In this paper we examine the general paraboloidal co-ordinate system, in which the normal surfaces are elliptic or hyperbolic paraboloids, including as special cases the “parabolic plate” and the “plate with a parabolic hole”. We then show that normal solutions of Laplace's equation in these co-ordinates are given as products of three Mathieu functions, and apply this to the solution of boundary-value problems for Laplace's equation in these co-ordinates. In a subsequent paper the corresponding treatment of the wave equation will be given.

This paper formulates a general solution, within the scope of classical elastostatic theory, for the problem of layered systems subjected to asymmetric surface shears. As an illustrative example the solution for the problem of an elastic layer supported on an elastic half-space is presented for the particular loading consisting of a surface shearing force uniformly distributed over a circular area. Numerical results are included indicating some displacement and stress components of interest.

The solution of the dual series relations

where {λn) is the sequence of positive zeros of the Bessel function Jν(αλ), arranged in order of increasing magnitude, þ and ν are real numbers (−1 <þ < 1, ν >0), the functions, f1(ρ), f2(ρ) being prescribed, is obtained by giving an integral representation of {αn} in terms of a single function g(t). The problem is reduced to that of solving a Fredholm integral equation of the second kind for the auxiliary function g(t).

The problem discussed is that of determining the sequence {an} such that

where (λn) is the sequence of positive zeros of the function λJν(λ) + HJν(λ), arranged in order of increasing magnitude, þ, ν and H are real constants (−I <þ < I, ν > −½) and f1(ρ), f2(ρ) are prescribed. By expressing the sequence {cn} in terms of a sequence of integrals involving a function g(t) the problem is reduced to the solution of a non-singular Fredholm integral of the second kind for g(t).

The methods developed in I, II of this series of papers are applied to a solution of a variety of dual series relations involving trigonometric series. In general the problem is reduced to one of solving (usually by numerical methods) a Fredholm integral equation of the second kind for an auxiliary function g(t), but for certain values of the parameters it is possible to obtain analytical solutions of the integral equations and these cases are considered in detail.

It is shown that by using the methods developed in papers I–III of the present series it is possible to reduce the problem of deriving the solution of a certain class of dual relations involving Jacobi polynomials to that of solving an integral equation of Schlömilch kind.

It is generally agreed that the long-range alpha particles of fission are set free before the fragment nuclei have acquired more than a small fraction of their final energy of separation, but whether the alpha particle is liberated before the instant of scission, at that instant, or from one of the fragment nuclei very shortly thereafter, has remained an open question. Each of these views has been seriously advocated. These various hypotheses are examined in relation to recently published information regarding the distribution of mass in low-eneigy ternary fission, and other considerations, and it is suggested that the hypothesis having the strongest claim to attention is that which assumes that the alpha particles originate in the heavy fragments exclusively, being liberated, very shortly after the instant of scission, with probability not much less than unity, from fragment nuclei of low yield and small neutron excess. Conclusions which would follow, if this hypothesis were accepted, are indicated, and possible experimental tests of these conclusions are suggested.