1. Let

be a numerical series. If for sufficiently small h > 0 the series

is convergent, we can form the upper and lower limits of J (h) as h → 0. These limits are called respectively the upper and lower sums (R, 1) of the series (1). For the purposes of the present paper it will be convenient to consider a more extended definition of these upper and lower sums. We shall suppose that for sufficiently small h the series J (h) is summable by Poisson's method. We denote the Poisson sum by PJ (h). The upper and lower limits of PJ (h) as h → 0 will be called the upper and lower sums (R, 1) of the series (1).

1. The analogue for trigonometric integrals of de la Vallée Poussin's classical theorem concerning the uniqueness of the trigonometrical development has recently been obtained by M. Jacob in the following form.

1. In a recent paper I have given some definite integrals involving Legendre functions which, as a limiting case, give known results involving Bessel functions. In another paper I have shown how some integrals involving Bessel functions can be obtained from Bateman's integral

and the well-known expansion

In a previous paper the writer has obtained various formulae for surfaces in higher space by means of Cayley's functional method; and recently the same method has been applied to surfaces on a quadric form in [5] with the object of investigating the properties of line congruences in [3]. The purpose of the present note is two-fold: to indicate a few applications of the formulae mentioned above, and to give direct and (it is believed) novel proofs of some results obtained elsewhere by the functional process. The demonstrations given here differ, save in two cases, from those based on Schumacher's four-dimensional representation which, as it stands, applies only to congruences without double rays and therefore lacks generality. It was James who first noticed that a congruence without singularities has in general a finite number of double rays and is accordingly defined by five, instead of four, independent characters. These characters have been considered in previous work; but for present purposes it is convenient to describe them afresh.

In this paper it is shown how all the regular polytopes (including the Kepler-Poinsot polyhedra and Hess's analogous star polytopes in four dimensions) arise as rational solutions of trigonometrical equations.

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.

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.

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.

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.

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.

1. The formulae usually given for the scattering of electrons by atoms are calculated without taking into account the radiative forces. It has been suggested by the author in a former paper that in the case of the large-angle scattering of electrons with velocity comparable with that of light (for which case the scattering is entirely nuclear), these forces might influence the scattering by an appreciable amount, which would explain the divergence between the theoretical formula and the results of Chadwick and Mercier, which are some 40 per cent. greater. In this paper we shall show that for electrons–of any velocity the influence of these forces cannot be greater than 2 or 3 per cent., and so cannot be invoked to explain discrepancies between theory and experiment.

It has been shown by Ratcliffe and White that the effective specific inductive capacity (ε) and the effective conductivity (σ) of soil placed, as dielectric, in a simple condenser vary in a marked manner with frequency. The range of frequency used was from 0·1 × 106 cycles per second to 10 × 106 cycles per second. It was found that at 6·1 × 106 cycles per second the specific inductive capacity was very large (about 40 e.s.u.) and that it decreased to a value of about 12 e.s.u. at 10 × 106 cycles per second. This latter value agrees with the value found in experiments on the propagation of short radio waves over the surface of the earth. The effective conductivity of the soil had a low value at 0·1 × 106 cycles per second but increased rapidly to a constant value for frequencies above 3 × 106 cycles per second.

The β-ray spectrum of radium C is remarkable for the large number of faint lines of about equal intensity which occur above Hρ 4900. These lines were first measured by Rutherford and Robinson and many of them were found again by Ellis in a reinvestigation of this spectrum. It was then stated that no attempt was made to check the existence of each faint line and that Rutherford and Robinson's measurements of the faint lines were taken over directly, with certain necessary adjustments to the absolute energies. Subsequently the intensities of the β-ray lines in this spectrum were measured by Ellis and Wooster and by Ellis and Aston and the faint lines were again treated in much the same way.

The purpose of this note is to amplify the discussion and conclusions of a recent paper by one of us. It has recently been shown by Webster, drawing largely from the work of Weiss and Forrer, that in single crystals of both iron and nickel there is a family of natural directions of micromagnetization defined by the observed directions of easy holomagnetization. The evidence presented by Webster is completely convincing, and we therefore may refine our picture of micromagnetization by asserting that the micro-elements are all (holo)magnetized in one or other of the directions of easy holomagnetization. These directions are the family of six directions (± 1, 0, 0) (0, ± 1, 0) (0, 0, ± 1) for iron and the family of eight directions (± 1, ± 1, ± 1) for nickel. We wish to comment on Webster's theory later in connection with the properties of cobalt.