That excellent Geometer the late Dr Matthew Stewart, as long ago as the year 1746, published his book of General Theorems, all of them, except the first five, without the demonstrations. As I do not find that any demonstration of them has ever been made public, I may perhaps flatter myself that what I now communicate to this Society will not be wholly unacceptable. The demonstrations given are of the propositions relating to the sums of the squares, and of the fourth powers of lines drawn in a certain manner, and are selecled from the rest, as most connected with one another. The theorems that respect the cubes and other higher powers, may afford materials for another paper, should this meet with the approbation of the Society.

Although it is very improbable that there remains to be discovered any new, and at the same time simple, fact connected with a question which has been elaborately treated by many of the greatest mathematicians of this and the preceding century, the employment of a new mathematical method may enable us to present some of their results in a more intelligible form, and with far less expenditure of analytical power than has hitherto been deemed necessary; and it may give us such an insight into the question, that we shall be able easily to discover the mutual relations among the various processes which have been already employed; so far, at least, as these differ in principle, and not merely in the peculiar co-ordinates assumed for the purpose of simplifying the equations. Such a method is that of Quaternions, which seems to be expressly fitted for the symmetrical evolution of truths which are usually obtained by the ordinary Cartesian methods only after great labour of calculation, and by modes of attack so indirect, and at first sight so purposeless, as to bewilder all but a very small class of readers. Quaternions afford so clear a view of the nature of the question they are applied to, that even the student, if he have some little knowledge of them, can often see why a transformation is made, whose object he would have been unable to discover had the problem been masked in the unnecessarily artificial difficulties of Cartesian geometry, or the outrageously repulsive formulæ of spherical trigonometry.

It is well known to the Society, that a mineral, found in the strata of Torbanehill, not far from Bathgate, in the county of Linlithgow, has lately been the subject of a keen lawsuit. Specimens of this mineral were early in the year put into my hands by the lessee, and sometime afterwards by the proprietor of Torbanehill, for the purpose of obtaining a mineralogical opinion on the nature of the mineral. I stated at the time to both parties, that I would carefully examine it before I could presume to offer a decided opinion. In fact, the mineral was not altogether new to me; for, in the two preceding years, I had received specimens of a mineral from Torbanehill, and from Boghead, which I was unable to refer to any described species; and I was therefore determined to give my opinion with due caution, especially as I understood that large interests were involved in the question.

The morphology of sterile and fertile structures (terminal strobili) of the Upper Devonian heterosporous lycopsid Kossoviella timanica Petrosjan 1984 from northern Russia (North Timan) is re-described: the axes are dichotomously branched; sterile leaves are narrow with smooth margins; the transition from sterile axes to strobili is gradual; the strobili are narrow and cylindrical, occasionally dichotomously branched; sporophylls are long, lanceolate, with crenulated margins; megasporangia with thin, mostly destroyed, sporangium walls contain one or two tetrads of large megaspores without a gula; numerous microspore tetrads are present in the microsporangia; both mega- and microspores are cavate, with a two-layered sporoderm; the outer layer of the sporoderm of both mega- and microspores consists of a net of intertwined cylindrical elements; the inner layer of the megaspore sporoderm is a basal lamina; and the inner homogeneous layer of the microspore sporoderm is split into multilamellate zones near the arms of the proximal scar. A comparison between abortive and fertile megaspores, some of which apparently were not completely mature, allows us to hypothesise that the enlargement and lateral stretching of structural units of the sporoderm, and the spaces between them, took place during the final stages of ontogenesis of megaspores along with the additional accumulation of amorphous sporopollenin. Both layers of the megaspore sporoderm, as well as the cavity between them, developed early in the ontogenesis. Although Kossoviella timanica was certainly a unique Late Devonian plant, it bears some resemblance to the Givetian heterosporous, bisporangiate lycopsid Yuguangia ordinata in having dichotomously branching axes, sporophylls with spiny margins and strobili with proximal megasporangia and distal microsporangia. Kossoviella timanica is also similar to the Famennian bisporangiate lycopsid Bisporangiostrobus harrisii in lacking a ligula and in having dichotomously branching strobili with proximal megasporangia and distal microsporangia.

Although the Spectrum whose linear record is now presented to the Royal Society, Edinburgh, is unfortunately not so perfect as it might have been with better apparatus (but which I did not possess)—yet it represents the labour and expense connected with two voyages in 1877–1878 to Portugal; and many weeks work there in both years, with the sun in a more favourable position for observing really solar, and not telluric, or atmospheric, phenomena, than is ever, at any time, obtainable in Great Britain.

In determining the solubility of salts and other bodies in alcohol, it is desirable to operate with a spirit wholly free from water. But anhydrous or absolute alcohol is formed with difficulty, even by the most improved process—that of Richter. In rectifying alcohol from chloride of calcium, as recommended by Richter, I have never obtained it under the specific gravity 0.798 at the temperature of 60°, by a single distillation; but upon rectifying this product again from new chloride of calcium, I generally succeeded in reducing it to 0.796, which is the specific gravity of the standard alcohol of that chemist. The following experiment illustrates this process.

The general identity, in idea and design, of the ancient monuments of Southern and Western Europe, with those of Hindostan, is so marked, as to appear to justify the inference that races of Asiatics proceeded westward at different ages, and established themselves along the shores of the Baltic and Mediterranean Seas, and part of the Atlantic Ocean; along which route they have left characteristic monuments, which resemble those of their original country.

Hydrocarbon migration mechanism into a reservoir is one of the most controversial in oil and gas geology. The research aimed to study the effect of supercritical carbon dioxide (СО2) on the permeability of sedimentary rocks (carbonates, argillite, oil shale), which was assessed by the yield of chloroform extracts and gas permeability (carbonate, argillite) before and after the treatment of rocks with supercritical СО2. An increase in the permeability of dense potentially oil-source rocks has been noted, which is explained by the dissolution of carbonates to bicarbonates due to the high chemical activity of supercritical СО2 and water dissolved in it. Similarly, in geological processes, the introduction of deep supercritical fluid into sedimentary rocks can increase the permeability and, possibly, the porosity of rocks, which will facilitate the primary migration of hydrocarbons and improve the reservoir properties of the rocks. The considered mechanism of hydrocarbon migration in the flow of deep supercritical fluid makes it possible to revise the time and duration of the formation of gas–oil deposits decreasingly, as well as to explain features in the formation of various sources of hydrocarbons and observed inflow of oil into operating and exhausted wells.

The area lying between the north of Britain and the Faeröe Islands is classical in the annals of Marine Zoology, not only from the fact that the first systematic deep-sea investigations undertaken by this country were carried out there, but also from the number of new and remarkable types of animal life which have been first found in that region. Speaking only of the Echinoderm fauna, in justification of these remarks, it will suffice to mention the discovery of such forms as Phormosoma and Porocidaris amongst Echinoids, and of Hymenaster, Korethraster and Zoroaster amongst Asteroids.

The St David's Head Intrusion, exposed in North Pembrokeshire, is a tholeiitic sill up to 570 m thick, comprising principally mafic gabbros, which are in part layered. Layering varies from the centimetre to the metre scale. A range of gabbroic compositions is present, defining seven major petrological types, which are cut by thin silicic (aplitic) veins.

Log–log plots of incompatible elements from the various lithological units indicate that all of the rocks in the intrusion are petrogenetically linked, although a variety of processes has been operative. Roach (1969) considered the quartz gabbros and dolerites, which form an envelope around the other units, to relate most closely to the parental composition. However, the least evolved compositions in the intrusion are from the xenolithic laminated olivine gabbros, although these mafic compositions are due primarily to the presence of abundant, mafic, cognate xenoliths. These xenoliths are thought to relate to an earlier episode of crystal accumulation in a high-level magma chamber. The various laminated gabbros reflect crystal accumulation in situ after magma emplacement, leading in certain layers to extreme enrichments in Fe, Ti, and V, related to high modal proportions of cumulus ilmenite. Further in situ crystallisation led to differentiation of the residual liquid, producing more silicic gabbros with well-developed granophyric textures, the granophyre reflecting the silicic residuum. Extreme differentiation, possibly combined with expulsion of silicic residual liquid during crystal accumulation and compaction, resulted in the cross-cutting aplite veins.

Three different types of layering are present in the intrusion. Firstly, preferred orientation of tabular minerals in the laminated gabbro units is thought to result from discrete sedimentation episodes from a convecting magma chamber. Secondly, macrorhythmic modal layering up to 1 m thick consists of an alternation of relatively ilmenite-rich and ilmenite-poor layers in the laminated gabbro units, although the reason for the modal variation is not certain. Thirdly, a centimetre-scale felsic–mafic microrhythmic layering is present in the envelope quartz gabbros and dolerites, which is similar to the inch-scale layering in the Stillwater Igneous Complex. This layering is thought to relate to metasomatic reaction in the gabbro in the presence of water at a late magmatic stage. Overall, these various lithological units themselves define a large-scale layering in the intrusion.

Combined, the petrological and geochemical data suggest that the St David's Head Intrusion was not emplaced in a single event. Rather, a series of magma pulses, of contrasting compositions but petrogenetically linked, was intruded. Some of the chemical variations now seen existed prior to emplacement, indicating the former presence of high-level crustal magma chambers, while other variations developed as a result of in situ crystallisation processes and related chemical differentiation.

The mapping of sub-drift topography is an important problem in this country and one which, under suitable circumstances, lends itself readily to solution by the method of gravitational survey with the Eötvös Torsion Balance. It has been pointed out in previous papers that structures such as anticlines, synelines, faults, and dykes may be determined by a linear traverse with the balance transverse to the strike; but in the case of buried topographic features a network of observations with subsequent plotting of the gradients and isogams, i.e. lines of equal gravity, is necessary to determine even approximately the shape and dimensions of rock-masses underlying a thick covering of drift.

We consider the reinterpretation of Bethia by Bassett et al. (2008) to be flawed in several regards. Details of the pedicle rootlets and wrinkles strongly imply a soft-tissue structure rather than calcified sheath. The ornament is unusual, but equally so under either an orthide or a plectambonitoidean model. The reinterpretation of the ‘deltial plates’ as ‘chilidial plates’ is likely correct, but argues against a plectambonitoidean affinity by implying that the pedicle emerges between the valves. Other arguments presented in favour of a plectambonitoidean affinity are also discussed; we consider them unpersuasive. Finally, we contend that Bethia is sufficiently well characterised to deserve a taxonomic name.

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.

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.

A mechanism may be defined as a combination of plates, bars, or flexible members jointed together, so that, while the parts may move relatively to each other, the relative positions of all the different parts are determinate for each given possible relative position of any two parts.

It follows immediately that the simultaneous relative displacements, velocities, and accelerations of velocity of all parts are also strictly determinate.

This determination, by accurate graphic means, forms the subject of this paper. In it those mechanisms alone are considered that are composed of rigid members, the motions of whose parts are all continually parallel to one plane, the constancy of the plane being defined relatively to one of the members of the mechanism itself.

I have explained at some length, in my “Reply to Prof. Boltzmann,” the circumstances under which the present inquiry originated and has been pursued. Of these I need now only mention two:—first, the very limited time which I can spare for such work; second, the very meagre acquaintance I possessed of what had been already done with regard to the subject. My object has been to give an easily intelligible investigation of the Foundations of the Kinetic Theory; and I have, in consequence, abstained from reading the details of any investigation (be its author who he may) which seemed to me to be unnecessarily complex. Such a course has, inevitably, certain disadvantages, but its manifest advantages far outweigh them.

Sir Isaac Newton's theory of the colours of natural bodies, is perhaps the most ingenious and lofty of all his speculations. It was devised, however, at a time when the doctrine of light had made comparatively but little progress, and before the discovery of various principles on which the colours of bodies must depend, or by which, at least, they must be extensively modified. The different dispersive powers of transparent substances;—the irrationality of the spectrum;—the action of striated surfaces;—the decomposition of polarised light;—the reflection of coloured light at the confines of equally refracting media;—and the absorption of common and of polarised rays,—are principles which embrace within their individual range a great variety of facts to which the Newtonian theory of colours bears no relation.

(1.) In February 1850, I laid before the Royal Society of Edinburgh a paper, in which the laws of the pressure and expansion of gases and vapours were deduced from the supposition, that that part of the elasticity of bodies which depends upon heat, arises from the centrifugal force of the revolutions of the particles of elastic atmospheres surrounding nuclei, or atomic centres. A summary of the results of this supposition, which I called the Hypothesis of Molecular Vortices, was printed in the Transactions of this Society, volume xx., as an introduction to a series of papers on the Mechanical Action of Heat; and the original paper has since appeared in detail in the Philosophical Magazine.

The bearing which the subject of the following communication has upon the much-vexed question of the unity of the human species, and the inferences which may be drawn from it in relation thereto, invest it with an importance which would not otherwise belong to it.

The position which the question I allude to at present occupies is this. On the one hand, the advocates of the view that all men are not of the same species, but that they compose a genus consisting of many species, maintain that the parasites which infest the different races of man are distinct; and because we usually find that distinct species of parasites are allotted respectively to the different species of the lower animals, they infer that the same rule must hold with man, and that therefore each different race possessing a distinct parasite must be a distinct species.