A circular disc of radius a, made of homogeneous, isotropic, linearly elastic material, contains a radial edge crack of length b(0 < b < 2a). The disc is in equilibrium in a state of generalized plane stress caused by loading the faces of the crack by a constant internal pressure. The problem of determining the resulting stress field throughout the disc is solved analytically in closed form. The principal results are that the stress concentration factor at the crack tip, the total strain energy W, and the opening U at the mouth of the crack, are given exactly by

where A is a constant whose value correct to 6 significant figures is

and , W0, U0 are normalising factors defined in section 6.

A generalization of the notion of Riemannian submersion is given for submersions of a Lorentzian manifold onto a Riemannian manifold of one lower dimension. These maps are called comersions. A comersion induces a surjection of the space of causal curves in the domain manifold onto curves of bounded length in the image manifold. Thus if the source space is causally 1-connected, the target space is 1-connected.

(i) J. Baumgartner has kindly drawn our attention to the fact that Theorem 2 as stated in (1) is false. A counter example is the case in which m = ℵ2; n = ℵ1; p = ℵ0. For by reference (3) of the paper (1) there is an almost disjoint family (Aγ: γ < ω1) of infinite subsets of ω̲ Put Aν = ω̲ for ω1 ≤ ν < ω2. Then, contrary to the assertion of that theorem, all conditions of Theorem 2 are satisfied. However, Theorem 2 becomes correct if the hypothesis

is strengthened to

In fact, Baumgartner has proved the desired conclusion under the weaker hypothesis

The general multipole solutions obeying the appropriate boundary conditions are presented for stationary non-axisymmetric electromagnetic fields in the Kerr background geometry.

In this paper we investigate a time harmonic plane strain problem in a homogeneous isotropic semi-infinite elastic solid with mixed boundary conditions on its surface. The boundary is free from applied stress apart from on an infinite strip where the tangential components of stress are zero, and the normal component of displacement is a prescribed function. By reducing the mixed boundary value problem to the solution of a Fredholm integral equation of the second kind, an existence theorem is proved and a simple low frequency asymptotic formula relating the normal stress to the prescribed displacement is derived rigorously. The results are applied to the forced heaving and rocking motion of a rigid cylindrical body in smooth contact with an elastic half space. Some further applications to scattering problems are suggested.

For an elastic solid the constitutive law can be written in terms of the deformation gradient α and its conjugate nominal stress s ≡ s(α), and also in terms of the right stretch u and its conjugate stress τ ≡ τ(u). It is shown that for isotropic elastic solids s(α) is invertible, in the local sense, for all u in the domain of u-space where τ(u) is locally invertible, with the exception of certain configurations which correspond to planes in τ-space.In the global sense a given s corresponds to four distinct τ's, and s is invertible to give four distinct α's when the corresponding τ's are uniquely invertible. That there are four branches of the inversion α(s) is of fundamental importance in that it clarifiesthe extent to which non-uniqueness of solution of boundary-value problems can be expected.

The implications of these results in respect of the complementary variational principle are discussed, and the controversy surrounding the use of nominal stress in this principle resolved.

Consequences of the required restrictions on τ(u) are examined and discussed in relation to inequalities which may be regarded as entailing physically reasonable response. It is intimated that τ(u) is invertible in the domain of elastic response.

In this paper, a free-boundary problem of the steady plane flow of an ideal fluid through a slot is solved. The fluid flows under the effect of gravity between rigid boundaries, and then out of a slot as a jet which becomes horizontal at infinity downstream. A constructive proof of the existence of solutions to a non-linear integral equation is given for a parameter range 0 < ε < ε* ≃ 1·02 (where ε, the inverse Froude number, measures the effect of gravity). Approximations to the solution are then found for ε → 0.

A laboratory model simulating aspects of oceanic or atmospheric circulationconsists of a rotating fluid annulus which is subject to a temperature gradient on the lower surface and is insulated on the inner and outer curved walls and on the top surface. A theoretical study of the resulting steady axi-symmetric fluid motion in the limit of small Ekman number, E, by Daniels (2) indicated that if an appropriate convective parameter is sufficiently large, oscillatory solutions are generated in the layer which is located on the hotter of the two side-walls, which do not decay into the geostrophic interior. The present paper resolves this difficulty by using the method of multiple scales to trace the development of the oscillatory solutions in the interior of the fluid. This leads to a consistent overall flow pattern throughout the annulus, except at an infinite set of discrete values of the convective parameter at which resonance is predicted.