1. Making use of the familiar abbreviation

let us adopt a contracted notation for the generalized hypergeometric function AFB(x) and write

where (a) denotes the sequence of parameters a1, a2,…, aA. It will be assumed throughout the present paper that there are A of the ‘a’ parameters, A′ of the ‘a′’ parameters, and so on. Thus [(a)]m is to be interpreted as

with similar interpretations for [(a′)]m, etc.

1. The object of this paper is to evaluate an infinite integral, involving the product of H-functions, generalized hypergeometric functions and confluent hypergeometric functions by means of finite difference operators E. As the generalized hypergeometric function and H-function are of a very general nature, the integral, on specializing the parameters, leads to a generalization of many results some of which are known and others are believed to be new.

The purpose of this paper is to present the concept of ‘modal subspaces’ for systems of coupled non-linear autonomous homogeneous second-order differential equations of the complex variables z1, z2. This development is an extension of the previous paper, entitled ‘Modal subspaces in the complex domain’, by Atkinson and Swann (6), which dealt with a pair of coupled non-linear autonomous homogeneous second-order differential equations of the form

Differentiation is with respect to a real variable t and ajk and bjk are constants which may be complex: n is any positive integer: f1 and f2 are the real and imaginary parts respectively of g1 and g2 are real and imaginary parts respectively of and zj = xj + iyj (j = 1, 2).

Let g0, gl, g2,…be a sequence of mutually independent, normally distributed random variables with mathematical expectation zero and variance unity. In this work, we obtain the average number of maxima (minima) of the random algebraic curves with the equations

This average is (½(3½ + 1)) log N + O((log N)⅔ (log log N)½), when N is large.

Over a period of many years there has developed an extensive literature on ballot problems. These, in effect, constitute a very special class of random walk problems, and their recent continued development has been justified by the apparent difficulty of reducing expressions given by the general theory down to the very simple ones that it is possible to obtain in an elementary fashion. In this short note we show that the obstacle presented by this reduction problem is actually a rather small one. For background to the above comments, together with a fairly comprehensive list of references to the ballot theory and its attendant applications, the reader is referred to Takács(2).

A Green function formulation of a system that describes the interaction of fields of arbitrary spin with the gravitational field andparticles is given. The Lagrangian considered is essentially the most general that admits of a description through Green functions.

In treating plane wave scattering by a periodic surface, Lord Rayleigh (10) assumed that the discrete, outgoing and evanescent plane wave representation for the scattered field was valid on the surface itself. Recently, this Rayleigh assumption has been questioned and criticized. For the surface y = b cos kx on which the total field vanishes, Petit and Cadilhac(8) have demonstrated its invalidity when Kb > 0·448. The present paper discusses scattering of a wave, incident from y > 0, by an analytic periodic surface y = f(x) ( – ∞ < x < ∞), and shows that the Rayleigh assumption is valid if and only if the solution can be continued analytically across the boundary at least to the line y = minf(x). Conformal mapping and results relating to the analytic continuation of solutions to elliptic partial differential equations reduce the problem to one involving the location of singularities and critical points of a potential Green's function. Provided that the perturbation of the surface from a plane is sufficiently gentle, the validity of the Rayleigh assumption is established. For the surface y = b cos kx it is shown that the assumption is valid if Kb < γ, where γ is a positive number no greater than 0·448, the precise value of which is unknown. Possible extensions of the analysis to different or more general situations are suggested.

The study of the slow motion of a solid body through a viscous fluid is of importance in many fields. We may instance viscometry and chemical engineering as two subjects in which a quantitative understanding of such motions is essential. Thus, in recent years systematic attempts have been made to calculate the drag force and couple on slowly sedimenting particles in a bounded viscous fluid. Using the ‘method of reflexions’ Brenner (1) has obtained formulae which give, for a wide class of sedimenting particles, an estimate of the effect of the container walls on the drag force, the error being O(c/l)3, where c is a characteristic particle dimension and l the distance from the centre of the particle to the nearest boundary wall. The reader is referred to the recent book by Happel and Brenner (2) for a complete account of these and other pertinent investigations; also to Williams (3) for an alternative integral equation approach to these results.

Maximum and minimum principles are developed for boundary-value problems with field equations T*Tφ + f(φ) = 0 in some region V, subject to φ = h(Tφ) on the boundary of V.

The problem stated in the title is studied for small values of the diffusivity ratio ε and the magnetic force coefficient N with the aim of finding the effect of rotation and the disc conductivity on the magnetic and velocity fields, currents, shear stresses, etc.

The elastic fields in an elastic circular inclusion and its surrounding infinite dissimilar elastic matrix, are determined when either the matrix or inclusion is subject to a concentrated force or edge dislocation.

Limiting solutions are derived for the flexure of simply supported many-sided regular polygons, as the number of sides is increased indefinitely. It is shown that these solutions are different from those for simply supported circular plates. For axisymmetric loading, circular plate solutions overestimate the deflexions and the moments by significant factors.