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

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.

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.

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.

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.

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.