This paper is concerned with a study of a class of stochastic processes that we shall call weakly non-deterministic.

The pressure distribution on the surface of a long, thin prolate spheroid is found under the conditions:

(1) The spheroid is illuminated by a harmonic point source of wavelength λ located on the axis of the spheroid.

(2) If a and b are the semi-major and semi-minor axes of the spheroid, then

(3) The boundary conditions on the spheroid are either the Neumann or the Dirichlet conditions.

The distribution is found from the asymptotic solutions of the Helmholtz equation using the inequalities in (2). The distribution is interpreted on certain regions of the surface in terms of travelling waves.

In a recent paper (1), Paria has discussed the propagation through a solid of a certain type of magneto-thermo-elastic plane wave. The analysis is essentially a reconciliation of the equations governing three fields: the electromagnetic field, the thermal field and the elastic field, which interact one with another. The principal result which was obtained was the dispersion equation connecting the frequency and the wavelength of waves of this type.

1. In many problems in theoretical physics one encounters situations in which it is desirable to be able to expand orbitals centred at one particle about another centre, frequently, but not always, another particle.

There have long been available methods for permitting this to be effected for bound states but no such procedures are available for the continuum.

Typical examples of physical situations in which such procedures are desirable are positron scattering from atomic and molecular systems, electron scattering from molecules, and nucleon scattering from light nuclei, to mention only a few.

The author's interest arose from a desire to establish a general theory of slow electron scattering from molecules, and an account of such a theory will appear shortly (1).

1. It was first observed by Rayleigh(17) in 1897 that low-frequency approximations to the solutions of the steady-state wave equation could be obtained from the solutions of the corresponding static problems. Rayleigh determined only the first term in the expansion of the solution in powers of the product of the wave number and a typical dimension. Many recent investigations have been concerned with deriving a systematic method of calculating the higher-order terms in the above expansion. Most of the problems which have been solved in this manner have been concerned with scattering by a disk or by a strip: the first systematic approach to these problems was that of Bouwkamp ((3), p. 71).

General methods for solving two-dimensional punch and crack problems in classical elasticity have been developed by Muskhelishvili and others. An alternative method for solving such problems is presented here. Use is made of suitable integral representations of the complex potentials and each problem is reduced to the solution of an Abel or Fredholm type of integral equation.

The problem considered is the oppositely charged, circular disk condenser when the disks are very close together. An integral equation due to Love(5) is used as the governing equation of the problem. This equation is solved asymptotically for small separations by splitting the field into regions, one being an annulus containing the edges, the other being the rest of the domain, and combining these solutions. Bounds for the error in the solution of the integral equation are obtained rigorously; the error is shown to approach zero as the separation approaches zero. The capacity of the system is deduced from this solution. The problem of finding the capacity has previously been attempted by various authors, whose results have differed. The present treatment establishes which of these results is correct.

Solving a problem of Erdős(5), I recently proved (4) that for n = 1, 2, …, the hypothesis implies that if the lines in the Euclidean space Ek (k ≥ 2) are distributed into n + 2 disjoint classes Li(i = 1, …,n + 2), then there exists a decomposition of Ek into sets Si such that the intersection of each line of Li with the corresponding set Si is finite. Without assuming any hypothesis other than the axiom of choice, I have also recently proved (3) that if di (i = 1,2,…) is any infinite sequence of mutually non-parallel lines in the plane, then the plane can be decomposed into N0 sets Si such that each line parallel to di intersects the corresponding set 8i in at most a single point.

In this paper, it is shown that certain quantities defined with reference to the rays of a rectilinear congruence, remain unchanged for a change in the director surface, provided that the spherical representation of the congruence is kept unchanged. These quantities may be called semi-invariants. Some known invariants have been expressed in terms of these semi-invariants. An attempt has been made to find how these semi-invariants alter when the parametric curves are changed in any manner.

The thermal boundary layer in the converging flow between non-parallel plane walls is studied. Analytical solutions of the boundary-layer equations are derived and the heat transfer across the wall is obtained from these solutions.

Let X be a compact differentiable manifold without boundary. A differentiable regular map of X into Euclidean N-space, dim X < N, is called an immersion (written X £ N). An immersion which is also one-one is called an embedding (written XN). A nonembedding theorem of the form: if X is a manifold with the property P then XN − 1, is best possible if there exists a manifold X with property P and an embedding XN. The purpose of this note is to show that two frequently used non-embedding theorems are best possible.

This is a continuation of (1) on the two-dimensional problem of the diffraction of elastic waves by irregular surfaces. The effect of an irregular surface with an isolated irregularity like a trough or ditch on incident plane harmonic P- and SV-waves is discussed. The maximum depth of the ditch is assumed small compared to the wavelength of the incident wave.

It is found that, when either a P- or an SV-wave is incident on such a boundary, besides the specularly reflected P- and SV-waves whose amplitudes are independent of the curvature of the surface there exist scattered waves travelling in various directions. In particular the diffracted zone contains the following second wave-types whose amplitudes are proportional to the depth of the ditch: (i) direct reflected P- and SV-waves, which at large distances appear to diverge from the point of intersection of the axis of symmetry of the ditch and the horizontal plane asymptotic to the boundary if the ratio of the wavelength of the incident wave to the half-width of the ditch is large. If the ratio is small these waves are reflected in the specular directions, (ii) A ‘secondary S-wave’ which finishes as P having travelled most of the way as an SV-wave. Its energy is confined to the neighbourhood of the free surface, (iii) A secondary P-wave which travels along the surface and finally emerges into the medium as an SV-wave at the critical angle for the medium, (iv) Rayleigh waves whose particle motion is in elliptic orbits.

After some manipulation from the general formula, Racah (1,2) obtained a closed formula for his V coefficient in the special case of all three mj- components zero. He also showed that the integral of the product of three Legendre polynomials followed at once from these special V coefficients. In this case the general formula ((2), eq. 10–14) becomes

where

and a, b, c are all non-negative integers. By changing the parameter of summation to (a+ b−c−p) it follows that is zero unless 2g = a+b+c is even (2).

ROAD TRAFFIC FLOW CONSIDERED AS A STOCHASTIC PROCESS*

By A. J. MILLER

I am grateful to G. F. Newell for pointing out an error which occurs at the bottom of page 322 of this paper. The distribution hf(u) is the spatial distribution of desired velocities and hence is the expected distribution of these velocities over a length of road at a random time. Hence in determining hf(u) in terms of hf(u), the relative times, rather than distances, travelled at restrained and unrestrained velocities should have been considered. (30) and (31) should be corrected to read

* Vol. 58 (1962), pp. 312–325.