Several years ago, when discussing the question of the distribution of the stars, a problem occurred to Professor Forbes, which, simple as it is, appears to have escaped notice prior to that time. Having been consulted as to its solution, I communicated my results to Professor Forbes, who has inserted one of them in his paper printed in the Philosophical Magazine for 1850, vol. xxxvii., p. 425. But for the very ingenious application which Professor Forbes has there made of it, the problem might probably not be worth recurring to. As it is, I have thought it would not be altogether uninteresting to give the complete solution.

The Problem is as follows:—There are n dice, each of which has p faces, p being not less than n; it is required to find the number of arrangements which can be formed with them, 1°, So that no two show the same face; 2°, That no three show the same face; 3°, That no four do so, and so on.

In approaching the subject of solar light, the first point is to endeavour to form an idea, not altogether indefinite, with respect to its quantity and intensity, as compared with some familiar standard of artificial flame. With this view were made, in the course of last summer, the observations now to be described.

After several trials, it was found, that the most convenient mode of procedure was first to compare a definite small surface, illuminated by solar light, with a like surface illuminated by the mere light of the sky, and then to compare the latter with a similar surface illuminated by the flame of a moderator lamp. The light of the sky thus affords a middle term between the extreme lights of the sun and the lamp, which are too diverse to be directly compared.

(1.) The inquiry which, with its results, I propose to lay before the Society, was suggested by the following passage in the very popular Treatise on Logic by Dr Whately, now Archbishop of Dublin.

“As in the case of two probable premises, the conclusion is not established except upon the supposition of their being both true, so in the case of two (and the like holds good with any number) distinct and independent indications of the truth of some proposition, unless both of them fail, the proposition must be true: we therefore multiply together the fractions indicating the probability of the failure of each—the chances against it—and the result being the total chances against the establishment of the conclusion by these arguments, this fraction being deducted from unity, the remainder gives the probability for it.

Twenty-two years have now elapsed, since Runge first published his remark, able experiments on coal naphtha, and it would, perhaps, be difficult to instance any chemical investigation which has formed the point of departure of a greater number of researches. When we consider the vast quantity of bodies which have, first and last, been obtained from coal-tar, it might appear that little more remained to be done,—that the mine was exhausted,—but so far from this being the case, the discovery of one substance has only served to pave the way for the isolation of others.

Among the bodies examined by Runge, there was one which apparently possessed comparatively few features of interest; indeed its very name (the first syllable derived from λευχός was intended to express its supposed inability to produce coloured reactions, a feature which, in the chemistry of the time, militated greatly against its claims to notice. I have used the expression “supposed inability,” because I shall show further on, that this substance is capable, under certain conditions, of affording extremely brilliant colorations. Eventually, Gerhardt, by acting on quinine, cinchonine, and strychnine, with hydrate of potash, obtained the same body. The first chemist who succeeded in procuring any of its compounds in a state of tolerable purity was Hofmann, whose analysis of the platinum salt is very nearly exact. But, at the time of that analysis, he was of opinion that the products obtained from coal and chinoline were essentially different, an opinion which he subsequently retracted. In the mean time, the alkaloid, as obtained from cinchonine was examined by Bromeis and Laurent, their results, however, not elucidating the composition of the basic fluid obtained in the manner alluded to.

It is well known that no satisfactory demonstration has ever been given of Fermat's celebrated theorem, which asserts that the equation an = bn + cn is impossible, if a, b, c, are whole numbers, and n is any whole number greater than 2. In Legendre's Théorie des Nombres, he demonstrates the cases of n = 3, n = 4, and n = 5, the latter only in his Second Supplement. In Crelle's Mathematical Journal, ix. 390, M. Dirichlet, a mathematician of Berlin, has demonstrated the case of n = 14, but I am not aware whether his demonstration is considered successful. Legendre informs us (Second Supplement, p. 3) that the Academy of Sciences, with the view of doing honour to the memory of Fermat, proposed, as the subject of one of its mathematical prizes, the demonstration of this theorem; but the Concourse, though prolonged beyond the usual term, produced no result.

Problem. If p be one of the roots of the equation xm − 1 = 0, (not 1,) then (1 − p)(1 − p2) …. (1 − pm − 1) = m, provided m is a prime number.

If m be not a prime number, and if , the same will hold for all roots p = p1α, where α is a number < m and prime to m. But for all roots p = p1α, where α, or one of its prime factors, is also a prime factor of m, the product (1 − p)(1 − p2) …. (1 − pm − 1) will be equal to 0.

The phenomena presented by the prismatic spectra of flames have occupied the attention of many and excellent investigators. In most instances, however, no attempt has been made to procure accurate measurements of the positions of the bright lines which many of the spectra exhibit; and much in this field of observation, therefore, remains to be accomplished. I purpose, from time to time as I shall have leisure, to make a series of observations, whose object shall be the actual numerical determination of the positions of the bright lines in the spectra of flames; and I have commenced the series with an examination of the spectra of the flames of compounds of carbon and hydrogen. In an investigation into the phenomena of flames, the compounds of carbon and hydrogen claim our first attention, as constituting the most important means of artificial illumination; for it is scarcely necessary to remark, that, with the grand exception of sun-light, the combustion of these substances is the source of nearly all the light and heat from which we derive such extensive benefits in the arts and in domestic economy. It will be found, moreover, that the spectra of carbohydrogen flames possess, in common, remarkable features, which seem as yet to have received little attention, but which promise to be of service in explaining the general phenomena of artificial light.

Having, several years ago, in the course of a prolonged investigation of the geological structure of the Appalachian chain of the United States, conducted partly in co-operation with my brother, Professor W. B, Rogers, as a purely scientific inquiry, partly by myself, in connection with a Government Survey of the State of Pennsylvania, discovered what we deemed important laws, applicable generally to all corrugated tracts of strata; and being prepared, by observations since made in the United States and in Europe, to extend their application, and give them a more general expression, I have thought that I could not select a more suitable subject for my first communication to the Royal Society of Edinburgh, than this portion of descriptive and dynamic geology, which has engaged much of my attention, theoretically and practically, for these many years. In presenting an outline of the views already arrived at, and published by us as a necessary part of the further generalizations since reached, I will refrain from repeating, in historical detail, what we have already written, but will give our conclusions in the form and with the brevity most compatible with clearness, referring to the printed papers and communications where the special topics included in this more general summary may be seen.