Consider bounded sets of points in a Euclidean space Rq of q dimensions. Let h(t) be a continuous increasing function, positive for t>0, and such that h(0) = 0. Then the Hausdroff measure h–mE of a set E in Rq, relative to the function h(t), is defined as follows. Let ε be a small positive number and suppose E is covered by a finite or enumerably infinite sequence of convex sets {Ui} (open or closed) of diameters di less than or equal to ε. Write h–mεE = greatest lower bound for any such sequence {Ui}. Then h–mεE is non-decreasing as ε tends to zero. We define

It is shown that anomalously large and anomalously small values of the constant A in the thermionic current formula for metals can be explained by the usual statistical equilibrium theory provided proper account is taken of the fact that free electrons may be shared by two competing overlapping energy bands. When this theory is developed there arises a comparatively large temperature variation of the statistical parameter ζ which occurs in the Fermi function, and when the exact formula is forced into the empirical form with a constant work function, the temperature factor appears as a multiplying factor of the current constant, this factor being greater or less than unity depending upon the nature of the overlapping bands. In particular the observed values of the constant A for nickel and for hafnium are well explained.

1. In a recent paper(1) expressions were found for the elastic stresses produced in a semi-infinite elastic medium when its boundary is deformed by the pressure against it of a perfectly rigid body. In deriving the solution of this problem—the ‘Boussinesq’ problem—it was assumed that the normal displacement of a point within the area of contact between the elastic medium and the rigid body is prescribed and that the distribution of pressure over that area is determined subsequently. The solutions for the special cases in which the free surface was indented by a cone, a sphere and a flat-ended cylindrical punch were derived, but no attempt was made to give a full account of the distribution of stress in the interior of the medium in any of these cases.

1. The reflexion of a simple harmonic train of surface waves by a rigid plane barrier inclined at an angle α to the undisturbed free surface of the water is considered in the particular case in which 2sα = π, where s is an integer. The solution if s = 2 has been given by Gourret(1), while the solution if s = 1 is well known; the solution for any integral value of s is given here.

The paper considers the application of Dirac's classical theory of radiating electrons to examine the motion of an electron in a uniform magnetic field. The non-relativistic equations of motion are solved exactly. It is shown that for the particular case when the motion is confined to a plane, the physical motion is such that the electron describes an equiangular spiral with velocity which steadily decreases as

exp {negative constant × t};

and in the non-physical motion the electron spirals outwards with velocity which steadily increases as

exp {positive constant × t},

and ultimately the electron escapes to infinity. The relativistic equations of motion are solved approximately as a series in ascending powers of the field. It is shown that in the physical motion the velocity begins to decrease, and hence after a time the motion is given correctly by the non-relativistic solution.

I take this opportunity to express my deepest gratitude to Prof. Dirac, who suggested this problem, for his patient guidance and supervision.

A collineation in a space of paths is defined as a point transformation which carries paths into paths. Such transformations were first studied by L. P. Eisenhart and M. S. Knebelman. Subsequently J. Douglas introduced the geometry of K-spreads and E. T. Davies has shown that the results for an affine space of paths can be extended to these more general spaces and written in a very elegant form by the aid of Lie derivation.

1. In a recent paper(1) the partial differential equation governing the symmetrical vibrations of a thin elastic plate was reduced to an ordinary differential equation by the use of the Hankel transform method. By the discussion of the solutions of the latter equation and by the use of the Hankel inversion theorem an account was given of the free and forced vibrations of the plate under symmetrical conditions.

1. The reflexion of waves on the surface of water by a thin plane vertical barrier is considered and the coefficient of reflexion (the ratio of the amplitudes, at a great distance from the barrier, of the reflected and incident waves) is calculated. If the top edge is at a depth a below the surface, it is found that the coefficient of reflexion is about ¼ when where T is the period of the incident waves, so that the condition that the coefficient may exceed ¼ is a .

The problem of the equivalence of binary forms is of great importance, both historically and intrinsically, and is also significant for the problems of the so-called canonical and automorphic forms. It consists in deciding whether two given forms are equivalent, i.e. transformable one into the other linearly, and, when they are, in finding all the linear substitutions transforming one form into the other.

1. I. N. Sneddon has recently considered(11) the two-dimensional problem of the stress distribution due to a force uniformly distributed along a strip in the interior of a semi-infinite elastic medium by integrating the distribution due to an isolated force. The method of solution for the isolated force problem differs from that originally used by Melan(9), but both writers employ real variables. A solution of this problem, using complex variables, has been given by Stevenson and a considerable saving of labour is revealed. A complex variable method of solution for problems of stress distributions in an elastic medium bounded by a plane face was, however, given earlier by Sen(10), although he only made applications to problems in which the force system was applied to the plane face. Sen's method is somewhat different from that used by Stevenson and gives results more directly for many problems, including the problem considered by Sneddon.

1. In the complete algebraic system of concomitants of a pair of quaternary quadrics given by Turnbull† there appear sixteen covariant line-complexes, eight of the second order and eight of the third. In a later paper‡, the same author gave some geometrical interpretations of these and other concomitants. None the less, no systematic geometrical account of the relations of these complexes seems to have been made. The purpose of the present paper is to discuss the geometrical relations systematically, and make their origin clearer. Incidentally, the syzygies which exist between the complexes are obtained explicitly.

A possible degree of heterogeneity in the structure of continua is investigated in this paper.