A relationship between the Hamiltonian of a system and its distribution function in phase space is sought which will guarantee that the average energy is the weighted mean of the Hamiltonian over phase space. This relationship is shown to imply the existence of a wave function satisfying the Schrödinger equation, and dictates the possible forms of time-dependence of the distribution function.

A quantum model of the scattering of a particle via absorption and re-emission at the origin is described. The model is capable of exhibiting an arbitrary number of resonances, and the associated unstable states are discussed. The emphasis is on obtaining exact results. It is found that the exponential decay law is of limited application, and that in all cases of non-zero coupling the decay goes ultimately as t−3. Finally, it is pointed out that the treatment of unstable states in this type of model depends only on a knowledge of the phase shift as a function of radial momentum, and a possible phenomenological method is suggested.

1. Introduction. In linear systems, the concept of ‘free vibrations in normal modes’ is well defined and fully understood. The meaning of this phrase is far less clear when it is applied to non-linear systems. It is the purpose here to define and examine the free vibrations in normal modes (and their stability) in certain non-linear systems composed of masses and springs and having a finite number of degrees of freedom. Of necessity, such a paper is in some degree conceptual in nature.

A simple expression is derived for the magnetic vector potential of current in a thin ring, in terms of the first derivative of toroidal functions of zero order. The axial and radial field components and the mutual inductance between two wire rings are obtained. These expressions are evaluated on a digital computer and the results are summarized in a series of graphs.

1. Introduction. The problems associated with low-frequency diffraction by a circular disc have attracted the attention of many research workers. A review article by Bouwkamp(1) gives an excellent account of the research that was done prior to 1953 and a report by Eggimann(3) provides a summary of the papers that were published between 1953 and 1960. References to work published since 1960 may be found in papers of Noble (10) and Williams (16), (17).

The theoretical studies of Chandrasekhar on the stability of Couette flow in a viscous, electrically conducting, fluid in the presence of a uniform axial magnetic field are extended to include cases of finite gap width between the cylinders, and cases in which the conductivity of the walls of the containing cylinders is finite. In addition, the non-axisymmetric modes of instability are discussed, and the results of numerical computations are presented.

1. Introduction. Little has been written on the effects of surface tension (or cohesion) on waves in liquid media; this is not surprising as in general its effects are small compared with those of gravity or compressibility, except in the case of ripple motion (Lamb (3), sections 265, 267). Bondi (1) has investigated the problem of a compressible fluid half space; he found that under cohesive action ‘faster-than-sound’ propagation along the surface is possible, and he studied it with the aid of an elegant image-function-space method; but he carefully points out the small size of the parameters involved.

The Prandtl boundary-layer theory is extended for an idealized elastico-viscous liquid. The boundary-layer equations are solved numerically for the case of two-dimensional flow near a stagnation point. It is shown that the main effect of elasticity is to increase the velocity in the boundary layer and also to increase the stress on the solid boundary.

This paper deals with the two-dimensional problem of a circular inclusion undergoing spontaneous dimensional changes in an infinite elastic medium with a circular hole. This provides a more realistic model for some physical phenomena where inclusions have found applications. The effect of a concentrated force acting at an arbitrary point of an infinite medium with a circular hole has been found as an auxiliary result. After the evaluation of the effect of a concentrated force, Eshelby's. point force approach (1) has been used to get an exact solution to the inclusion problem.

1. Introduction. The disturbance produced in an elastic half-space by different types of buried sources have been studied by various authors. However, the only exact solutions of problems for the half-space in which the disturbance is produced by impulsive surface tractions are due to Pekeris (9) and Chao (4), though for the half-plane such problems have been studied by de Hoop (5), Ang (1) and Mitra (8). Pekeris (9) evaluated the surface displacement produced by a normal point load on the surface using a method of inversion of the operational solution developed earlier (Bateman & Pekeris (2)). A finite pressure-area, however, represents physical situations better than a point load. In this paper, the displacement produced in the half-space by uniform impulsive pressure acting over a circular portion of the surface has been obtained in terms of definite integrals. On the surface, the displacement has been split up into a number of terms some of which represent the P-wave contribution, while others give the Rayleigh-wave contribution to the displacement. The method of solution involves the use of integral transforms and Cagniard's (3) method; Pekeris's method is not applicable in this case since the appropriate transformation of an integral along the real axis to one along the imaginary axis cannot be carried out.