1. Write down the multiplicand in the usual manner.

2. Write on a separate slip the multiplier with its figures reversed (as in contracted multiplication), being careful to space the figures so that they can be exactly placed over those of the multiplicand.

We have now to give some examples of compound benefits, which are those consisting of two or more simple benefits; but the combinations which may be formed of these being obviously very numerous, it would be beside our present purpose to attempt giving a complete list of them. Our object will be, in selecting a few of them for illustration, to indicate the method of dealing with the more complicated cases, and also to prepare the way for the most general application of the Commutation Tables, which application will form the subject of the concluding portion of this paper. A very complete list of the formulae for the more elementary of these benefits is contained in Professor De Morgan's first paper on the subject; and as it is hoped that little difficulty will be experienced with these, after the illustrations to which our space limits us, we shall not scruple, as we have occasion, in the solution of any of the problems with which the present paper will be occupied, to refer to any of the learned gentleman's formulæ which we may not have deduced for ourselves. Our references will be made in the following manner, which is rendered necessary in consequence of the formulae not forming one consecutive series. Formula 10, on page 16, for example, will be denoted thus, [16,10]; formula 72, on page 18, thus, [18, 72]; and so on.

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.

In a paper read a short time ago before the Institute, I endeavoured to show the inexpediency of using, for the purposes of valuation, tables constructed on the principle which characterizes the Northampton table. I did not intend to pursue this particular branch of the subject further; but, finding that more difference of opinion exists in reference to it than I was aware of, and considering that the practice of making valuations by the Northampton table, and by tables framed upon similar principles, is still carried on to a great extent, I am led to offer a few more observations upon the matter previously to entering upon the consideration of some other questions, the correct solution of which materially depends upon a right understanding of this one.

To find the value of a life annuity on (x), of which the first payment is to be £1; the second, £2; the third, £3; and so on, increasing annually, by the amount of the first payment, to the end of life.

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.

Public attention has for some time past been earnestly directed to the introduction of the decimal system in our weights, measures, and coins. The nation is generally convinced, that the adoption of such a system would prove of immense benefit—that it would afford great facilities for calculations of all kinds, that it would shorten the work of education, that it would economise labour, and that it would diminish the chances of error. The Society of Arts, and other scientific societies, have investigated the subject in all its phases and bearings, and we have been expecting the speedy adoption of some practical plan which would be certain to confer so great a boon. Unfortunately, the Russian war, the Indian mutiny, and other political events, have rendered it necessary to put aside the consideration of many social reforms, and this, among the rest, shared the same fate. We have bestowed, also, far too much attention to the pound and mil scheme, as if upon it rested the entire question of decimalisation, and thus years have passed without a single step of a definite character being taken.

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.