It seems strange that a principle so fundamental and so widely used as Lagrange's Rule for Solving a Linear Differential Equation should hitherto have been almost invariably provided with an inadequate demonstration. I noticed several years ago that the demonstrations in our current English text-books were apparently insufficient; but, as the method by which I treated Linear Partial Differential Equations in my lectures did not involve the use of them, it did not occur to me to analyse them closely with a view to discovering in what the exact nature of the defect consisted. The consideration of certain special cases recently led me to examine the matter more closely, and I was greatly surprised to find that most of the general demonstrations given are vitiated by a very obvious fallacy, and in point of fact do not fit the actual facts disclosed by the examination of particular cases at all.

When preparing to make a night journey over one of the plains of South Africa, in the month of June 1843, a friend called my attention to the peculiar appearance of the sky in the west, as offering a very decided proof, “agreeably with theory,” that there was no “Solar atmosphere” to be seen at that season of the year.

On looking in the direction mentioned, the last portion of the twilight was just visible, and forming a peculiarly level line above the place where the sun had set, for an extent in azimuth, of perhaps 40°, and at a height of about 5°. All the gorgeous colours which had attended the setting of the sun had long since vanished, and there only remained sufficient light within the flattened arc described, to make the space included between it and the horizon appear light blue, while all the rest of the sky had attained a deeper colour, nay was almost black, and thickly spangled with small as well as large stars.

While the external infiltration of water has been identified from modern geothermal and/or fossil hydrothermal systems through stable isotopes, the physicochemical boundary conditions like the initial oxygen isotopes of water $( {{\rm \delta }^{ 18}{\rm O}_{\rm W}^{\rm i} } ) $ and rock as well as alteration temperature were implicitly presumed or empirically estimated by the conventional forward modelling. In terms of a novel procedure proposed to deal with partial re-equilibration of oxygen isotopes between constituent minerals and water, the externally infiltrated meteoric and magmatic water are theoretically inverted from the early Cretaceous post-collisional granitoid and intruded Triassic gneissic country rock across the Dabie orogen in central-eastern China. The meteoric water with a $ {{\rm \delta }^{ 18}{\rm O}_{\rm W}^{\rm i} } $ value of −11.01 ‰ was externally infiltrated with a granitoid and thermodynamically re-equilibrated with rock-forming minerals at 140°C with a minimum water/rock (W/R)o ratio around 1.10 for an open system. The lifetime of this meteoric hydrothermal system is kinetically constrained less than 0.7 million years (Myr) via modelling of surface reaction oxygen exchange. A gneissic country rock, however, was externally infiltrated by a magmatic water with $ {{\rm \delta }^{ 18}{\rm O}_{\rm W}^{\rm i} } $ value of 4.21 ‰ at 340°C with a (W/R)o ratio of 1.23, and this magmatic hydrothermal system could last no more than 12 thousand years (Kyr) to rapidly re-equilibrate with rock-forming minerals. Nevertheless, the external infiltration of water can be theoretically inverted with oxygen isotopes of re-equilibrated rock-forming minerals, and the ancient hydrothermal systems driven by magmatism or metamorphism within continental orogens worldwide can be reliably quantified.

The non-archosauriform archosauromorph Dinocephalosaurus orientalis was first described from the Upper Member of the Guanling Formation (late Anisian, Middle Triassic) of Guizhou Province by Li in 2003 on the basis of a complete articulated skull and the first three cervical vertebrae exposed in dorsal to right lateral view. Since then, additional specimens have been discovered in southwestern China. Here, five newly discovered specimens are described for the first time, and redescriptions of the holotype IVPP V13767 and another referred specimen, IVPP V13898, are provided. Together, these permit the description of the complete skeleton of this remarkable long-necked marine reptile. The postcranial skeleton is as much as 6 metres long, and characterised by its long tail and even longer neck. The appendicular skeleton exhibits a high degree of skeletal paedomorphosis recalling that of many sauropterygians, but the skull and neck are completely inconsistent with sauropterygian affinities. The palate does not extend back over the basisphenoid region and lacks any development of the closed condition typical of sauropterygians. The arrangement of cranial elements, including the presence of narial fossae, is very similar to that seen in another long-necked archosauromorph, Tanystropheus hydroides, which at least in part represents a convergence related to an aquatic piscivorous lifestyle. The long and low cervical vertebrae support exceptionally elongate cervical ribs that extend across multiple intervertebral joints and contribute to a ‘stiffening bundle of ribs’ extending along the entire ventral side of the neck, as in many other non-crocopodan archosauromorphs. The functional significance of the extraordinarily elongate neck is hard to discern but it presumably played a key role in feeding, and it is probably analogous to the elongate necks seen in pelagic, long-necked plesiosaurs. Dinocephalosaurus orientalis was almost certainly a fully marine reptile and even gave birth at sea.

All taxa of Orthoptera known from the latest Eocene deposits of the Isle of Wight are redescribed or discussed, based on the study of their type material. The following new taxa are described: Ontogryllus gen. nov., O. rossi sp. nov. (type species), Achetomorpha gen. nov., A. irregularis sp. nov. (type species), and Proeneopterotrypus gen. nov. The latter genus is here proposed for Pteroplistes danicus Rust, described from the Palaeocene/Eocene of Denmark, as this find is important for understanding the systematic position of the genus Eneopterotrypus Zeuner. Remarks about the taxonomy of the taxa studied, age of these deposits, and ecology are given.

Changes in soil chemistry in response to varying climatic regimes can alter the equilibria of soil systems and result in different clay minerals. Variations in phyllosilicate clay composition can reflect temporal and spatial climatic changes, such as summer/winter monsoon cycles. The objective of this research was to investigate the mineralogy of the clay fractions as a proxy for determining variations in the East Asian monsoon climate, based on a section at Chaoyang in China spanning the last 0.423 Ma BP. The clay mineralogy record in the Chaoyang section was compared with other proxies as recorded in this section and with other palaeoclimatic indicators, including oxygen isotopes from oceanic sediments and loess–palaeosol sections on the Chinese Loess Plateau (CLP). The results from clay mineralogy and related climatic studies show that the summer monsoon has a trend of four increased stages and four decreased stages; whereas the winter monsoon displays the opposite trend. During the last 0.423 Ma BP, the strongest winter monsoon occurred around 0.243–0.311 Ma BP. During this period, which included an intense winter monsoon, the soil in the section had the least illite, one of the smallest kaolinite and illite/Chlorite (I/C) indices and an overall decreasing clay content. The period 0.225–0.243 Ma BP had the strongest summer monsoon over the last 0.423 Ma BP. This period had the greatest amount of illite, the highest I/C index, greater overall clay content and the strongest magnetic susceptibility signal. Additionally, this section contained the smallest mean grain size. The multi-monsoon climate cycles of alternating cold-dry and warm-moist conditions as recorded in the Chaoyang section corresponded well with multiple glaciation cycles based on deep sea sediments. This indicates that the Chaoyang section provides a record of palaeoclimate changes in northeast China that can be linked to mineralogical suites to assist in reconstructing the palaeoclimate over the Late Middle Pleistocene, and complements the global palaeoclimate records in the CLP.

Old Red Sandstone sediments from Arran in western Scotland yield an array of Pb isotope compositions whose slope corresponds to an age of 1830 ± 400 Ma. These compositions are not consistent with a significant contribution of material from the Southern Highlands, Midland Valley, or Southern Uplands, but are similar to the Torridonian sandstones of NW Scotland. Bearing in mind the probable Laxfordian provenance of the Torridonian assemblage, this points to a source for the Old Red Sandstone in a crustal block of early Proterozoic age corresponding to the age of the Laxfordian episode. Since such a source is not presently in a position to yield the Old Red Sandstone, strike-slip movement along the Highland Boundary fault is a strong possibility.

The object of this investigation was not to obtain fresh data on the average strength of timber when subjected to destructive stresses. This has been done in great detail by Bauschinger, Lanza, Johnson, and others, and is still being carried on, in America, on a large scale under the Bureau of Forestry. The intention was rather (a) to find the effect of the medullary rays when timber was stressed in compression, tension, shear, or by cross-bending; and (b) to endeavour to connect up in a satisfactory way the maximum stresses induced in cross-breaking with those obtained by direct compression, tension, or shear. For this purpose, then, it was not necessary that the number of experiments should be excessive, but it was essential that the results obtained should be comparable one with another. To obtain this, a single tree of each variety of wood to be tested was procured, the test pieces were sawn out of the best part of the trunk, carefully stacked, and allowed to season for at least twelve months.

Since Lord Kelvin showed that the resistance of the magnetic metals was influenced by the state of magnetisation of these substances, a good deal of work has been done in investigating more carefully the extent and nature of the relation. It does not appear, however, that any attempt has been made to discover how change of temperature affects this influence. Twelve years ago in Japan I set one of my Japanese students to the construction of an apparatus suitable for making this research. A few preliminary experiments were made; and leaving Japan shortly after, I brought the apparatus with me in the hope of continuing the work in this country. Not till this summer, however, did I find leisure to return to the problem. The results obtained I wish now to lay before the Society.

The resin to which the following paper refers was obtained by Professor Bayley Balfour from a species of Dracæna which he discovered in Socotra during his visit to that island in the year 1880, and to which he has given the name of Dracæna Cinnabari. In outward appearance and in general physical properties the resin differs from the varieties known commercially under the name of dragon's blood. With the view of ascertaining if these differences depend upon difference of chemical composition, we undertook, at Professor Balfour's request, to make an examination of the resin.

From the results obtained by Berzelius in his experiments with tungsten, the number 189·28 is calculated as the atomic weight of that metal. Many later investigations have been made, in which uniformly a lower figure was arrived at, in the majority of cases very nearly 184, which has therefore been regarded as the atomic weight.

On a moonless night, whenever clouds of an ordinary elevation in the atmosphere appear upon, or pass across, the star-spangled sky behind them, they exhibit themselves, as a rule, dark, sometimes even black, in comparison therewith. And no wonder, when every part of the open sky from visible star to visible star therein must be lit up to some, though doubtless a very small, extent by the faintest general and cumulative radiance of those myriads and myriads of lesser stars, which only a large telescope can show to be individually existent as actual stellar points of light, but in their aggregate more nearly eternal, and still more constant from age to age, than our gigantic Sun itself.

Details of the anatomy of the heart and the great vessels are recorded in the following genera of prosimian Primates, Loris, Arctocebus, Perodicticus, Galago, Galagoides, Hapalemur, Lemur, Lepilemur and Propithecus, and comparison made with previously published data on Nycticebus. Special attention is drawn to the proportions, shape and mode of branching of the systemic aorta, which is of typical lemurine pattern in all but Loris and Propithecus. In the heart, attention is paid particularly to the valvular arrangements in the right atrium, where much variation has been found; to the number of pulmonary veins entering the left atrium, also a variable feature; to the internal and external morphology of the ventricles and the valvular arrangements therein. The presence of an os cordis is recorded for the first time in any Primate (viz. in Loris), and the occurrence is mentioned of “brown” (glandular) fat sub-epicardially in Loris. The significance is discussed of the principal anatomical features.

1. The first series of these researches, in the exact form in which they are printed, were laid before the Royal Society on the 19th January 1835. The whole of the experiments there described were made, and the paper written and printed, within a space of time little exceeding two months. This haste, unfavourable to composition, I considered as a less evil than postponing for a period of time, which must have been considerable, the publication of a class of facts which might be said almost to embrace a new science. The professional duties which pressed upon me during the whole continuance of those experiments, then called imperatively on my attention, and during the remainder of the Session my time was devoted to them. The summer I devoted to a foreign excursion, some of the results of which I afterwards digested; and it was not till the commencement of the winter session which has just closed, that I prepared, with a fresh stock of health and spirits, to reinvestigate the whole subject of the Polarization of Heat, and to assign numerical values to the effects, whose existence I had before been contented to prove.

In a preliminary note read before the Society on July 4, 1904, I drew attention to the fact that a number of line-series, forming a group which includes the first series of Hydrogen, can be represented by an equation of the form

where v denotes the wave-frequency of any line of the series, v∞ that of the so-called “tail” of the series (m = ∞), and a1, b1 constants depending on the nature of the emitting substance; the frequencies of successive lines being obtained by substituting successive integers for m. We see at once that this equation is a generalisation of Balmek's formula, into which it is transformed by equating b1 to zero. In the same note I also pointed out the existence of another group represented by an equation of the same form, if (m + ½) is substituted for m. As a special case (b1 = 0) this group contains the second Hydrogen series discovered by Professor Pickering in the spectrum of ζ Puppis. Subsequent investigations convinced me, however, that, although a considerable number of line-series may be classified into these two groups, there are numerous instances where the more general formula

must be employed, in which μ represents various fractional numbers.