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).

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

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.

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

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.

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.

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.

A singular integral equation is derived for the three-dimensional Fourier transform of the source function, when the scattering atmosphere is contained in a finite convex volume.

This equation is shown to reduce to the usual equation in the case of an isotropic point source in a finite spherical atmosphere of radius R0, and is used to solve the same problem when the source is anisotropic.

It is shown that in the latter case an expansion in Legendre polynomials results, in which the coefficients are obtained from the integral equations of a similar construction to those for an isotropic source. The error in taking the dominant part is now however of order 1/R0 and not e–R0.

The integral equation for the three-dimensional Fourier transform of the source function, derived in a previous paper, is applied to scattering in a spheroidal atmosphere. This problem is not amenable to solution by any of the general methods that have been successful for problems of finite atmospheres of a more regular shape.

It is shown that if the dimensions of the spheroid are large the solution can he obtained by iteration as an expansion in powers of 1/R0, where R0 is a typical length. The leading term is shown to be that for the infinite atmosphere, which result would be anticipated on intuitive grounds.

A pair of identical circular discs, held at equal and opposite potentials, forms a condenser whose capacity C depends on the ratio ε of separation against diameter. The determination of an asymptotic expansion for C when ε is small poses an axisymmetric boundary-value problem for harmonic functions that has engaged the attention of numerous investigators over a long span of time. It is a simple matter to construct a Fredholm integral equation of the first kind for the charge density ± σ on the discs, in terms of which the potential field and the capacity are implicitly determined, but the equation is unsuitable if ε ≪ 1. Integral equations of the second kind and of the dual variety have also been proposed as a means of securing a more manageable formulation of the boundary-value problem. An elementary approximation follows from the hypothesis that the charge density is almost the same as though the discs were of infinite extent, except for a region close to the edges, and leads to the result C ∼ l/8ε as ε → 0. Kirchhoff considerably improved on this crude estimate by suggesting a plausible edge correction which yields two further terms for C, of orders log ∈ and a constant, respectively, and his results have been rigorously established by the more refined analysis of Hutson. In the present work an integral equation of the first kind for the distribution of potential off the discs is derived and utilized to obtain an approximation for C when ε is small, reproducing the result of Kirchhoff and Hutson. Furthermore, an estimate of the error provides explicit details regarding the next term in the asymptotic expansion of C, which is of the order ε(log ε)2.