A glance at the expression

is sufficient to verify the fact that it is symmetric with respect to a, b, c, d, and also with respect to α, β, γ, δ. It is likewise, although not quite so evidently, an alternating function with respect to the interchange

that is to say, if a and α be interchanged, and at the same time b and β, c and γ, d and δ, the function is not altered in magnitude, but merely changes sign. With a little trouble, indeed, the expression can be transformed into

or say

Dispersed spore assemblages have been recovered from the Am Binnein Sandstones from the upper part of the ‘Lower Old Red Sandstone’ sequence on the island of Arran, Scotland. The spore assemblages belong with the Emphanisporites annulatus–Camarozonotriletes sextantii (AS) Spore Assemblage Biozone (SAB), indicating an Early Devonian, Emsian (but not earliest Emsian or latest Emsian) age. This is the first reliable age constraint for the ‘Lower Old Red Sandstone’ of Arran, and enables correlation with the more extensive sequence developed on the mainland in the Midland Valley of Scotland. The Am Binnein Sandstones are confirmed as correlatives of the Strathmore Group.

The genus Orthobittacus was established by Willmann (1989) and is characterised by a long Sc vein and the unusually developed medial sector for the Bittacidae. Four Jurassic species have been described in this genus to date: O. abshiricus (Martynova, 1951) from Kirgizia; O.desacuminatus (Bode, 1953) from Braunschweig (Germany); O. polymitus Novokshonov, 1996 from Karatau (Kazakhstan); and O. maculosus Liu, Shih, Bashkuev & Ren, 2016 from the Jiulongshan Formation of Daohugou (China). The fifth congeneric and second species from China, O. suni sp. nov., is described herein. The importance of the genus Orthobittacus for the phylogeny of Bittacidae, as the most plesiomorphic genus, is discussed.

The presence of archaeocyath-bearing clasts from Cenozoic tills and Cambrian Mount Wegener Formation reveal erosion of a hidden Cambrian carbonate platform in Shackleton Range, Antarctica. We provide microfacies, paleontological, diagenetic and tectonically induced fabric data from carbonate clasts which, in addition to available geochemical and geochronological data from Shackleton Range, allow the paleoenvironmental reconstruction of a lost Cambrian Series 2 mixed siliciclastic–carbonate platform that was developed and eroded during the Ross orogeny. Carbonate production was dominated by non-skeletal grains in possibly restricted platform-interior and oolitic shoal complex settings, while open subtidal sub-environments (calcimicrobe carpets, calcimicrobe–archaeocyath patch reefs, muddy bottoms) were dominated by a diverse calcimicrobe assemblage and/or by secondary to accessory heterozoan assemblage (archaeocyaths and other sponges, chancelloriids, hyoliths, coralomorphs, trilobites, echinoderms). We describe a Botoman assemblage with 34 archaeocyathan species among 12 existing archaeocyathan genera. A new archaeocyath family Shackletoncyathidae is proposed. New species (Rotundocyathus glacius sp. nov., Buggischicyathus microporus gen. et sp. nov., Paragnaltacyathus hoeflei, Shackletoncyathus buggischi gen. et. sp. nov., Santelmocyathus santelmoi gen. et sp. nov., Wegenercyathus sexangulae gen. et sp. nov.) and Tabulaconus kordae coralomorph are reported from Antarctica for the first time. Archaeocyathan fauna share few species with contemporary fauna of South Australia (9) and even fewer with the Antarctic platforms of the Shackleton Limestone (2) or the Schneider Hills limestone (1). Similarity is greater with Antarctic allochthonous assemblages of Permo-Carboniferous tillites from Ellsworth Mountains (2), Cenozoic deposits from King George Island (4) or Weddell Sea (1). The Shackleton Range lost/hidden platform shows a distinct entity related with its tectonosedimentary evolution, in a possible back-arc basin on the Mozambique seaway during the E and W Gondwana amalgamation, which distinguishes it from those developed on the palaeo-Pacific margin of the E Antarctic craton.

The early tetrapod Eoherpeton watsoni is known from the mid- to late Carboniferous (late Viséan to Namurian, approximately 346–313 Ma) of Scotland. The holotype is made up of a nearly complete but crushed skull with postcranial fragments. The skull anatomy of Eoherpeton was first described over 40 years ago; however, many details are obscured due to deformation of the specimen, including internal bone surfaces, the palatal bones and dentition, and suture morphology. Most phylogenetic analyses place Eoherpeton as an embolomere/reptilomorph on the lineage leading to amniotes, making it a key taxon for understanding anatomical changes during the fish-tetrapod transition. In this paper, we scanned the holotype using micro-computed tomography and digitally prepared the specimen. Based on these data, we present a revised description of the skull, including sutural morphology, that supplements and amends previous descriptions. New anatomical findings include the presence of a previously unknown tooth-bearing vomer, additional information on the shape of the basipterygoid processes and jaw joint, the ability to visualise the full extent of the pterygoid, and confirmation of the arrangement of the coronoid series. We also note the size of the pterygoid flange, which is larger than previously described for Eoherpeton. The pterygoid flange is widely considered to be characteristic of amniotes and serves as the origin of the medial pterygoideus muscle. The differentiation of the adductor muscles and appearance of medial pterygoideus are thought to have permitted a static pressure bite in amniotes, potentially resulting in greater bite forces and increased dietary range. Thus, the presence and extent of the pterygoid flange in Eoherpeton suggests this feature (and associated changes in feeding mechanism) may have evolved earlier than previously thought. Finally, the skull was digitally repaired and retrodeformed to create a new, hypothetical three-dimensional reconstruction of the skull of Eoherpeton.

It is a singular fact in the history of science, that, after all the attempts of the most eminent modern mathematicians, to obtain a simple and satisfactory demonstration of the fundamental property of the lever, the solution of this problem given by Archimedes, should still be considered as the most legitimate and elementary. Galileo, Huygens, De la Hire, Sir Isaac Newton, Maclaurin, Landen, and Hamilton, have directed their attention to this important part of mechanics; but their demonstrations are in general either tedious and abstruse, or founded on assumptions too arbitrary to be recognised as a proper basis for mathematical reasoning. Even the demonstration given by Archimedes is not free from objections, and is applicable only to the lever, considered as a physical body.

The history of the newer sciences presents many instructive examples of the progress of the human mind in developing the germs of natural knowledge, and building on a single observation entire departments of science. Few pursuits indeed are more interesting, even to the student of immaterial nature, than in the perusal of such a history to trace the footsteps of the several investigators, and to mark how far, and by what means,—whether by new methods or by greater patience of research,—each successive observer advanced the inquiry. We see the human mind, as it were, set free from the trammels of time, and developing its resources on a large and continuous scale, not limited by the powers of one intellect, the length of one life, or the means of one station. We see, at the same time, what varied gifts and opportunities are necessary for the elucidation of a single subject; how these gifts, though not all imparted to one man or to one generation, are yet wisely and beneficently bestowed on the entire species; and how all are thus enabled to co-operate, either in unfolding abstract truth, or in drawing forth those practical results which directly conduce to the amelioration and comfort of all.

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.

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 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.

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.

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.

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.

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.

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 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.

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.