Many writers have studied special cases of the Saint-Venant torsion and flexure problem for an isotropic beam of constant cross-section. An excellent summary of this work has been given by Stevenson (4) who also greatly extended the field of application of the theory by showing that the general Saint-Venant problem can be reduced to the solution of boundary problems involving six simple ‘canonical’ flexure functions. Miss Rosa Morris (2, 3) has applied the Saint-Venant theory, as extended by Stevenson, to the solution of the flexure and torsion problem for beams possessing two fairly general types of cross-section, her results including many previous solutions as special cases.

1. In the first paper of this series I have explained a method by which the complete irreducible system of combinants of a pencil of quadric surfaces may be obtained, and have determined explicitly the combinantal invariants and covariants. The present paper deals with combinantal contravariants. The notation used is that of I, a knowledge of which will be assumed.

The drag on a fragment moving through the air at supersonic velocity tends to be proportional to the projected area of the fragment perpendicular to the line of flight and to be independent of the Reynolds number, depending only on the Mach number. Also the minimum velocity at which the fragment perforates a target is a function of the projected area at the moment of strike. Hence the probability that a fragment, with a given initial velocity, perforates the target at a given distance, depends on the mode of rotation and the distribution of projected areas of the fragment. It is customary to assume that the fragment rotates randomly so that we require to know the distribution of the projected areas when all orientations are equally likely. The distributions for fragments in the form of cylinders and rectangular parallelepipeds are derived in this paper.

It is pointed out that the equations of motion for any field obtained by varying a Lagrangian subject to auxiliary conditions are exactly equivalent to a certain set of canonical equations and that the commutation relations between the dynamical variables for the latter equations are Lorentz-invariant. By extending the theory to Lagrangians containing higher derivatives of the field quantities, it is shown that any given set of field equations can be put into the canonical form, though it is not derived from variational principles. The question of Lagrangians with missing momenta is also considered. It is shown that if the Lagrangian is ‘gauge-invariant’, some of the p's must be missing and the corresponding Eulerian equations can be replaced by equations containing no q and then can be replaced by initial conditions. The commutation relations between gauge-invariant quantities are Lorentz-invariant. For Lagrangians which are not gauge-invariant but are such as to have missing momenta, the passage to quantum theory will in general give rise to non-Lorentz-invariant commutation relations. In both cases, the equations of motion can be cast in canonical forms.

1. Since many writers, among them G. I. Taylor (1) and W. L. Bragg (2), have based theories of the strength of metals upon the assumption that there is some type of dislocation in the metal, it has seemed desirable to give an account of a dislocation in which displacement and stress are given, according to the classical theory of elasticity, by simple closed formulae.

The ordinary one-dimensional wave equation

has special integrals of the form

which satisfy the first-order equations

respectively, and are often called progressive waves, or progressive integrals, of (1·1). The straight lines

in an xt-plane are the characteristics of (1·1). It follows from (1·2) that progressive integrals of (1·1) are constant on some particular characteristic, and are characterized by this property.

The possibility of the propagation of disturbances along a free surface with a velocity exceeding the velocity of sound in the liquid is examined. Simple necessary conditions for the occurrence of such ultrasonic propagation are found in terms of the boundary condition. It is shown that the conditions are not satisfied in the case of gravity waves, and a detailed examination is made of the pressure and surface waves due to a localized disturbance. Capillary waves are shown to satisfy the conditions for ultrasonic propagation, the existance of a ‘faster-than-sound’ wave is proved, and its main properties are examined.

An account is given of the transformation of coordinates and of absolute axial frames in Euclidean space and Galilean space-time. The connexion with Eddington's group of E-numbers is shown. The geometrical properties of Dirac's wave equation and of analogous equations in three dimensions are discussed in terms of absolute axial frames.

A simple and direct variational method is described for finding both complex and real eigenvalues of the wave equation of anomalous propagation in a horizontally stratified atmosphere. It may be looked upon as an extension of Rayleigh's method to complex eigenvalues. In this paper it is illustrated by an example, taken from the duct theory of super-refraction, in which the refractive index of the air varies with height according to a power law. Numerical agreement in the values for the lowest order eigenvalues with those obtained by the differential analyser is better than ½%.

The method for the solution of problems of surface waves on water with which this note is concerned can be sufficiently illustrated by a particular example given by Lamb (1). In Lamb's example it is supposed that a steady stream in which the x, y components of the velocity are U, 0, where U is constant, is slightly disturbed by the application of a constant pressure ρPU2 to a band of finite width 2a of the surface of the stream; ρ denotes the density of the water and P a non-dimensional constant. In the solution Lamb, using an artifice due originally to Rayleigh and widely used since, assumes that ‘the deviation of any particle of the fluid from the state of uniform flow is resisted by a force proportional to the relative velocity’, and states that this ‘law of friction does not profess to be altogether a natural one, but it serves to represent in a rough way the effect of small dissipative forces; and it has the great mathematical convenience that it does not interfere with the irrotational character of the motion’.

The object of this paper is to determine all the solutions of the wave equation

which are of the simple form

where F denotes an arbitrary function. It will be shown that, in addition to the obvious cases of plane or spherical progressive waves, such solutions exist only when the wave fronts

are certain algebraic surfaces of the fourth order, the cyclides of Dupin. These include, as degenerate cases, the sphere, the plane, the cylinder, the cone, and the torus.

After solving the equations of motion of Dirac's self-accelerating electron, a physical picture of it is formed by plotting graphically the surfaces of constant scalar potential when the electron has built up a velocity close to the velocity of light.

In this paper we are concerned with a material which can support a finite stress elastically without flow and which flows with constant mobility(1) (or plastic fluidity) when the stresses are sufficiently great. Following Bingham(1) and Houwink(2), such a material is called a Bingham solid and the type of flow (purely) plastic. The transition from elastic to plastic behaviour takes place at the yield point.

The paper contains a theoretical discussion of the scattering by a ship of the wave sent out from a radar transmitter. An expression is derived for the power returned to the receiver of the radar installation, on the assumption that the ship is composed of a large number of scattering elements which scatter energy non-coherently. Calculations based on this expression are compared with the results of experiment and shown to be in good agreement as regards the variation of signal strength with range, provided that a suitable value for the effective radius of the earth (a quantity which changes with meteorological conditions) is taken.

An experiment is described in which the absolute level of the received signal from a ship was measured by comparing it with that from a metalized sphere flown from a balloon. The result was found to be in satisfactory agreement with theoretical expectations.

In this paper the two-dimensional reflection of surface waves from a vertical barrier in deep water is studied theoretically.

It can be shown that when the normal velocity is prescribed at each point of an infinite vertical plane extending from the surface, the motion on each side of the plane is completely determined, apart from a motion consisting of simple standing waves. In the cases considered here the normal velocity is prescribed on a part of the vertical plane and is taken to be unknown elsewhere. From the condition of continuity of the motion above and below the barrier an integral equation for the normal velocity can be derived, which is of a simple type, in the case of deep water. We begin by considering in detail the reflection from a fixed vertical barrier extending from depth a to some point above the mean surface.

In relativistic field theories derived by a variation principle from a Lagrangian, the problem arises of finding a symmetric tensor of rank 2 which has vanishing divergence in virtue of the field equations and is such that taken over a space-like section is equal to the corresponding integral of the so-called canonical energy-momentum tensor. It is well known that the latter condition is satisfied if the difference between the two tensors is the divergence of an antisymmetric tensor of rank 3.

The rheological equations of state for an isotropic Bingham solid may be written in tensor form:

where

The notation is explained fully in a previous paper (1). The frame of reference is essentially Eulerian. Primes denote deviatoric components of the stress, strain and rate of strain tensors, pik, εik and eik (for example, ). The dilatation is denoted by Δ. The rigidity and bulk moduli μ and κ are, if not constants, essentially scalar functions of the stress tensor. The variable viscosity η involves two constants, the yield value ϑ and the reciprocal mobility η1.

The heat of adsorption of particles which occupy two sites on a surface is examined by determining the statistical equilibrium. In particular, equations which are appropriate to immobile films are obtained. The heats of adsorption calculated from these equations are compared with the experimental values obtained for the heat of adsorption of hydrogen on a tungsten surface, and with the values which were obtained using an empirical method.

A classical theory of a spinning particle with charge and dipole moment in an electromagnetic field is obtained by working symmetrically with respect to retarded and advanced fields, and with respect to the ingoing and outgoing fields. The equations are in a simpler form than those of Bhabha and Corben or those of Bhabha, and involve fewer constants. On the assumption that the spin angular momentum tensor θμν satisfies the equation θ2 ≡ θμν θμν = constant, the value of the dipole moment Zμν is chosen to be Cθμν, where C is a constant. The theory is generalized to the case of several particles with charge and dipole moment. By using a suitable Hamiltonian equation, the classical equations of motion, obtained on the assumption that θ is a constant, are put into Hamiltonian form by means of the ‘Wentzel field’ and the λ-limiting process. The passage to the quantum theory is effected by the usual rules of quantization. The theory is extended to the case of particles with charge and dipole moment in the generalized wave field by defining the Wentzel potential in terms of the generalized relativistic δ-function.

A short discussion of steady rectilinear plastic flow of Bingham material between rigid cylinders of general shape in relative motion parallel to their generators was given in a paper of the same main title(1). It was shown that, for sufficiently great relative velocity V of the boundaries, plastic flow in the whole region would be expected, as for a Newtonian liquid. But for slower relative motion, part of the material might remain elastically deformed (i.e. stressed below the yield point) with a limited region of plastic flow surrounding the inner boundary. In the paper referred to, the problem was illustrated by the case of eccentric circular cylindrical boundaries. The choice of circular boundaries introduced a great simplification when the region of plastic flow did not extend to the outer boundary, namely axial symmetry in the flow pattern in the plastic region. It is the main purpose of the present paper to illustrate how the methods already given for determining velocity distributions may be modified to suit more general problems, by consideration of confocal elliptic cylinders as boundaries.