Let θ > 0 and α ≠ 0 be real numbers, and let θ be irrational. Khintchine has shown, by the use of continued fractions, that there is an infinite number of pairs of positive integers (p, q) which satisfy the inequality

for any given K > 5−½; and, more recently, Jogin has shown the same is still true with K = 5−½. The condition that p and q shall be positive is, of course, essential, as otherwise there is the classical result K = ¼ due to Minkowski.

1. We write L for the class of integrable functions in (− ∞, ∞), V for the class of functions of bounded variation, and define A, A to be the classes of functions F(x) which may be expressed in the forms

respectively.

1. We recall the meaning attributed by F. Hausdorff to the statement: ‘The (linear) set S is of dimension [h(x)].’

1. The problem investigated in this note was suggested by a study of the mathematical basis of change ringing. Its campanological application is discussed in § 4.

Bagnera(1) shows that when a certain primitive finite group of order 576, G576, is represented as a group of collineations in three-dimensional space, it contains twenty-four harmonic inversions with respect to points and planes, and he shows how to calculate the coordinates of the points and planes of the inversions. The purpose of this note is to discuss the configuration of these points, which we shall call vertices, to show that the whole group is generated by the twenty-four harmonic inversions, termed projections after Prof. Baker (2) and Dr J. A. Todd (4), and to enumerate the conjugate sets of the group.

In my book The Theory and Applications of Harmonic Integrals (hereinafter referred to as H. I.), Chapter iv is devoted to the application of the theory to algebraic varieties. In order to introduce harmonic integrals on an algebraic variety I have to assign a metric to the variety; this metric is not related to the variety in any invariant sense, and indeed its introduction is extremely artificial. Nevertheless it turns out that we can deduce from the properties of the harmonic integrals a number of properties of the variety which are birationally invariant, that is, do not depend on the choice of metric. This remarkable fact seems to indicate either that the invariant properties in question should be obtainable directly without the introduction of the harmonic integrals, or that the metric introduced is not so artificial as it first appeared to be. This note is intended to examine this question.

1. In this paper we find explicitly the base for the prime ideal associated with any irreducible Vd−1 on a Segre variety, or a Veronesean variety, Vd. This work extends that of an earlier paper (1) in which the base was found when the Vd−1 is a complete intersection of Vd with one primal. As particular examples we can write down the base for any irreducible curve on a quadric surface, a Veronese surface or a Del Pezzo surface. We also show how the base for any prime ideal changes under a Veronesean transformation.

If the probability differential of a set of stochastic variates contains k unknown parameters, the statistical hypotheses concerning them may be simple or composite. The hypothesis leading to a complete specification of the values of the k parameters is called a simple hypothesis, and the one leading to a collection of admissible sets a composite hypothesis. In this paper we shall be concerned with the testing of these two types of hypotheses on the basis of a large number of observations from any probability distribution satisfying some mild restrictions and their use in problems of estimation.

The production of a ‘wake’ behind solid bodies has been treated by different authors. S. Goldstein (1) and S. H. Hollingdale (2) have discussed the laminar wake behind a flat plate, while the wake of a two-dimensional grid has been treated for the turbulent case alone using L. Prandtl's ‘Mischungsweg’ theory by E. Anderlik and Gran Olsson (3), (4).

In three papers (1, 2, 3) Bhabha has set up a new theory for relativistic particles of any spin, and has suggested that a particular wave equation of half-odd spin, classified as (, ½), may be applicable to the proton. It is therefore of interest to study this equation and to calculate results which could perhaps be compared with experiment; since in the non-relativistic limit the particle obeys Dirac's equation, it is sufficient to consider the behaviour at energies large compared to the rest mass. Here the important problem is the calculation of scattering cross-sections for cosmic-ray processes.

The relativistic field theories of elementary particles are extended to cases where the field equations are derived from Lagrangians containing all derivatives of the field quantities. Expressions for the current, the energy-momentum tensor, the angular-momentum tensor, and the symmetrized energy-momentum tensor are given. When the field interacts with an electromagnetic field, we introduce a subtraction procedure, by which all the above expressions are made gauge-invariant. The Hamiltonian formulation of the equations of motion in a gauge-invariant form is also given.

After considering the Lagrangian L as a scalar in a general relativity transformation and thus a function of gμν and their derivatives, the functional derivative of

with respect to gμν (x) at a point where the space time is flat is worked out. It is shown that this differs from the symmetrized energy-momentum tensor given in the above sections by a term which vanishes when certain operators Sij are antisymmetrical or when the Lagrangian contains the first derivatives of the field quantities only and whose divergence to either μ or ν vanishes.

In this paper an approximate formula is developed for the phase shifts required in the calculation of the scattering of particles according to wave mechanics. It is shown to be more accurate than the current approximations and it is not difficult to handle. The same method can also be used to compute any quantity which is sensitive to the form of the wave function near the centre of force (e.g. a matrix element).

A method is described in which the rate at which fast neutrons cross unit area is measured by counting protons projected into a small solid angle in the forwards direction from thin and thick layers of polythene. The protons are detected by triple coincidences between three proportional counters mounted coaxially behind the hydrogenous layers. The method is applicable to neutrons of energies from 1 MeV. upwards, and can be used in the presence of intense γ-rays. The flux of these neutrons is calculated in terms of the rate of detection of the protons, the solid angle for proton collection, the mass per unit area of the polythene layer, and the neutron-proton-scattering cross-section. A study of the behaviour of proportional co-axial counters, used in this manner, has been made. A determination of the angular distribution of fast neutrons produced by a deuteron-deuterium source has been made by our coincidence method, and the results compared with those obtained by the ionization chamber method by Bretscher and French. Absolute values agree to within 10%.

During the past few years, several methods for the measurement of fast neutron flux have been developed in the Cavendish Laboratory. A critical inter-comparison is given in a forthcoming paper by Allen, Livesey and Wilkinson. Some of the methods depend on the counting of protons which are projected by the neutrons from a film of hydrogen-rich material. Polythene is usually employed because its hydrogen content is high and accurately known. If the thickness of the film is comparable with the range of the fastest proton produced by the incident neutrons, it becomes necessary to know the range-energy relation for protons in polythene. This is particularly so in the ‘thick film chamber’ method to be described by Allen and Wilkinson. Another method, the ‘homogeneous ionization chamber’ method to be described by Bretscher and French makes use of the well-known principle that the average density of ionization obtaining in a chamber whose walls have an atomic composition identical with that of the gas which fills the chamber is the same as that in an infinite extent of that gas. This principle itself depends on the identity of mass stopping power of solid and gas of the same atomic composition. The chamber usually used has polythene walls and is filled with ethylene.

The yields of 139Ba per fission in natural uranium by unslowed neutrons from a mixed radium-beryllium source and by C-group neutrons were compared. In each case the fission rates measured with an ionization chamber were correlated with the β-ray activity of 139Ba, chemically extracted from a mass of uranium oxide exposed to an identical neutron spectrum. The ratio of the two fission yields is found to be 0·84 ± 0·03. The possibility of using 139Ba as a local or integrating indicator of fission in extended uraniferous media, and of distinguishing between the contributions by fast neutrons and by slow neutrons is discussed.

Equipment has been designed and constructed for the interruption of the ion beam in a high-voltage accelerating tube and for the detection and measurement of the period of short-lived radioactive elements. This method has been applied to the measurement of the half-life of 12B which has been found to be 27 ± 2 msec. The reaction 15N(n, α)12B has been detected.