The limited district of country which I am about to describe, is one of those which may rank amongst the least frequented in the civilized part of Europe, yet which might justly claim for France the character of romantic beauty which travellers on her beaten highways commonly, and not without reason, deny to her.

The modern department of the Ardêche, corresponding in part to the ancient province of the Vivarais, includes country of very dissimilar features, the southern and eastern part, forming the right bank of the Rhone near Viviers, being comparatively flat; whilst the north-western boundary is the irregular chain of the Cevennes, including the localities more immediately to be described. This chain is not so remarkable for its absolute height, although that be considerable, rising at the Mont Mezenc, in the neighbouring department of the Haute Loire, to an elevation of 5750 English feet above the sea, as from forming the separation of a remarkably elevated tract stretching to the north and west, and which suddenly subsides, at the point of which we now speak, into the wide champaign country of the Lower Rhone, possessing a very different aspect, soil, climate, and population.

The facilities which are afforded by the introduction of the function into certain classes of Differential expressions are well known. This function has effected the combination and generalization of Problems, which, although found to be capable of solution in particular cases, were regarded rather as isolated and exceptional forms, than as integral parts of some comprehensive expression. But the subject is far from exhausted. Some of the most important Differential Equations have never as yet been solved by a general method. The present Memoir is intended to supply this defect. The process employed differs little from that which I have previously exhibited; but the range of Problems which it embraces is much more extensive, and the Problems themselves are of a more important character.

During the last few years, great progress has been made in the study of the organic alkalies, and the discovery of methods by which these substances can be artificially produced, and the long train of investigations by which it has been followed, has greatly extended our previous information, and afforded us some definite ideas regarding their constitution. The advance made has, however, related entirely to the volatile bases produced by artificial processes, and our knowledge of the natural fixed alkaloids stands very much where it did some years since, and is still very imperfect, and in regard to many entirely fragmentary; so much so, indeed, that of all the alkaloids of this class described in chemical works, there are not perhaps a dozen of which the constitution can be considered as definitely fixed, and not half that number of which we know the products of decomposition. The fact is, that the interest attaching to the artificial bases has altogether diverted the attention of chemists from the natural alkalies, which have not hitherto proved a very productive field of inquiry; at least the researches to which many of them were subjected ten or fifteen years since, proved comparatively unfruitful in their results.

There are few parts of mechanics in which theory has differed more from experiment than in the theory of elastic solids.

Mathematicians, setting out from very plausible assumptions with respect to the constitution of bodies, and the laws of molecular action, came to conclusions which were shewn to be erroneous by the observations of experimental philosophers. The experiments of Œersted proved to be at variance with the mathematical theories of Navier, Poisson, and Lamé and Clapeyron, and apparently deprived this practically important branch of mechanics of all assistance from mathematics.

The attention which has of late years been paid to the elucidation of the manners and arts of the ancient inhabitants of America, has been productive of the most convincing proofs of the communication between the Eastern and Western Continents at remote but unknown epochs. The learned and highly-interesting researches of Humboldt on the antiquities of the New World, have irresistibly led him to this conclusion, which has farther been strengthened by the researches of later travellers. The comparison of the idioms of the Asiatic and American tongues, has hitherto not afforded very direct proof; because the philologist has not yet been put in possession of a sufficient number of materials to make the comparison with advantage. Our ignorance of the languages and customs of Central Asia is a great bar to such studies, and needs not any other illustration than the fact that a highly-polished nation, with a literature and arts hitherto almost unknown in Europe, should have existed for ages in Central Asia. Our countryman, Dr Gerard, stimulated by the humane desire of extending the blessings of vaccination to Thibet, has been for some time in that country, and has discovered in its language an Encyclopædia in forty-four volumes, of which the medical part alone fills five volumes; and he finds, that the art of Lithography, so new in Europe, has been practised from time immemorial in Kinnaour, a principal city in Thibet, where he found it employed to display the anatomy of the human body.

While the physical appearances of comets have ever excited such intense curiosity and interest, all the theories concerning them are generally confessed to be insufficient to explain them; and, certainly, if we may judge from the various views advocated by different writers, and the anomalous forces gratuitously brought in to support the different hypotheses—it is so.

This unsatisfactory state of things, so different from that in which is the theory of the motions of comets,—seems to be owing partly to the difficulty of making the necessary observations by reason of the undefinable nature of the bodies themselves, and partly from the untoward circumstances under which the observations must be made, as well as the rareness of any opportunities offering. Hence, theories are built upon accounts handed down from old astrological times when men's prejudices would have prevented them, even if their means had been ample, which they were not, from giving any satisfactory and trustworthy accounts of the phenomena displayed by the heavens of their day.

The ensuing paper forms part of a series of researches respecting the consequences of an hypothesis called that of Molecular Vortices, the object of which is to deduce the laws of elasticity, and of heat as connected with elasticity, by means of the principles of mechanics, from a physical supposition consistent and connected with the theory which deduces the laws of radiant light and heat from the hypothesis of undulations. Those researches were commenced in 1842, and after having been laid aside for nearly seven years, from the want of experimental data, were resumed in consequence of the appearance of the experiments of M. Regnault on gases and vapours.

The investigation which I have now to describe, relates to the mutual conversion of heat and mechanical power by means of the expansion and contraction of gases and vapours.