Uniqueness theorems for analytic functions of one and of several complex variables By L. A. Rubel and B. A. Taylor Proc. Camb. Phil. Soc.64 (1968), 71–82

Proposition 15, on page 80 should read

Let E be an arbitrary set, a σ-algebra of subsets of

the Borel sets of F. We write A × B for the product of the sets A and B, and

for the product σ-algebra of and (i.e. the σ-algebra generated by the rectangles A × B with A ∈ and B ∈ ).

Axially symmetric static fields produced by one extended body are considered. The metric outside the body is the Weyl metric which can be written in so-called Weyl coordinates in the form ds2 = a2(dr2 + dz2) + b2dϕ2 – c2dt2 (a, b, c functions of (r, z) and r > 0) with b2 = r2c−2. Criteria are found under which these Weyl coordinates can be extended through the whole of the body so that ds2 keeps the form above (although in general b2 ǂ r2c−2 inside the body). In the spherically symmetric (⇒ axially symmetric) case it is shown that the Weyl coordinates cover the whole of spacetime only if one allows r < 0 in a certain part of the body.

In Part I of the work that follows solutions for the flow around a semi-infinite oscifiating plate are obtained, one by an operational method using parabolic coordinates and one in the form of a series solution. The latter is used to obtain an estimate of the skin friction distribution about the edge of the plate and hence the excess skin friction over the infinite oscilliating plate value. A valuable check on this result is provided by an equivalent consideration of the operational solution. Having demonstrated the usefulness of the series solution we proceed to extend its application to a wedge of arbitrary angle in Part II and eventually use the results to derive a general expression for the skin friction on an infinite cylinder of arbitrary cross-section oscifiating rapidly parallel to its length.

Let be a Dirichlet series. Then it is known that if is divergent, the abscissa of convergence σ1 is given by

and that if is divergent, the abscissa of absolute convergence σ2 is given by

The flow of a rarefied gas between parallel plates has been studied via the linearized Boltzmann transport equation. Using a general boundary condition, which includes an arbitrary ratio of specular to diffuse reflection from the wall, we have derived an integral equation for the mass flow velocity. The integral equation is solved by using a replication property of the kernel and application of the method of Muskelishvili.

The total volumetric flow rate is obtained and a slip boundary condition is deduced for use with the hydrodynamic equations.

Certain aspects of the eigenvalue spectrum associated with the Boltzmann equation are discussed.

A generalized Euler–Lagrange equation is presented. It provides a unified approach to boundary value problems in potential theory, diffusion, magnetostatics, and integral equations.

Dipolar fluids have been discussed by Bleustein and Green in quite a recent paper (3). They have obtained explicit constitutive equations for homogeneous incompressible linear dipolar fluids. An application of this theory is made here to the problem of Couette flow. Monopolar and dipolar stress fields are determined. Numerical values for velocity and rate of shear for different values of radius in the annulus are given. Graphs are also drawn to compare our results with the classical ones. It is found that the rate of shear does not change sign within the annulus.

An alternative definition is proposed for the kernel used by Feynman. This definition involves a functional integration in a q, , p space, treating these variables as independent. The equivalence of this definition to the Feynman one and to the one using the variables q, p is exhibited.

An exact solution in terms of analytic functions is given for the Stokes stream function and forces acting on two spheres in contact when the spheres move uniformly with the same velocity along their line of centres. For the case of equal sized spheres, the values of the forces agree with the Faxén limit of the Stimson–Jeffery forces for separated spheres when the minimum clearance between the spheres tends to zero. Numerical values of the forces for arbitrary sized spheres are tabulated and compared with corresponding values when the spheres are separated over a range of values of the minimum clearance. Comparison of the theoretical results with those obtained experimentally for the case of equal sized spheres is also given.

A systematic study of the stress distribution in an infinite elastic solid when time dependent and time independent body forces act on certain regions of the solid is made by Eason, Fulton and Sneddon(l). They have stated the importance of studying problems concerning moving point charges in electrodynamics. Mukhopadhyay(2) has discussed the disturbance created by the propagation of torsional disturbance in an infinite elastic solid due to time dependent rotating body forces.

In this paper we use a method due to Williams(1) to discuss the scattering of surface waves of small amplitude on water of infinite depth by a fixed vertical plane barrier extending indefinitely downwards from a finite depth.

It is shown that a revision is needed of certain transformation equations of the Planck–Einstein formulae. In particular it will be shown that the formulae

must be replaced by

where T and Q are respectively the absolute temperature and heat of the system. Furthermore, it will be shown that both formulae for the energy

are correct, but they refer to different cases.

Complementary variational principles are derived for a class of electrostatic and magnetostatic problems using the pairs of adjoint operators (grad, – div) and (curl, curl). This theory unifies the principles of Dirichlet and Thomson in electrostatics and of Schrader in magnetostatics. The results are illustrated by deriving upper and lower bounds for the capacity of a surface, and it is shown how such bounds can be systematically improved by Ritz procedures.