The object of the present paper is to introduce certain surfaces, which will be shown to play the same part in the general electromagnetic field as Faraday's tubes of force do in electrostatic and magnetostatic fields.

The value of Faraday tubes in electrostatics and magnetism has led many investigators to seek for them a wider application, in connection with variable electromagnetic fields.

In a paper on “The Scattering of Light in the Refractive Media of the Eye” published in the Philosophical Magazíne, for November 1919 (p. 568), I discussed the explanation of the luminous effects observed when a small brilliant source of light is viewed directly by the eye against a dark background, and especially of the marked difference between the cases in which the source emits white light and highly monochromatic light respectively. In both cases the source appears to be surrounded by a diffraction-halo: but the structure of the halo is markedly different in appearance.

Prehensility is defined as the slope at the origin of an adsorption isotherm.

A method of measuring prehensility is described, and results given for various adsorbents at liquid air temperature.

It is shown that from the prehensility the evacuating power of a substance may be calculated. In evacuating any given volume, the weight of charcoal required to yield a required reduction of pressure may be computed. The degree of vacuum obtained in a Dewar liquid-air container is discussed.

The plumstone charcoal used by the author had a higher evacuating power than cocoanut charcoal.

Reference is made to a colloidal silica of appreciable evacuating power, though, at –190° C. over four times as much of it is required (by volume) as of plumstone charcoal to attain the same result, and it acts more slowly.

The very high degree of vacuum procurable by using a succession of charcoal bulbs is discussed, and it is shown that with a given weight of charcoal the reduction of pressure obtainable by division of the mass among a number of bulbs does not indefinitely increase with that number, but eventually reaches a maximum.

It became evident during the War that some men who are organically sound are nevertheless incapable of supporting heavy exertion; and among the older groups that were called to the Colours in 1918 the proportion of such individuals was considerably more. The effect of forcing these men to carry out the duties falling to the lot of an “A” recruit was in many cases harmful to them, and in extreme instances resulted in permanent disability and sometimes in death. The Army Medical Department therefore sought to obtain a method of physical examination, which would supplement the ordinary medical examination of a recruit, and which could be applied to men who proved to be unserviceable material in the hands of the drill sergeant—a test which would, in fact, enable the malingerers to be distinguished from those who were truly incapable of sustained labour.

This paper summarises the results of an attempt to extend the theory upon which the relationship between linear differential equations and integral equations is based. The case in which the nucleus K(x, s) of the integral equation arises as a Green's function is well known; the nucleus is there characterised by its having discontinuous derivates when x = s. The method here dealt with is virtually an extension of Laplace's and analogous methods for solving linear differential equations by definite integrals, and leads to nuclei which are continuous and have continuous derivates for x = s.

The Myriapods recorded in this paper were collected in Mesopotamia—mostly at and around Amara on the Tigris—and in the neighbouring uplands of N.W. Persia, by W. K. Evans, B.Sc., Edinburgh, late Captain R.A.M.C.

They are referable to seventeen forms: of these eight are new and three could not be identified owing to want of appropriate material. Amongst the novelties two specimens deserve special mention as belonging to an lulid group, the known representatives of which are very few. The new genus Calyptophyllum has been erected for these.

During the past four years I have been conducting surveys of the colour vision of students in the University of Glasgow. The first survey was made by a colour-perception spectrometer very similar to Dr Edridge-Green's instrument, and embraced 79 observers. The second survey was made by Dr Edridge-Green's bead test, and embraced 100 observers. The third survey, carried out in collaboration with Miss Margaret A. Dunlop, was made by an original method, called here for short the microscope test, and embraced 1000 observers. At present there are two other surveys under progress. The object of these surveys is to find a numerical method of specifying goodness of colour vision; to see, by the application of statistical methods, whether the colour blind fall naturally into groups or are merely outliers of a homogeneous population; to find whether colour blindness is a Mendelian characteristic for men and merely an extreme case of normal variation for women; and to throw light on the subject of colour vision generally. Consequently, the normal have been investigated with as much care as the colour blind. But in the course of the four years I have made the acquaintance of many trained observers with abnormal colour vision, and have been possessed with an ever-growing desire to know exactly, irrespective of all theory, what was the matter with their colour vision. In spite of the vast literature on the subject, the tests generally have been of a very superficial nature, and unsatisfactory to the man with mathematical instincts. As these abnormal cases were beginning to leave the University, I addressed myself last spring to the problem of finding a method of testing which would describe their condition independent of theory, and, indeed, independent of words. This paper describes how the problem was solved, and gives data for twenty-three cases of colour blindness, four of normal colour vision, and one case of exceptionally good colour vision.

In Whittaker and Watson's Modern Analysis, chap, xvi, the asymptotic expansion of the confluent hypergeometric function Wk.m(z) is established for the region—π < amp z < π, z ≠ 0. The object of the first part of this paper is to show that this expansion is valid in the extended region –3π/2 < amp z < 3π/2, z ≠ 0.

In 1890 I published a theory of ferromagnetic induction in which it was suggested that the equilibrium of Weber's elementary magnetic particles was due only to magnetic forces. It was shown that when the elementary magnets are made to turn by applying a magnetising force which is progressively increased, the conditions of equilibrium must be such that there is first a small amount of stable (reversible) deflection, then a break away with irreversible deflection into a new position of stability, and finally a reversible approach to the position of complete parallelism which corresponds to saturation. A model was constructed showing that these conditions could be satisfied by a purely magnetic control. It was made up of little magnets, pivoted on fixed centres which were uniformly spaced. The magnets were all free to turn, but controlled one another by their mutual magnetic forces. They tended to form rows, and when an external field was applied the rows broke up and fresh rows were formed more nearly in the direction of the field. With this simple model the known characteristics of the magnetising process were, qualitatively, well reproduced. A recent study of the stability of such rows of magnets has, however, led me to abandon this model, and to design instead a model in which each atom forms a magnetic system comprising a Weber element capable of turning, but controlled by the magnetic forces exerted on it by other parts of the atom which are taken as fixed. As in the old model, the control is wholly magnetic. Various forms of the new model will be described in a later part of this paper, but in the first place it may be useful to give some account of the investigations of stability which have convinced me that the old model fails, quantitatively, to represent the process.

It is now well established experimentally that when an atom is caused to emit radiation of frequency v by collision with an electron, the amount U of the kinetic energy of the electron which is absorbed by the atom is given by the equation

where h denotes Planck's quantum of Action : an electron whose kinetic energy before the encounter is less than hv is incapable of stimulating the atom to emit the radiation, and is merely repelled from the atom without any loss of energy.

In Professor Whittaker's highly interesting and suggestive application of my model the postulated structure is a system of magnets rigidly connected like the spokes of a wheel, having poles of the same name at the circumference, the corresponding poles of contrary name being all at the centre. He shows that when a bombarding electron, moving in the direction of the axis, approaches such a wheel it sets the wheel turning, the rotation of the poles constituting what he calls a “magnetic current.” If the initial velocity of the bombarding electron falls short of a certain limiting value, the encounter is perfectly elastic, and the electron is reflected back without loss of energy; the point at which its direction of motion becomes reversed may be on either side of the plane of the wheel, depending on the electron's initial velocity of approach. If the initial velocity exceeds the stated limit, the electron passes clear through the system and escapes, but with a definite reduction of velocity, corresponding to a quantum of energy which it has given up to the wheel. When the encounter is elastic, the angular velocity which the wheel acquires during the approach of the electron is lost during the return of the electron. When the encounter is not elastic, and the electron has passed clear through, the wheel is left with an angular velocity ω; and the energy which that represents is the source of the consequent radiation.

The characteristic properties of quadratic differential forms, involving two or more independent variables, have been investigated from the days of Gauss onwards. Initially, the discussion arose for the case of two variables; and, in its most general trend, it was concerned with a form

associated with surfaces, E, F, G being integral functions of p and q. But the relation does not, by itself, define a surface completely. When a surface is deformed in any manner, without stretching and without tearing, the quantity ds2 preserves its measure unchanged; the measure is of fundamental importance. Consequently, the measure must remain unchanged whatever changes of the variables are admitted. Further, changes of the variables, of any kind, allow the existence of covariant concomitants which therefore persist through these changes. In particular, there is one function, of E, F, G and of their derivatives up to the second order inclusive, which persists unaltered; it is the Gauss measure of the curvature of the surface.

Calamoids—the four-dimensional tubes of electromagnetic force introduced by Professor Whittaker in a communication in November 1921—are not referred to in his work “On the Quantum Mechanism in the Atom” (see pp. 129–142), in which he is content to deal with three dimensions; but there is at least one point of contact between the two papers. The four-dimensional tubes include as particular cases both kinds of Faraday tubes (electric and magnetic), and magnetic forces are treated as on an equality with electric forces. In seeking for a mechanism to elucidate the quantum, a magnetic model is employed; and although Professor Whittaker is careful not to insist on the magnetic structure suggested, careful examination of his paper shows that some form of magnetic element is required both for the absorption of energy by the atom from the electron and for the transformation of the absorbed energy into the radiant form.

It seems desirable to check Professor Whittaker's model numerically; algebraical expressions when evaluated sometimes have the wrong order of magnitude.

Let us suppose that initially the magnet system is at rest, and that the electron starts from the origin with velocity u.

The atom, even in its simplest form, must possess a most complicated structure. The aim of physical science is to find a hypothetical structure, and mode of its functioning, which will give a good account of atomic phenomena. When the phenomena are complicated, and only very partially known, hypothetical investigations are most uncertain and provisional, and make little attempt to work otherwise than by analogy or illustration. In such cases help is often obtained through the discovery of a general law which unifies a wide range of phenomena, and which may be followed by many diverse structures in their activities. Well-known examples are found in the law of conservation of energy, the law of least action, and, recently, the law of quantum action.

The method of describing a field of force by means of lines or tubes of induction, which originated with Faraday, was given a quantitative form by Sir J. J. Thomson, and further discussed by N. Campbell in his book Modern Electrical Theory. Since Maxwell himself looked on his work as a mathematical theory of Faraday's lines of force, one is tempted to examine the original physical theory for hints as to the modification of the Maxwellian theory to suit certain modern requirements.

It is proposed in this paper to consider the spectral lines which would arise in the case of an electron describing an orbit about two centres of attracting force. The assumptions of the Quantum Theory as generalised by Wilson and Sommerfeld are made. The centres of attracting force are assumed, as in the simpler applications of the Quantum Theory, to be at rest. Such a spectrum would arise in the case of a hydrogen molecule which had lost one electron and in which the remaining electron described an orbit about the positive nuclei. Novikoff has shown that this configuration would be stable (vide Whittaker's Analytical Dynamics, 2nd ed., p. 407, ex. 1). It follows that, if at any stage an electron were withdrawn from a molecule, the remaining electron would describe a stable orbit. Much experimental work has been done on the secondary spectrum of hydrogen, but I am not aware that any theoretical explanation has been given of the immense number of lines found by experiment. Dufour in Ann. de Chimie et de Physique, 1906, gives a very complete account of the work done on this spectrum and the various controversies which it has evoked. Dufour's conclusion is that the spectrum of the atom is the primary spectrum, and that the secondary spectrum is due to the molecule. H. E. Watson in Proc. Roy. Soc., vol. lxxxii, 1909, states the main points at issue, and gives the most modern measurements of the lines and compares his results with wavelengths found by Hasselberg (Mém. Acad. Imp. St Petersburg, vol. xxxi, 1883), Ames (Phil. Mag., vol. xxx, 1890), and Frost (Astrophysical Journal, vol. xvi, 1902).

Before the advent of the aeroplane and airship the gyroscope derived its practical importance from its applications to the torpedo and the gyroscopic compass, and neither of these applications originated in this country. With the introduction of aircraft the applications of the gyroscope become very numerous and of great importance. Navigation of such craft must be made precise by the provision of gyroscopic sextants; photography from aeroplanes, to be accurate, must be carried out by means of special cameras stabilised so that the photographs are true vertical productions; war aeroplanes must be steered and controlled in accordance with accurate dynamical principles, and in this respect the claims of the gyroscope cannot be ignored; bombing from aeroplanes must be rendered accurate by designing the bombsight as part of an accurate stabiliser; battleships must be provided with anti-aircraft devices controlled by stabilisers, and the guns on such ships must be controlled both with respect to the vertical and the horizontal, and in azimuth, by means of instruments of the greatest precision.

In seeking the expansion of the equation of the centre, and the logarithm of the radius vector in a planetary orbit, in terms of the mean anomaly, the following expressions are obtained:—

where g is the mean anomaly, and

Mr Innes remarks: “It is shown for the first time, I believe, that the expansion for the true anomaly in terms of the mean (equation of the centre) can be derived quite easily from the addition of the two simple expansions

and the two series have most simple recurrence expressions.”

The MS. containing the proofs of these expressions, with further developments, is in the hands of the Society.

In an earlier note communicated last session the action of self light and fatigue was partially considered. The question of the origin of these effects is one which can only be settled by observation or experiment. The question of the mathematical form of the laws regulating the effects also requires an observational or experimental basis. In the absence of self light, Fechner's well-known law that the change in sensation is proportional to the fractional change in the stimulus forms a close approximation to results of experience throughout a wide range of intensity. Fechner's modification of this law, by the addition of a constant term to the denominator of the fraction, was shown by Helmholtz to account broadly for the action of self light; and he showed also that the introduction of three such constants into three such independent fractions enabled a good description to be given of various phenomena of colour vision in light of different intensities. But, from the phenomena of vision, it is certain that the numerical values of the parameters are influenced by antecedent as well as present illumination. Thus the parameters are really functions which are variable in more than one way, and the form of these functions can only be found by observation. This is the subject of the note referred to above.