simple matters of this kind, little harm is likely to result from assuming as self-evident, propositions which admit of proof. But the case is very different when we come to more complicated and obscure questions; and not the least advantage to be derived from a mathematical training, is the gradual forming of the habit of mind which requires strict proof to be furnished of many important propositions which are usually taken for granted. I purpose, then, instead of taking for granted the use of mathematical studies, to examine exactly why and how they are of service to the actuary; and I hope that the conclusions I shall arrive at, may be of some assistance to those among you who wish to extend their mathematical reading beyond the point that is by common consent considered indispensable to the actuary.

There is nothing immutable in this world, and certainly life companies are no exception to the general rule; for in them, from year to year, we find fluctuations of every kind—in amount of new assurances—in expenses of management—in value of assets—in realized interest—in the death claims—in fact, in every branch and item of their business. Of all these variations there is none so important as that concerning the mortality experienced. The fluctuations in the other elements of a life office's affairs are trifling as compared with this, and can without difficulty be prepared for. But in the perturbations of the death claims from year to year, we are dealing with a matter that cannot be calculated on in advance —an element which, if violent, may endanger the safety of the company itself; and, even when this is not the case, will often prove a serious inconvenience to it. I have called violent mortality fluctuations inconvenient to a life office. To men acquainted with the practical as well as the theoretical side of assurance, the word will not seem exaggerated. They, at least, know how important it is that the bonus to policyholders and the dividend to shareholders, do not retrograde. The moment the bonus tide begins to ebb, there is an outcry; and variations in the mortality may thus create disgust in those already connected with an office, and at the same time add another difficulty to the already sufficiently difficult task of its agents and canvassers.