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.

The paper presents an analytical solution for the dispersion of a solute in an electrically conducting fluid flowing between two parallel plates in the presence of a transverse magnetic field. It is shown that the solute is dispersed relative to a plane moving with the mean speed of the flow with an effective Taylor diffusion coefficient which decreases with increase in magnetic field.

Consider a queueing system GI/G/∞ in which (i) the inter-arrival times are distributed with distribution function A(t) (A(O +) = 0) (ii) the service times have distribution function B(t) such that the expected value of the service time is β(>∞).

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.

In this paper we give a characterization theorem for a subclass of the exponential family whose probability density function is given by

where a(x) ≥ 0, f(ω) = ∫a(x) exp (ωx) dx and ωx is to be interpreted as a scalar product. The random variable X may be an s-vector. In that case ω will also be an s-vector. For obvious reasons we will call (1) as the linear exponential family. It is easy to verify that the moment generating function (m.g.f.) of (1) is given by

In Part I of the paper, a simple theory of infinite systems of linear equations of the type L2 is presented. In Part II, these results are applied to the elastostatic problem of determining the stress distribution in an axially loaded bar, held in position by a centrally placed circular bolt. Some numerical results are obtained and these are compared with experimental observation.

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.

Let {Xn: n ≥ 1} be a sequence of independent random variables and write Suppose that the random vairables Xn are uniformly bounded by a random variable X in the sense that

Set qn(x) = Pr(|Xn| > x) and q(x) = Pr(|Xn| > x). If qn ≤ q and E|X|r < ∞ with 0 < r < 2 then we have (see Loève(4), 242)

where ak = 0, if 0 < r < 1, and = EXk if 1 ≤ r < 2 and ‘a.s.’ stands for almost sure convergence. the purpose of this paper is to study the rates of convergence of

to zero for arbitrary ε > 0. We shall extend to the present context, results of (3) where the case of identically distributed random variables was treated. The techniques used here are strongly related to those of (3).

The determinantal factor on p. 512 of Proc. Cambridge Philos. Soc. 61 (1965), should have the off-diagonal element in position (s, t) of the array multiplied by bst. This slip was pointed out to me by Dr Donald Knuth. When the rooted-labelled tree is not necessarily ‘chromatic,’ so that the bss's can be 1, the diagonal terms of the matrix become

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.

The exterior metric of an infinitely long cylinder whose parameters vary periodically along its length is obtained as a Weyl type solution in terms of Bessel functions, and extended into the interior of the source. The question of inductive energy transfer between two co-axial cylinders of this nature is considered.

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).

In a recent paper Srivastav (2) considered the solution of certain two-dimensional mixed boundary-value problems in a wedge-shaped region. The problems were formulated as dual integral equations involving Mellin transforms and were reduced to the solution of a Fredholm integral equation of the second kind. In this paper it will be shown that a closed form solution to the problems treated in (2) may be obtained by elementary means.