The object of the present paper is the geometrical study of the groups of rotation and reflexion of the regular polytopes in higher space, and the extension to these configurations of known results in the cases of the ordinary regular polyhedra. It will appear from the work that the groups can be defined abstractly in terms of a certain number of operations, the relations connecting which have a particularly simple form. For a polytope in n dimensions this number is n − 1, if we consider the group composed simply of the rotations of the polytope, while if we consider the extended group, which includes the possible reflective symmetries of the polytope, n operations suffice. It appears, further, that with one exception all the groups so obtained possess the property that their operations are expressible in terms of two, so that the entire group can be generated by two operations. The relations connecting these, however, are in general complicated, and the symmetrical forms involving more operations are more convenient to use.

An abstract continuous group, G, is an abstract space whose points have a continuous associative multiplication law, with division. The object of this note is to sketch a proof, which will appear in full elsewhere, that if the space G is locally Cartesian and compact, and the group G is Abelian (commutative), then G is the ordinary closed translation group in n dimensions: i.e. the space is that obtained from the “cube”

1. In a note in these Proceedings, Mr S. Verblunsky proves the following result:

Theorem A. Let C be a continuous arc joining the points (x1, y0) and (x2, y0), where x2 > x1. Suppose that C does not cross the line y = y0. Then, given any positive h < x2 − x1, there is α yh such that some two points (ξ1, yh), (ξ2, yh), in which y = yh cuts C, satisfy the relation ξ2 − ξ1 = h.

The application of the operational method of Oliver Heaviside to the solution of linear differential equations has been fully described in a recent Cambridge Tract by Dr H. Jeffreys.

Configurations of points in higher space have been known and studied for some time. In what follows we shall classify those configurations of points lying on the hyper-sphere in [4] which may be said to possess the group property. In this classification we shall seek to enumerate only those configurations which are essentially different from this point of view, in order to bring out any geometrical differences which might throw light on the fundamental problems of the theory of linear groups.

1. The perturbations of a focal elliptic orbit are as easily found by Quaternions as the orbit itself.

The following considerations form the basis of the work on generalised integrals with which I have been engaged for some years. Their intimate connection with Alexander's new notation for combinatory topology encourages me to publish them separately.

A mechanical method of integrating a second-order differential equation, with any boundary conditions, is described and its applications are discussed.

1. It has been shown, in a well-known paper by Born and Oppenheimer, that in evaluating the energy levels of a molecule, the nuclei may be treated as fixed, or slowly vibrating, centres of force. If M be the average mass of the nuclei, and m the mass of an electron, the error involved in doing this is of order of magnitude (m/M)½. The possibility therefore of treating the nuclei as fixed depends on their great mass, compared with that of the electron.

1. This note is a sequel to a previous one. Our first object is to rectify some mistakes in I. For these corrections I am indebted to the kindness of Prof. J. R. Wilton. In the enunciation of Theorem A (I, p. 159) the equation

should be altered to

The mistake occurs at the end of I (p.169) where it is erroneously assumed that

the correct being

1. Let ℱ be a family of functions X (t) defined in (0, 2π). The functionals to be considered are of a particular class, namely those which define a correspondence between each function X of ℱ and a number which we denote by F(X). It is convenient to refer to F(X) as the functional.

When we treat an atom containing a number n of electrons by the method of the self-consistent field, we assume that each electron has its own particular “orbit,” specified by a wave function (q|r) in four variables q. These four variables are usually taken to be the three coordinates of the electron together with a variable describing the spin, but according to the transformation theory of quantum mechanics, they may be any four independent commuting functions of the coordinates, momenta and spin variables.

Let (X1, …, Xn) be the coordinates of the centre of a unit cube C, in n-dimensional space, whose (n − 1)-dimensional faces are parallel to the axes of coordinates. Further let the X's be integers. Let εi = ± 1 for r (≤ n) values of i, 1 ≤ i ≤ n, and let εi = 0 for the remaining n − r values of i. Then (X1 + ε1, …, Xn + εn) gives the centre of a cube C′, which touches the cube C along an (n − r)-dimensional edge or face. The cube C and the cubes C′, for all possible arrangements of the ε's, which are subject to the above conditions, form a symmetrical arrangement of cubes. This paper discusses the possibility of completely filling space by means of the packing together of such sets of cubes.

The problem of the transmission of power over a transmission line from an alternating current generator at one end to a synchronous motor at the other is easily solved when the line is uniform or made up of a small number of uniform line segments. The object of the present note is to apply the theory of integral equations to the general case of the non-uniform line. The solution obtained for the current at any point on the line in the form of a rational function of the frequency of the two alternators is of advantage when resonance effects are being considered; the latter effects are hard to trace in the case of the uniform line by the usual methods. It seems probable that numerical methods and computing devices can be developed for the solution of the integral equations given. The effect of armature inductance on the power limit of the system may then be more easily studied than at present.

1. A locus Vn, of dimension n, in [2n + 1], for example a curve in [3] or a surface in [5], has ∞ 2n chords, of which a finite number pass through a general point of the space.

The magnetisation of a single crystal of a ferromagnetic substance is accompanied by a distortion of the crystal which depends on the intensity and direction of magnetisation. Webster has shown that the magnetostriction in unsaturated states may be accounted for in an entirely satisfactory manner by making the assumption (for which the evidence is entirely convincing) that at ordinary temperatures an apparently unmagnetised crystal consists of small regions magnetised to saturation in various directions, so as to give no resultant magnetisation to the whole crystal, and that these regions possess natural directions of easy magnetisation.

This paper deals with the calculation of the inter-atomic force at large distances for the cases of hydrogen and helium, and is based on the method used in calculating the polarizability of helium.

1.1. In a linear space Sr, of r dimensions, we may consider involutory transformations determined by linear systems of primals whose freedom is r and whose grade, that is, the number of free intersections of r primals of the system, is two; we shall be concerned only with complete systems, systems of all the primals (of given order) satisfying certain fundamental linear conditions, for example, containing a point, or a curve, or touching a plane at a given point. The primals of such a system Φ that pass through a point P form a system ∞r−1; any r linearly independent primals of this subsidiary system will meet in one other point Q outside the base elements, and Q is common to every primal of Φ through P; further, all the primals of Φ that contain Q must contain P. We thus have an involutory transformation of the space Sr.

Experiments on the emission of secondary electrons from hot nickel surfaces bombarded by Cs+ ions at normal incidence show that the emission is a function of bombarding voltage and target temperature. The emission is negligible below 300 volts and increases steadily with further increase of the bombarding voltage, rising to 4% at 4000 volts with the target at 950° C. Increase of temperature causes the secondary emission to diminish. If positive ions are allowed to evaporate from the hot target the number of Cs+ ions leaving the surface may, under certain conditions, be equal (with an accuracy greater than 1%) to the number arriving in the incident beam.

I much appreciate the kindness shown me in many ways by Lord Rutherford. The experiments were commenced at the instigation of Dr M. L. E. Oliphant and I gratefully acknowledge that this paper represents to a great extent the results of his advice and assistance. The work was made possible by a grant from the Department of Scientific and Industrial Research.

The following calculations were made in connection with recent work on the kinetics of the attack on platinum by iodine vapour at low pressures. A platinum filament was heated in a glass bulb to which iodine was admitted at a pressure of 0·027 mm. of mercury from a reservoir of iodine kept at 0°C. The tap to the reservoir was opened for a period of two minutes, which was evidently sufficient to allow the iodine to saturate the walls of the bulb, since on closing the tap the pressure remained constant. On heating the filament, reaction ensued, causing a steady fall in pressure. This does not however truly represent the “clean up” due to the reaction, as the gas initially adsorbed on the walls of the bulb is continuously desorbed as the pressure falls. To obtain the true pressure decrease due to the reaction, a quantity representing the gas desorbed must be subtracted from the pressure reading at any moment.