A simple approximate formula is obtained for the capacity of the condenser formed by a “small” conductor placed inside a much larger one. The formula involves a constant whose choice is, to a certain extent, arbitrary and it is shown that, for problems involving spheroids inside cylinders and between parallel plates, the constant may be found in a simple fashion so as to give very accurate results. A similar formula is obtained for the loss in potential energy due to a crack or cavity in a circular beam or a thick plate. For the particular cases of the boundary value problems considered which have been treated by other means very close numerical agreement is obtained between those results and ones deduced in the present paper.

The object of this note is to state and prove two theorems of the nature of Montel's Limit Theorem for a function which is regular and bounded in a region G, but involving as hypothesis the limit of a mean value of the function instead of the limit of the function itself. Theorem 2 below (with b = b′ = 1), in which G is a half-strip, was stated several years ago in a letter to A. J. Macintyre and myself from J. M. Whittaker, who added that it could be proved by integrating the inequality in Lemma 3 below (which is due to Dr Whittaker). As far as I can discover, such a proof still has not appeared in print; hence the present one.

In a recent paper (1) Portnov used a form of Poisson integral to find the exact solution for the temperature distribution in a freezing semi-infinite slab occupying the region x > 0, and having an arbitrary time dependent temperature applied at the face x = 0. Previously, Boley (2) had used a method based on Duhamel's theorem to find solutions for problems involving melting, in both finite and semi-finite regions, caused by time dependent heat fluxes. Steady-state solutions have been investigated by Landau (3), Masters (4) and others (5).

In a well-behaved homomorphism theory for a class of algebraic systems certain “closed objects” relative to a given G ∈ are distinguished which act as kernels of homomorphisms. For example, if is the class of groups then the closed objects relative to a given group G are the normal subgroups of G; if is the class of semigroups with zero element then one can devise a homomorphism theory in which the closed objects relative to a given S ∈ are the ideals of S[cf. Rees (3)]; in the class of groupoids one may define the closed objects relative to a given groupoid G to be the congruence relations on G, that is, subsets π⊆G×G which are equivalence relations having the property that (x1y1, x2y2) ∈ π whenever (x1, x2), (y1, y2) ∈ π. Given such a closed object N relative to G there exists a “factor” system G/N and a (canonical) homomorphism η: G→G/N characterised by the property: If σ G→H is a homomorphism with kernel N then there is a unique homomorphism : G/N→H such that . η = σ and the kernel of is trivial in the sense that the kernel of is the unique smallest closed object relative to G/N.

Let f(z) be represented on its circle of convergence |z| = 1 by the Taylor series

and suppose that its sole singularity on |z| = 1 is an almost isolated singularity at z = 1. In the neighbourhood of such a singularity f(z) is regular on a sufficiently small disk, centre z = 1, with the outward drawn radius along the positive real axis excised. If also in this neighbourhood |f(z)| e−(1/δ)ρ remains bounded for some finite ρ, where δ is the distance from the excised radius, then the singularity is said to be of finite exponential order.

1. The problem discussed is the temperature distribution in a semi-infinite solid, initially at zero, when the surface is suddenly raised to and maintained at a temperature which is an arbitrary function of the distance from some line which lies in that surface.

In this paper we give a method for the solution of the dual integral equations

where Jv and Yv are Bessel functions of the first and second kind, −½≦α≦½, f1(ρ) and f2(ρ) are known functions and ψ(ξ) is to be determined. Such equations arise in the discussion of boundary value problems for half-spaces containing a cylindrical cavity. For example, let us take the problem of finding a potential function φ(ρ, θ, z) which satisfies Laplace's equation for

subject to the usual regularity conditions and the following boundary conditions:

A new finite integral transformation (an extension of those given by Sneddon (1)), whose kernel is given by cylindrical functions, is used to solve the problem of finding the temperature at any point of a hollow cylinder of any height, with boundary conditions of radiation type on the outside and inside surfaces, with independent radiation constants. It is to be noticed that all possible problems on boundary conditions in hollow cylinders can be solved by particularising the method described here.

In my paper on the continuous (α + β) – theorem (these Proceedings 12 (1961), 209–211) the proof of Lemma 6 assumes that if A and B are closed sets in the relative topology as subsets of the space P of positive reals then C = A + B is closed.