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.

The double integral equation, which takes the place of the Milne equation in the one-dimensional problem, is derived from the governing partial differentio-integral equations. An analytical solution of the problem of a distribution of point sources on a plane, when the other boundaries are at infinity, is then found. The possibility of more complicated boundary conditions is discussed.

The class of stationary point processes known as ‘doubly stochastic Poisson processes’ was introduced by Cox (2) and has been studied in detail by Bartlett (1). It is not clear just how large this class is, and indeed it seems to be a problem of some difficulty to decide of a general stationary point process whether or not it can be represented as a doubly stochastic Poisson process. (A few simple necessary conditions are known. For instance, Cox pointed out in the discussion to (1) that a double stochastic Poisson process must show more ‘dispersion’ than the Poisson process. Such conditions are very far from being sufficient.) The main result of the present paper is a solution of the problem for the special case of a renewal process, justifying an assertion made in the discussion to (1).

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.

Gani and Prabhu (1) have presented a general theory of storage for dams subject to a steady release. At time t ≥ 0, let a dam of infinite capacity have a continuous random content Z(t) ≥ 0; the input X(t) ≥ 0 which has entered it during time t follows an infinitely divisible distribution, while the dam is subject to a steady release at constant unit rate except if it is empty, when the release ceases. A procedure is obtained whereby various functions describing the transient behaviour of Z(t) can be calculated in principle.

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.

The problem of causality in non-relativistic scattering where a bounded region of non-linear interaction occurs is re-examined in the context of a paper by Ebel. In the model considered it is shown that the acausal behaviour cannot be accounted for in the straightforward manner that is possible for an energy-dependent but linear potential which gives rise to a similar R-matrix.

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).

A cross-point switch is a device for connecting members of a set of links L with members of another (disjoint) set of links R. Each link of either set has possible connexions with every member of the other set, but in any actual state each link is connected either with no other links or with exactly one link in the other set. Unconnected links are called idle and connected links busy. A switch with p L-links and q R-links is called a,{p × q}-switch.

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.