Green's function for SH-line source inside a wedge shaped medium with its apex removed by a circular cylindrical surface has been obtained. Exact location of the image sources facilitates the determination of the reflexion and diffraction at the circular and also at the plane boundary of the multiple phases clearly. The scattered field from the corner is indicated. The connexion of this solution with that of complete wedge has been shown with suitable contour integrations.

An analysis is made of the stability of a layer of an elastico-viscous liquid flowing down an inclined plane in the presence of two-dimensional disturbances. The modified Orr-Sommerfeld equation is solved by a regular perturbation technique for disturbances of large wavelengths. It is shown that in the absence of surface tension, the layer is more unstable as compared with that for an ordinary viscous liquid if Q1 > Q2, Q1 and Q2 being stress relaxation and strain retardation parameters respectively.

We recall that a square matrix (finite or infinite) with non-negative elements and with each row-sum and column-sum equal to 1 is called doubly-stochastic (d.s.). If each row-sum and column-sum of a non-negative square matrix is less than or equal to 1, the matrix is called doubly-substochastic (d.s.s.).

The two fourth-order partial differential equations giving the transverse displacement and the stress functions for shallow shells are reduced to a sixth-order differential equation in one variable which has been solved, and the values of displacements, stress resultants and couples are all expressed in terms of Bessel Functions of imaginary argument. Numerical values of the displacement presented for points on the shell on different meridian lines, showed that on the symmetrical line deflexion is maximum at the point of loading but the point of maximum deflexion shifts gradually towards the pole.

The part of combinatorial analysis which deals with systems of representatives draws on a variety of different sources. Thus the theory of graphs, the theory of flows in networks, linear programming, and integral linear programming all provide relevant methods and arguments. Nevertheless, it is in all likelihood possible to give a unified set-theoretic account of the subject, based in the main on Hall's criterion for the existence of transversals and on a certain generalization of the Schröder–Bernstein theorem. This approach has the added advantage of greater flexibility since it enables us sometimes to cope with transfinite situations where other methods may be only applicable to finite cases.

A simple formula is derived for the connexion between the scattered fields due to a sound-soft, a sound-hard and a perfectly conducting circular disc respectively.

Some features of the linear theory of an elastic Cosserat plate are discussed and special attention is given to a bending theory which corresponds to the bending of a transversely isotropic three-dimensional plate. A useful representation for the solution of the system of differential equations characterizing the bending of an elastic Cosserat plate is deduced and the results are applied to the problem of an infinite Cosserat plate with a circular hole, the plate being in a state of uniform bending at infinity.

Consider the two-dimensional irrotational steady flow of an incompressible inviscid fluid under no body forces between two infinite horizontal planes. When a vertical wall which intersects the upper plane is introduced in the flow, the fluid will flow under the wall and will be bounded above by a free streamline along which the pressure is constant. This configuration is called a sluice gate. At infinity the flow will be uniform and parallel with input velocity V and exit velocity U.

The non-linear equations governing the convective flow of heat, beyond marginal stability, in a fluid sphere containing a uniform distribution of heat sources, and cooled at its surface, are expressed in the Boussinesq approximation as a perturbation of the steady-state conduction solution. A steady-state axisymmetric poloidal flow is assumed, and the equations for an approximate solution, involving the first spherical harmonic only, are obtained using a special case of the general evolution criterion of Glansdorff and Prigogine. The solution is found numerically using an iterative procedure, the results showing good agreement with those obtained by Stuart's ‘shape assumption’ for values of the Rayleigh number just above the critical value for the onset of marginal stability.

The question has arisen in the theory of queues whether the product of two independent exponentially distributed random variables is infinitely divisible. In this note it is proved that the product of any non-negative random variable with an independent exponentially distributed variable is infinitely divisible. Such random variables are members of a wider class of infinitely divisible random variables, which will be exhibited first.

It is argued that the zero Reynolds limit of the steady incompressible axially symmetric viscous fluid motion interior to a sphere due to a Landau source at the centre is a Stokes flow. The first three terms of the perturbation expansion are determined and the order of magnitude of the general term not derivable from the Landau source is established. Comparison of the convection terms with the diffusion terms for each order of the Reynolds number demonstrates self consistency at each stage of the expansion.

The growth of the laminar compressible boundary layer on a moving flat wall is considered analytically for the case of zero velocity in the free stream outside the boundary layer. The results of this analysis are compared with other published results for the cases in which the free stream has some finite velocity. In all the cases considered in the present paper, the boundary layer is taken to originate at some stationary point on the moving wall. This type of boundary-layer flow occurs behind moving shock waves and it is argued that the case of particular interest in the present paper, that of the stationary gas outside the boundary layer, provides bounding values of such parameters as displacement and momentum thicknesses for shock-induced laminar boundary-layer flows.

An approximate formula is found for the field produced on the boundary of an obstacle in an inhomogeneous medium by a source which is close to the boundary. Both harmonic and aperiodic waves are discussed.

The laminar compressible boundary layer induced by the passage of a plane shock wave over a flat wall is examined in detail. Use is made of the empirical viscosity-temperature relationship μ ∝ Tω. The boundary-layer equations are solved numerically for various values of the index ω, Prandtl number and shock strengths. The resulting solutions are then used to construct simple semi-empirical relationships for some of the more important boundary-layer parameters.

Energy of the exterior and the interior fields of a particle, charged or otherwise, embedded in an empty space is calculated, using Møller's tetrad formulation of the energy-momentum complex in general relativity. It is found that the exterior field contains positive gravitational energy, and in addition, the electromagnetic energy if the particle is a charged one and that the sum total of the energies of the interior and the exterior fields is equivalent to the Newtonian mass of the particle.

The solution by Powell (6) of the diffraction of a complete expansion wave by a corner is extended to an incomplete wave. It is shown that the pressure is discontinuous across one of the boundaries and becomes logarithmically singular at a point on it. Graphs of the pressure distribution on the fixed boundary are given.