1. The main object of this note is to evaluate two definite integrals involving the product of Meijer's G-function with Gauss's hypergeometric function. The first result established in this paper is the extension of the results recently given by Saxena ((3), page 490) in these proceedings and includes both of his results given there as particular cases.

In 1955 Noll (7) gave a definition of an anisotropic fluid. Noll's idea is developed in more detail and the results of applying the theory to some simple flows are compared with those obtained by Ericksen (1) from his theory of transversely isotropic fluids.

Formulae are obtained for the Laplace transforms of the first and second partial derivatives of a function of time and position which, together with its first partial derivatives, is discontinuous on a moving surface. These formulae are then used in a discussion of the application of the Laplace transform method to the solution of mixed initial and boundary-value problems in linear thermoelasticity. Particular attention is given to thermoelastic disturbances involving plane shock waves, and it is pointed out that no properly posed formulation of problems of this type has yet been found.

1. Introduction. In this note, we consider some estimates for the modulus of infinite products of the form

where the lv are near to an arithmetic progression of common difference unity. In a subsequent note, an application will be given of these estimates to the study of a class of Cauchy exponential series which includes, as a special case, the series of Korous(l).

Numerical integration of the partial differential equations shows that rigid boundaries promote the development of a shock from a finite-amplitude disturbance. A simple step-by-step method agrees with one that also utilizes the characteristics.

1. Introduction and summary. There has been considerable interest in the problem of expanding a Coulomb potential l/r12 in spherical polar coordinates defined relative to two centres A(0, 0, − d) and B(0, 0, + d) where 2d = D = rAB.

1. Let

be a positive definite binary quadratic form with determinant αβ − δ2 = 1. A special form of this kind is

We consider the Epstein zeta-function

the series converging for . The function Zh(s) can be analytically continued over the whole s-plane and it is regular except for a simple pole with residue π at s = 1.

1. Introduction. The problem considered here derives its motivation from a paper by Friedlander (8) on the propagation of small disturbances in a compressible, conducting fluid in the presence of a uniform magnetic field (see also Courant and Hilbert (3), VI, §3a). In this the displacement current and energy dissipation by viscosity, heat conduction and Joule heat are neglected and a system of linear partial differential equations is obtained, which generalizes the equations of motion of the theory of sound. Their solution is in general the superposition of an arbitrary incompressible Alfven wave and a magneto-acoustic disturbance. This latter was considered by constructing a Green's function by means of suitable combinations of plane wave solutions and it was found that there are fast and slow wave fronts diverging from a point disturbance. The latter are conoidal in shape and have a singularity at their vertices which propagate along the field line in either direction from the source.

The unsteady two-dimensional boundary-layer equations, linearized as by Lighthill are studied. A unified method is developed, from which the results for the stagnation flow, the flow along a flat plate, the flow in a converging canal, etc., can be derived as special cases. Solutions are obtained in two systems, one when the main stream is in unsteady motion and the wall is at rest and the other when the main stream is in steady motion and the wall is in an arbitrary motion. The stagnation flow has been done by Glauert and generalized by Watson. The flow along a flat plate and the flow in a converging canal are considered in detail.

We shall be concerned in this paper with a class of temporally homogeneous Markov processes, {R(t), X(t)}, in discrete or continuous time taking values in the space

The marginal process {X(t)} in discrete time is, in the terminology of Miller (10), a sequence of random variables defined on a finite Markov chain. Probability measures associated with these processes are vectors of the form

where

We shall call a vector of the form of (0·2) a vector distribution.

In a previous paper we discussed a uniform algebraic method of solution of problems in which a prescribed real rational function (or polynomial) g(x) was to be approximated in a given finite interval by a real rational function (or polynomial)f(x) with prescribed numerator and denominator degrees, the approximation being Tchebysheffian, i.e. such as to make the ‘deviation’ of f, max |f − g| in the interval, as small as possible.

Our previous discussion ((4), (5)) is here extended to an ellipsoid of arbitrary conductivity. Again, the flow is ultimately cylindrical and the magnetic field undisturbed. An ellipsoidal insulator allows fluid to pass through its circumscribing cylinder, but also forces some to go around. An ellipsoidal conductor behaves like a very tall insulator, irrespective of its own height or conductivity: all the fluid goes round. If it is very wide an unexpectedly large amount of fluid is carried with it.

One of the most important techniques in the theory of non-linear differential equations is the direct method of Lyapunov and its extensions. It depends basically on the fact that a function satisfying the inequality

is majorized by the maximal solution of the equation

Using this comparison principle and the concept of Lyapunov's function various properties of solutions of differential equations have been considered (1–11).

1. Introduction. The stability of a system of queues in series was investigated in (1) but the results obtained were not quite complete, for the situation in which all component queues are substable and at least one is critical was scarcely touched upon. The only reference to this situation, in fact, was an example on page 513 showing that necessary and sufficient conditions are not obvious. (The ideas and terminology of (1) will be used freely throughout this paper.)

Formulae for the elastic stress distribution in an infinite slab due to symmetrical pressure distributions on the upper and lower faces have been given by Filon (two dimensions) and by Sneddon (axial symmetry). Few numerical values have been published except for the limiting cases of line and point loads. The stresses in the mid-plane are given here for a uniform pressure acting over circles and bands of different sizes, and the published values for a point load are shown to be wrong.

The replacement of the second space derivative by finite differences reduces the simplest form of heat conduction equation to a set of first-order ordinary differential equations. These equations can be solved analytically by utilizing the spectral resolution of the matrix formed by their coefficients. For explicit boundary conditions the solution provides a direct numerical method of solving the original partial differential equation and also gives, as limiting forms, analytical solutions which are equivalent to those obtainable by using the Laplace transform. For linear implicit boundary conditions the solution again provides a direct numerical method of solving the original partial differential equation. The procedure can also be used to give an iterative method of solving non-linear equations. Numerical examples of both the direct and iterative methods are given.

It is assumed initially that only the increase of entropy defines locally the sense of advance of time. This assumption, together with the feature of global coherence of statistical processes, which is provided by the cosmic line-element, imply that there is a unique way of associating the standards with the cosmic statistical processes. There is hence a connexion between the sense of divergence of geodesics of fundamental observers and the cosmic time-sense. This is in keeping with the unmodified conservation laws of gravitation and the analogous adiabatic conservation law quite generally without dependence on the explicit form of the line element. These remarks correspond exactly to consequences of the ‘elliptic’ model in the case of de Sitter space, obtained by entirely separate geometrical considerations. A remark is made with regard to the irreversibility associated with quantum measurement.