The problems referred to under the above designation are those which involve the double survivorship contingency of A living after the death of B before C. The following solutions of these cases are quite general, and consequently applicable to any mortality table whatever.

I have taken the problems in the order in which they appear in the work of Mr. Milne, whose notation I have also followed. By means of an obvious relation which exists between the value of the compound survivorship assurance and the corresponding annuity, the whole of these cases are reduced to the single fundamental annuity value combined with the values of simple survivorship assurances. Consequently, if we should ever possess a complete table of the values of the simple survivorship assurance and also of the values of the annuity we shall be able to calculate the numerical value in any case with the greatest facility.

When a mass of results are to be calculated and added up, it generally happens that a small fraction is of little consequence in each item, except as the little of which many make a mickle. The money calculator knows nothing of the principle that it is very hard, next to impossible, practically quite impossible, to get a mickle out of positives and negatives, when both are sure to occur in equal numbers in the long run, and with equal quantities. Generally speaking, the items of money calculations are determinable, and accuracy to the penny below the true result is a matter of contract. But in life contingencies, in which a certain amount of inaccuracy is known to lurk in the fundamental tables, no real truth is gained by tacking shillings, pence, and farthings, on to hundreds of pounds. If the nearest pound be always taken, granting that the calculated result shall be thereby a little altered, no man alive can say whether that alteration shall be approach to, or recession from, what we should call absolute truth, if our tables were perfect: it is as likely to be one as the other.

Policies of life assurance are of many kinds—subject to the payment of premiums in various ways, and to risks of different descriptions.

A policy has several values, according to the purposes to which it is applied, the necessities of the holder, and the intention of the purchaser.

The following pages are intended as a guide to those who, not having made the nature and uses of a policy a subject of study, may yet have occasion to form an estimate of its value for some special purpose.

The difficulty of dealing equitably with assurers whose prospects of longevity are considered by the medical referee to be below the average of healthy lives of the same age will not, I think, be disputed. The practice of charging a higher premium corresponding to an advanced age, which has hitherto generally been adopted, is, of course, open to the double objection, that the judgment of the medical officer may have been at fault in placing the life in the second or third class at all, and, supposing he were right in his decision thus far, that the premium at which he has assessed the risk may not be in accordance with its precise magnitude. It is true that in after years reductions may be made if the person's health improve; but this plan is not without trouble both to the Office and the assured, and it has the further disadvantage of leaving the latter in a state of uncertainty as to what his future premiums may be.

The change which has come over the condition of this enterprise within the last few years is very remarkable. Only so recently as 1851 we were led to make some observations on its general aspect which were suggested by the state of affairs at that time, and which we now in part repeat with a view to show how curiously that state contrasts with the one at which we have now arrived.

In the Diary for 1850 the following; mathematical query was proposed by the Rev. T. P. Kirkman, of Croft, near Warrington:—

Two answers were printed in the Diary for 1851; one by the proposer and another by Messrs. Samuel Bills, of Hawton; Thomas Jones, of Chester; Thomas Wainman, of Burley; and W. H. Levy, of Shalborne.