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.

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.

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.

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.

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 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.

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.