1. Definition. Let λ ≡ λ(ω) be continuous, differentiable and monotonic increasing in (0, ∞) and let it tend to infinity as ω → ∞. A series is summable |R, λ, r|, where r > 0, if

where A is a fixed positive number(3).

In a series of recent papers Verma and Upadhyay (7,8,9) developed the theory of basic hypergeometric series with two bases q and q½. These investigations were made in an attempt to discover a summation formula for a bilateral basic hypergeometric series 2Ψ2 analogous to that for a 2H2 (cf. Bailey (2,3)) and in finding relations between certain q-infinite products. In one of their papers they mentioned that it did not seem possible to develop the corresponding general theory for basic series with two unconnected bases q and q1. A recent paper by Andrews (1) indicates that transformations between basic hypergeometric series with two unconnected bases can be very interesting and useful in the study of ‘mock’ theta functions and their extensions. Besides this interest, such a theory also enables one to extend the entire existing transformation theory of the generalized basic hypergeometric series.

1. This paper investigates the boundedness, as t → ∞, of the solutions of the real differential equation

where α is a constant and φ2, φ3, ψ depend on the arguments shown with φ2, φ′3, ψ continuous.

We consider in this paper solutions of the equation

where the primes indicate differentiation with respect to s, and a, b, c are constants.

This paper describes a method whereby iteration can be carried out simultaneously with two or more trial vectors in order to obtain the latent roots of largest modulus of a symmetric matrix and their corresponding latent vectors. The convergence of any particular vector is automatically accelerated by making use of the current approximations to those vectors subdominant to it, the dominant latent vectors being obtained quite rapidly. The method is particularly suitable where a few of the dominant latent roots and vectors are required of a large symmetric matrix.

This paper is a contribution to the study of set functions which, although additive, fail to be measures by not being countably additive. It is motivated by the suggestion (to be found, for instance, in (2), section 2.3) that such functions might be used instead of measures in the theory of probability, and by speculation about the effect which the adoption of such a suggestion would have on the theory of stochastic processes.

In the theory of finite state, discrete time, non-homogeneous Markov chains, different notions of ergodicity have been introduced in the literature. These notions are concerned with the long-run behaviour of chains and with their tendency to get some stability properties after a sufficiently long period of time. The aim of this paper is the study of non-homogeneous Markov chains with a denumerable number of states. It will be shown that some theorems which are valid in the finite case are also valid for chains with a denumerable number of states as well. Moreover, a new notion of stability is introduced and it is shown to be satisfied for some chains. Although the paper is self-contained some familiarity with the theory of finite state non-homogeneous Markov chains is desired. Without any attempt of completeness we list for the interested reader the papers: (1–3, 8, 9).

The generalization of the Dirac and quaternion algebras to Riemannian spaces is outlined. The components of various elements of the algebras are interpreted as physical quantities (tensors) and their symmetries and algebraic properties are linked with the properties of the algebra. The generalization of the quaternion algebra is of particular interest in that it resolves the anomalies that arise in the usual identification of quaternions with rank two spinors. Algebraic expressions for the electromagnetic energy momentum tensor, the Ricci tensor and Einstein tensor are obtained in both E-number and quaternion form. Extension of the principles to EF-numbers yields a proof of the symmetry properties of Bel's tensor and a simple expression for its divergence.

Recently Hoyle and Narlikar(2) have put forward a conformal theory of gravitation in which they obtain an explicit solution of the field equations using certain approximations and assumptions of symmetry. To obtain the solution, the metric is assumed to have a special form. In the present paper a more general form of the metric is considered, and it is shown that many of the features of the special solution are shared by the solution in the more general case. In particular, the implications of the special solution as concerns the sign of the gravitational constant remain valid in the more general case under certain assumptions.

A simple sufficient condition is given for the existence of a lacuna in two-dimensional wave propagation governed by an equation of the second order. This work was inspired by Petrowsky's very general work (Petrowsky (3)) but for waves in two space dimensions we do not need the sophisticated machinery developed by him.

Green's function for SH-line source inside a wedge shaped medium with its apex removed by a circular cylindrical surface has been obtained. Exact location of the image sources facilitates the determination of the reflexion and diffraction at the circular and also at the plane boundary of the multiple phases clearly. The scattered field from the corner is indicated. The connexion of this solution with that of complete wedge has been shown with suitable contour integrations.

A simple formula is derived for the connexion between the scattered fields due to a sound-soft, a sound-hard and a perfectly conducting circular disc respectively.

The non-linear equations governing the convective flow of heat, beyond marginal stability, in a fluid sphere containing a uniform distribution of heat sources, and cooled at its surface, are expressed in the Boussinesq approximation as a perturbation of the steady-state conduction solution. A steady-state axisymmetric poloidal flow is assumed, and the equations for an approximate solution, involving the first spherical harmonic only, are obtained using a special case of the general evolution criterion of Glansdorff and Prigogine. The solution is found numerically using an iterative procedure, the results showing good agreement with those obtained by Stuart's ‘shape assumption’ for values of the Rayleigh number just above the critical value for the onset of marginal stability.

The growth of the laminar compressible boundary layer on a moving flat wall is considered analytically for the case of zero velocity in the free stream outside the boundary layer. The results of this analysis are compared with other published results for the cases in which the free stream has some finite velocity. In all the cases considered in the present paper, the boundary layer is taken to originate at some stationary point on the moving wall. This type of boundary-layer flow occurs behind moving shock waves and it is argued that the case of particular interest in the present paper, that of the stationary gas outside the boundary layer, provides bounding values of such parameters as displacement and momentum thicknesses for shock-induced laminar boundary-layer flows.

The laminar compressible boundary layer induced by the passage of a plane shock wave over a flat wall is examined in detail. Use is made of the empirical viscosity-temperature relationship μ ∝ Tω. The boundary-layer equations are solved numerically for various values of the index ω, Prandtl number and shock strengths. The resulting solutions are then used to construct simple semi-empirical relationships for some of the more important boundary-layer parameters.

The solution by Powell (6) of the diffraction of a complete expansion wave by a corner is extended to an incomplete wave. It is shown that the pressure is discontinuous across one of the boundaries and becomes logarithmically singular at a point on it. Graphs of the pressure distribution on the fixed boundary are given.

The Linear Theory of an Elastic Cosserat Plate

By A.E. Green and P. M. Naighdi

(Received 23 June 1967)

We are grateful to Dr R. B. Osborn for the following corrections:

Equation (3·19)1 should read

Equation (5·11)3 should read

Owing to an unusual definition of stress couples an error has occurred in one boundary condition at r = a in section 5. The following changes are necessary:

Interchange M12 and M21 in (5·1) and (5·2).

Replace Mrθ by Mθr in (5·12).

In equations (5·26), (5·28), (5·29), (5·30) and (5·31) replace γ by −γ but leave 1 + γ unaltered.

Replace (5·32) by

a result independent of γ.

Replace (5·33) by