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.

The kernel of the integral equation for the source function in a three-dimensional homogeneous atmosphere possesses the properties of a Green's function. These properties are used to transform the integral equation into a singular integral equation for the kernel. The particular case of a homogeneous plane parallel atmosphere is discussed and a solution to the kernel equation is obtained at all points of the atmosphere.

The formula

gives the conditional probability of an (r1 × r2 × … × rk) k-variate contingency table (where is the entry in the cell identified by the k indices im), given all the marginal totals

and the hypothesis that the k random variables tabulated are mutually independent. This formula appears in (1) and more restricted forms can be found elsewhere in the literature. Certain criticisms have been made of the logic and clarity of the available proofs of (1); it is hoped that the alternative derivation given here will dispel any residual doubts, and will provide better insight into the nature of the problem.

The method of matched (or of ‘inner and outer’) asymptotic expansions is reviewed, with particular reference to two general techniques which have been proposed for ‘matching’; that is, for establishing a relationship between the inner and outer expansions, to finite numbers of terms, of an unknown function. It is shown that the first technique, which uses the idea of overlapping of the two expansions, can be difficult and laborious in some applications; while the second, which is the ‘asymptotic matching principle’ in the form stated by Van Dyke(13) can be incorrect. Two different sets of conditions sufficient for the validity of the asymptotic matching principle are then established, on the basis of assumptions about the structure of expressions which approximate to the desired function f(x,∈) for all relevant values of x. Finally, it is noted that in four classes of singular-perturbation problems for which complete and rigorous asymptotic theories exist, uniform approximations to the solutions have a structure which is a particular form of the general one assumed in this paper.

It is shown that Eddington's Fundamental Theory admits only four basic operators. The symmetry properties and the reality conditions of the E-frame of reference are formulated in an abstract way.

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).

Complementary variational principles associated with the boundary-value problem ∇2φ = f(φ) in ∇,φ = φ0 on the boundary of ∇, are presented. The theory is applied to the Thomas–Fermi equation in quantum mechanics and the Liouville equation in equilibrium statistical mechanics.

The prescription for forming composite series, and a modified form of that procedure, are used systematically and extensively to derive uniform asymptotic approximations to various given functions which cannot be approximated uniformly by expansions of classical (Poincaré) type. The formal method of matched expansions is then applied to a boundary-value problem for an ordinary differential equation with a turning point. It is proved with the help of the uniform approximations found earlier that in this problem a restricted form of the asymptotic matching principle is valid, even when it is applied to truncated inner and outer expansions which do not overlap to the order of the terms being matched.

An exact solution is given for the slow viscous flow caused by the translation of two equal spheres in contact with equal velocities perpendicular to their line of centres. An asymptotic theory is presented for solving the problem when two equal spheres of radius a almost in contact rotate with equal and opposite angular velocities. The problem when the spheres touch is shown not to be well posed; the forces acting on the spheres are shown to be O(1) and the couples of the form α log∈ + β where α and β are independent of the minimum clearance 2∈α between the spheres and have been determined explicitly. The relevance of the results to the free settling of two spheres in a viscous fluid under the influence of gravity is discussed.

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.

The formal method of matched expansions is applied to two further examples. The first concerns the magnetic field induced by a steady current in a thin toroidal wire. The second, which involves a non-linear ordinary differential equation of the fourth order, has been chosen to resemble the problem of flow past a circular cylinder at small Reynolds numbers. The results of the formal procedure are proved in each case to be expansions of the exact solution.

The pressure fluctuations in a compressible fluid in statistically steady isotropic motion are studied. These depend crucially on the scale of the turbulent region. If the region is small enough, the mean square acoustic pressure varies as the linear scale of the region, and as the fourth power of the turbulence Mach number. In the limit of infinite scale, however, diffusive effects limit the otherwise infinite pressure fluctuations as first shown by Lighthill(6). The mean square acoustic pressure then varies as the Mach number and as the turbulence Reynolds number. The pressure fluctuations above a sheet composed of statistically uniformly fluctuating sources are also examined. Here the mean square pressure diverges, initially, in proportion to the logarithm of the scale of the sheet, until viscous effects again become significant in limiting the pressure.

An analytical study is made of the boundary-layer approximation for the steady laminar flow of a viscous incompressible fluid past a paraboloid of revolution. The functional form of the vorticity distribution is found and shown to involve exponential decay with distance from the surface of the body, and inequalities are established concerning the displacement thickness.

The exact results are compared with various linearizations. The Oseen linearization (12) turns out to be incorrect except in the limit of vanishing Reynolds number; this result is connected with the infinite drag of any paraboloid of non-zero thickness. An alternate linearization, related to a method of Burgers (3) and allowing for the displacement effect of the body and of its boundary layer, is shown to give the correct functional form for these quantities. Numerical results are included which bear out these conclusions.

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.

Basis functions are derived and tabulated for the double-valued representations of the triclinic, monoclinic, orthorhombic, trigonal, tetragonal and hexagonal crystallographic point groups using the method of projection operators.

The theory of laminar boundary layers is employed to determine the rate of decay of small amplitude oscillations of a non-homogeneous viscous fluid which completely fills a stationary rectangular container. An application of Green's formula is invoked and an expression is obtained in terms of the (general) density and viscosity distributions.

Using a technique developed by Lighthill, surface waves are studied when a concentrated pressure point oscillating with a constant frequency moves along OX in a rotating frame O X Y Z on the free surface of rotating liquid bounded below by a horizontal plane. The effect of rotation is to split the coincident gravity modes of the non-rotating case and further to produce a new system of wavelets (countably infinite) which do not have any counterpart in the non-rotating case.

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.