In a paper which I communicated to this Society in 1815, and which was published in the Seventh volume of their Transactions, I described a new species of coloured fringes, produced between two plates of parallel glass. From a consideration of the theory of this class of phenomena, it was obvious that analogous, though much more complicated, systems of rings should be produced between plates with curved surfaces, but it was not till 1822 that I succeeded in detecting them; and so completely are these rings concealed by the superposition of similarly situated images, that, in consequence, of having forgotten my method of observation, I have experienced the greatest difficulty in rediscovering them.

My earliest experiments were performed with a double achromatic object-glass, made by Berge, having a diameter of inches, and 30 inches in focal length. The inner surfaces of the crown and flint glass lenses were ground to different radii, as shewn in the section of it at AB, CD, Fig. 5, Plate IX.; and the outer surface of the flint-glass lens was concave, so that there was left between the lenses a meniscus of air A 2, B 3 A.

The restoration of the ancient books of geometry would have been impossible, without the coincidence of two circumstances, of which, though the one is purely accidental, the other is essentially connected with the nature of the mathematical sciences. The first of these circumstances is the preservation of a short abstract of those books, drawn up by Pappus Alexandrinus, together with a series of such lemmata, as he judged useful to facilitate the study of them. The second is, the necessary connection that takes place among the objects of every mathematical work, which, by excluding whatever is arbitrary, makes it possible to determine the whole course of an investigation, when only a few points in it are known. From the union of these circumstances, mathematics has enjoyed an advantage of which no other branch of knowledge can partake; and while the critic or the historian has only been able to lament the fate of those books of Livy and Tacitus which are lost, the geometer has had the high satisfaction to behold the works of Euclid and Apollonius reviving under his hands.

Nearly six years ago, I presented to the Society two Memoirs on the subject of Differentiation, with fractional indices. The method which I adopted to extend the signification of a differential coefficient consisted in assuming that the function , which enters into the value of the coefficient deduced from a particular hypothesis, is limited only by the definition . This generalization appears to be perfectly satisfactory, and promises to offer, if not the only, certainly the best extension of the Differential Calculus. Considering the length of the interval which has elapsed since the publication of my former Memoirs, it is remarkable that so little addition has been made to our knowledge of this branch of analysis. With the exception of one or two papers in Liouville's Journal, and a few remarks by Professor De Morgan, in his Treatise on the Differential Calculus (pp. 598–600), I am not aware that anything has been written on this subject since that time. Seeing, therefore, that others are not willing to enter on this very promising field, I consider it not improper that I should make known a number of extensions of this science to which I have been subsequently led, many of which have been in my possession a considerable time.

Musket-Bullets are occasionally found inclosed in ivory, and every anatomical museum contains specimens of this kind. Why bullets should be so frequently met with in this situation, it is not easy to say; the head of the animal appears to be generally aimed at, and foreign bodies when they enter the tusks, instead of being removed in the usual manner, are retained by the process, an investigation of which is to form the subject of the present paper.

For many years geologists have recognised the occurrence of at least two boulder-clays in the British Islands and the corresponding latitudes of the Continent. It is no longer doubted that these are the products of two separate and distinct glacial epochs. This has been demonstrated by the appearance of intercalated deposits of terrestrial, freshwater, or, as the case may be, marine origin. Such interglacial accumulations have been met with again and again in Britain, and they have likewise been detected at many places on the Continent, between the border of the North Sea and the heart of Russia. Their organic contents indicate in some cases cold climatic conditions; in others, they imply a climate not less temperate or even more genial than that which now obtains in the regions where they occur. Nor are such interglacial beds confined to northern and north-western Europe. In the Alpine Lands of the central and southern regions of our Continent they are equally well developed. Impressed by the growing strength of the evidence, it is no wonder that geologists, after a season of doubt, should at last agree in the conclusion that the glacial conditions of the Pleistocene period were interrupted by at least one protracted interglacial epoch. Not a few observers go further, and maintain that the evidence indicates more than this. They hold that three or even more glacial epochs supervened in Pleistocene times. This is the conclusion I reached many years ago, and I now purpose reviewing the evidence which has accumulated since then, in order to show how far it goes to support that conclusion.

At the close of the preceding session, the Council of the Society did me the honour of entrusting me with the chemical examination of several articles sent not long ago to the Society by Mr Swinton, Secretary to Government at Calcutta. The articles in question are, 1. Specimens of the black varnish used in different parts of Hindostan and the Burmese territories, with specimens of the juices of which these varnishes are said to be compounded; 2. Specimens of naphtha from Persia, and of petroleum from Rangoon; 3. Specimens of wood-oil, a variety of fluid turpentine; 4. Specimens of crude caoutchouc, and of solutions of it in wood-oil.

The only one of these articles which has hitherto yielded results of such interest as to induce me to lay them before the Society, is the petroleum of Rangoon, which appears to contain a compound inflammable principle hitherto unknown.

In May 1891 I communicated a paper to this Society “On a Method of Observing and Counting the Number of Water Particles in a Fog,” and in July of the same year a paper “On the Solid and Liquid Particles in Clouds.” One conclusion to which the observations contained in these papers pointed was, that there existed a relation between the thickness or density in clouds and fogs and the number of water particles present. Though the figures did not show that this relation was very close, yet in all the fogs and clouds tested there was a rough relation between the thickness of the air and the number of water particles observed.

The discoverer of Nova Aurigæ is the Rev Thomas D. Anderson of Edinburgh, D.Sc. in Classical Philology. Dr Anderson is “almost certain” that he saw the star at 2 a.m. on January 24 of this year; it was then slightly brighter than χ Aurigæ. Unfortunately, he mistook it for 26 Aurigæ, which it precedes by about 6m 39s, merely remarking to himself that the star was brighter than he had previously thought it to be. Twice in the following week he made the same observation at about the same hour of the night. At last, on the morning of January 31, it flashed upon him that, after all, the star was not 26 Aurigæ, and that 26 Aurigæ had a much greater right ascension. He consulted a small star-map, and the discovety was made. Regretting that he had not earlier compared the map with the heavens, and thinking that the star might be well known to astronomers, Dr Anderson wrote an anonymous postcard to me on the same morning bearing the words:—“Nova in Auriga. In Milky Way, about two degrees south of χ Aurigæ, preceding 26 Aurigæ. Fifth magnitude, slightly brighter than χ.” I may add that Dr Anderson's plant consists of a small hand spyglass adapted to astronomical purposes by removing the front pair of lenses from the eyepiece. In this state it magnifies about ten times, and, of course, gives inverted images. Dr Anderson hopes that amateurs, although provided with only the most modest appliances, may, by his unexpected success, be induced to persevere in their observations.

§ 1. The general problem in elasticity, as usually presented for solution, supposes the elastic substance to pass initially from a state without strain. But important cases exist where it must be conceived to start from a state already under considerable stress. When this is the case, the constitution of the solid undergoes great change, as is shown by the fact that strained glass loses its isotropic property, and becomes doubly refractive. This subject was long ago attacked by Cauchy, who, by means of the theory of molecular actions, deduced the existence, in the expressions for the stresses due to the secondary strains, of terms proportional to the initial or primary stresses. The problem has been since discussed by MM. De St Venant and Boussinesq, who have applied to it Green's expression for the energy stored up during the strain. But the question, in their hands, still retains traces of Cauchy's hypothetical element, inasmuch as their expression for the potential energy was deduced by means of the molecular theory. M. De St Venant even considers it a strong argument for the truth of the latter, that it is indispensable in the discussion of this problem. These authors have also failed to see in what way the remaining part of the potential depends on the original strain.

Section VI.—A Review of the Fundamental Principles of the Mechanical Theory of Heat; with Remarks on the Thermic Phenomena of Currents of Elastic Fluids, as illustrating those Principles.

(Article 46.) I have been induced to write this Section, in continuation of a paper on the Mechanical Action of Heat, by the publication (in the Philosophical Magazine for December 1852, Supplementary Number) of a series of experiments by Mr Joule and Professor William Thomson, on the Thermal Effects experienced by Air in rushing through small Apertures. Although those authors express an intention to continue the experiments on a large scale, so as to obtain more precise results; yet the results already obtained are sufficient to constitute the first step towards the experimental determination of that most important function in the theory of the mechanical action of heat, which has received the name of Carnot's Function.

At the end of 1886 a method occurred to me of rapidly recording the positions of the lines of the solar spectrum, which I thought might be used with advantage for determining the faint “Telluric dry-gas lines” near D, mentioned in Professor Piazzi Smyth's maps of The Visual Solar Spectrum in 1884. The fundamental idea of the recording apparatus is that of magnifying by some mechanical means the motion of the grating, or prism, to such an extent that it can be recorded on a continuous fillet of paper. The viewing telescope being then firmly clamped, the exact positions of the grating can be pricked off on the strip of paper as the lines are successively brought to the fixed cross in the field of view.

For the satisfactory use of the method two conditions suggest themselves as desirable. First, the punctures should not be less than a good-sized pin hole, and the interval between the closest lines, which the spectroscope is able to separate, should be represented on the paper strip by a space of several tenths of an inch.

Several of my friends have suggested to me the propriety of publishing something of the kind now offered to the Society, observing, that the great brevity with which the Antecedental Calculus is written, and the very concise form in which it is delivered to the public, may lead some to form erroneous opinions respecting the principles on which it is founded. In compliance partly with their request, I have drawn up this short paper, which I hope will remove even the possibility of misconception on that head, and convince every intelligent reader, that the antecedental calculus has the same geometrical principles for its ground-work, that the formulæ in the Universal Comparison themselves have, from which I originally derived it more than twenty years ago.

The Reverend Dr William Richardson, late F. T. C. D., having sent to Dr Hope a collection of specimens from the northern coast of Antrim, with a catalogue, and observations, the specimens were exhibited, and the observations were read in the Royal Society, March 1803.

In the month of November 1830, a small portion of a magnificent fossil stem was discovered in the quarry of Craigleith, in this vicinity, and more and more of it was brought into view as the working of the quarry proceeded.

In December following, I communicated to the Natural History Society of Northumberland a description of that part of the petrifaction which had then been uncovered. This description will appear in their Transactions.

At that period a considerable part of the stem had been brought into view, and since that time the remaining portion, and the roots, have been exposed.

This investigation on the morphology of the Fore-brain or Prosencephalon of Elasmobranchii was suggested to me by Professor G. Elliot Smith, F.R.S., and I am indebted to him for introducing me to an interesting if somewhat difficult and perplexing subject. A glance at the literature on the morphology of the prosencephalon will readily convince one of two things—firstly, of the endless field for work in endeavouring to ascertain the true homologies of the neurone areas and fibre tracts in the brain throughout the Phylum Vertebrata; and secondly, of the somewhat unnecessary obstacles which have been introduced into the problem by the more recent additions to a terminology, already overburdened and encumbered, of new terms for equivalent parts. It would appear to be very advantageous if mutual agreement could be arrived at among workers on this subject regarding the use of descriptive terms, and, if such is at present impracticable, that each new worker should refrain as far as possible from applying a new term to a structure already previously described. In one case, namely, that of the tuberculum olfactorium, five different terms have apparently been applied to the same area.

A small lot of fossil whale barnacles from the Upper Pleistocene of California and the Middle Pleistocene (Chibanian) of Oregon (United States West Coast), described in a 1972 unpublished MA thesis, are formally described and illustrated herein. In that thesis, a new genus and species name were proposed; however, according to the International Code of Zoological Nomenclature, they have no taxonomic standing and are thus unavailable. Based on our reappraisal, two specimens in this lot belong to a new, extinct species that can be assigned to the purportedly extant genus Cetopirus. Cetopirus polysyrinx sp. nov. differs from congeners in that its secondary T-shaped flanges are multitubiferous internally, that is, they are perforated by a high number of irregularly-sized and irregularly-spaced tubules that result in a spongy aspect in transverse section. Whether or not this peculiar condition had any adaptive significance is difficult to determine. Considering that Cetopirus is currently known as an obligate epibiont of right whales (including the North Pacific form Eubalaena japonica (Lacépède 1818)), the host of C. polysyrinx sp. nov. was E. japonica or some other species of Eubalaena. The Plio-Pleistocene deposits of the Pacific coast of North America have yielded a rather idiosyncratic fossil whale barnacle fauna, inclusive of the genera Cetolepas, Cryptolepas and now Cetopirus, which seemingly contrasts with all other coeval assemblages worldwide, the latter being in turn dominated by Coronula spp.

Identifying the relative importance of urban and non-urban land-use types for potential denitrification derived N2O at a regional scale is critical for quantifying the impacts of human activities on nitrous oxide (N2O) emission under changing environments. In this study we used a regional dataset from China including 197 soil samples and six land-use types to evaluate the main predictors (land use, heavy metals, soil pH, soil moisture, substrate availability, functional and broad microbial abundances) of potential denitrification using multivariate and pathway analysis. Our results provide empirical evidence that soils on farms have the greatest potential denitrifying ability (PDA) (10.92±6.08ng N2O-N·g–1 dry soil·min–1) followed by urban soil (6.80±5.35ng N2O-N·g–1 dry soil·min–1). Our models indicate that land use (low vs. high human activity), followed by total nitrogen (TN) and heavy metals (Cu, Zn, Pb, Cd) was the most important driver of PDA. In addition, our path analysis suggests that at least part of the impacts of land use on potential denitrification were mediated via microbial abundance, soil pH and substrates including TN, dissolved organic carbon and nitrate. This study identifies the main predictors of denitrification at a regional scale which is needed to quantify the impact of human activities on ecosystem functionality under changing conditions.

Twelve specimens of early Toarcian Mecoptera and Diptera from the vicinity of Wolfsburg were investigated for the present study. The material was found during house building activities in the 1980s at the locality Große Kley in Mörse, an urban district of the city of Wolfsburg, Lower Saxony, Germany. The specimens were found in calcareous nodules of the Harpoceras falciferum Zone that occur within the Liassic black shales (Posidonia shale). Six specimens of Mecoptera, five belonging to the family Orthophlebiidae and one belonging to the Bittacidae, and six representatives of the following Diptera families were identified: Ptychopteridae, Limoniidae, Anisopodidae and the superfamily Mycetophiloidea. The fossil fauna of Wolfsburg is similar to that of other early Toarcian sites in Germany, described by Handlirsch (1906, 1939), Bode (1905, 1953) and Ansorge (1996) from Braunschweig, Dobbertin and Grimmen. Two new species are described, Mesorhyphusulrichi sp. nov. (Anisopodidae) and Archipleciomima germanica sp. nov. (Mycetophiloidea).

Drillholes on shells provide a useful way to investigate prey and predator relationships. The current study documents predator–prey interactions exemplified by a faunal assemblage of the fossil gastropod Turritella cingulifera from the Pleistocene Szekou Formation in Hengchun Peninsula, Taiwan. All recognisable skeletal and shell fragments that are larger than 3 mm in size were collected and recorded. Processed bulk sediments (5.24 kg) contained 1462 molluscan shells, including 824 specimens of T. cingulifera, and 27 non-molluscan invertebrates. In the current study, approximately 41.6% (609/1462) of molluscs are drilled with at least one hole. Drilling intensities (DIs) regardless of shell completeness in all gastropods, bivalves and the turritelline gastropod T. cingulifera are 0.546, 0.060 and 0.413, respectively. DI on turritellids is significantly lower than that on other gastropods (χ2= 21.039, P < 0.001). Furthermore, the percentage of drillholes that occur in multiply drilled specimens is 34.7% (95/275) for turritelline gastropods based on complete to nearly complete specimens (n = 588). Our study shows no significant preference of drillhole position either on the suture or on the whorl (χ2= 0.055, P = 0.814). Most drillholes are located in whorls two to four proximal to the aperture. Drillhole diameters of the shells with one drillhole and ones with multiple drillholes are 1.0 and 0.5 mm on average, and the results of Mann–Whitney tests indicate that they are significantly different (P < 0.001). The first turritelline gastropod shell with an incomplete drillhole from Taiwan is documented here. The dominant drilling predators were naticids based on the drillhole morphology and the presence of naticids in the same assemblage. No apparent prey size selectivity is observed, so a ‘size refugium’ does not exist for the turritellids in the current study.