My first duty is to express my cordial thanks to the Council and Fellows of this Society for electing me as President in succession to the late Professor Geikie. I have accepted the honour with great reluctance, for I am conscious of the fact that others have far stronger claims, and have waived their claims under very exceptional circumstances. But I may take this opportunity of assuring the Fellows that since they have been pleased to confirm the recommendation of the Council, I shall take an active interest in the affairs of the Society and do my best to promote its welfare.

The list of papers read during the past session shows that important contributions relating to different branches of science have been made to the Society. Notwithstanding the strain caused by the war, the output of original work is noteworthy. In this connection reference may be made to some researches having a direct bearing on the war. Last session the Council appointed a Committee, composed of leading scientists, to conduct investigations in connection with submarines, aeroplanes, asphyxiating gas, and high explosives.

It is well known that, when examined with a modern high-power spectroscope, the bright lines in the spectra of the different elements exhibit an individual character. Some are diffuse, some are sharp, some are diffuse on the one side and sharp on the other, and some are accompanied by fainter lines in their neighbourhood. To the latter the name of satellites has been given. The most celebrated satellites are those of the green line of mercury, which have been investigated very often owing to the ease with which the mercury spectrum can be produced. They are much fainter than the main line, and at least six can be seen. But there are cases occurring in which a satellite is so strong that it is hardly possible to say which is the main line. Thus the red line of hydrogen is a close doublet with a separation of ·14 A.U., the two components of the doublet being of approximately equal brightness.

Emile-Hilaire Amagat was born in 1840. When Professor of Physics at l'Ecole Normale at Cluny, he began in 1867 his investigations of gases under high pressures. Later at Lyons, where he was Professor at the Catholic University, he utilised the tower of one of the churches as the site for a manometer giving pressures up to 80 atmospheres, and later constructed in one of the coal mines of St Etienne a Boylean tube measuring up to 430 atmospheres. In some of his experiments he applied with great success a suggestion of Tait's for recording volumes by means of mercury coming in contact with platinum wires at different heights in the tube. His results in the gaseous laws for hydrogen, nitrogen, and carbonic acid and other vapours are recognised as the most authoritative we have.

In a recent communication (Proc. Roy. Soc. Edin., 1914–15, vol. xxxv, p. 22) measurements of the optical rotation and cryoscopic behaviour of the following sugars, viz. β-l-arabinose, l-xylose, α-d-glucose, α-d-galactose, d-mannose, d-fructose, α- and β-lactose, dissolved in (α) formamide, (b) water, were described. The investigation has been carried a step further by the addition of β-d-ghicose, β-d-galactose and maltose to the above list. Thus the change in rotation or mutarotation in formamide solution has now been measured, starting from both the α and β-form of d-glucose, d-galactose and lactose. As in the case of the water solutions of these sugars, the constant rotation shown when there is equilibrium between the two modifications is found to be the same whether the startine-point be the α- or the β-modification.

When an elastic body is constrained in any manner whatsoever, it is susceptible to vibration, by virtue of its elasticity, when disturbed from its position of equilibrium by an externally applied force. The period and amplitude of such vibration are dependent upon the mass and inertia of the system, the rigidity of the constraints, and upon the nature of thedisturbing force.

When a beam of commercial section is loaded centrally, and subjected to vibrations, the frequency of transverse vibration can be readily determined from a knowledge of the dimensions of the beam, its modulus of elasticity, and the conditions of loading. On the other hand, where the loading is eccentric, the transverse vibration is accompanied by a torsional vibration the frequency of which is very much lower than is indicated by the ordinary elastic theory, due to the inefficiency in torsion of beam sections other than circular. In practice it is not always possible to eliminate the eccentric loading of beams, as for instancewhere power is transmitted through countershafts supported from structural steel-work, and it is with the problem of the torsional vibration of such eccentrically loaded beams that itis proposed here to deal.

The literature of the sugars would appear to show that no sugars have hitherto been observed to sublime with the exception of glycolose, CH2OH CHO, which is described as being “perceptibly volatile with water and alcohol vapour under diminished pressure, especially from a pure, concentrated solution” (Lippmann, Chemie der Zuckerarten, p. 4). This substance, being the first of the sugar series, might be expected to have properties somewhat different from the other members of the series, which are usually looked upon as non-volatile. Experiments with rhamnose and fructose appear to show that under diminished pressure they do sublime.

A new method for the mechanical analysis of soils and deposits has been developed. It consists in weighing, from time to time, on a specially constructed balance, the amount of sediment accumulated on a circular disc suspended from the balance near the bottom of a vessel containing an aqueous suspension of the sample. From the “accumulation-curve” thus obtained one finds by a series of mathematical operations a “distribution-curve,” i.e. a curve showing how the amount of particles of a certain size varies with the latter quantity.

This method has been applied to the study of some deep-sea deposits from the Challenger Office. The distribution-curves thus obtained show marked differences for the different samples, and reveal a surprising lack of very fine particles in the deposits from the largest depths, besides suggesting interesting problems for new research.

Geologists who have made a special study of petroleum and the conditions under which it occurs have not infrequently to deal with the kindred subject of oil-shales. These subjects, it is true, have always been considered as entirely separate and different, and for all practical purposes they are so, since the mining of oil-shale and the drilling for oil have nothing in common from the engineering point of view; yet the final products of the distillation of natural petroleum and oil-shale are to some extent identical, and it has more than once been suggested that some relation may exist between an oil-rock and an oil-shale.

Colin Maclaurin, the celebrated mathematician, was born in 1698 at Kilmodan in Argyllshire, where his father was minister of the parish. In 1709 he entered Glasgow University, where his mathematical talent rapidly developed under the fostering care of Professor Robert Simson. In 1717 he successfully competed for the Chair of Mathematics in the Marischal College of Aberdeen University. In 1719 he came directly under the personal influence of Newton, when on a visit to London, bearing with him the manuscript of the Geometria Organica, published in quarto in 1720. The publication of this work immediately brought him into prominence in the scientific world. In 1725 he was, on the recommendation of Newton, elected to the Chair of Mathematics in Edinburgh University, which he occupied until his death in 1746.

The following note presents in a simpler form the essence of Dr Sven Odén's mathematical discussion of the fall of small particles through a column of liquid, as given in his paper (immediately preceding) “On the Size of Particles in Deep-sea Deposits.”

It has the further advantage of solving the problem without taking account explicitly or implicitly of Stokes's law (or modification thereof) as to the relation connecting the time of fall with the size, form, and density of the particles considered. It is important, I think, to refrain as long as possible from making the assumptions involved in such a law of fall, and to recognise how far we may carry the investigation before introducing these assumptions.

My last paper on the “Theory of Circulants” appeared four years ago (Proc. Roy. Soc. Edin., xxxii, pp. 136–149), and brought up the history of the subject to 1880. I have now completed the research up to the close of the nineteenth century—the goal originally aimed at. It is considered that from the beginning of the present century the indexing of mathematical writings is sufficiently full and detailed to make it comparatively easy for a worker to follow up for himself the recent lines of development.

In the course of a systematic inquiry, conducted in the Physiology Department of Edinburgh University, into the morphological and physiological changes produced in animals by thyroidectomy and thyroid-feeding—especially in rats—certain striking morphogenetic changes in the pancreas have come under observation. The full results of the investigations of which these observations form a part will be published subsequently, but it has been thought well to bring the effects herein described before the notice of this Society without waiting for the completion of every part of the investigation.

The value of the continued-fraction, as a means of representing an analytic function f(x), is now fully recognised. Other representations, such as power-series or Fourier series or Dirichlet series or series of inverse factorials, converge in general only over limited regions of the x-plane, and fail to converge over the rest of the plane, whereas the representation of the function by a continued-fraction converges (in a large class of cases at any rate) over the whole x-plane, with the exception of certain singular curves.

In 1900 I communicated to this Society a paper on the above subject. Since that date a great deal of information has been obtained by means of free balloons carrying instruments which recorded the temperature, humidity, and pressure of the air up to great elevations. Much of this new knowledge seems to contradict our previous ideas, and does not seem to fit into the old convectional theory that cyclones are formed by the rising of the hot, moist air from the surface of the earth; their energy being due to their temperature and to the heat liberated by the condensation of the water vapour in them. We are told by those who have studied the bearing of the new knowledge on our atmospheric circulation that the old theory is “utterly untenable.” Their reasons for this conclusion are, first, that the recent investigations show that the air is colder in cyclones than in anticyclones; second, that the isothermal layer is lower than the mean over cyclones, while it is higher than the mean over anticyclones. At first sight these discoveries seem to shatter the convectional theory, but before we come to any conclusion I should like to present certain facts which it appears to me will require to be considered before we scrap our old ideas.

The variation in the sugar content of the blood in certain pathological conditions has created the need for modes of estimation which require only a small amount of blood and at the same time give sufficiently reliable results. The more recent methods of estimation fall under three types: (1) those in which the method of Rona and Michaelis for the precipitation of the protein is combined with Bertrand's method for sugar estimation, (2) MacLean's more recent method, and (3) the “micro-method” of Ivar Bang.

The earthquakes of Scotland, as compared with those of England and Wales, are notable on several accounts:—(i) They are more numerous. During the twenty-six years from 1889 to 1914, 357 earthquakes riginated within the area of Great Britain–51 in England, 27 in Wales, and 279 in Scotland. If area be taken into account, the inequality is still greater. For every shock occurring in England, there were during this period four in Wales and nine in Scotland, (ii) They are invariably simple earthquakes, twin earthquakes (which include the strongest of all British earthquakes) being confined to England and Wales, (iii) As simple earthquakes, they are due to movements along strike-faults, and, for the most part, along well-known faults, the majority of which belong to the Caledonian system, the only exceptions known to me being those connected with the Loch Broom earthquake of 1892 and the two Glasgow earthquakes of 1910.

The lavas of the Scottish Carboniferous are predominantly basaltic. True andesites and rhyolites are conspicuous by their absence, whilst trachytes and allied rocks are present in quite subordinate quantity. This association of basalt and trachyte is in accordance with the fact that the basalts have a slight alkaline cast, which is evidenced by an alkali content greater than that of the average basalt, and by the occasional presence of nepheline and analcite amongst their constituents. The transition from basalts to more acid types is accomplished, not by way of the andesites, but through the mugearites, volcanic rocks with chemical affinities to the essexites. All transitions from the more felspathic types of basalts, the Jedburgh and Markle types, can be traced through mugearites to trachytes or allied rocks.

In a paper* read before this Society in 1889, Captain Weir gave a diagram for solving graphically, by means of a chart and parallel ruler, the polar spherical triangle whose vertices are the pole, zenith, and a star, so as to obtain the Azimuth when the declination, hour-angle, and latitude are known. The purpose of this note is to show that this diagram is in reality equivalent to the later straight-line nomogram, and to deduce a similar diagram to Weir's for another problem in spherical trigonometry.

§1. It is well known that Laplace's equation ∇2V = 0 possesses certain classes of solutions of the form V = F(X, Y) where F satisfies a partial differential equation with X and Y as independent variables and X and Y are real functions of x, y, and z. Such solutions have been called binary potentials. They are of some interest in the problem of finding cases in which the equations of motion of an electron in a steady magnetic field are readily integrable. There are also classes of solutions of the same type, except that X and Y are complex quantities and the coefficients in the partial differential equation for F may also be complex. In particular, there are solutions of the form V = Yƒ(X) where ƒ is an arbitrary function with continuous second derivative.