Significance tests for several hypothetical discriminant functions have been developed by Williams (7,8) and considered further by the author (6). The test criteria consist of the factors in certain factorizations of the residual likelihood criterion, when the effect of the hypothetical discriminant functions has been eliminated. The independence and distributions of the factors can be seen by geometrical considerations, to be a consequence of the manner in which the factors are constructed in the sample space. In the case of a single hypothetical discriminant function Kshirsagar (5) has produced analytic proofs, by means of matrix transformations, of the independence and distributions of the factors. In this paper we shall give analytic proofs of the independence and distributions of the factors, given in sections 4 and 5 of the authors' paper (6), by extending Kshirsagar's proof to the case of several hypothetical discriminant functions.

Let J denote the half line 0 ≤ t < ∞, Rn the Euclidian n-space and R+ = [0, ∞). Let ║x║ denote the Euclidian norm of x ∈Rn, and S denote the set [x: ║x║ < ρ].

Motivated by discussion on practical stability in LaSalle and Lefschetz (3), Weiss and Infante (5), have discussed various notions of stability over finite time interval of a given differential system. This theory of stability differs from the usual stability theory mainly by the preassigned limits to which any given solution of the differential system must adhere. Sufficient conditions for these notions of stability are given in (5) in terms of certain Lyapunov-like functions satisfying some appropriate differential inequalities. The purpose of this article is to introduce some complementary notions of finite time stability and weaken the conditions on the differential inequalities involving Lyapunov-like functions by the use of comparison principles.

It is well known that a spherically symmetric space-time is, in general, of class two. A necessary and sufficient condition for a spherically symmetric space-time to be of class one has been obtained in terms of the Riemann curvature tensor. By means of a transformation property of s.s. space-time, three distinct cases are shown to exist. The incompatibility of class one spherically symmetric space-times with Rainich algebraic conditions is established in these three cases. It is concluded that spherically symmetric electromagnetic fields cannot be embedded in a flat space of 5-dimensions.

The spinor representations of the rotation group in a six-dimensional space with indefinite metric are shown to be four-component spinors, which become the usual Dirac spinors when the formalism is restricted to a four-dimensional subspace. Eriksson's work on the five-dimensional Lorentz group is found to result from a restriction of the six-dimensional treatment to a five-dimensional subspace, and the algebraic significance of Eriksson's work is thereby clarified.

It is shown that Huygens's principle holds for the solutions of Maxwell's equations for p-forms of all degrees in a gravitational plane wave space, while the solutions of the wave equation for 1, 2, and 3-forms, however, may have tails.

The quantum-mechanical energy-level density g(E) is given as a functional of the quantum-mechanical kernel K(q″, q′, t″ −t′). On taking the kernel K in the Feynman's form, one obtains the function g(E), without solving a Schrödinger equation. As an example, the embedding of a particle in the one-dimensional square well with infinitely high walls is analysed. The functions K(x″, x′, t−t′) and g(E) are represented as sums of terms corresponding to classical paths of different types. By an adequate choice of some terms due to the ‘most important’ paths, one may construct partial sums giving approximations of the function g(E). The utilization of such approximations for estimation of energy levels is demonstrated.

In this paper the effect of surface tension is included in a well-known problem in the theory of two-dimensional infinitesimal water waves. The problem is that of the reflection of waves from a fixed vertical barrier immersed to a depth a into deep water. It is shown how the solution for the velocity potential may be determined uniquely when simple assumptions are made concerning the behaviour of the free surface near the barrier. In particular, expressions are derived for the reflection coefficient, defined as the ratio of the amplitude of the reflected wave to that of the incident wave, at infinity, and the transmission coefficient, defined similarly. It is shown how these coefficients, for small values of the surface tension force, tend to the values obtained by Ursell (4) when surface tension is ignored. The related problem of a completely immersed vertical barrier extending to a distance a from the surface may be solved in a similar manner. Expressions for the reflection and transmission coefficients for this case are given.

The wave function φ(x, y) satisfies the equation

in the infinite region

and satisfies the boundary condition Kφ + φy = 0 on the two line segments (y = 0, −∞ < x < −R) and (y = 0, R < x < ∞). We are not concerned with the properties of φ(x, y) inside the circle r = R. It is shown that outside the circle r = R the wave function φ(x, y) can be written as the sum of a wave source, a wave dipole, regular waves and wave-free potentials. (These basic functions are defined in section 2.) The proof makes use of analytic continuation into the image region (r > R, y ≤ 0) cut along the negative y-axis.

Two methods are given for this step. The first method uses Schwarz's symmetry principle to define the auxiliary wave function Φ = Kφ + φy throughout r > R, and φ can then be found by solving differential equations in the y or x direction. The second method uses Green's theorem. Applications in the theory of water waves are suggested.

In considering the problem of waves on a sloping beach, little regard seems to have been given to the effect of surface tension. Wehausen and Laitone (7) tend to attribute this to the fact that the additional force is small. This does not, of course, preclude the possibility that the effect may be appreciable in certain regions, and Longuet-Higgins (3), for example, has shown this to be the case near the crests for waves on the point of breaking. They also add, which is probably rather more pertinent, that difficulties arise when a solid boundary pierces the surface, since an additional boundary condition is required at the intersection, but give no indication as to what the boundary condition should be.

In this paper the effect of surface tension on water waves is considered. The usual assumptions of the linearized theory are made. A uniqueness theorem is derived for the waves at infinity for a general class of bounded two-dimensional obstacles in a free surface by means of an energy argument. It is shown how the wave amplitude at infinity depends on the prescribed angle at which the free surface meets the normal to the obstacle. The particular case of a heaving half-immersed circular cylinder is considered in detail, and an expression obtained for the velocity potential in terms of a convergent infinite series, the coefficients of which may be computed.

This paper is dedicated to the question of solving unsteady thermal boundary-layers in the case of two-dimensional low-speed flows, provided that the difference between the temperature of the stream and that of the wall is not too great (so that the density is sensibly constant) and that the change in the wall temperature Tw (x, t) takes place at the same instant as the body is set into motion. The velocity boundary-layer is uncoupled from the thermal one and can be considered separately. In paper (2) is given the method for obtaining the velocity field (u, v). The temperature field T depends on the velocity field and can be obtained only after this one. The method (2) being available for this purpose assuming the difference between the temperature of the wall Tw (x, t) and that of the main-stream T∞ in the form of

where S(x) and θ(t) are arbitrary functions of x respectively t, satisfying certain conditions required by the method.

Certain simple steady electrically driven flows under transverse applied magnetic fields, for which exact solutions are available, are considered. Electric potentials are prescribed on fixed perfectly conducting electrodes and the resulting Lorentz force j × B0 gives rise to a velocity field in the fluid. It is assumed that the magnetic Reynolds number of the flow is small (Rm ≪ 1) so that the induced magnetic field may be neglected. In particular, we examine in detail the simplest flat plate configuration which is correctly posed as a two-part mixed boundary value problem of the Wiener–Hopf type. At large Hartmann number, the fluid vorticity and the current flow transverse to the applied magnetic field are suppressed. Thin current transition layers, which emanate from an electrode edge and are aligned with the applied magnetic field, provide a smooth transition from one region of almost constant current to another. The thickness of these layers is found to be of the same order as the thickness of certain velocity transition layers occurring in other situations in magnetohydrodynamics.

We discuss non-linear thermodynamical theories of rods and shells using the three-dimensional theory of classical continuum mechanics as a starting point. The three-dimensional theory is reduced to a two-dimensional theory for a shell, or plate, and a one-dimensional theory for a rod by employing an exact expansion for the displacement but an approximation for the temperature. For elastic rods and shells a method of approximation is suggested which brings the respective theories into correspondence with those of Green and Laws (1) and Green, Naghdi and Wain-wright(2).

In a recent paper (1), we derived the solutions to the equilibrium equations for linearized plane strain of an isotropic mixture of two elastic solids. The solutions were given in terms of four complex potential functions. Here we examine the properties of these solutions and evaluate the forces and couples across an arc in the mixture.

In the course of this paper series expressions are obtained for the displacement vector so that the stresses take prescribed values on the bounding surfaces z = ±h of the plate. Since no conditions are satisfied on the curved surface that forms the bounding edge of the plate these expressions or particular integrals cannot, in general, be solutions to three-dimensional plate problems. They are, however, complementary to the work of Green in (1).

On the non-absolute summability of a Fourier series and the conjugate of a Fourier series by a Nörlund method

(Proc. Camb. Phil. Soc. 63 (1967), 407–411)

By R. Mohanty and B. K. Ray

In the proof of Theorem 2 there is the tacit assumption that χ(t) is of bounded variation in (½π, π)—an assumption altogether lost sight of in its statement.