A solution of a triad of integral equations involving Bessel functions is given. This, like earlier ones, is in the form of a pair of Fredholm integral equations, which may be solved by iteration in certain cases. In spite of a slightly more general formulation of the problem, the kernels of these equations are simpler than those given in earlier solutions. Certain extensions are considered and a formal solution given. Application is made to the problem of incompressible inviscid flow normal to an annular disc, and to the flow due to the slow rotation of such a disc in a viscous fluid.

In recent years considerable attention has been given to problems of thermal stress in isotropic materials. Much of this work has been devoted to statical problems although there has been some work on problems with time dependence, for example the quasi-static solutions obtained by Sternberg (1) and Eason and Sneddon (2). A good deal of interest has also been shown in statical thermal stress problems when the material is anisotropic. For the type of material considered here statical problems have been investigated by Grechushnikov and Brodovskii (3) and Sirotin (4) among others. Little attention has been given, however, to time dependent thermal stress problems when the material is anisotropic.

In problems in the mathematical theory of elasticity related to the symmetric deformation of an infinite elastic solid with an external crack we encounter the problem of determining an axisymmetric function φ(ρ, z) which is harmonic in the half-space z>0 and satisfies the mixed boundary conditions

on the plane boundary z = 0, where it is assumed that f(ρ) is continuously differentiable in [1, ∞). Further φ→0 as √(ρ2+z2)→∞.

The integral evaluated in this note was suggested by the famous one connected with the Poincaré polynomials of the classical groups (see (1)).

Let X be an n × n matrix whose elements depend on k parameters. Denote by a manifold in Euclidean space of dimension n2, with the property that if X ∈ , then so does XI−i for 1≦i≦n, where I−i is the unit matrix I altered by a minus sign in the (i, i)th place. Suppose further that there exists on a measure which is invariant under the transformation X→XI−i. Such manifolds and measures exist. For example (see (2), § 5), the set of all proper and improper n×n orthogonal matrices H is such a manifold, the H depending on ½n(n−1) parameters because of the orthogonality and normality of the columns of H. Since the set of all H is a compact topological group, an invariant measure exists.

One of the simplest three part boundary value problems is the electrostatic problem for the circular annulus and, at present, there seems to be no method available for obtaining the solution in a closed form. It has recently been shown by the author (1) and Cooke (2) that this problem can be reduced to the solution of a Fredholm integral equation of the second kind. The equation obtained in (1, 2) is fairly simple and is suitable for obtaining a numerical solution but, unfortunately, it cannot be solved iteratively to give a simple form of solution valid for small values of the ratio (inner radius/outer radius).

A representation of degree 3 is used in § 4 to establish the necessary and sufficient condition

for the finiteness of the group (3, p | q, r) defined by

If f(z) is analytic in a suitable domain, it is shown how the integral of f(α1x1+…+αnxn) over the interior of a simplex may be reduced to the evaluation of a contour integral, in fact to an exercise in partial fractions.

The contour integral is expressed in two ways, according as the simplex is given in terms of its vertices or faces.

We present in this paper a straight-forward method of deducing integral transform pairs directly from second order differential equations. In (1) this subject has been very thoroughly treated and in view of this we believe that the method described here has its greatest justification in its simplicity.

In the present paper we determine the Laplace transforms of the modified Bessel function of the second kind Kn(t±mx), where m is any positive integer. The Laplace transforms are given in (2) and (4) below, p being the transform parameter and having positive real part.

L. J. Mordell has recently considered (1) the squared modulus of a complex difference-product, namely

under the conditions

and also under the quite different condition

He proves that under (2) the maximum of δ is nn, and is attained when and only when the zr are vertices of a regular n-gon on the unit circle.

This paper concludes the investigation of axially symmetric stress distributions in elastic solids containing penny-shaped cracks, commenced in previous papers (1), (2), by considering the stress distribution in a circular beam containing a crack opened by internal pressure or by uniform tension. The method of analysis, developed in the previous papers, is to first seek a representation of the displacement at a point of the beam as a sum of two terms, one of which is a representation of the displacement due to the crack in an otherwise unbounded infinite solid whilst the second is a general representation of the displacement in an undamaged beam, and then to show that this representation satisfies the conditions on the crack and the curved surface of the beam provided an unknown function occurring in it is the solution of a certain Fredholm integral equation. This equation holds whatever the ratio of the radius of the crack to that of the beam, but is most readily solved by iteration when this ratio is small, this solution being a perturbation on that for a crack in an infinite solid.

Let pn denote the nth prime and let ε be any positive number. In 1938 (3) Ishowed that, for an infinity of values of n,

where, for k≧1, logk+1x = log (logk x) and log1x = log x. In a recent paper (4) Schönhage has shown that the constant ⅓ may be replaced by the larger number ½eγ, where γ is Euler's constant; this is achieved by means of a more efficient selection of the prime moduli used. Schönhage uses an estimate of mine for the number B1 of positive integers n≦u that consist entirely of prime factors p≦y, where

Herer x is large and α and δ are positive constants to be chosen suitably.

Let

and

be, respectively, the upper and lower central series of a group G. Our purpose in this note is to extend known results and find some information as to which of the factors Zk/Zk−1 and Γk/Γk+1 may be infinite. Though our conclusions about the lower central series will be quite general we assume in the other case that the group is f.n., i.e. an extension of a finite group by a nilpotent group. The essential facts about f.n. groups are to be found in P. Hall's paper (4). We also refer to (4) for general notation; we reserve the letter k for positive integers.

For positive integral n and positive real x1, … xn let

Where

and let

The purpose of this article is to describe two problems which involve drawing graphs in the plane. We will discuss both complete graphs and complete bicoloured graphs. The complete graphKn with n points or vertices has a line or edge joining every pair of distinct points, as shown in fig. 1 for n = 2, 3, 4, 5, 6.