This paper is an attempt to apply a simple series solution of the wave equation to the reflexion of a sound pulse. The reflector is a paraboloid of revolution, and the incident pulse is spherically symmetrical and comes from the focus of the reflector, so that the wave fronts of the reflected pulse are planes at right angles to the axis of symmetry of the reflector. If the incident pulse consists, at any point, of a discontinuous pressure rise followed by constant excess pressure, the series reduces to a power series in the time counted from the onset of the reflected pulse; for other forms of the incident pulse it can be interpreted as the result of the superposition of elementary pressure pulses which are constant in a small time interval and vanish outside it. No convergence proof is given, so that the interest of the investigation is physical rather than mathematical; but the numerical results indicate that the convergence of the series is unsatisfactory except near the vertex of the reflecting paraboloid. The coefficients of the series are obtained with the aid of recurrence formulae, and the first seven coefficients have been calculated. The calculations become more laborious at each successive stage. A detailed numerical investigation of the reflexion of the ‘simple rectangular pulse’ referred to already (for which the series reduces to a power series) reveals that initially the maximum pressure on any plane section at right angles to the axis of the paraboloid, due to the reflected pulse alone, occurs at the axis and has the same value everywhere on it; but after some time a secondary pressure maximum is established over a circular ring at some distance from the axis. A consideration of the initial pressure gradient of the reflected pulse suggests that a similar state of affairs exists at all distances from the vertex, but the actual calculations only extend to a plane section whose distance from the vertex is four times the focal length. The unsatisfactory convergence of the series precludes the investigation of subsequent changes in the distribution of pressure. It is finally pointed out that these results apply to a certain extent to a finite parabolic mirror.

The author's general variational method is applied to the case of a particle for which second moments are important but third and higher moments are negligible. Equations of motion are obtained for the angular momentum and for the centre of mass, equations (12·35) and (12·41), with arbitrary external forces X.

The angular forces are then calculated for a charged particle with electric and magnetic moments moving in a general electromagnetic field, on the assumption that the effect of a certain part of the energy tensor, Tiii of (15·17), is negligible. This leads to the equations of angular motion, (17·13), from which it is inferred that, in order that the magnitude of the angular momentum may be integrable, the angular momentum, electric and magnetic moments must all be parallel in a frame of reference in which the particle is instantaneously at rest.

The linear forces are then calculated for the case of no electric moment, leading to the equations for linear motion (18·10). From these it is inferred that, in order that the mass may be integrable, the ratio of the magnetic moment to the angular momentum must be constant.

There are many types of problem in the study of molecular structure in which integrals involving two or more centres of force arise. We may mention the calculation of molecular energy levels, the Stark effect in molecules, the van der Waals forces between molecules, and, more recently, the internal arrangement of neutrons and protons in nuclei. During the last few years the writer has been accumulating a table of these integrals, and the explicit forms for many of them are given in this paper.

It is shown that the theorem stated in Born's paper, and proved for the case of a linear lattice of N equal particles under certain restrictions concerning the forces between the particles, that macroscopic stability (stability for long waves) implies microscopic stability, may be extended to three dimensions for the particular case of a face-centred cubic lattice, where the effects of all neighbours, other than the first twelve neighbours, are neglected.

I take this opportunity of expressing my sincere thanks to Prof. Born for much valuable advice.

In a note in the April 1942 issue of these Proceedings, Hodge gives a formula for the number of independent terms in what he calls a k-connex, but while conjecturing its general validity he proves it only for very restricted cases. Its truth is here demonstrated in the general case.

For a direct comparison of the individual attractive and repulsive terms of an intermolecular potential determined by the inductive analysis of themodynamic data with the same terms calculated by quantal methods it is desirable to carry out the analyses, in the first approximation, with an intermolecular potential of the form ø(R) = Pe−aR − A1/R6 − A2/R8. For mathematical convenience, in place of the above expression, two potential functions,

and

are considered, the first being taken to be adequate in the range of values of R between 0 and R0 (the minimum of the potential function) and the second, in the range from R0 to ∞. By dividing the problem in this way it is possible to find substitutions which permit the integration of the classical expression for the second virial coefficients (and other appropriate thermodynamic data) directly in terms of fairly simple series in | ψ0 |, R0, a and r. Finally it is pointed out that for such simple atoms or molecules as the rare gases, oxygen, nitrogen and methane r may be taken as 0·15 throughout, which considerably simplifies the application of the method to the experimental data.

The main purpose of the paper is an investigation of the stability of a certain class of Bravais lattices, namely, those with a rhombohedral cell of arbitrary angle. The potential energy is assumed to consist of two terms, each proportional to a reciprocal power of the distance. In the continuous series of lattices obtained by changing the rhombohedral angle, there are included the three cubic Bravais lattices, the simple (s), the face-centred (f) and the body-centred (b) lattices. It is shown that (f) and (b) correspond to a minimum of the potential energy, and (s) to a maximum. A method for calculating the potential energy for the intermediate rhombohedral lattices is developed, and, with the help of a certain characteristic function, it is shown by numerical calculation that the (f) lattice corresponds to the absolute minimum of potential energy, and that no extrema, other than (f), (s) and (b), exist. In the last section, the case of a compound (non-Bravais lattice) is considered, and it is shown that the equilibrium and stability conditions for the law of force assumed can be divided into one set for change of volume, and an independent set for change of shape.

We take this opportunity of expressing our sincere thanks to Prof. Born for his interest in our work, and for much valuable advice.

The theory of lattice deformations is presented in a new form, using the tensor calculus. The case of central forces is worked out in detail, and the results are applied to some simple hexagonal lattices. It is shown that the Bravais hexagonal lattice is unstable but the close-packed hexagonal lattice stable. The elastic constants of this lattice are calculated.

Calculations have been made of the momentum distribution and the shape of the Compton profile for gaseous methane, using the Dirac transformation theory and three distinct approximations (molecular orbital, electron-pair and self-consistent-field) to the true molecular wave functions. The bearing of the results upon the validity of the original wave functions is discussed; it is concluded that the exponents in molecular wave functions should be approximately 1·1 times greater than in corresponding atomic wave functions. Comparison is made with experimental electron scattering results.

Our thanks are due to Prof. Massey for providing us with the numerical tables of the self-consistent-field for methane in advance of publication, and to Mr E. G: Phillips of the Mathematics Department of Univeristy College of North Wales, for the loan of a calculating machine.