Let ξ, η, ζ be linear forms in u, v, w with real coefficients and determinant Δ ≠ 0. A conjecture of Minkowski, which was subsequently proved by Remak, tells us that for any real numbers a, b, c there exist integral values of u, v, w for which

and the constant ⅛ on the right is best possible.

This paper is a sequel to a previous one(1) with the same title which dealt with the general solution of equations of the type

We consider here the more general equation

where g(x) is a given function. We are interested particularly in the existence and uniqueness of solutions of the latter equation and show how these are related to the closure and completeness properties of sets of functions {xneωnx} derived from the kernels kr(y).

In this paper a periodic solution of the gravitational equations is examined. The solution, which is valid throughout all space, is such that the internal and external fields and their derivatives are continuous at the boundary of a spheroid rotating about an axis other than that of symmetry. At great distances from the system the solution has the same form as the field due to a rotating rod and can be applied to problems like the determination of the velocity of propagation of gravitational waves and the loss of energy of a rotating cohesive system.

The approximate value of the world invariant at a point in a plane containing a rotating cohesive system such as a rod is calculated for a point at great distances from the system. It is found that the invariant has a stationary value when the rod is seen to be pointing towards the point in question. It is concluded that the speed of propagation of gravitation is the same as that of light.

The relativistic second quantization of free bosons is extended to fermions. To know how relativistic creation and annihilation operators operate on bra and ket vectors, it is necessary to have a relativistic scalar product which in the case of free particles can be constructed. For particles interacting with each other or capable of emitting and absorbing particles of the same kind, it is pointed out that adequate wave equations for an indefinite number of particles each taking a separate time coordinate have not yet been found and so no scalar product can be found. Thus creation and annihilation operators can be defined only by their operation on a bra vector or a ket vector but not by both. With the help of these, the relativistic wave equations referred to above are proposed and their consistency conditions studied. For the particular case of scalar particles, an illustration is given that such a theory may admit certain expressions as probabilities.

It is pointed out that the equations of motion for any field obtained by varying a Lagrangian subject to auxiliary conditions are exactly equivalent to a certain set of canonical equations and that the commutation relations between the dynamical variables for the latter equations are Lorentz-invariant. By extending the theory to Lagrangians containing higher derivatives of the field quantities, it is shown that any given set of field equations can be put into the canonical form, though it is not derived from variational principles. The question of Lagrangians with missing momenta is also considered. It is shown that if the Lagrangian is ‘gauge-invariant’, some of the p's must be missing and the corresponding Eulerian equations can be replaced by equations containing no q and then can be replaced by initial conditions. The commutation relations between gauge-invariant quantities are Lorentz-invariant. For Lagrangians which are not gauge-invariant but are such as to have missing momenta, the passage to quantum theory will in general give rise to non-Lorentz-invariant commutation relations. In both cases, the equations of motion can be cast in canonical forms.

1. Since many writers, among them G. I. Taylor (1) and W. L. Bragg (2), have based theories of the strength of metals upon the assumption that there is some type of dislocation in the metal, it has seemed desirable to give an account of a dislocation in which displacement and stress are given, according to the classical theory of elasticity, by simple closed formulae.

A simple and direct variational method is described for finding both complex and real eigenvalues of the wave equation of anomalous propagation in a horizontally stratified atmosphere. It may be looked upon as an extension of Rayleigh's method to complex eigenvalues. In this paper it is illustrated by an example, taken from the duct theory of super-refraction, in which the refractive index of the air varies with height according to a power law. Numerical agreement in the values for the lowest order eigenvalues with those obtained by the differential analyser is better than ½%.

The paper contains a theoretical discussion of the scattering by a ship of the wave sent out from a radar transmitter. An expression is derived for the power returned to the receiver of the radar installation, on the assumption that the ship is composed of a large number of scattering elements which scatter energy non-coherently. Calculations based on this expression are compared with the results of experiment and shown to be in good agreement as regards the variation of signal strength with range, provided that a suitable value for the effective radius of the earth (a quantity which changes with meteorological conditions) is taken.

An experiment is described in which the absolute level of the received signal from a ship was measured by comparing it with that from a metalized sphere flown from a balloon. The result was found to be in satisfactory agreement with theoretical expectations.

The heat of adsorption of particles which occupy two sites on a surface is examined by determining the statistical equilibrium. In particular, equations which are appropriate to immobile films are obtained. The heats of adsorption calculated from these equations are compared with the experimental values obtained for the heat of adsorption of hydrogen on a tungsten surface, and with the values which were obtained using an empirical method.

The optical properties of reflecting echelette gratings, used in the infra-red region of the spectrum, are discussed theoretically. Expressions are derived for various important characteristics including the wave-length relationship, the dispersion and the resolving power of spectrometers using such gratings. The intensity distribution is discussed and illustrated by reference to a particular spectrometer fitted with a grating of 1200 lines to the inch. The relative importance of other factors such as the variation of available energy with wave-length, the dispersion of the preliminary optical system, etc., is considered and tabulated. Particular consideration is given to the properties of such gratings when used in much higher orders of interference than are common at present. With large aperture spectrometers, this possibility of using comparatively coarse gratings with a high order of interference is of great practical importance.

The hysteresis losses and demagnetizing coefficients of rectangular Fe-Si strips of various dimensions have been investigated. The hysteresis loss is approximately independent of the value of N, in agreement with a simple theoretical argument. The value of the demagnetizing coefficient is in fair agreement with the value calculated for the inscribed ellipsoid.

A more detailed analysis has been carried out of earlier results obtained in the scattering of X-radiation by liquid He II. When certain necessary corrections have been applied to these results and a comparison made with theoretical intensity distribution functions to be expected for certain ordered arrangements of atoms in a liquid, it is found that the arrangement of atoms in liquid He II most probably follows either a diamond or a simple cubic configuration. Considerations of energy and atomic volume render the former the more likely arrangement.

The production of by deuteron bombardment of uranium has been established and the radiations of this product have been examined by the absorption method. The normal mode of disintegration involves β-particles having an upper energy limit of 1·20 ± 0·02 MeV. and a low-energy γ-ray of 76 ± 3 keV. quantum energy. Evidence has been found for electrons of less energy (about 0·1 per disintegration) and more penetrating γ-rays of relatively low intensity. Quantum radiations corresponding broadly to the L X-radiations, to be expected to follow the internal absorption of the low-energy γ-ray, have also been detected.

The excitation function for the process has been studied, and the conclusion reached that the Oppenheimer-Phillips mechanism is responsible for the larger part of the yield.

With the help of certain inequalities concerning the elements of the dispersion matrix of a set of statistics, and of the information matrix, the following results have been proved. Some of these inequalities are extensions of results given by Fisher (1) in the case of a single parameter.

(i) Efficient statistics are explicit functions of the minimal set of sufficient statistics.

(ii) Functions of the minimal set of sufficient statistics, satisfying the property of uniqueness defined in the text, are best unbiased estimates. Under certain conditions estimates possessing exactly the minimum possible variance can be obtained by the method of maximum likelihood.

(iii) In large samples maximum likelihood estimates supply efficient statistics in the case of several parameters.

(iv) The importance of replacing the sample by an exhaustive set of sufficient statistics (referred to in this paper as the minimal set) as a first step in any methodological problem has been stressed by R. A. Fisher in various articles and lectures. The above discussion supplies a formal demonstration of this view so far as the problem of estimation is concerned.