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.

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.

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.

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. 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.)

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.

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):