In the year 1861 I laid before the Royal Society of Edinburgh a theorem concerning the time of descent in a circular arc, by help of which that time can be computed with great ease and rapidity. A concise statement of it is printed in the fourth volume of the Society's Proceedings at page 419.

The theorem in question was arrived at by the comparison of two formulæ, the one being the common series and the other an expression given in the “Edinburgh Philosophical Magazine” for November 1828, by a writer under the signature J. W. L. Each of these series is reached by a long train of transformations, developments, and integrations, which require great familiarity with the most advanced branches of the infinitesimal calculus; yet the theorem which results from their comparison has an aspect of extreme simplicity, and seems as if surely it might be attained to by a much shorter and less rugged road. For that reason I did not, at the time, give an account of the manner in which it was arrived at, intending to seek out a better proof. On comparing it with what is known in the theory of elliptic functions, its resemblance to the beautiful theorem of Halle became obvious; but then the coefficients in Halle's formulæ are necessarily less than unit, whereas for this theorem they are required to be greater than unit.

My attention was called to the subject of this paper in May 1862, during a short stay at Elie on the coast of Fife. Close to the friend's house with whom I lived a part of the sea-bank had been laid open by the waves, and among other deposits I found a bed of clay containing fossil shells, such as now live only in the Polar seas. An account of this I had the honour of laying before the Society on the 2d of March 1863. During the following autumn, while residing at Bridge of Earn, I found a similar deposit, with the same species of shells, at Errol on the Tay, and a notice of this I also laid before the Society on the 2d of May 1864, intimating that at some future period I should again ask their attention to the facts thus ascertained and the inferences to be drawn from them. The delay that has taken place has arisen from other occupations, which leave little time for such pursuits, but it has not been wholly without advantage. The cuttings of the East of Fife Railway were carried past the outskirts of Elie, and I had an opportunity of examining the series of beds, while laid open for the time, in a very remarkable way. This last autumn also, while residing on the spot for a few days, I examined with some care a transverse section, nearly at right angles to the two former; and now, in this paper, I shall endeavour first to state in detail the facts connected with these separate localities, and then to bring into one view the general results.

39. In the first part of this paper, read to the Royal Society of Edinburgh in April 1862, and published in their Transactions, I explained the principles of a method devised by me in 1850 for ascertaining the absolute conducting power of substances capable of being formed into long bars; and I also stated the general results of experiments made in 1851 on the Conductivity for heat of wrought Iron.

40. I explained in Art. 14 of that paper, that the publication of the results had been for ten years withheld, partly in consequence of the state of my health which completely interrupted the experiments, but still more from the defective graduation of some of the thermometers used, which made it necessary to submit the instruments to a careful scrutiny, and to repeat with the duly corrected numbers the whole of the elaborate projections of the curves and calculations from them, on which the accuracy of the final results of course depends.

Having already published in the Transactions a detailed report on the Hourly Meteorological Register kept at Leith Fort, at the expense of the Society, during the years 1824 and 1825, it is unnecessary to enter into any recapitulation respecting the origin and history of this class of observations.

The singular and unexpected results obtained from these Registers, and the rapid approximation to general laws which some of these results exhibited, attached a great interest to the observations of future years; and it is satisfactory to find that the results for the two following years of 1826 and 1827 are almost perfectly coincident with those for 1824 and 1825, not only in their general relations, but even in their numerical laws.

The following communication is an attempt to eliminate the geological history of our Scottish Peat Mosses. So much, however, has already been done in this matter, that the reconsideration of phenomena, for the most part well known, may appear almost a superfluous task. But, notwithstanding the essays of Walker, Rennie, Anderson, and others, in this department of geological inquiry, there is still probably much to be gathered from the same source, which shall greatly increase our knowledge of the condition of these latitudes in the ages that followed upon the close of the glacial epoch. At present, I mean to give only an outline of the subject, reserving for some future occasion a fuller statement of the facts on which the conclusions arrived at are based.

In the year 1812 I described an apparatus, by which the light of the sun, or of any luminous body concentrated in the focus of an improved lens, could be returned by reflection from a spherical mirror into the same focus, thus increasing the light and heat produced by the cone of refracted rays. The apparatus contained also a double system of lenses and plain mirrors, by which additional beams of light could be concentrated in the same focus.

In 1827 I described the very same apparatus as applied in the dioptric system of lights, the whole of the light which issues from a lamp being thrown into one wide and parallel beam, constituting what has been called a holophote, now in use in every part of the world.

Though it is generally admitted that the cuticle performs an important part in retarding and regulating evaporation from the surface of the body, yet I am not aware of any inquiry hitherto made to determine the fact with exactness.

On account of the importance of the subject, I have been induced to engage in it. The experiments instituted for the purpose have all been of a very simple kind and easily made. They were on the similar parts of dead animals, detached immediately or very soon after the animals had been killed. From one specimen in each instance, the cuticle with the cutis, or the cuticle alone, was removed; whilst from the other these parts of the integuments were left entire. Each was carefully weighed, and then suspended, exposed to the air, side by side. Day after day, with occasional interruptions, or hour after hour, the weighing was repeated, and the result in the loss sustained was noted down. The experiments were made, when not otherwise mentioned, as just described, in a room in which, except in the height of summer, there was commonly a fire by day, its temperature during the day and night varying from about 50° of Fahr. to 55° and 58°. The animals affording the subjects of the trials were the trout, frog, toad, hare, rabbit, pig, thrush, common fowl, blue tit. Even at the risk of tediousness, I shall give the results of the weighing in some detail,—exactness in such trials being the first thing necessary.

The dynamics of natural labour have been the field of very little successful study or investigation. The object of the present paper is to make a contribution to this subject. I purpose to show what amount of pressure per square inch is sustained by the ovum in the easiest class of natural labours, and thence to estimate the propelling power exerted in such cases.

It is well known that natural births are ever and anon occurring, in which the ovum is expelled whole, the membranes containing the liquor amnii continuing entire. Into this category many more cases would enter, were it not a generally-followed rule for the attendant to rupture the bag should it advance entire as far as the external parts. Again, as Dr Poppel has pointed out, the attentive observer of a series of easy natural labours has no difficulty in arriving at the conclusion, that in not a few cases the same force which ruptures the bag of membranes is able to, and actually does, complete the delivery.

During the summer vacation, Mr Henry Perigal of London proposed to me the following problem:—

“To determine the proportions of an epicycloid of which the loops touch each other.”

The solution of this problem contains some points of interest to the general analyst, and exhibits relations between certain trigonometric formulæ and algebraic equations. I, therefore, offer an outline of it to the attention of the Royal Society.

Mr President and Gentlemen,—I beg to thank you for the favourable opportunity which you have kindly afforded, and the facilities you have granted, for enabling me now to try to lay before you some short and simple account of my chief employments last winter at the Great Pyramid of Jeezeh.

My object in going there, was not to excavate, nor to collect antiquities, but merely to inquire instrumentally, and by my own individual labour, into the very discrepant, and sometimes mutually contradictory, accounts which have been published of the form and detail of that ancient monument by writers of almost all nations; and as one chief source of their unfortunate variations seems to have been the usually short and hurried character of their visits, my first care was to apply for leave, from His Highness the Viceroy of Egypt, to occupy the ground at the Pyramid with a permanent establishment, and to stay there as long as might be necessary for the work in hand. His Highness, as I am extremely happy to confess, and with the best of my thanks to acknowledge, was most liberal in his condescensions; conveyed our party, at his own expense to the Pyramids; lent us tents for the period of our stay; and sent a force of twenty men for a month to clean out the interior of the Great Pyramid, or otherwise prepare its more than classic walls for the examination to be made, and which lasted from that time uninterruptedly for a period of four months.

In a paper “On the Phenomena of Thin Plates of Solid and Fluid Substances exposed to Polarised Light,” published in the “Philosophical Transactions” for 1841, I had occasion to notice certain motions and colours which I had observed upon films of some of the volatile oils; but as they were unconnected with the subject I was then investigating, I made no attempt to discover their nature and origin. Their apparent similarity, however, to the molecular movements and colours upon the soap-bubble, induced me to resume the subject, and to examine them as exhibited upon films of various evaporable liquids, stretched over apertures differing in size, form, and substance.

In 1861, in a part of the province of Otago, New Zealand, not previously botanically explored, I made, among other botanical collections, one of Lichens and Fungi. The number of new species and varieties proved to be considerable, amounting to about 20 per cent. of the whole lichens, and 40 per cent. of the whole fungi, collected. Since my return home, I have submitted (with a view specially to the study of the minute anatomy of the reproductive organs and their contents) the new species (and varieties) in question—some of them repeatedly—to microscopical examination: the results whereof are contained in the notes which follow.

The problem which bears the name of the Italian geometer Malfatti, by whom it was first proposed and solved, has long attracted the attention and exercised the ingenuity of mathematicians, and has been made the subject of many careful and elaborate researches.

The great attention which has been bestowed upon this problem has arisen partly from its intrinsic difficulty, but chiefly on account of the extreme simplicity of the solution finally obtained by Malfatti, which seemed to open new views of geometrical research, and gave reason to hope that simple solutions might in like manner be found of many other geometrical problems usually accounted very difficult or insoluble.

In the beginning of the year 1865, I received from the Rev. Alexander Robb, one of the missionaries of the United Presbyterian Church, residing at Creek Town, Old Calabar, Western Africa, a package containing some zoological specimens, preserved in spirits; and in a letter dated 28th October 1864, referring to its contents, he states, there “are also two or three small eel-like fishes.” Some time passed before I was able carefully to examine the package, and I was then at once attracted by these small fish, which I saw belonged to the very distinct and interesting Order of the Ganoid fishes, so abundant in a fossil state in the rocks of our earlier geological epochs, but of which so few representatives are now to be found as living inhabitants of the present waters of our globe.

One of the most remarkable phenomena in our recent historical literature is a tendency to whitewash all characters which had previously presented a black appearance; to prefer the intellectual divination of a subtle modern professor to the plain testimony of a sober old chronicler; and generally to unsettle all things that we had in previous ages been taught to look on as settled. That this tendency, dating from the gigantic excavations of Niebuhr and Wolf, had its origin in an honest love of truth, and a searching scrutiny of evidence, cannot be doubted. That its results have in the main been beneficial is equally certain; but, on the other hand, it is not to be denied that it has sometimes run into the most wanton excesses, and that it has tainted some of the most notable historical productions of our age with a vice which will render it necessary for a future generation to repeat the work now done from a broader point of view, and with a juster criticism.

It has always appeared to me that the difficulties which present themselves in investigations concerning the Frequency of Error, and the deduction of the most probable result from a large number of observations by the Method of Least Squares (which is an immediate consequence of the ordinary “Law of Error”), are difficulties of reasoning, or logic, rather than of analysis. Hence I conceive that the elaborate analytical investigations of Laplace, Poisson, and others, do not in anywise present the question in its intrinsic simplicity. They seem to me to be necessitated by the unnatural point of view from which their authors have contemplated the question. It is, undoubtedly, a difficult one; but this is a strong reason for abstaining from the use of unnecessarily elaborate analysis, which, however beautiful in itself, does harm when it masks the real nature of the difficulty it is employed to overcome. I believe that, so far at least as mathematics is concerned, the subject ought to be found extremely simple, if we only approach it in a natural manner.

1. Dr Matthews Duncan having requested me to point out to him some simple method of comparing the fertility of different races, I endeavoured, as a preliminary step, to represent by formulæ some of the chief results which he has obtained in his very lucid and elaborate papers recently read to this Society, and printed in their Transactions for 1863–4 and for the present session. Some of the formulæ which I have obtained are so simple, and accord so well with the tables, that I have thought them worth bringing before the Society.

One of the grandest steps which has ever been made in Dynamical Science is contained in two papers, “On a General Method in Dynamics” contributed to the Philosophical Transactions for 1834 and 1835 by Sir W. R. Hamilton. It is there shown that the complete solution of any kinetical problem, involving the action of a given conservative system of forces, and constraint depending upon the reaction of smooth guiding curves or surfaces, also given, is reducible to the determination of a single quantity called the Characteristic Function of the motion. This quantity is to be found from a partial differential equation of the first order, and second degree; and it has been shown that, from any complete integral of this equation, all the circumstances of the motion may be deduced by differentiation. So far as I can discover, this method has not been applied to inverse problems, of the nature of the Brachistochrone for instance, where the object aimed at is essentially the determination of the constraint requisite to produce a given result. It is easy to see, however, that a large class of such questions may be treated successfully by a process perfectly analogous to that of Hamilton; though the characteristic function in such cases is not the same function (of the quantities determining the motion) as that of the Method of Varying Action.

The first observations of the diurnal variation of magnetic declination, made near the equator, seem to have been those of Mr Macdonald, who observed in 1794–95 at Fort Marlborough, Sumatra, 3° 46′ S., and at St Helena. Two conclusions seem to have been deduced from these observations—1st, That near the equator the range of the diurnal variation was much smaller than in Europe; 2d, That the needle moved in opposite directions south of the equator and in Europe. This latter conclusion was made use of by M. Arago, in his report made in 1821, on the “Voyage de l'Uranie,” as the base of a hypothesis that there must be a line betwixt the two hemispheres on which the magnetic needle moves neither east nor west—that is, remains stationary. M. de Freycinet's observations showed that this line was not the terrestrial equator, and M. Arago supposed it must be the magnetic equator.

Since reading this paper, I have seen Sir Benjamin Brodie's paper on “The Calculus of Chemical Operations,” read before the Royal Society of London.

The two papers resemble one another, merely in being applications of mathematical language to chemistry. They entirely differ in method, object, and result.

I may here mention, what I have omitted to state explicitly in the paper, that I have no idea of attempting to substitute a functional notation for that in common use. I only propose to use a functional notation to express certain general and serial relations in those cases where the common atomic notation is inconvenient or obscure.