Experiments are described in which the stopping power of liquid water relative to air has been re-determined for 5 MeV. α-particles, using only a limited portion of the range. The results confirm those of earlier experimenters, who concluded that liquid water did not obey the Bragg law.

Dirac has suggested that the quantization of electric charge could be explained by the existence of magnetic monopoles. In view of this hypothesis, this paper investigates what theoretically would be the behaviour of such monopoles in a Wilson cloud chamber. The treatment, which for simplicity is basically classical, closely follows Bohr's work on the decrease of velocity and ionization properties of α- and β-particles, and expressions are derived for the rate of decrease of energy and the number of ion-pairs produced per centimetre by a monopole passing through a gas. These expressions are then discussed with particular reference to the case of heavy particles, and the main differences between them and the corresponding expressions for α-particles both as to range and ionization are indicated; these differences can be summarized by saying that monopoles have much shorter paths, but create many more ion-pairs per centimetre than α-particles. Also, the very sharp increase in the ionization at the end of the path of an electric particle is missing, the ionization for the monopole decreasing to a small amount near the end of the path.

The inequality is

which is established by an elementary argument and is shown to lead directly to the evaluation of the integral

and to the expression of sin x as an infinite product.

1. The problem of the partition of numbers, first investigated in detail by Hardy and Ramanujan (1), has in recent years assumed importance on account of its application by Bohr and Kalckar (2) in evaluating the density of energy levels in heavy nuclei. A ‘physical approach’ to the partition theory has been made by Auluck and Kothari (3), who have studied the properties of quantal statistical assemblies corresponding to the partition functions familiar in the theory of numbers. The thermodynamical approach to the partition theory, apart from its intrinsic interest, draws attention to aspects and generalizations of the partition problem that would, otherwise, perhaps go unnoticed. Thus we are led to consider restricted partitions such as: partitions where the summands are repeated not more than a specified number of times; partitions where the summands are all different; partitions into summands which must not be less than a specified value; partitions into a prescribed number of summands, and so on. The generalization that seemed to us to be the most interesting is the extension of the partition concept to include partitions into non-integral powers of integers.

A general theory of the interaction of two charged identical colloidal particles of arbitrary shape is developed. An expression for the Helmholtz free energy of the electric double layers is obtained by the methods of statistical mechanics. The condition that there is equilibrium between the ions adsorbed on the surfaces of the colloidal particles and those dissolved in the dispersion medium is accounted for by requiring that the free energy of the whole system be a minimum with respect to variation of the ionic density on the surfaces.

The theory presented here is a further development of the work of Verwey and Overbeek. The conclusions of this paper are that in dilute sols, to which the present investigations are restricted, the results of these authors require extension in two directions. First, there is a correction to the mutual energy of two particles, due to the Coulomb interaction of the ions in the bulk of the solution. Secondly, no special assumption concerning the relation between the surface potential (or charge) and interparticle separation need be introduced. The equations set up to determine the free energy of interaction at the same time yield the ‘adsorption isotherm’ for the ion type which is common to the solution and the surface of the particle.

An extension is proposed to the theory of Verwey and Overbeek on the force between two colloidal particles in a dilute sol. The effect of the Coulomb interaction of the ions in the dispersion medium is considered. It is shown that this contributes an additional repulsion term to the mutual free energy of the two particles. This correction term behaves as an inverse power and is the predominant effect at large separations, although it is a second-order effect at small separations. For parallel plates, this power is the fourth, and for two spheres, the sixth. A tentative calculation shows that, at least for thin plates and low electrolyte concentrations, our new repulsion will outweigh the attractive London-van der Waals force considered by Verwey and Overbeek, and more recently by Casimir and Polder, when the plates are far apart.

Two of the strongest emission lines in the spectra of the planetary nebulae were identified by Boyce, Menzel and Payne (1) as due to forbidden transitions in the ground configuration of doubly ionized neon. The transitions are of wave-length 3967 and of wave-length 3869 in the configuration Ne III 1s22s22p4. These lines are excited by inelastic collisions of electrons with the neon ions. In order to investigate these transitions in detail a knowledge of the collision cross-sections is required. As a first step in this direction, wave functions have been calculated for the 3P state by the method of the self-consistent field, including the effect of exchange. In later papers these wave functions will be used in the calculation of collision cross-sections, in a study of the departures from the Landé rule in the triplets of the p2 and p4 configurations and, it is hoped, in the calculation of line strengths for certain permitted lines of astrophysical importance in the spectrum of Ne II.

It was shown by Sas (1) that, if K is a plane convex body, then it is possible to inscribe in K a convex n-gon occupying no less a fraction of its area than the regular n-gon occupies in its circumscribing circle. It is the object of this note to establish the n-dimensional analogue of Sas's result, giving incidentally an independent proof of the plane case. The proof is a simple application of the Steiner method of symmetrization.