The two-dimensional flow in a jet, falling under gravity from a slot in a horizontal plane, is studied. The fluid is considered to be incompressible and inviscid; the flow is taken to be irrotational; and the reciprocal ε of the Froude number is considered to be small. By taking the complex potential as the independent variable we overcome the difficulty that the boundary geometry is not known in advance. The method of matched asymptotic expansions is applied. The first two terms of an inner asymptotic expansion and the first three of an outer one are found: the inner expansion is valid above and near the slot, but is inappropriate far downstream, while the outer expansion is valid far downstream, but fails to satisfy the conditions upstream. The two expansions are matched and ‘composite’ approximations, covering the whole flow field, are derived.

A method of finding explicit expressions for the roots of a certain class of transcendental equations is discussed. In particular it is shown by determining a canonical solution of an associated Riemann boundary-value problem that expressions for the roots may be derived in closed form. The explicit solutions to two transcendental equations, tan β = ωβ and β tan β = ω, are discussed in detail, and additional specific results are given.

We shall consider a non-stationary Markov chain on a countable state space E. The transition probabilities {P(s, t), 0 ≤ s ≤ t <t0 ≤ ∞} are assumed to be continuous in (s, t) uniformly in the state i ε E.

By the use of a semi-martingale property of the Kolmogorov supremum, the results of Pyke (6) on the weak convergence of the empirical process with random sample size are simplified and extended to the case of p(≥1)-dimensional stochastic vectors.

Positive definite functions on metric spaces were considered by Schoenberg (26). We write σk for the unit hypersphere in (k + 1)-space; then σk is a metric space under geodesic distance. The functions which are positive definite (p.d.) on σk were characterized by Schoenberg (27), who also obtained a necessary condition for a function to be p.d. on the it sphere σ∞ in Hilbert space. We extend this result by showing that Schoenberg's necessary condition for a function to be p.d. on σ∞ is also sufficient.

An approximation is found to the solution of the partial differential equation

in the region −1 ≤ x ≤ 1, t > 0, where u satisfies a general linear boundary condition on x = ± 1. This approximation is a polynomial in x, and is an exact solution of a perturbed form of the differential equation. By choosing the perturbation appropriately, this approach is mathematically equivalent to some recent methods for solving the differential equation in the form of a Chebyshev series. Better approximations to the required solution (and particularly to the eigenvalues) are obtained by choosing the perturbation to satisfy a least squares criterion.

The variational derivative of Lagrange densities which are functions of the metric tensor and its first and second derivatives is considered. This tensor is generally of fourth order in the derivatives of the metric tensor. If derivatives of any order are not present in the variational derivative then the Lagrange density is said to be degenerate in these derivatives. An explicit expression for the variational derivative of Lagrange densities which are degenerate in third and/or fourth derivatives is displayed in tensor form.

1. An isotropic tensor is one the values of whose components are unaltered by any rotation of rectangular axes (with metric σi(dxi)2). Those up to order 4 in 2 and 3 dimensions have many applications. The results suggest a general theorem for tensors of order m in n dimensions, that any isotropic tensor can be expressed as a linear combination of products of δ and є tensors, where δij = 1 if i = j and 0 otherwise, and is 0 if any two of the i1 to in are equal, 1 if i1…in is an even permutation of 1, 2, 3, …,n, and – 1 if it is an odd permutation.

We consider the ground state of a system of three interacting particles of equal mass. An integro-differential equation is obtained for the optimum pair function f in the product wavefunction Ψ(123) = f(12)f(13)f(23). The solution for harmonic forces reproduces the known exact ground state. Approximate analytic solutions are obtained for inverse-square forces, and for a general force law in the semiclassical limit.

The numerical and experimental results given in Madsen and Mei(16) are predicted using asymptotic methods and some knowledge of the Korteweg-de Vries (K-dV) equation. This is accomplished by first deriving, using formal asymptotic expansions, the K-dV equation valid over a variable depth. The depth is chosen, in the first instance, to slowly vary on the same scale as the initial (small) amplitude of the motion. The appropriate form of the Kd-V equation is then

where H(X,ξ) describes the surface profile and d(σX) is the changing depth. The rest of the paper is devoted to a study of this equation.

Waves in a rotating, stratified fluid of variable depth are considered. The perturbation pressure is used throughout as the dependent variable. This proves to have some advantages over the use of the vertical velocity. Some previous three-dimensional solutions for internal waves in a wedge are shown to be incorrect and the correct solutions presented. A WKB analysis is then performed for the general problem and the results compared with the exact solutions for a wedge. The WKB solution is also applied to long surface waves on a rotating ocean.

The structure of a wide range of Hopf algebras over fields is well known. Corresponding theorems for Hopf algebras over graded rings do not exist. This note is devoted to a Decomposition Theorem for certain Hopf algebras over rings (of finite characteristic) of the types which occur in topology, and a study of homological properties of the factors which therein arise.

The similarity solution for a laminar boundary layer on a continuous, moving, porous surface is obtained using a series of exponential terms. The convergence of this series which was previously used by Ackroyd (1) is investigated by a technique due to Weyl (4) and it is summed by a novel technique.

This paper is concerned with the flow of viscous fluids around and through porous bodies. Previous boundary conditions that have been used are discussed and a generalization of a boundary condition adopted by Beavers and Joseph, for plane boundaries, is proposed for curved surfaces. Using this condition the problem of slow viscous flow past a spherical shell is solved and several special limiting cases are considered.

In earlier papers of the same main title Jones and Lewis(1) (here-after referred to as I†) and Jones (2) studied, using ‘virtual body force’ methods, the axially symmetric flows of elastico-viscous liquids caused by the rotation of surfaces of revolution of arbitrary section. Particular attention was paid in I to the (oblate and prolate) ellipsoid geometry, but, for reasons of mathematical facility, the analysis was restricted to that of small eccentricity, the interesting and degenerate scheme of maximum eccentricity being outside its range of validity. It is the main purpose of the present note to broaden the scope of the analysis of I to cover a wider class of liquids and to present an exact analytical solution (in simple closed form) for the ellipsoid geometry.

The two-dimensional indentation of an elastic half space by a rigid punch under a slowly applied normal load is considered, for the case in which there is a finite coefficient of friction μ between the surfaces. The contact area is then divided into an inner adhesive region −c < x < c in which the surface displacements are known, surrounded by regions c < |x| < 1 in which the friction is limiting and the lateral displacement (which must increase in proportion to the overall load) is not known in advance. The problem is formulated in terms of a coupled pair of singular integral equations for the normal and shear stresses σ and τ at the surface; these are combined to give a single homogeneous Fredholm equation with positive kernel for a quantity π proportional to the difference τ – μσ in the adhesive region. The largest eigenvalue of this equation, for which π > 0, gives the adhesive boundary c in terms of μ and Poisson's ratio ν. A similarity transformation shows that c has the same value for both flat-faced and power law punches.

An isotropic infinite plane containing a circular inhomogeneity is subjected to an arbitrary loading condition. Assuming that the unperturbed elastic field is known, it is proved that the disturbed Papkovich potentials are expressible in terms of the potentials for the homogeneous plane. This knowledge can save considerable labour in the actual construction of the elastic field for the composite plane. The results are applied to some particular problems which include that of an arc displacement discontinuity in the composite plane. This example has as yet not been solved explicity by other methods and reveals that the displacement field at the cavity surface is independent of the elastic constants.

The solution for a simply supported many-sided polygonal plate does not agree with that for the corresponding circular plate. This paper describes the earlier work of Rao and Rajaiah on polygonal plates and then explains why best convergence of series solutions occurs when the boundary conditions are defined as