1. Introduction: In (4) Moran considers two independent random variables X and Y taking non-negative integral values to give a characterization of the Poisson distribution. He establishes that the conditional distribution of X, given the total X + Y, is binomial for all given values of X + Y and there exists at least one i so that P(x = i) > 0, P( Y = i) > 0 if and only if X and Y have Poisson distributions. A slightly improved version of this result is given by Chatterji (1). For a comprehensive bibliography on the Poisson distribution the reader is referred to (3).

A quantum mechanical theory of the action-at-a-distance electrodynamics of Wheeler and Feynman is given using an S-matrix approach. The response of the universe is introduced, and a perturbation expansion leads to the usual expression for the spontaneous transition rate between atomic energy levels, an effect normally attributed to quantized field oscillators. The Feynman propagator is then recovered, leading to the familiar self-energy formulae. Finally, a comparison of the formal structure of the new theory with the conventional is shown to establish a complete mathematical equivalence to all orders in the expansion.

We denote by tm(n) the number of partitions of the positive integer n into non-decreasing parts which are positive or zero powers of a fixed integer m > 1 and we call tm(n) ‘the m-ary partition function’. Mahler(1) obtained an asymptotic formula for tm(n), the first term of which is

Mahler's result was later improved by de Bruijn (2).

Proc. Cambridge Philos. Soc.63 (1967), 1273–1280

Our object here is to refine the theorem proved in (3), and we use the notation of that paper. Let Z1, Z2, …, Zn, where Zr = (Xr, Yr), be independent random variables in two dimensions with zero first-order moments and finite third-order moments; and et the covariance matrix of Zr be

We write

Let (x, y) be the distribution function of the sum and let (x, y) be the normal distribution function having the same first- and second-order moments as (x, y).

Analytical reflexion coefficients are obtained for horizontally and vertically polarized electromagnetic waves respectively, obliquely incident from below two recently devised models of horizontally stratified isotropic media. It is shown that both models are totally reflecting and the phases of the reflexion coefficients may be expressed in terms of arg Γ(z).

In (1) and (2), Gould has discussed some generalized Vandermonde-type convolution identities, for C coefficients satisfying

where

and

Complementary variational principles are developed for boundary-value problems with field equations φiv = f(φ) in (0, 1) subject to φ(0) = φ(1) = 0, φ′(0) = φ′(1) = 0. The theory is applied in the non-linear case f(φ) = – keφ and accurate variational solutions, which are simple polynomials in x, are obtained.

Geometric structure of classical field theory in Lagrangian formulation is investigated. Symmetry transformations with generators depending on higher-order derivatives are considered and the corresponding conservation laws are obtained.

In any V4 two quadratic invariants of the Riemann tensor, J4 and K4, say, are known which are functionally constant, i.e. the Hamiltonian derivatives of which vanish identically. Certain natural generalizations, J2n and K2n, of degree n of these are considered, J2n being defined in any Vm but K2n only in a V2n. It is shown by methods having purely local character that J2n is functionally constant in any V2n, and that K2n vanishes identically when n is odd or when n is even and the V2n is either conformally flat or admits a hypersurface-orthogonal Killing vector; whilst in a V4n it is functionally constant. These results follow upon a prior spinorial treatment of the special case n = 3 which leads to an explicit expression for a pseudo-vector density of which the scalar density 6 is the divergence. Finally, it is shown that for appropriate values of n there exist functionally constant invariants other than J2n and K2n

When Rossby number is small but Ekman number is very much smaller, study of the flow field far from a body moving at right angles to the axis of rotation of a large body of fluid indicates that the region of influence should not be a Taylor column parallel to the axis, but a trailing Taylor column, bent backwards on both sides of the body at a small angle (proportional to Rossby number) to the axis. The paper reviews the physical significance of, and experimental evidence for, this conclusion.

A static metric is considered. A meaningful definition of gravitational force is given and the potential, which is the norm of the Killing vector ξa, is studied. For the case that the metric is asymptotically flat, the following is shown: The equi-potential surfaces are closed 2-dimensional surfaces S lying in the rest space V3, which is the hypersurface orthogonal to ξa. All the surfaces S enclose matter, and the gravitational force points into S towards the enclosed matter. A test particle starting at S will be pulled into the domain bounded by S and will never leave this domain.

The title problem is investigated with the view of clarifying the nature of the separation in an adverse pressure gradient when the suction υ0 is very strong. A linear model is investigated, but it is argued that this must contain many of the essential features of the exact (non-linear) problem. With a free-stream U = 1 − x the similarity solution u = (1 − x)f(y) is appropriate when (υ0x − y) ≫ 1, x = O(1), 1−x = O(1). There is an O(1) layer centred on y = υ0x in which more complex x-dependence first appears and then, when (y − υ0x) ≫ 1, the solution assumes its final asymptotic behaviour. The y = υ0x layer is non-uniform in x and exhibits a marked change in behaviour when (1 − x;) = O(l/υ0). The solution in this narrow rectangle is singular like (1 − x)−2 at the trailing edge. It is these singular terms that induce separation and reversed flow in an exponentially small neighbourhood of the trailing-edge.

It is shown that a stationary C3 vacuum space-time has an analytic structure with respect to which the metric gab is an analytic tensor field.

This paper is concerned with the unsteady hydromagnetic boundary-layer flow past a semi-infinite flat plate when the oncoming free stream is perturbed by an arbitrary function of time and the applied magnetic field is parallel to the plate far away from it. Following Lighthill, the two-dimensional boundary-layer equations are separated into those representing steady and unsteady parts of the flow and they have been solved in sequence. For the unsteady part of the motion two types of solutions are obtained, one for large times and the other for small times. Also the quasi-steady solution is obtained in terms of the steady solution. The skin friction and the tangential magnetic field at the plate are calculated.

Steady motion of a viscous incompressible fluid in a rotating circular cylinder with a sloping bottom is investigated at low Ekman number. The flow is driven by a lightly faster rotation of the top, and non-linear inertia terms are neglected.

A solution is found for a shallow container of small bottom slope. The side-wall boundary layer is shown to have an almost axi-symmetric component as well as the asymmetric layer found by Pedlosky and Greenspan (3). A further asymmetry in the interior flow is produced by the presence of the second component of the side-wall boundary layer.

The axisymmetric potential problem for a plane circular electrode of radius a in a concentric hole of radius b in a coplanar earthed sheet is formulated in terms of triple integral equations for the Hankel transform of the potential, and reduced to a single Fredholm equation by use of the Erdélyi-Kober fractional operators.

In the limit of small gap width (b − a)/b, the equation takes the form

which is solved by applying the Wiener-Hopf technique to the Mellin transform of f(x). This leads to the asymptotic expression

for the capacity of the disc; for the opposite limit the expression

is derived. Numerical integration of the governing Fredholm equation has been carried out for a range of intermediate values of b/a.