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.

It is shown that the solution of the Rikitake two-discdynamo system may be described by an orbiting point which, for sufficiently large time, lies arbitrarily close to a limit surface of bounded area. Reversal of the field currents of the dynamo coils are shown to occur in the juxtaposed regions of the two sheets of the limit surface. The two singular points of the system are shown, by Liapounov's direct method, to be unstable foci. An asymptotic theory is developed for the case of small difference in dynamo velocities; the theory is applied to cases of small dissipation in which orbits tend to become nearly periodic with a single oscillation between reversals.

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.

The internal labelling problems for the group-subgroup combinations Sp (2n) ⊃ Sp (2n − 2) × Sp (2) and O(n) ⊃ O(n − 1) are solved by the method of elementary multiplets.