In the last Number of this Journal I examined and compared the solutions of this question given by Milne, Baily, and Griffith Davies in their standard works. The first of these writers cannot be considered to have treated the question in a satisfactory manner. The two latter have investigated the problem more rigorously. They both proceed upon the supposition commonly adopted, viz. that the deaths occurring in each year of age are distributed uniformly over the year. They differ, however, in their views as to the interest to be allowed for fractions of a year, Davies throughout adopting Milne's view, by which the present value of £1 due in six months' time is , while Baily takes the value as . The latter, as I have already stated, appears to me the preferable view. Davies, however, has also given an investigation of the value of an annuity payable m times a year, on the supposition that interest is convertible into principal m times in the year; and his solution, as I have shown in the last Number of the Journal, is substantially the same as Baily's.

Milne, in his Treatise on Annuities, chapter vii., has discussed this subject with much clearness and elegance; yet his method of investigation does not seem to be the best possible, nor do I think that his results are given in forms well adapted to numerical applications. It may therefore be not altogether useless to consider the subject in a somewhat different manner, and so to exhibit the formulas as best to meet the wants of the computer.

In the course of last year a very important document was published by the American Government, under the direction of the Secretary of the Interior, entitled “Statistics of the United States (including Mortality, Property, &c.) in 1860; compiled from the original Returns, and being the final Exhibit of the Eighth Census.” It contains an introduction by J. M. Edmunds, Commissioner of General Land Office, in charge of census, and a still fuller introduction to the mortality statistics for the year ending June 1st, 1860, by Dr. Edward Jarvis, a very able writer, who was a delegate to the Statistical Congress, when it was held in London, and is a Corresponding Member of the Statistical Society of London.

In 1842-43 a Mr. Henson had proposed what he called an aeronaut steam-engine, and a Bill was brought in to incorporate an “Aerial Transit Company.” The present plan is altogether different, the moving power being the explosion of mixed hydrogen and air. Nothing came of it—not even a Bill. What the final destiny of the balloon may be no one knows: it may reasonably be suspected that difficulties will at last be overcome. Darwin, in his Botanic Garden (1781), has the following prophecy:—

“Soon shall thy arm, unconquered Steam! afar

Drag the slow barge, or drive the rapid car;

Or, on wide-waving wings expanded, bear

The flying chariot through the fields of air.”

Darwin's contemporaries, no doubt, smiled pity on the poor man. It is worth note that the two true prophecies have been fulfilled in a sense different from that of the predictions. Darwin was thinking of the suggestion of Jonathan Hulls, when he spoke of dragging the slow barge: it is only very recently that the steam-tug has been employed on the canals. The car was to be driven, not drawn, and on the common roads. Perhaps, the flying chariot will be something of a character which we cannot imagine, even with the two prophecies and their fulfilments to help us.