Sir,—You have communicated to me, that the Royal Society of Sciences in Edinburgh has done me the honour to elect me as a corresponding member. I beg you to render my humble thanks to the Society, and to assure, that it shall be my earnest wish to fulfil every task in my power which the Royal Society should demand.

That this letter may not reach your hands without any scientific communication, I subjoin the following:—From November 1822 to April 1824 inclusive, I observed the height of the barometer in Christiania, and found the mean reduced to 0°R., and to the level of the sea =757m·763 = 335‴·913 lign. de Paris. As the mean height of the barometer observed at Paris by Bouvard, and reduced to 0°, and the level of the sea is = 337‴·53, I was surprised at the great difference of l‴·62 between Paris and Christiania. If p denotes the pressure of the atmosphere at the level of the sea, m and h the density of the mercury and its height in the tube, g the force of gravity, we have p = mgh, and, in another place, p′= mg′h′. If p′ = p is gh = g′h′, or h′ = . If, in the first place, the latitude is = ϕ, in the second, = ϕ′, we have ; h−h′ = h, 0·0025911 (cos 2 ϕ – cos 2 ϕ′). Taking ϕ = 0′, ϕ′ =90°, we have h−h′ = l′′′·74; and when ϕ = 48°·50′ (Paris), ϕ′ = 59°·55′ (Christiania), we have h−h′ = 0′′′·32. But the observations have given for Paris and Christiania h−h′ = l′′′·62; consequently, the mean pressure of the atmosphere is not the in different latitudes (“Magazin for Naturvidensk.” 1824, page 282–291).

Nearly six years ago, I presented to the Society two Memoirs on the subject of Differentiation, with fractional indices. The method which I adopted to extend the signification of a differential coefficient consisted in assuming that the function , which enters into the value of the coefficient deduced from a particular hypothesis, is limited only by the definition . This generalization appears to be perfectly satisfactory, and promises to offer, if not the only, certainly the best extension of the Differential Calculus. Considering the length of the interval which has elapsed since the publication of my former Memoirs, it is remarkable that so little addition has been made to our knowledge of this branch of analysis. With the exception of one or two papers in Liouville's Journal, and a few remarks by Professor De Morgan, in his Treatise on the Differential Calculus (pp. 598–600), I am not aware that anything has been written on this subject since that time. Seeing, therefore, that others are not willing to enter on this very promising field, I consider it not improper that I should make known a number of extensions of this science to which I have been subsequently led, many of which have been in my possession a considerable time.

It may be remembered that, in the paper formerly laid before this Society on this subject, I endeavoured to establish the principle still disputed by some physiologists, that the laws which regulate the chemical relations, as well as those which regulate the visible movements of the particles of matter, undergo a certain determinate modification or change in living bodies, which is essential to the commencement and to the maintenance of the organization of those bodies; and farther, that I undertook the task of attempting to deduce, from the numerous but somewhat discordant experiments and observations lately made on the subject, certain inferences which appear to be well ascertained, although not generally admitted, as to the essential nature of this change, i. e., as to laws which regulate those chemical actions which are peculiar to the state of life, and essential to the maintenance of organization, both in vegetables and animals.

1. The symbol is called an impossible or imaginary quantity, because, in analogy with the received laws of algebraic symbolism, it must mean such a quantity as, being multiplied into itself, gives for a product –1. Assuming, then, that every quantity must be either plus or minus, it follows that the square of every real quantity must be plus; and hence , which gives its square minus, is called an imaginary or impossible quantity.

In a former communication to the Society, I noticed a reaction presented by all spring, well, and river waters which I had examined, that, even after being boiled, they yielded, with acetate of lead, a precipitate readily soluble, in whole or in great part, in acetic acid. This easy solubility in acetic acid shewed that the precipitate was neither a sulphate nor a phosphate, and the comparatively slight action of nitrate of silver proved that it was not a chloride. There seemed, therefore, to remain only the conclusion that it was either a carbonate, or was due to organic matter. The former alternative, of course, depended on whether the solution in acetic acid was attended with effervescence or not; and as this seemed usually not to be the case, and as, on decomposing some of the precipitate by sulphuretted hydrogen, some organic matter in solution was obtained, the conclusion seemed to be, that the appearance was caused by organic matter, probably of the nature of the crenic and apocrenic acids of Berzelius. I have since, however, found that by very careful observation, effervescence may be noticed during the solution of the precipitate more frequently than I at first supposed. It is not so easy as might be imagined to determine this point.

Numerous researches have established as a general rule that the products of the decomposition of organic substances vary with the circumstances of the experiment, and the nature of the agents under the influence of which it is performed. If, for instance, we examine the action of heat alone, we find it causing a set of decompositions specially characterised by the evolution of carbonic acid, formed by the union of part of the carbon of the substance with the whole or part of its oxygen; and this action is rendered more definite, and the number of the products circumscribed by all circumstances facilitating the formation of carbonic acid, such as the presence of a base, which will even cause its evolution when heat alone is incapable of producing decomposition. Acids, on the other hand, have a precisely opposite effect, they, in some instances, altogether prevent the formation of carbonic acid, and cause the oxygen to exert its action on the hydrogen of the compound, and to eliminate one or more atoms of water which do not generally exist ready formed in it.

According to the theory devised by Huygens, to explain the phenomenon of double refraction in Iceland spar, a pencil of light transmitted through that substance is divided into two pencils; the index of refraction for the one being constant, while for the other it varies with the inclination of the transmitted light to the optical axis of the crystal.

Dr Wollaston, in 1802, verified the spheroidal form of the wave of light, which Huygens had assumed to account for the refraction of the extraordinary pencil, by a careful experimental investigation, conducted by means of his elegant instrument for “examining refractive and dispersive powers by prismatic reflection.” In 1810, Malus, in his Théorie de la Double Réfraction, also demonstrated experimentally the accuracy of the Huygenian law for the extraordinary pencil. I have not had an opportunity of consulting the memoir of Malus, so as to know the precise nature of his experiments, with reference to the refraction of the ordinary ray; but the object of Dr Wollaston's researches was simply to prove the law of extraordinary refraction, and the constancy of the index of refraction for the ordinary ray, is therefore tacitly assumed by him.

My Dear Sir,—I have taken the opportunity of a short visit to England to bring with me a complete copy, to the end of 1845, of the Observations of Terrestrial Temperature, made at Trevandrum (of which I have already sent you a partial account), and which I have now the pleasure to forward to you herewith, to be dealt with in any way you may think proper. Should you think them worthy of presentation to the Royal Society of Edinburgh for publication in their Transactions, it will be gratifying to me. The observations made with the 12 feet thermometer are now at an end, in consequence of the fracture of the instrument during a high wind last year; but, after rejecting those made immediately on the introduction of the instruments, say all those of 1842, there will still remain three entire years of trustworthy readings, which I hope may be found to be sufficient for any purpose for which such observations in a tropical region may have been considered desirable.

There are few questions in geology which have given rise to so many theories, and so much speculation, as the origin of the parallel roads in the valleys of Lochaber.

In the year 1817, the late Dr MacCulloch gave an elaborate description of them, in a paper read before the Geological Society of London. In the year 1818, Sir Thomas Dick Lauder read before the Royal Society of Edinburgh a paper, full of equally interesting details. Both of these observers suggested, in explanation of the shelves which mark the mountain sides of these valleys, that they had been occupied by lakes, which, by earthquakes or other violent convulsions, had been drained. This theory was generally received, until, in the year 1839, Mr Darwin, so justly celebrated as a geologist, and an accurate observer, published his views, and pronounced the shelves to have been formed by the sea; an opinion which, besides being rested on proofs derived from the locality, he enforced, also by his observation of similar appearances in South America.