In Sadowsky and Sternberg(1), elliptic integral representations of axially symmetric flows suggested essentially by Weinstein's work (2) on axially symmetric flows of an ideal incompressible fluid have been considered. The physical and practical significance of their investigation was twofold. First, a more transparent analytic description was obtained than that afforded by the representation through discontinuous integrals of Bessel functions originally used by Weinstein. Secondly, by superposition of such axially symmetric flows and appropriately chosen uniform streams, a variety of technically significant flows around solids, annular bodies and half-bodies of revolution can be constructed. On the other hand, in an attempt to utilize the symmetry properties of the potential field in reference to a spherical boundary, Weiss(3) has derived an (exterior) ‘sphere theorem’ by applying Kelvin's transformation and using the potential function. Butler (4) later obtained, by means of the Stokes stream function, a sphere theorem applicable to axially symmetric flows only. Ludford, Martinek and Yeh(5) found then the ‘interior sphere theorem’ as well as a theorem satisfying the general radiation condition. A general sphere theorem was consequently conceived, valid for all linear boundary conditions, and was recently published by Yeh, Martinek and Ludford (6). The significance of these theorems again lies in their application to physical problems. They often give closed form expressions of the disturbance potential in terms of higher transcendental functions whenever the undisturbed potential is given by means of transcendental functions. Furthermore, when the singularities (discrete or distributed) he near the perturbing spherical boundary the usual treatment by expansion in spherical harmonics leads to solutions in the form of infinite series which are, because of slow convergency, unsuited for numerical computation. For this situation the sphere theorems provide a remedy in the form of neat formulae readily adaptable to numerical work.

This paper depends upon the method developed by Kolossoff and Muskhelishvili for problems of plane elasticity and later extended to plate problems by Lechnitzky. Exact solutions in closed forms are obtained for the problem of a thin circular plate supported at several interior or boundary points and normally loaded over the area of an eccentric circle, the load being symmetrical with respect to the centre of the circle and the boundary of the plate being free. Explicit formulae for the deflexion, the bending and twisting moments and shearing stresses are given at any point of the plate. As limiting cases plates in the form of the infinite plane and half plane are also considered.

Following the method outlined in a previous paper, exact solutions in finite terms are obtained for the problem of an infinitely large plate with outer edge free and an inner free circular boundary, the plate being supported at any number of interior or boundary points and normally loaded over a circle. The load considered is symmetrical with respect to the centre of the circle.

This paper is concerned with the small transverse displacement of an infinite thin plane isotropic plate due to the application of a transverse force applied at an arbitrary point of the plate. The plate has its outer edge free, and is clamped along and bounded internally by a closed curve that can be mapped onto the unit circle by means of a polynomial. Three polynomials are considered and in each of these cases the deflexion is obtained in finite terms. Circular and elliptic holes as well as curvilinear polygonal holes are included as special cases.

The observation of faint radio sources is often limited by the angular resolving power of the radio telescope, and not by its sensitivity. Continuous recording of the output from the receiving apparatus then produces a trace which always shows the sum of the effects of a number of sources in the reception pattern, instead of showing sources passing through the reception pattern one at a time. The apparent intensities of sources found on the records are affected by the presence of several other, fainter, sources in the reception pattern, and even the existence of many of the faint sources is doubtful. Counts of numbers of sources at given intensity levels are therefore unreliable.

Any assumed number-intensity law for radio sources can, however, be tested by a method free from the uncertainties due to confusion between adjacent sources, using a direct relation between the number-intensity law and the frequency distribution of deflexions of various sizes on the observed records. The present paper shows the derivation of the relation between M (I) dI, the mean number of radio sources per steradian in the intensity interval dI, and P(D)dD, the probability that the deflexion D recorded at an arbitrary instant lies in the interval dD. Expressions for P(D) are worked out for some specific models of M (I); a comparison with observation has been given in an earlier paper (6).