The principle of similar structures was first enunciated by Galileo. Applying it mechanically, it appears that the strength of a structure varies as the square of the linear dimensions, and the weight as the cube. This principle and its mechanical applications have been widely illustrated by reference to the bodies of animals, and many of their peculiarities are necessary consequences of its effect in the course of their evolution. For instance, the columnar legs of the elephant or of the moa are held to be the inevitable sequel to the large size and consequent weight of those animals, while the thin arched legs of insects are only possible where the body itself is small and light.

§ 1. When solved for the second derivatives, the Lagrangian equations of motion for a system in which there are no extraneous forces have the form

or, disregarding the parameter t,

being the second Christoffel symbol † of the matrix associated with the kinetic energy. If there are extraneous forces Fk and, denoting t by xn+1, we add to the set of equations, the equations of motion are

Extreme supporters of the principle of relativity find no place for an æther. Thus Lord Haldane in discussing the theory of Einstein writes: “Space, as a physical thing with unvarying geometrical properties, is to be banished, just for the same sort of reasons as the æther was banished before it. Only observable things are to be recognised as real in the new system of modern physicists.” On the other hand, many followers of Faraday and Maxwell have regarded æther as the primary real substance; all mass, momentum, or energy being mass, momentum, or energy of the æther. Without entering into a discussion of the significance of physical “reality,” we may, with most physicists, agree to use the concept of æther as giving a model, inadequate though it is at present, for the interpretation of physical phenomena.

The four transverse galvanomagnetic and thermomagnetic effects were determined in the case of iron, nickel, cobalt, silver, copper, zinc, cadmium, aluminium.

The results were considered in the light of the electron theory of conduction in metals as formulated by G. H. Livens, and were found to be in fair accord with the theory as regards the ratios of the effects, but they could not be reconciled with the expressions given for the effects themselves.

It is generally accepted that the position and development of any one tissue in a plant is to be studied from the point of view of function: nevertheless the physiological need for it may alter as the plant develops. The endodermis is an illustration of this. Its primary function (1) is to restrict the passage of water and soluble substances to certain definite channels: it follows that its best development is in those parts nearest to the source of supply, e.g. in roots and young stems. The cell-walls are at first relatively thin, with the characteristic suberised strip on the radial and transverse walls. Its cells are in uninterrupted contact one with another, there being no intercellular spaces. This leads incidentally to its being also a barrier to gaseous interchange. The ventilating system of the cortex is limited by it internally, and cut off from such intercellular spaces as may lie within.

Fechner's law states that change of visual perceptivity is proportional to the fractional change in the intensity of the light. At weak intensities a term, regarded as constant, has to be added to the intensity of the external light on account of the self-luminosity of the eye. By integration over the whole stimulated part of the retina, Helmholtz obtained an expression for the perceptivity which agreed with observation in so far as the general nature of the relation between perceptivity and external stimulus is concerned. Close correspondence can be obtained by assuming that the self-luminosity term in Fechner's expression is itself a simple function of the external stimulus, rising rapidly to a maximum and thereafter slowly falling to a steady value.

Early investigators regarded the soil as an inert framework of soil grains of various sizes covered with a continuous film of water, and the properties which in theory it should possess under such a hypothesis were found not to accord too well with the results of experiment. But when the existence of soil colloids was understood, the differences between theory and experiment tended to disappear one by one, as shown by the work of Bouyoncos in America in 1915 and of Keen in England in 1914, 1919, and 1920.

The shear influence line is a line the ordinates of which give the values of the shearing forces at any one point in a beam or bridge as a load, or a series of loads, pass over it. There is thus, for any one beam, an influence line for every point in it.

This memoir is devoted to the study of certain new functions, which may be regarded as limiting cases of the “hypergeometric functions of two variables” discovered by Appell.

The experiments here described were made on behalf of the Oxygen Research Committee of the Scientific and Industrial Research Department, and the paper is given by permission of the Department. The fullest acknowledgment is due to Dr J. A. Harker, F.R.S., and to his co-workers, Professor G. W. Todd and Mr S. H. Groom, who have, in a series of able memoirs presented to the Oxygen Committee, analysed the nature of the heat-transfer from the outer atmosphere to the interior of metal vacuum bottles; but for their memoirs the writer's experiments would not have suggested themselves.

(1) In a “Note sur une formule d'intégration indéfinie” of the year 1874, Cayley has occasion to use a determinant of a very peculiar structure, whose value when of the (n + l)th order is

[a]r in the development of this standing for

PART IV.—This paper concludes the authors’ account of the Vascular Cryptogams found in the Rhynie deposit. Restorations of the four plants, Rhynia Gwynne-Vaughani, R. major, Hornea Lignieri, and Asteroxylon Mackiei, are given. A few additional features, supplementary to the descriptions of these plants in the preceding papers of the series, are described and illustrated. The hemispherical projections of Rhynia Gwynne-Vaughani are shown to have originated underneath stomata. A comparison is made between them and certain intumescences in existing plants. Areas of necrosis and marked wound-reactions of the tissues around them are described for both species of Rhynia. The apex of a stem of R. major is figured. For Asteroxylon additional figures are given of a large rhizome, of the leaf-arrangement and immature structure of the stem in the region of a shoot-apex, and of the longitudinal markings on the epidermal cells resembling those found in Rhynia Gwynne-Vaughani. The discussion summarises the authors’ views on the main bearings of the facts described in Parts 1–4 on various problems in plant morphology.

PART V.—The Thallophyta occurring in the peat bed; the succession of the plants throughout a vertical section of the bed, and the conditions of accumulation and preservation of the deposit.

1. Experiments made at 15° C. with various adsorbents and gases (chiefly nitrogen and hydrogen) show that a cylinder filled with adsorbent granules has a capacity for dry gas under a given pressure which is generally greater than its capacity when containing no adsorbent. For example, a cylinder charged with nitrogen at 35 atmospheres has its capacity increased by 66 per cent. by filling it with cocoanut charcoal.

2. Sudden outbursts of firedamp in coal-mines are the result of releasing immense quantities of gas adsorbed under pressure in coal.

3. The logarithmic relation derived by Williams is shown to apply to the adsorption isotherms of gases above their critical temperature up to pressures of 100 atmospheres, providing a correction be applied for the gas in the capillaries which is not adsorbed, but which exists under simple compression.

In the course of research on Mine Rescue Apparatus under the Department of Scientific and Industrial Research it was found necessary to investigate the conditions of utilisation of solid caustic soda for absorbing carbon dioxide, and particularly the effects of variations of temperature and water vapour upon the reaction.

The conditions determining the stable flow of a viscous fluid in a uniform channel and in a uniform circular pipe were investigated experimentally for water by Osborne Reynolds in a searching series of papers.

The surface of the Queensferry Road, from about the bend above Blackhall, past Marchfield, and on towards Cramond Bridge, was remade in the spring 1919.

It was made in the modern fashion for a motoring road, with road metal and liberal supplies of bitumen, and small pieces of quartz scattered on top of the bitumen, the whole being rolled by steam roller.

The following investigation was undertaken to verify by the American Army Tests a result previously obtained by the use of Mr Cyril Burt's tests of intelligence (Journal of Experimental Pedagogy, vol. vi, No. 1), i.e. that there appears to be some likelihood that boys born in the spring months are slightly less intelligent on the average than those born in other months. Of these American tests (see Mental Tests in the American Army, C. S. Yoakum and R. M. Yerkes, pp. 220–230) the groups of tests marked “form 6” were used; but “test 8” was modified to suit British conditions by using test 8 of group 9 (p. 274, loc. cit.) as a basis, and replacing tests 1, 3, 7, 8, 9, 11, 13, 19, 35, 36, 38 by 8,1; 6, 3; 7, 39; 8, 39; 6, 35; 7, 36; 7, 38; 6, 39, and by three others of local interest.

A Pith ball suspended from the centre rod of a charged Leyden jar provides us with an interesting and very delicate method of demonstrating the ionised atmosphere surrounding flames and incandescent bodies. In an ionised atmosphere of sufficient intensity the ball, provided that the suspending fibre is of high insulating quality, rapidly loses its charge and falls back toward the centre rod of the jar, from which it receives a fresh charge and is violently repelled, only to repeat the performance; thus an oscillation is set up the rapidity of which is governed chiefly by the length of the suspending fibre.