Mr President,—It has been a practice from the foundation of the Royal Society of Edinburgh, to commemorate its deceased distinguished members by memoirs or biographical notices, read at the ordinary meetings of the Society. Some of these have been printed in the Transactions; and our published volumes are enriched by papers of Dugald Stewart, Professor Playfair, Sir John MacNeil, and Dr Traill, on the characters and writings of Adam Smith, Dr Hutton, Professor Robison, Sir Charles Bell, and Dr Hope. A biographical notice is now due to the memory of a distinguished countryman, late Vice-President of the Royal Society; and the following remarks will, in attempting that object, make a deviation from those more severe discussions with which the time of the Society is usually occupied, in connection either with pure mathematics, natural philosophy, or natural history.

It is unnecessary to detail to this Society the various ingenious hypotheses which have been proposed to account physiologically for the accommodation of the eye to distinct vision at different distances. In later years, these different theories have been so circumstantially and correctly recapitulated in systematic works (as for instance in Young's Lectures and in Müller's Physiology), that it would be a waste of time to copy and recite them here. I will only do so, then, so far as may be necessary to justify the attempt I have now to make, and to strengthen my views by those of others, as far as they bear upon them.

The general properties of numbers, considered without reference to the notation in which they are expressed, have been very fully investigated by several of the most distinguished mathematicians. Little attention, however, has been paid to the particular properties resulting from the principle of the modern notation, which is the expression of every number in a series, a + bn + cn2, &c. where a, b, c, are the digits, and n the local value or root of the notation. Having been led to examine some of these results, and to account for them, I am now desirous of laying them before the Society. I do not flatter myself that they possess any great practical importance; but as I have reason to believe that they are new, I trust the Society will not think them entirely unworthy of their attention.

There is an important geometrical problem which proposes to find a curve having a given relation to a series of curves described according to a given law. This is the problem of Trajectories in its general form.

The series of curves is obtained from the general equation to a curve by the variation of its parameters. In the general case, this variation may change the form of the curve, but, in the case which we are about to consider, the curve is changed only in position.

This change of position takes place partly by rotation, and partly by transference through space. The rolling of one curve on another is an example of this compound motion.

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).

The chemistry of the colouring matters has, perhaps, scarcely as yet met with the extended and complete investigation which the importance of the subject, in a theoretical and practical point of view, appears to deserve. The attention of chemists has been almost exclusively directed to the study of a comparatively small number of these substances, such as indigo, logwood, and the colouring matters of the lichens, which have been well and completely investigated; while the remaining, and by far the more extensive class, has received only a very partial and imperfect examination. To the latter, however, belong some of the most important of our dyes; and among others, the most valuable and indispensable of all, madder namely, the chemistry of which forms a problem as yet very far from being solved, but which chemists have shewn little disposition to submit to a thorough and searching investigation; and this disinclination seems to continue, notwithstanding that ground has been broken on the subject by the important observations of Robiquet, Kuhlmann, Runge, and others, which, though extremely incomplete, serve at least to indicate the importance of the results it is likely to afford, and to clear away the preliminary difficulties by which the commencement of such an investigation is surrounded.

The following communication is intended to be the first of a series, in which I propose to consider the results of observations made at Makerstoun, near Kelso, Roxburghshire, in the Observatory of the President of this Society. These observations, and the tabular results, will be found ultimately in the volumes of Makerstoun Observations, constituting volumes of the Society's Transactions.

While examining, in polarised light, the form and structure of the numerous crystals which I had discovered in the fluid cavities of Topaz, my attention was particularly called to certain optical phenomena exhibited in other parts of the specimen. These phenomena, when first presented to me, were very indefinite in their character, and very imperfectly developed; but after a diligent examination of nearly 900 specimens of topaz, I succeeded in obtaining the most satisfactory exhibition of them under various forms, and in various degrees of intensity.

1. The presence of heat may be recognised in every natural object; and there is scarcely an operation in nature which is not more or less affected by its all-pervading influence. An evolution and subsequent absorption of heat generally give rise to a variety of effects; among which may be enumerated, chemical combinations or decompositions; the fusion of solid substances; the vaporisation of solids or liquids; alterations in the dimensions of bodies, or in the statical pressure by which their dimensions may be modified; mechanical resistance overcome; electrical currents generated. In many of the actual phenomena of nature, several or all of these effects are produced together; and their complication will, if we attempt to trace the agency of heat in producing any individual effect, give rise to much perplexity. It will, therefore, be desirable, in laying the foundation of a physical theory of any of the effects of heat, to discover or to imagine phenomena free from all such complication, and depending on a definite thermal agency; in which the relation between the cause and effect, traced through the medium of certain simple operations, may be clearly appreciated. Thus it is that Carnot, in accordance with the strictest principles of philosophy, enters upon the investigation of the theory of the motive power of heat.

The object of this short notice is merely to submit to the Society some astronomical results which were recently communicated to me in a letter from my friend Captain Jacob, as they appeared not only to be of a highly interesting nature in themselves, but imperatively to require being followed up farther, and as the observer has lately been obliged by bad health to resign his situation in India, it seemed advisable, for the purpose of procuring attention to the subject elsewhere, to make its peculiarly interesting features as generally known as possible amongst scientific men ; and as a Centauri is already in a manner identified with Scotland, through the researches of the late Professor Henderson, and his determination of the parallax, no medium can be more appropriate than the Transactions of the Royal Society of Edinburgh.

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.

In 1823 and 1826 I communicated to the Society two papers on the nature and properties of two immiscible fluids, which I discovered, in contact with each other, in the cavities of topaz and other minerals. Although the facts contained in these papers were of so extraordinary a nature as to be received with scepticism by some, and with ridicule by others, yet I am not aware that, during the twenty years which have elapsed since their publication, any person has either repeated my observations, or advanced a single step in the same path of inquiry. In shewing to strangers some of the leading phenomena of the two new fluids, my attention has been frequently recalled to the subject; but it was not till last spring, when I discovered cavities in topaz filled with the most beautiful crystals of various form, that I was induced to undertake a new investigation of their nature and properties.

Hauy, and other mineralogists, observed the two colours which are visible in several varieties of fluor-spar. He regarded the two tints as complementary, and explained them, as he did every other analogous phenomenon, by a reference to the colours of thin plates. In describing a species of dichroism noticed by Dr Prout in the purpurates of ammonia and potash, Sir John Herschel ascribes the green reflected lights “to some peculiar conformation of the green surfaces producing what may be best termed a superficial colour, or one analogous to the colour of thin plates, and striated or dotted surfaces.” And he adds—“A remarkable example of such superficial colour, differing from the transmitted tints, is met with in the green fluor of Alston Moor, which, on its surfaces, whether natural or artificial, exhibits, in certain lights, a deep blue tint, not to be removed by any polishing.”

Last year, my friend Captain Cockburn, R. N., brought to my notice the very lax method which is usually pursued at sea in determining the strength and direction of the wind ; and said, that he had for many years been trying to contrive some sort of anemometer that might be useful on board, as well as an easy method of eliminating the effect of the motion of the ship on the true character of the wind, but hitherto without success. I undertook, therefore, to endeavour to supply him with these two desiderata. He thought that they would be useful, in a practical point of view, in seamanship; and as I considered that they might be of importance in meteorology, I was the more ready to lend my assistance.

Some time ago my brother, Professor William Thomson, pointed out to me a curious conclusion to which he had been Ied, by reasoning on principles similar to those developed by Carnot, with reference to the motive power of heat. It was, that water at the freezing point may be converted into ice by a process solely mechanical, and yet without the final expenditure of any mechanical work. This at first appeared to me to involve an impossibility, because water expands while freezing; and, therefore, it seemed to follow, that if a quantity of it were merely enclosed in a vessel with a moveable piston, and frozen, the motion of the piston, consequent on the expansion, being resisted by pressure, mechanical work would be given out without any corresponding expenditure; or, in other words, a perpetual source of mechanical work, commonly called a perpetual motion, would be possible. After farther consideration, however, the former conclusion appeared to be incontrovertible; but then, to avoid the absurdity of supposing that mechanical work could be got out of nothing, it occurred to me that it is necessary farther to conclude, that the freezing point becomes lower as the pressure to which the water is subjected is increased.

In forming designs of marine works, the engineer has always a difficulty in estimating the force of the waves with which he has to contend. The information on such a matter, which is derived from local informants, who, although intelligent in the departments of trade which they follow, are, nevertheless, more or less prejudiced from being constantly on the spot, is not satisfactory; and it has, therefore, often occurred to me that it would be most desirable if the engineer could be enabled, to some extent at least, to disregard the prejudiced statements of others, and the vague impressions left by them on his own mind, and really to ascertain, by direct experiment, what force, expressed in pounds per square foot, the sea actually exerts upon the shores where his buildings are proposed to be erected.

The careful study of the products of destructive distillation has enriched organic chemistry with an extensive series of results of unexpected interest and importance. These results have affected, in no inconsiderable degree, the recent progress of the science; and their influence has been of a twofold character, both general and particular, exerted in the former case in the development of some of the more remarkable general doctrines of organic chemistry; in the latter, in the important light thrown by their investigation on the constitution of the substances from which they are derived, and the facilities they have afforded of following out connections, which the examination of the original substance either does not at all present to our view, or, at least, indicates only in an imperfect or dubious manner. Added to this, we have the remarkable fact of the appearance among these products of substances in some cases identical with those occurring in organised beings; and in others, presenting analogies of the very closest character with the actual products of vital affinity, which, taken together, afford abundant reason for pursuing the investigation of substances which have already afforded results of so remarkable a character.

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.