Article 1. The introduction of the notion of a function of a variable quantity into the mathematics, without any regard to its particular form, has given vast extension to the science, and been the germ of some of its most important theories. The doctrine of curve lines, no doubt, produced that of functions, for the former may be made the visible expression of the latter: thus, either of the coordinates of a curve being taken as the representation of the variable, the other co-ordinate is a function of the variable; so also are the arc of the curve, and its area. Indeed, in contemplating functions, and discussing their properties, it is convenient to substitute in our reasonings the geometrical representation for the abstract notion of the function.

The preparation of my Lectures on Medical Jurisprudence involved a consideration of the means of diminishing the chances of successful forgery, and again engaged me on a subject to which, many years ago, my attention had been very painfully turned by the frequency of executions for that crime. This will scarcely appear exaggeration, when it is stated, that, in the year 1809, there were no less than thirteen executions for forgery in the county of Lancaster, where I then resided: and when it is recollected that, in the fourteen years preceding 1819, two hundred and four individuals perished on the scaffold for that offence in England and Wales, every means of discouraging so fertile a source of misery and crime must be allowed to be a subject of no trifling importance.

This mineral is found embedded in small crystals in prehnite, at Bishop town, in Renfrewshire. It had been long supposed by mineralogists to be a variety of zinc-blende; but it was first distinguished from that mineral by Lord Greenock, who communicated his opinion to Professor Jameson; and two small crystals, together less than a grain in weight, were sent to me for chemical examination by the latter, who concurred in the supposition that it was a new mineral.

In the year 1830, I succeeded in establishing a Register of the Thermometer at the Bonally Reservoir, which formerly supplied the city water-works, being at a distance of five miles in a direction south-west from Edinburgh. This station is on the northern acclivity of the Pentland Hills, at a height of 1100 feet above the sea. The following year I obtained corresponding observations at the village of Colinton, situated a mile and a half north of the preceding station, and above 700 feet lower. Although this difference of level be not very considerable, yet, as these comparative registers have been kept for nearly five years with pretty uniform results, some confidence is evidently due to the conclusions, even although considerable difficulties opposed themselves to obtaining registers quite free from exception. The interest attaching to them is the greater, that, although registers have been kept at Leadhills and other elevated stations, I do not recollect any strictly comparative observations in Scotland, perhaps not even in Great Britain, at two stations near one another, and differing considerably in level, from which the important meteorological element of the decrement of temperature in the atmosphere could be deduced.

That the facts which I communicate regarding the natural history of the salmon in its earlier stages, may not appear altogether undeserving of consideration, I may premise that my remarks have not proceeded from hasty or imperfect observation, but from the experience of many years sedulously devoted to the subject, the whole of my life, with the exception of a few seasons, having been spent on the banks of streams where salmon are in the habit of depositing their spawn, and where of course the parr is likewise abundant. My opportunities of observation have thus been as ample, as my efforts have been unremitting and laborious, to discover the true history of this invaluable species. I shall here present a brief abstract of my earlier proceedings in relation to the subject.

In a former memoir on this subject, it was my endeavour to exhibit the principles of the science of General Differentiation in a simple, at the same time in a general, point of view. I endeavoured to deduce, from one general formula, results easy of application in all instances; and thus to exhibit the unity of the different parts of the science, and the completeness of its fundamental formulæ, shewing at the same time the facility of their adaptation to particular and varied cases. With the exception of certain expansions by means of a theorem analogous to the series of Taylor, I gave no application of the principles to problems of any kind. It is my intention in the present memoir to supply this branch of the subject, without which, indeed, however interesting may be the details, as a portion of pure analysis, they will offer little to interest any but those who attach themselves to the study of analytical combination. We hope, by the exhibition of a few simple mechanical problems, solved by this process, to give to our subject an interest in the eyes of all, derived not from its intrinsic beauty, but from its use as a medium of demonstration. It is well known that considerable difficulty hangs over several very simple inverse mechanical problems; from the generality of their statement, a direct solution is sometimes impossible by the ordinary methods. We shall shew that by our process such solutions are attainable with the greatest readiness. By this means we hope to give a value to our subject as a branch of knowledge, independent of that value which it must possess from its curious and elegant structure.

Previously to the 25th and 26th November 1838, there had prevailed in Great Britain and Ireland, for more than a week, a steady wind from the NE., accompanied with frosts, a progressively rising barometer, and tolerably clear weather. The same sort of weather existed on the Continent, and over a large portion of northern Europe, both on sea and land.

This state of things was changed, by the arrival of two storms from southern latitudes, which passed over the British isles during the last week of November. These two storms, until they reached this part of the globe, were separate. The first one reached the British seas, about thirty-six hours before the other. But the second moved with about double the velocity of the first, and overtook the first somewhere about the north of Ireland and south-west of Scotland. Accordingly, in the southern parts of England, there were distinct indicia of two different storms, each having its own period of arrival, veering, and cessation;—whilst towards the north, these indicia became gradually less distinguishable, and were at length significant of only a general gale.

In the beginning of July last, I obtained, from the Frith of Forth, a species of bream, which has not been mentioned by naturalists as inhabiting the British seas. A few days after, I procured, from the same quarter, a second specimen of the same species, each exhibiting a conspicuous dark violet-coloured spot at the base of the upper part of the pectoral fins. (See Plate VI.)

On consulting the continental works on Ichthyology, I find this bream to agree best with the description Baron Cuvier has given of the Pagellus acarne, an inhabitant of the Mediterranean; but, as no figure of the fish accompanies his description, the discrimination of the species is rendered somewhat uncertain.

The following paper contains a continuation of the experiments on the action of the voltaic pile on alcohol, and some other liquids, of which experiments a considerable number was described to the Royal Society in a former memoir. At present it is intended, in the first place, to shew the perfect analogy between the electric action on pyroxylic spirit, and on alcohol, thereby confirming the interesting analogy already known to exist between these fluids in other respects: in the second place, to adduce a few farther illustrations of secondary voltaic actions in aqueous solutions; in the third place, to examine the nature of the changes produced in alcoholic solutions, under galvanic agency; in the fourth place, to inquire whether electric action does not throw light on the state in which the haloid salts are dissolved by water; and, lastly, to endeavour to suggest as a general law, regulating the electric decomposition of solutions of binary combinations of elementary substances in the principal solvents, that the dissolved body is not directly decomposed, but only the solvent, if itself an electrolyte.

1. The Council of the Royal Society of Edinburgh having, on my application in 1832, entrusted me with Hansteen's Magnetic Intensity Apparatus, in their possession, I feel it to be my duty to communicate to the Society the results then and subsequently obtained with it.

2. The instrument consists of a mahogany box 5 inches long, 4 broad, and 2 deep, with sides and top of glass, having also a wooden tube, screwing into the top, for containing a silk-worm's fibre about 5 inches long, by which the magnetic needle is suspended so as to place itself horizontally, and after being caused to deviate from its point of rest, the time of any given number of oscillations in a horizontal plane is measured,—whilst a graduated circle in the bottom of the box indicates its arc of vibration.

The series l(1+ x) = M (x—½ x2 + ⅓ x3—¼ x4+&c.), discovered by Mercator, seems to be the origin from which, directly or indirectly, all the series may be derived which are usually employed in the computation of logarithms. A series, which affords remarkable facilities for such computations, and which lately occurred to me, may be investigated in the following manner.

The object of the following paper is to put at rest a question which has been long agitated in Surgical Pathology, and which is intimately connected with some important points of Practical Surgery. An apology may seem due to the Society for bringing under its consideration a subject, which, though not exclusively professional, is still little studied except by those physiologists whose views are directed to surgery; but as the inquiry into which I propose to enter is neither long nor tedious, while it is quite intelligible without any previous knowledge of its details, I trust the patience of the members will not be exhausted; and if the question shall, as I hope, be decided to the conviction of those members who are conversant with surgical discussions, the prevailing diversity of sentiment relative to the point at issue will be more effectually composed than if I attempted to combat it through any other channel.

It is not a little remarkable, that in an age which assumes to itself the character of devotion to science, in anatomy, a science which embraces the best interests of humanity, this question should remain unanswered; What is the meaning of the nerves of the spinal marrow being in regular order and perfectly symmetrical: whilst the ten nerves arising from the brain present no similarity one to another, and agree neither in origin, size, nor distribution?

It is plain that we must be in the dark, not only with respect to the knowledge of the nervous system, but of the animal frame generally, whilst such a question is open and courts inquiry, and yet remains without an effort being made towards its solution. We must, I fear, attribute this neglect in part only to the difficulty of the inquiry, and much to the indifference to all that does not tend directly to profit; on which account it has the better demand on the attention of a learned and philosophical Society.

There is no branch of natural science about which we know so little as that which relates to the structure of crystalline bodies. By assuming the form of an integrant molecule, crystallographers have found no difficulty in building those geometrical solids which minerals and artificial crystals present to our observation. They conceive that these molecules unite by their homologous sides in the formation of the primitive crystal, and by supposing that they arrange themselves in plates on the faces of that crystal, each plate successively diminishing in size by the abstraction of a certain number of these molecules in lines of a given direction,—all the secondary forms of the crystal may be easily deduced.

I am not aware of any account having been published of the Coal-fields in East and Mid Lothian, or of any attempt to institute a geological survey of the country in which they are situated. Sinclair, the author of a well known work intituled “Satan's Invisible World,” published also in 1672 a treatise on Hydrostatics, in which he takes notice of the Prestongrange coals, and of the whinstone-dike that intersects them. Williams, in his “Mineral Kingdom” (published) in 1810, gives some information regarding the direction of the Gilmerton and Loanhead coal-seams. But the information contained in both these works, even respecting the coal-strata,—which alone they professed to treat of, is extremely vague, and generally very erroneous. Dr Hibbert was the first geologist who with a scientific eye entered on the district, in order to describe with fulness and accuracy any of its rocks. His discovery of the Saurian remains in the limestone-quarries of Burdiehouse, led him to a minute inspection of the strata in which they were imbedded, and to a consideration of the relative position of these particular strata in the Mid-Lothian coal-field.

Although a great number of Fishes have been described by naturalists as inhabiting the coast of Scotland, and though much has been done to increase its Fauna in other respects, yet in a field so extensive much must still remain to be done. Many lakes and estuaries in Scotland are still unexplored, and many fishes being merely local, a more thorough and accurate examination of these lakes and estuaries must take place, before the Ichthyology of Scotland can be fully ascertained.

The history of the newer sciences presents many instructive examples of the progress of the human mind in developing the germs of natural knowledge, and building on a single observation entire departments of science. Few pursuits indeed are more interesting, even to the student of immaterial nature, than in the perusal of such a history to trace the footsteps of the several investigators, and to mark how far, and by what means,—whether by new methods or by greater patience of research,—each successive observer advanced the inquiry. We see the human mind, as it were, set free from the trammels of time, and developing its resources on a large and continuous scale, not limited by the powers of one intellect, the length of one life, or the means of one station. We see, at the same time, what varied gifts and opportunities are necessary for the elucidation of a single subject; how these gifts, though not all imparted to one man or to one generation, are yet wisely and beneficently bestowed on the entire species; and how all are thus enabled to co-operate, either in unfolding abstract truth, or in drawing forth those practical results which directly conduce to the amelioration and comfort of all.

Sir Charles Blagden concludes a memoir in the Philosophical Transactions of London, vol. lxxviii., for the year 1788, “On the Effect of various substances in lowering the Point of Congelation in Water,” with the account of an experiment to determine the effect of salt upon the expansion of water by cold.

From that experiment, he imagined that he had reason to conclude, as far as one experiment goes, that the combination of a salt with water has no other effect upon its quality of expanding by cold, than to depress the point at which that quality begins to be sensible, just as much as it depresses the point of congelation.

The scientific world appear to have admitted this general deduction of Blagden, and, trusting to the result of a single experiment, extended it so far as to lay it down as a general law, that, as pure water has its maximum of density at 7½° above its freezing point, so every saline solution has a maximum of density at a temperature above its point of congelation, and that this temperature will be exactly 7½° distant from the freezing point of the solution.