It was the opinion of Malus, and adopted by Arago, Biot, and other philosophers, that the surfaces of regularly crystallised bodies acted upon light in the very same manner as the surfaces of ordinary bodies, whether solid or fluid; or, in other words, that the reflecting forces extended beyond the limits of the forces that produced double refraction and polarisation. Having been led to question this opinion, I undertook an extensive series of experiments on crystalline reflexion, as exhibited in calcareous spar, a crystal peculiarly fitted for this purpose, from its perfect transparency and great double refraction; and I published the results of these experiments in the Philosophical Transactions for 1819.

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.”

By ὀνοματοποιια the Greek grammarians understood that principle, or tendency in the growth of language, according to which certain words are formed by an imitation of the sounds which they signify. Thus, ὀγχ, the root of the Greek word ὀγχᾶσθαι, to bray, may be considered to have been formed of a human mimicry of that animal to which human beings of the lowest cerebral capacity are peculiarly compared; and in the same way, laogh, the Gaelic for a calf, seems to contain a sound to which only the throats of Highland calves, Highland chieftains, and Highland crofters are competent. The word onomatopœia, like some other technical terms of the old grammarians, is not particularly happy, for it means only and generally word-making, or rather name-making, and says nothing of the principle by which the special class of words in question is made. Instead of this term, therefore, I should prefer to speak of the imitative or pictorial principle in the formation of human speech; and I should contrast the whole class of words in which the operation of this principle can be traced, with another class, derived from ideas or notions about the thing to be named in the mind of the word-maker. Thus, the modern Greeks call a cock πετεινό, that is, the fowl, or flying animal, from πέτομαι, to fly; and the Latin word, equus, a horse, if it comes, as Professor Müuller says, from the Sanscrit root â'su, swift, will be another word formed on the same principle. The roots of these words I propose to call notional roots, as contrasted with the onomatopoetic class of roots, which I propose to call pictorial roots, or roots formed by phonic imitation.

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.

My attention was called to the subject of Cataract, in consequence of having, about forty years ago, experienced an incipient attack of that complaint, and studied its progress and cure.

While engaged in a game at chess with Sir James Hall, who was a very slow player, I amused myself in the intervals with looking at the streams of light which radiated from the flame of a candle in certain positions of the eyelids. In one of these observations I was surprised by a new phenomenon, of which I did not immediately see the cause. The flame of the candle was surrounded with lines of light, of an imperfectly triangular form, some parts of which were deeply tinged with the prismatic colours. Upon going home from the chess club, this optical figure was seen more distinctly round the moon, and of course it appeared, with more or less brightness, round every source of light.

Since the publication of my “Observations on the Polarisation of the Atmosphere,” a long and elaborate Memoir on the same subject, by Dr R. Rubenson, has appeared in the Acts of the Royal Society of Sciences of Upsal. The observations which it contains were made with the finest instruments, and with a degree of accuracy which had not been attempted by previous observers. They were begun at Upsal in 1859, and carried on at Rome between the 6th of June and the 5th of August 1861, at Segni in the Campagna, between the 6th and the 27th of August 1861, and at Rome from the 5th of October 1861 to the 27th of July 1862.

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 affection of Half-vision, or Half-blindness as it has been called, was first distinctly described by Dr Wollaston, in a paper “On Semidecussation of the Optic Nerves,” published in the Philosophical Transactions for 1824. “It is now more than twenty years,” he says, “since I was first affected with this peculiar state of vision, in consequence of violent exercise I had taken for two or three hours before. I suddenly found that I could see but half the face of a man whom I met, and it was the same with every object I looked at. In attempting to read the name Johnson over a door, I saw only son, the commencement of the name being wholly obliterated from my view. In this instance, the loss of sight was towards my left, and was the same, whether I looked with my right eye or my left. This blindness was not so complete as to amount to absolute blackness, but was a shaded darkness, without definite outline. The complaint lasted only about a quarter of an hour.” In 1822, Dr Wollaston had another attack of hemiopsy, with this difference, that the blindness was to the right of the centre of vision, and he has referred to three other cases among his friends; but in these, the affection was accompanied with headache and indigestion.

In a former paper I described the variation of the fecundity of women according to age, and arrived at the conclusion that the climax of fecundity in women was at or near the age of 25 years. Researches, completed since that paper was read, regarding the variations of length and weight of children according to the mother's age, and regarding the mortality of childbed as influenced by the mother's age, have been published in the “Edinburgh Medical Journal.” The results of these investigations seem to illustrate and confirm the statement made as to the age of the climax of fecundity, for I have found that, about that age, women produce the bulkiest children, as measured by length and weight; and about the same age of the mother there is the smallest mortality in childbed. As still further adorning the age of 25, I may add, that several sets of observations, including some made in St George's-in-the-East, London, and published in the eleventh volume of the “Journal of the Statistical Society,” show a greater amount of survival and rearing among children born of women about that age than at any other; and recently Dr Arthur Mitchell has published a collection of cases of idiocy, with the respective ages of the mothers at the time of the idiots' births, and these also show a smaller proportion born of women about the age of 25 years than at greater and lesser ages.

On a fluid of so much importance as the blood, observations with any pretension to accuracy can hardly be too often made and repeated, more especially when we consider its great instability, its little uniformity, and the differences of opinion entertained by physiologists respecting some of its most remarkable properties.

Such is the persuasion which has influenced me in engaging in the present inquiry, and in submitting its results to the Society.

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.

Before commencing the discussion of the subject, it is necessary to make some definitions, with a view to avoiding the confusion which extensively prevails, from the neglect of the all-important definition of terms. I might be even more exact than I shall be, and excuse myself from adopting such a seeming improvement, on the ground that further refinement of definition would itself cause confusion in the present stage of advancement of our knowledge.

Absolute sterility, I shall hold to mean the condition of a woman who, under ordinary favourable cirumstances for breeding, produces no living or dead child, nor any kind of abortion.

Sterility, I shall hold to mean the condition of a woman who, under ordinary favourable circumstances for breeding, adds not even one to the population, or produces no living and viable child.

Relative sterility, I shall hold to mean the condition of a woman who, while she may or may not be absolutely sterile, while she may or may not be sterile, is, under ordinary favourable circumstances for breeding, sterile in relation to the circumstance of time; or, in other words, in relation to her age, and the duration of her married life.

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.

The following series of theorems is given as an illustration of the modern method of trilinear co-ordinates, having been wrought out after perusal of Mr Ferrar's very lucid and elegant treatise on that subject. The demonstrations present no difficulty, requiring nothing more complicated than the formation of determinants of two and three places. Accordingly, after exhibiting the method of proof in a few instances, I have merely given the enunciations of the remaining propositions. As the series of theorems advances the manipulation becomes, of course, a little more complicated; but the co-ordinates and co-efficients always appear in such symmetry as very greatly abbreviates the task, and guarantees its accuracy. Two, or perhaps three, of these seventy-two theorems are known mathematical truths; but that so many new consecutive propositions should be so easily found, and so easily proved, is a convincing evidence of the simplicity, fertility, and power of this new and beautiful method.

In a paper on Hemiopsy, published in the present volume of the Transactions (p. 15), I have mentioned the remarkable fact, that the parts of the retina which are insensible to visual, are sensible to luminous impressions, the light being occasioned by irradiation from the adjacent parts of the retina. The parts thus affected in hemiopsy extend irregularly from the foramen centrale to the margin of the retina; but the space which they occupy is so small, their distribution so irregular, and the time of their continuance so short, that it is difficult to make such observations upon them as would establish a general property of the retina.

The idea of chemical structure, as founded on that of atomicity (or the equivalence of atoms), enables us to divide any molecule, whose chemical structure is known, into radicals. The number of ways in which this may be done increases with the complexity of the molecule. Each of these modes of division corresponds to a series of conceivable reactions, some of which have been observed. Any one of these series may be made the basis of classification; but it is obviously most convenient to select for this purpose the most characteristic reactions, and those which are common to such substances as form natural groups. In studying these, we find that each series implies the presence of a particular radical, within which the reactions in question take place. We may call such series of reactions the Generic reactions, and the corresponding radicals Generic radicals. These are sometimes residues of double decomposition, but very frequently this is not the case, and this may account for the fact, that the importance of these generic radicals has been very much overlooked.

A short paper of mine on Fagnani's theorem, and on Confocal Conic Sections, was inserted in the twenty-third volume of the Transactions of the Royal Society. Some of the conclusions of that paper can, however, be obtained more simply, as I will now proceed to show.

I will, in the first place, resume the problem—

“To find the intersection of a confocal ellipse and hyperbola.”

Since the curves have the same foci, and therefore the same centre, let the distance between the centre and focus be called unity, since it is the same for both curves. Let a, b, be the axes of the ellipse, A, B, those of the hyperbola. Then we have 1 = a2 − b2 = A2 + B2, which equation expresses the condition of confocality.

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).

The observations which I have now the honour to submit to the Society were made chiefly with the intent to endeavour to ascertain whether, in the instance of the egg of the common fowl, that which may be presumed to be vital action can for a while be arrested, and yet be capable of renewal. Whilst this was the main object kept in view in conducting the trials, a secondary one was to observe, however cursorily, the changes which take place in the contents of the egg when vital development has been prevented.

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.