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.

The following attempt to investigate the real nature of Symbolical Algebra, as distinguished from the various branches of analysis which come under its dominion, took its rise from certain general considerations, to which I was led in following out the principle of the separation of symbols of operation from those of quantity. I cannot take it on me to say that these views are entirely new, but at least I am not aware that any one has yet exhibited them in the same form. At the same time, they appear to me to be important, as clearing up in a considerable degree the obscurity which still rests on several parts of the elements of symbolical algebra. Mr Peacock is, I believe, the only writer in this country who has attempted to write a system of algebra founded on a consideration of general principles, for the subject is not one which has much attraction for the generality of mathematicians. Much of what follows will be found to agree with what he has laid down, as well as with what has been written by the Abbé Buee and Mr Warren; but as I think that the view I have taken of the subject is more general than that which they have done, I hope that the following pages will be interesting to those who pay attention to such speculations.

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.

We owe to Leibnitz the first suggestion of Differentiation, with fractional and negative indices, but no definite notion of the theory was attained until Euler expounded it in the Petersburgh Commentaries for 1731. Still Euler wrote only a few pages on the subject, so that the theory could scarcely be said to have come into existence, until Laplace, in his Théorie des Probabilités, and Fourier, in his Théorie de la Propagation de la Chaleur, shewed how general differential coefficients might be deduced by means of definite integrals, provided we assume or prove, by means of some elementary definition, that the differential coefficient of a circular or of an exponential function has a certain form. The formula given by M. Fourier is a very simple one; and our astonishment is great, when we reflect on the time which elapsed from its announcement to the first application that was made of it. This took place in 1832, in a memoir by M. Liouville, entitled Questions of Geometry and Mechanics resolved by a new analysis, which memóir is followed by two others on the more immediate theory of the analysis itself. Although M. Liouville regards his analysis as a new invention, we have no doubt that the idea is due to Fourier; but still to M. Liouville belongs the honour of moulding it into a shape capable of being made use of in the solution of problems.

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.

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.

In following out the principle formerly laid down—that the study of the organization and functions of the part to which the nerve is distributed, will explain the peculiarities of its origin and connections—I have in this paper entered on a subject of great extent and difficulty.

As the Facial Nerve is one of a distinct class, it will be necessary to shew in what that class is peculiar—that it essentially belongs to the act of breathing—that the act of respiration being, in its ordinary condition, independent of the will, there are nerves appropriated to that function. At the same time, it must be shewn, as the apparatus of breathing is made subservient to other purposes in the economy, by what relations it is brought under the influence of the will.

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.

Interesting as theoptical properties of the eye have been to philosophers in every age, there are conditions of this organ which are no less curious, and which have not had their share of attention.

In the year 1823, I introduced the subject to the Royal Society of London, nearly in the terms I am now using, but there is much more in the subject than I then conceived, although I see no reason to change the mode of contemplating it.

The eight muscles of the eye, and the five nerves, exclusive of the optic nerve, which pass to them, imply the complex nature of the apparatus exterior to the globe, and I fear it is too plain that the subject has not been satisfactorily treated.

It is chiefly with respect to the protecting motions of the eye that the difficulty occurs, for I hope the dependence of the proper organ of vision on the voluntary muscles of the eye, has been proved and acknowledged.

Permit me to draw the attention of the Society to what appears a very simple piece of anatomy, the circular muscle which closes the eyelids, orbicularis palpebrarum.

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.

Avant de parler des dykes contenus dans la portion d'Arran au midi de la baye de Lamlash, portion que, vu l'augmentation du nombre des dykes et le peu de temps que j'avois à donner à leur étude, je n'ai pu examiner que très rapidement et incomplètement, je vais rassembler ici en une sorte de résumé, les résultats généraux de l'examen des faits consignés dans les tableaux suivants.

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.

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.

The ancient Greeks have laid claim to the earliest discovery of the method of manufacturing iron, but it will appear that the art was known in Persia at least as early as among the Greeks. The method of producing malleable-iron by a single process directly from the ore, is not indeed quite unknown at the present day, but it is believed to be altogether disused in Great Britain and throughout Europe; but there is no doubt that, in Britain, particularly at Castle Cough, Glamorganshire, and at Furness, near Ulverston, in Lancashire, as well as elsewhere, malleable-iron must have been known long before the discovery of cast-iron. In the 17th century, malleable-iron appears to have been made directly from the ore, in preference to the method now practised. In the Philosophical Transactions (for 1693, vol. xvii. p. 695), there is the following short notice by Mr Sturdy, of the method as then practised at Milthorpe-forge in Lancashire. “The forge is like a common blacksmith's, with a hearth made of sow-iron, in which they make a charcoal fire, and put in ore, first broken into pieces like a pigeon's egg; it is melted by the blast, leaving the iron in a lump, which is never in a perfect fusion; this is taken out and beaten under great hammers, played with water, and, after several heatings in the same furnace, it is brought into bars. They get about one hundredweight of metal at one melting, being the produce of about three times as much ore; no limestone or any other flux is used.” It has been doubted by an intelligent author (Farey on the Steam-Engine, p. 271), whether, by the process here described, the iron was really made directly from the ore, or only from pig metal. The existence, however, of a similar process at the present day in Persia, evidently the same which has been practised in that country from a very remote period, will make it appear not the least improbable that iron may have been thus produced from the rich hematite or fibrous red iron-ore of Lancashire.