Let X denote the general point with coordinates (x1, x2, x3) in 3-dimensional space; and let P(X) be the function defined by

Let ξ, η, ζ be linear forms in u, v, w with real coefficients and determinant Δ ≠ 0. Then there exists a number ℳ such that, corresponding to any real numbers a, b, c, there exist rational integers u, v, w for which

The problem of evaluating the non-homogeneous critical determinant of the plane region |y − x2| ≤ 1 is solved as a particular case of a result concerning the non-homogeneous properties of convex regions proved in the author's thesis (not yet published). The search for an alternative method of proof led to a proof of the existence of an infinite sequence of minima analogous to the Markoff chain. This is the subject of the present paper.

Suppose that dυ and dυ′ are two volume elements situated at points P and P′ respectively in a three-dimensional right circular cylinder, that y is the distance PP′, that z(y) is a given function of y, and that we wish to evaluate the sixfold integral

taken over all pairs of points P, P′ within the cylinder. We observe that z(y) is a function of y only; so that the sixfold integral can be expressed as a single integral

that is to say a weighted mean of z(y) over the relevant values of y, where the weight function is evidently given by

If V is an irreducible variety and W is an irreducible simple subvariety of V, then one of the properties of the quotient ring of W in V is that it is a unique factorization domain. A proof of this theorem has been given by Zariski ((2), Theorem 5, p. 22), based on the structure theorems for complete local rings, and the fact that the local rings which arise geometrically are always analytically unramified. Here the theorem is deduced from certain properties of functions and their divisors which will be established by entirely different considerations. The terminology which will be employed is that proposed by A. Weil in his book(1), and we shall use, for instance, F-viii, Th. 3, Cor. 1, when referring to Corollary 1 of the third theorem in Chapter 8. Before proceeding to details it should be noted that Weil and Zariski differ in then-definitions, and that in particular the terms ‘variety’ and ‘simple point’ do not mean quite the same in the two theories. The effect of this is to make Zariski's result somewhat stronger than Theorem 3 of this paper.

In this paper I investigate some geometrical properties of a system of primals which arose a few years ago in the study of a purely algebraic problem: to parametrize completely the group of automorphic transformations of a given bilinear form. This problem is classical, and there exists a large literature on the subject, but the algebraists never succeeded in finding a complete parametrization. Indeed, the trend was to move away from those transformations not covered by the known parametrization; and Weyl, for example, writing about the orthogonal group in his book on the Classical Groups remarks ‘unfortunately Cayley's parametric representation leaves out some of the orthogonal matrices, and a good deal of our efforts will be spent in rendering these exceptions ineffective’. In another paper I shall show how to solve this problem of complete parametrization, via a geometrical approach; but here I confine my attention to some preliminary geometrical results.

The object of this paper is to obtain some properties of non-degenerate quadric primals in the projective geometry in [s] over a Galois field of order n. It will be shown that, if s is even, there is only one type of quadric, but that, if s is odd, there are two types of quadric. The number of points on a quadric of each type, and the number of quadrics of each type, will be found. Finally, a possible application to statistics will be indicated.

The Grassmannian, of lines of S5 and also its sections by general spaces of 13, 12, 11 and 10 dimensions were shown to be rational by Severi. The section by a general S9 is, by a result of Fano, birationally equivalent to a non-singular cubic primal of S4 and hence is probably not rational.

A transform method for the solution of linear difference equations, analogous to the method of the Laplace transform in the field of linear differential equations, has been described by Stone (1). The transform u(z) of a sequence un is defined by the equation

1. This paper is concerned with certain properties of the solutions of differential equations of the type

where f, g have continuous partial derivatives up to the second order satisfying Lipschitz conditions in some bounded domain and are periodic in t with period 2π, ω is a positive constant and k is a small parameter. If f = 0, (1·1) is equivalent to

an equation representing the forced vibrations of a quasi-linear system of one degree of freedom.

We consider n − 1 points xj (j = 1, …, n − 1) selected independently at random from the interval (0, 1), the distribution of any xj being the rectangular distribution

The points xj divide the interval (0,1) into n intervals. Let λ be the sum of the lengths of the k largest of these n intervals. It is the purpose of this paper to give the exact sampling distribution of λ for any given n and k (see equation (5)). Fisher (1) in 1929 solved the problem for k = 1. We first investigate the distribution of μ = 1 − λ.

During the Oxford Conference of the Econometric Society in September 1936, Ragnar Frisch proposed a problem in regression theory. A partial solution was found in 1938 by Miss H. V. Allen (1). A more complete solution was given by C. R. Rao (6) in 1947, and in the same year the present author (5) obtained a solution as a particular case of a more general result. These last two papers contained a flaw, and a correct solution was provided by Miss E. Fix (2). This last solution still leaves a part of the problem unanswered, and in the present paper a result of P. Lévy's (4), is used to complete the solution. At the same time further generalizations of the problem are considered and, in the cases of most practical importance, complete solutions are obtained. It is advisable, both from the point of view of rigour and simplicity of analysis, to use a general definition of the conditional expectation of a random variable. Accordingly, the paper begins with a summary of the relevant definitions. These notions were introduced by Kolmogoroff (3). It has been thought worth while giving the definitions here, in forms which are slightly different from Kolmogoroff's and seem more suitable for applications, in order to explain the notation and nomenclature used. The relevant consequences of these definitions are also stated in the form in which they are used.

It is shown that a mass of fluid bounded by fixed surfaces and by a free surface of infinite extent may be capable of vibrating under gravity in a mode (called a trapping mode) containing finite total energy. Trapping modes appear to be peculiar to the theory of surface waves; it is known that there are no trapping modes in the theory of sound. Two trapping modes are constructed: (1) a mode on a sloping beach in a semi-infinite canal of finite width, (2) a mode near a submerged circular cylinder in an infinite canal of finite width. The existence of trapping modes shows that in general a radiation condition for the waves at infinity is insufficient for uniqueness.

This paper considers the correlation P(r) between the fluctuating pressures at two different points distance r apart in a field of homogeneous isotropic turbulence. P(r) can be expressed in terms of the fourth moment of the velocity fluctuation, which is evaluated with the aid of the hypothesis that fourth moments are related to second moments in the same way as for a normal joint distribution of the velocities at any two points. The experimental evidence relevant to this hypothesis (which cannot be exactly true since it gives zero odd-order moments) is examined. The alternative hypothesis made by Heisenberg, that the Fourier coefficients of the velocity distribution are statistically independent, has identical consequences for the fourth moments of the velocity, although it does not lead to such convenient results.

The pressure correlation is worked out in detail for the important special case of very large Reynolds numbers of turbulence; the mean-square pressure fluctuation is found to be . The mean-square pressure gradient is evaluated, from the available data concerning the doublevelocity correlation, for the cases of very small and very large Reynolds numbers, and a simple interpolation between these results is suggested for the general case. Finally, the relation between the mean-square pressure gradient and rate of diffusion of marked fluid particles from a fixed source is established without the neglect of the viscosity effect, and the available observations of diffusion are used to obtain estimates of which are compared with the theoretical values.

The nature of the turbulent motion in a boundary layer with zero longitudinal pressure gradient has been investigated with the techniques of hot-wire anemometry which have been developed for the study of shear flow in wakes. Measurements have been made of the intensities of the turbulent velocity components, the turbulent shear stresses, the rates of transport of turbulent energy by diffusive movements, the intensities and flattening factors of the down-stream spatial derivatives of the three velocity components, and spectra of the down-stream component of the velocity fluctuation, at traverses through the boundary layer at three stations where the Reynolds numbers (based on the displacement thickness and free stream velocity) were respectively 3630, 4360 and 5080. Over the range of measurement, which did not include the laminar sublayer, the turbulent motion was similar at all three stations, and could be described in terms of universal functions. By considering the turbulent energy balance, it is shown that, except in the outer part of the layer where the turbulence resembles strongly the turbulence in a wake, there is a strong flow of energy directed toward the wall and transported by the action of turbulent pressure gradients. It is concluded that, most probably in contact with the laminar sublayer, there must be a ‘dissipative layer’ within which most of the turbulent energy dissipation takes place, and that the bulk of the eddies are, in a sense, attached to the wall and have very high rates of shear in that region. In agreement with this view of the structure of the turbulence, length scales derived from the apparent eddy viscosity and the local turbulent intensity are found to be comparable with distance from the wall. Length scales in the direction of the mean stream are much larger, and it is believed that the typical eddy is very elongated in this direction, and has its vorticity directed roughly parallel to the direction of maximum positive mean rate of strain.

Sack (3) and Sneddon (4) have considered the problem of a penny-shaped crack in an infinite medium under tension normal to the face of the crack. The same problem has been considered more recently by Green (1) by a different method. In this note we consider the case of the same shaped crack with the medium subjected to a uniform shearing stress parallel to the plane of the crack. It is found that the solution is analogous to that of the corresponding two-dimensional problem considered by Starr (5).

Three-dimensional stress distributions in hexagonal aeolotropic materials have recently been considered by Elliott(1, 2), who obtained a general solution of the elastic equations of equilibrium in terms of two ‘harmonic’ functions, or, in the case of axially symmetric stress distributions, in terms of a single stress function. These stress functions are analogous to the stress functions employed to define stress systems in isotropic materials, and in the present note further problems in hexagonal aeolotropic media are solved, the method in each case being similar to that used for the corresponding problem in isotropic materials. Because of this similarity detailed explanations are unnecessary and only the essential steps in the working are given below.

The equation of steady rectilinear motion of a material of variable viscosity is transformed, when the pressure gradient vanishes, into a second-order differential equation which is linear in the dependent variable. A general solution of the linearized equation can be obtained in the case of a wide class of plastic solids and viscous liquids. By making use of the idealized materials which lend themselves to this method of solution, a theoretical comparison can be made of the behaviour of plastic solids of different types under the same conditions. As an illustration, a plastic solid which yields very sharply when the yield point is first exceeded is compared with one which yields only gradually at the same yield value; the flow caused by the longitudinal motion of an (approximately) elliptic cylinder in an infinite mass of material is investigated in each case.

Both the Ising theory of ferromagnetism and the theory of regular solutions are concerned with systems arranged on a lattice and make the assumption that each system interacts only with its nearest neighbours. Mathematically, there is a close parallel between the two problems (see, for instance, Rushbrooke (1)). In the first half of this present paper the partition functions for these two problems are examined in some detail. Power series expansions of the partition function of the Ising model, valid for low and high temperatures, are obtained. The terms obtained in the power series have been analysed and approximate numerical results obtained. It is hoped to publish these in a second paper.

The object of the present paper is to discuss the Fourier expansion of the Riesz potential. For this purpose a new definition of the electromagnetic potentials, depending upon an arbitrary parameter α is given. It is shown that this definition is a generalization of the Wentzel potentials in the α-plane, whereas that given by Fremberg (3) is a generalization of the Maxwell potentials. The analysis is applied to the problem of eliminating, in a straightforward way, the longitudinal part of the potential describing the electromagnetic field. The problem of the quantization of the field, based on its Fourier expansion, will be considered in another paper. The recent work of Tomonaga, Schwinger and Dyson, and the regularization process of Pauli has lifted the theory of quantum electrodynamics to a much higher level of rigour and fruitful applicability. All the same, a further study of Riesz potential seems to us of some interest in this field.