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).

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.

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 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 stability of the flow between concentric, rotating cylinders is discussed when the gap is small and the cylinders are rotating in the same direction for a class of anisotropic fluids in which the fluid has a preferred direction. An important conclusion of the analysis is that a steady-state solution of the equations has previously been considered unstable on false grounds.

A long thin plate of finite width (a finite dock) extends over part of the otherwise free surface of a semi-infinite body of water under gravity and is given a forced heaving motion of small amplitude about this position. The solution of the boundary-value problem for the velocity potential of the resulting forced (two-dimensional) surface-wave motion may be reduced to that of an integral equation of the second kind for the value of the potential on the dock by means of an appropriate Green's function. There is an infinite number of choices for such a Green's function, and we show below how to construct one for which the kernel of the corresponding integral equation tends to zero with the ratio wavelength/dock-width. The integral equation may then be solved by iteration to calculate the short-wave asymptotics of the original wave-making problem.

The generalized wave-making problem for the forced high-frequency oscillations of a finite dock is solved through use of the Green's function obtained in the first part of this paper. The specific cases of heave and roll are considered with particular reference to the forces on the dock and the amplitude and phase of the radiated waves. These results are then utilized to solve the problem of transmission of short waves under a fixed dock.

1. It is well known that solutions of the Cauchy problem for the wave equation represent disturbances obeying the laws of geometrical optics. Specifically a solution ψ of the wave equation

for which ψ = δψ/δt = 0 initially outside a surface C0, vanishes at time t in the exterior of a surface Ct parallel to and at a normal distance ct from C0 (see e.g. (l), page 643). Analogous results hold for the solutions of any linear hyperbolic second-order partial differential equation with boundary-value conditions of the Cauchy type. Boundary conditions of the type representing reflexion have been treated by Friedlander(2). He showed that as well as the incident and reflected wavefronts, there sometimes exists a ‘shadow’ where diffraction occurs, and that the diffracted wave fronts are normal to the reflecting surface, the corresponding rays travelling along the surface and leaving it tangentially. The purpose of this paper is to extend these results to refraction, where instead of a purely reflecting surface we have an interface between two different homogeneous media.

The purpose of this note is to obtain an improved approximation to the torsional rigidity of tubes of constant normal thickness which may be applied when Bredt's first approximation(l) is not sufficiently accurate. (The reference to Bredt's approximation is taken from(2), page 299.)

Some Comments on Single-Service Queuing Methods and Some New results By Julian Keilson (Pages 237–251 of the present volume)

As the result of a printing mishap, the following typographical errors appeared on pages 240 and 241 of this paper.

(i) Page 240, third line from the bottom: ‘Paplace’ for ‘Laplace’.

(ii) Page 240, bottom line:

(iii) Page 241, line 10:

(iv) Page 241, equation (5·3):