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.

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.

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).

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.

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.

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.

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

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.

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.

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.