The presence of a non-uniform distribution of temperature in an elastic solid gives rise to an additional term in the generalized Hooke's Law connecting the stress and strain tensors and to a term involving the time rate of change of the dilatation in the equation governing the conduction of heat in the solid. The present paper is concerned with the effects produced by these additional terms in two simple situations. In the first, the elastic solid is regarded as being of infinite extent and the distribution of temperature in the solid is produced by heat sources whose strength may vary with time. In the second, the solid is supposed to be semi-infinite and to be deformed by prescribed variations in the temperature of the bounding plane and by heat sources within itself.

If the temperature in an elastic rod is not uniform and if it varies with time, dynamic thermal stresses are set up in the rod. This paper is concerned with the calculation of the distribution of temperature and stress in an elastic rod when its ends are subjected to mechanical or thermal disturbances. Simple waves in an infinite rod are first discussed and then boundary value problems for semi-infinite rods and rods of finite length. The paper concludes with an account of an approximate method of solving the equations of thermoelasticity.

The roots of the equation zez = a are of importance in several theories. Various authors have studied certain of their properties over more than a century. Here we solve the equation, in the sense that we define the sequence {Zn} of roots and, except for a small, finite number of values of n, find a rapidly convergent series for Zn. The terms in this series are alternately real and purely imaginary and so the series is very convenient for calculation. For the few remaining roots, we give practicable methods of numerical calculation and supply an auxiliary table.

The main results of this article have been announced without proof or details in Wright 1959.

Let Q(x1 …, xn) be an indefinite quadratic form in n variables with real coefficients. Suppose that when Q is expressed as a sum of squares of real linear forms, with positive and negative signs, there are r positive signs and n—r negative signs. It was proved recently by Birch and Davenport that, if

then for any ε > 0 the inequality

is soluble in integers x1, …, xn, not all 0.

Let λ(n) be Liouville's function denned by

where v is the number of prime factors of n, repeated factors being counted according to their multiplicity. Alternatively, λ(n) may be denned by the relation

where ζ(s) is the zeta function of Riemann.

If K is a convex body in n-dimensional space, let SK denote the closed n-dimensional sphere with centre at the origin and with volume equal to that of K. If H and K are two such convex bodies let C(H, K) denote the least convex cover of the union of H and K, and let V*(H, K) denote the maximum, taken over all points x for which the intersection is not empty, of the volume

of the set . The object of this paper is to discuss some of the more interesting consequences of the following general theorem.

The present paper is an attempt to develop and illuminate the foundations of structure theory as presented in a previous paper [8] which will be referred to as I.

Our approach is based on an unorthodox view of physical theory, largely due to Eddington ([5], [6], [7]), that leads us to expect that at least some (and perhaps all) physical laws are derivable from a consideration of the intrinsic nature of measurement. This is discussed in the Introduction of I. A theory with this approach will be called “pre-empirical”, in contrast with orthodox physical theories, which are postempirical.

Our object in this note is to construct rings such that

More generally we prove the following embedding theorem (Theorem 5.1) from which rings satisfying (I) are easily constructed:

Any algebra over a field F which contains no zero-divisors or unit-element may be embedded in an algebra over F satisfying (I).

Some examples of the bending of a plane elastic plate by transverse forces applied at isolated points are first considered. The plate is infinite and is bounded internally by a circular edge along which it is clamped; simple expressions are found for the displacement.

The analogous hydrodynamical problem is that of the steady flow of viscous incompressible liquid past a fixed circular cylinder; the equation for the stream function is in the same form as the equation for the displacement of a plate due to a distributed force of amount Z per unit area. The inertia terms in the hydrodynamical problem correspond to Z. In slow motion there is no stream function with the correct form at infinity, because in this case the inertia terms are ignored so that in the plate problem Z = 0. The effect of a transverse force system can be most simply illustrated by supposing that the forces are concentrated at isolated points; in the corresponding stream functions a simple form of allowance for inertia is therefore made, and they display some of the features of steady flow past a cylinder.

Suppose throughout that l, an (n = 0, 1, …) are arbitrary complex numbers, that α is a fixed positive number and that x is a variable in the interval [0,µ]. Let

In this note we consider a problem which is suggested by a paper of W. Feller. A plane set A lies in a circle C of centre O and radius 1, and is such that the linear measure of the intersection of A with any straight line does not exceed 21, where 0 < l < 1. To find the upper bound of the plane measure of A. Both linear measure and plane measure are taken in the sense of Lebesgue, and they will be denoted by m and m2 respectively.

A method of solving Oseen's equations for the flow of viscous fluid past a cylinder was devised by Bairstow, Cave and Lang [1], who found a doublet solution of the equations and reproduced the flow by a distribution of these doublets over the surface of the cylinder. The same method is used here to deal with axisymmetric flow and an integral equation is given to determine the density of the distribution. The particular case of the flow past a sphere at low Reynolds numbers is solved by this method.

An expression for the Stokes's stream-function for a slow steady axisymmetrical motion of viscous fluid due to a hydrodynamical distribution in the presence of a sphere has been given by the author [1] in terms of the stream-function for the motion due to the same distribution in unbounded fluid, this latter motion being assumed irrotational. The author has since been able to obtain a more general expression for the stream-function by not assuming the motion in unbounded fluid to be irrotational, this result being given in his Ph.D. dissertation [2]. Hasimoto [3] has since given these results, the second in a different form from that of the author. Since the general expression derived by the author is considerably simpler than that found by Hasimoto, it is given in this note. A corresponding result for the motion in a region bounded by a plane is also given.

In a short and little known paper, Jacobi [1] gives conditions for the cubic residuacity of small primes q = 2, 3, ..., 37 to a prime p in terms of the quadratic partition

in the form L ≡ ±µM (mod q), and LM ≡ 0 (mod q).

In all that follows, E denotes a separated locally convex space, E' its topological dual, and <x, x−> the bilinear form expressing the duality. We consider the differentiation of a function f:t → f(t) of a real variable t which takes its value in E; the domain of f will be an open interval which, without loss of generality, may be taken to be the entire real axis. There are various senses in which the derivative may or may not exist, and it is proposed to consider some relations between these senses.

Let Q(x1, …, xn) be an indefinite quadratic form in n variables with real coefficients. It is conjectured that, provided n ≥ 5, the inequality

is soluble for every ε > 0 in integers x1, …, xn, not all 0. The first progress towards proving this conjecture was made by Davenport in two recent papers; the result obtained involved, however, a condition on the type of the form as well as on n. We say that a non-singular Q is of type (r, n—r) if, when Q is expressed as a sum of squares of n real linear forms with positive and negative signs, there are r positive signs and n—r negative signs. It was proved that (1) is always soluble provided that