In the present paper a general solution of the equations of elasticity in complete aeolotropy is found under the assumption that the stresses and therefore the strains are linear in the third cartesian coordinate. This solution is applied to the elastic equilibrium of a completely aeolotropic cylinder, under a distribution of tractions on the lateral surface and resultant forces and couples on the end sections of the cylinder. The problem of extension of a completely aeolotropic cylinder by longitudinal lateral loading and end forces is solved with an application to the elliptic cylinder. The writer hopes to present in later communications applications of this general solution to the following problems: (i) Bending of a completely aeolotropic cylinder by longitudinal lateral loading and end bending couples, (ii) Torsion of a completely aeolotropic cylinder by transverse lateral loading and end twisting couples, (iii) Flexure with shear of a completely aeolotropic cylinder with free lateral surface. Particular cases of (ii) and (iii) were considered by Luxenberg [1] and Lechnitzky [2, 3].

In its simplest form, the theorem of Ascoli with which we are concerned is an extension of the Bolzano-Weierstrass theorem: it states that, if X and Y are bounded closed sets, of real or complex numbers, andis a sequence of equicontinuous functions mapping X into Y, thenhas a uniformly convergent subsequence. As is well known, this result has played a fundamental part in the development of several theories; it has also been widely generalized, by processes of abstraction and localization, and a very useful version of the theorem runs as follows (cf. [4], 233–234):

(A) Suppose that X is a locally compact regular space, and that Y is a Hausdorff space whose topology is determined by a uniform structure. Let YX be the space of all functions that map X into Y, with the topology of locally uniform convergence with respect to. Then a closed setin YX is compact if, at each point x of X, (i) the set(x) is relatively compact, in Y, and, (ii) is equicontinuous with respect to.

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

For each index β in a set J let Aβ be a group. The free product

is by definition a group generated by the Aβ in which any two distinct reduced words represent distinct elements. Here a reduced word is either the expression 1 or an expression

with each “syllable” ai ≠ 1 and each βi ≠ βi+1.

The composite area made up of a semi-infinite strip and a semicircle is represented on a half-plane. Different transformations are in effect used for the two sub-areas and there is accordingly a continuity condition at the line of separation. This condition is satisfied only to an approximation, but a fairly accurate solution is found without much difficulty. The transformation can be used to find the steady two-dimensional flow of perfect fluid in a channel that turns through 180°.

The problem considered is that of determining the stress distribution in an infinite plate which contains a straight crack terminated at one end by a circular hole, and is in a state of plane strain or generalized plane stress under given loads. The problem has some relevance to the engineering practice of “stress relief”, in which holes are made at the ends of a crack with the object of reducing the concentration of stress. When the diameters of the holes are small compared with the length of the crack, the distribution of stress near one hole will not be greatly influenced by the presence of the other, and will then be approximately the same as in the simpler case considered here.