The phenomena and colours of the soap-bubble have been the subject of more frequent observation than any other facts in science. For nearly two centuries they have afforded amusement and instruction to the young, and have exercised the genius of the most distinguished philosophers. Hook and Newton ascribed the beautiful colours of the soap-bubble to the different degrees of thickness of the attenuated film, and this opinion has been implicitly adopted by every optical writer down to the present day. In the latest and best Treatise on Light, Sir John Herschel expresses the current theory more distinctly than others, when he says, “that the brilliant colours which appear on soap-bubbles consist of a regular succession of hues disposed in the same order, and determined, obviously not by any colour in the medium itself, in which they are formed, or on whose surfaces they appear, but solely by its greater or less thickness.…It is at first uniformly white, but as it grows thinner and thinner, by the subsidence of its particles, colours appear to begin at its top, where thinnest, which grow more and more vivid, and arrange themselves in beautiful horizontal zones about the highest point, as a centre.”

In repeating some of the beautiful experiments of Professor Plateau, on the Equilibrium of Liquid Films, contained in seven Memoirs, published in the “Transactions of the Royal Belgic Academy,” and in prosecuting my experiments on the colours of the soap-bubble, I observed several new phenomena which may have escaped the notice of the Belgian philosopher.

In plunging a wire cube in a solution of soap, and lifting it up vertically, Professor Plateau found that there was formed within it a polyhedron, as shown in Plate XXXIV. Fig. 1, consisting of twelve similar liquid films adhering by capillary attraction to the twelve wires which compose the cube, and a small quadrangular film suspended in the middle of them. In many cases M. Plateau found that the vertical quadrangular film was often horizontal, as in Fig. 2; and M. Van Rees discovered, that by blowing very lightly upon one of its sides it was reduced to a simple line, and then reproduced in a horizontal position, from which it could be blown again into a vertical position, as in Fig. 1.

That part of universal arithmetic which relates to the changes of variable quantities, and which is known under the titles—Fluxions, The Higher Calculus, The Infinitesimal Calculus, The Theory of Functions, has been divided into two branches, called respectively the Differential and the Integral Calculus; the one of these being regarded as the converse of the other; and every problem connected with variation has been supposed to require either or both of the processes known as differentiation and integration.

The subject of the following paper is the first proposition in the Calculus of Primaries. The business of that calculus is to discover the relation between the primary variable and its function, when the relation subsisting between the function and its derivative is known. The simplest relationship between two variable quantities is proportionality when they are heterogeneous, or equality, when they are of one kind; and the case of proportionality can always, by a change in the unit of measure, be brought to an equality of the representative numbers; so that our first proposition becomes this: “To investigate the nature of those functions which reappear among their own derivatives.” Since this reappearance must necessarily be periodical, I shall use the name Functions with Recurring Derivatives.

Aristotle has a saying, which he has frequently repeated and which is often quoted, to the effect that the same degree of precision is not attainable in all branches of inquiry, and that it would be just as absurd to exact demonstration from a politician or an orator, as to accept probable reasoning from a mathematician. It is a saying full of truth and acuteness. To the cultivators of ethical and political philosophy, for whom it was intended, it is invaluable both as an encouragement and a warning; and yet, in behalf of the latter more especially, I often wish that it had never been said. Proceeding from such a master, I am persuaded that it has often tempted them to rest satisfied with a degree of success far short of the limits which the nature of their subjects really imposed; whilst, on the other hand, it has afforded an apology for excluding social and political philosophy from the meditations of learned bodies like this. I do not mean that they have been formally excluded. I know that the constitution of this, and of most similar societies, has always embraced the social as well as the physical sciences. But so rarely have those of us who were occupied with the former availed ourselves of the privileges of Fellowship, that it has come to be regarded almost as a matter of admission on our part, that our subjects defy scientific treatment: that when we talk of tracing out laws of social wellbeing or progress, we use words which either have no meaning at all, or which indicate a very faint analogy between the methods which we affect to follow and those really employed in the physical sciences: and that pretty nearly all that can be done is to hand us and our subjects over to the companionship of party politicians and popular declaimers.

It is well known, that whenever the three roots of a cubic are all real, the solution of the equation by Cardan's rule becomes illusory. This is the more remarkable, because, à priori, one might have expected that the rule would only fail when the roots were imaginary. Numerous researches have been made by mathematicians on this subject; but they have not succeeded in removing this obstacle; and the only mode of finding the roots of a cubic, when all three are real, has been, by successive approximations, or the use of trigonometrical tables, or (in the case of one root being a whole number), by tentative methods and trials (which often succeed without much difficulty, when the coefficients of the equation are small numbers).

One of the theorems of a paper which Professor Kelland did me the honour to read to the Society, in March 1865, opens up a field of geometrical investigation so interesting and fertile, that I venture to ask attention to some of the results of a partial examination of it in the following series of propositions. I think it right to explain, that I do not venture to expect attention to them on account of any importance attaching to them individually, but on account of their number and somewhat elegant relations. Considered individually, they may be of little importance, having no claim to rank, so to speak, among propositions of a planetary magnitude. But a system of moons, however diminutive, may become interesting if they present elegant relations among their mean motions and longitudes; and an orbit that would be grudged to a pigmy planet may be willingly accorded to a host of planetoids. If this is still too exalted language in which to speak of the following results, I can at least confidently affirm that they indicate a direction in which an analyst of very moderate attainments may easily discover for himself a shower of meteors.

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.

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.

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.

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.

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.

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.

Professor Agassiz, in his “Poissons Fossiles,” vol. ii. p. 78, has mentioned a species of Pygopterus as occurring in the carboniferous shales of Wardie, near Edinburgh, and which he has named P. Greenockii, in honour of Lord Greenock, the first collector of ichthyolites from that locality. He has, however, neither given a figure of this species nor any description of it, beyond saying that the known fragments consist of hardly anything but heads, with the anterior portion of the trunk, and that the scales covering this part of the body are higher than broad—a circumstance distinguishing them from the scales of all the other species of the genus.

In 1855, the Professor of Materia Medica in the University of Edinburgh, in a paper read before this Society, directed the attention of physiologists to some of the remarkable properties of the Calabar bean. In 1862, I presented a graduation thesis to the University of Edinburgh on the “Characters, Actions and Therapeutic Uses of the Ordeal Bean of Calabar.” The principal results I had obtained at that time were that this substance causes death by either syncope or asphyxia, the latter being due to an effect on the spinal cord and on the respiratory centres; that the symptoms resemble those of cardiac or pulmonary embarrassment, according to the quantity of the poison administered, and to its rate of absorption; and, also, that the topical application of this agent to the eyeball, or to its neighbourhood, produces a marked and rapid contraction of the pupil and various disturbances of vision. Since then, and more especially because of the peculiarity of the last of these conclusions, a lively interest has been taken in this substance. Its actions on the eye have been investigated by nearly all the leading ophthalmologists of Europe and of America, and its general physiology has occupied the attention of many distinguished students of biology. Nor have these labours been barren of practical results. Ophthalmic medicine has adopted this agent as one of its important remedies, and there can be little doubt that general medical practice will soon include in its Pharmacopœia a drug of so great energy.