The problem of constructing an n-dimensional metric differential geometry based on the idea of a two-dimensional area has given rise to several publications, notably by A. Kawaguchi and S. Hokari (1), E. T. Davies (2), and R. Debever (3). In this geometry the area of a two-dimensional plane element is defined by a fundamental function L(xi, uhk), where the xi are point coordinates and the uhk are the coordinates of the simple bivector representing the plane element. L is supposed to be a positive homogeneous function of the first degree with respect to the variables uij, and to possess continuous partial derivatives up to and including those of the fourth order. With these assumptions the problem of the construction of the metric differential geometry splits into two problems; the first of these is the problem of constructing a metric tensor gij(xr, uhk), and the second is the problem of constructing an affine connexion. We deal with the first problem only in this paper.

Recent investigations by F. Yates (1) in agricultural statistics suggest a mathematical problem which may be formulated as follows. A function f(x) is known to be of bounded variation and Lebesgue integrable on the range −∞ < x < ∞, and its integral over this range is to be determined. In default of any knowledge of the position of the non-negligible values of the function the best that can be done is to calculate the infinite sum

for some suitable δ and an arbitrary origin t, where s ranges over all possible positive and negative integers including zero. S is evidently of period δ in t and ranges over all its values as t varies from 0 to δ. Previous writers (Aitken (2), p. 45, and Kendall (3)) have examined the resulting errors for fixed t. (They considered only symmetrical functions, and supposed one of the lattice points to be located at the centre.) Here we do not restrict ourselves to symmetrical functions and consider the likely departure of S(t) from J (the required integral) when t is a random variable uniformly distributed in (0, δ). It will be shown that S(t) is distributed about J as mean value, with a variance which will be evaluated as a function of δ, the scale of subdivision.

The problem considered in this paper is the fluid motion arising from a thin semi-infinite plate started to move impulsively from rest (in viscous incompressible fluid at rest) with a velocity, subsequently maintained uniform, parallel to the edge. Two solutions are given, one obtained in polar coordinates is in the form of an infinite series, whilst the other, derived operationally in parabolic coordinates, leads to a single integral for the velocity distribution. The former is convenient for computation in the vicinity of the edge, the latter being more convenient elsewhere.

A hypothesis originally introduced by Rayleigh after he had discussed the corresponding flow for an infinite plane has been used here to draw deductions about the steady flow past a quarter-plane whose leading edge is normal to the direction of flow and also to obtain approximate expressions for the effect of the edges on the skin friction of a sufficiently broad rectangle whose length is parallel to the incident stream.

This paper is mainly concerned with the distribution of stress near a flat elliptical crack in a body of infinite extent under a uniform tension at infinity perpendicular to the plane of the crack. After an analytical solution of this problem was found the authors received a copy of a paper by Sadowsky and Sternberg (3) in which they solved the more general problem of the stress concentration around a tri-axial ellipsoidal cavity in an elastic body of infinite extent, the body at infinity being in a uniform state of stress whose principal axes are parallel to the axes of the cavity. The method of solution adopted by the present writers for the special case of the elliptical crack is somewhat different from the more general work of Sadowsky and Sternberg, and is, moreover, surprisingly simple, so it seems to be of value to present this solution here.

In this paper, Fourier integrals are used to solve some elastic problems of generalized plane stress and small transverse displacements in infinitely long, rectangular, isotropic plates stressed only at their edges. The Airy stress function and the transverse displacement satisfy the two-dimensional bi-harmonic equation, and the basic mathematical problem is to solve this equation subject to different sets of boundary conditions. Little attention has been given hitherto to problems in which some of the boundary conditions depend directly upon displacements. Here the general problem is solved when one long edge is fixed, and stresses or displacements are arbitrarily prescribed at the other, with no stresses and displacements at infinity. The problem of a concentrated edge force is discussed in detail and numerical values of the stresses at the fixed edge are given.

In a recent tract (Baker (1)) there is described in considerable detail a configuration of forty-five points which are nodes of a quartic primal in four dimensions. The geometry of this primal is very fascinating; among its interesting properties is the fact that a number of well-known geometrical configurations, which usually arise as unrelated phenomena, here all appear in connexion with the one figure. The interest of the primal, of course, lies chiefly in the large number of collineations which leave it invariant. The group G* of these collineations is considered in a paper by Burkhardt (2), in which are given explicit expressions for five algebraically independent functions of the five variables, which are left invariant by the operations of the group. The simplest of these invariants is of the fourth order, and when equated to zero represents the quartic primal which is the subject of Baker's tract.

1. The questions considered in this note are suggested by the elementary topology of the trajectories of systems of non-linear differential equations. Such a system may be assumed in the form

and the values of the dependent variables x1, x2, …, xn at ‘time’ t can be represented by a point P(t) in a ‘phase space’ . As t varies, P(t) describes a curve in , which is a trajectory of (1). Now it often happens that contains a subspace E (usually of lower dimension) with the following properties: (i) by considering the trajectories generated by points P(t) which are, for t = 0, in E, all the trajectories of (1) are obtained; (ii) if P(0) is in E, then P(t) is not in E for 0 < t < c, where c is a constant independent of P(0) in E; (iii) if P(0) is in E, then the trajectory meets E again for some finite t at a point P(T) (T is not necessarily the same for all points of E). By considering P(T) as the image of P(O), a mapping of E into itself is defined which is associated with the system (1), and the topology of the trajectories of (1) can be studied conveniently by discussing this mapping. When the functions fi in (1) satisfy the continuity and Lipschitz conditions of the classical existence-and-uniqueness theorem, the mapping is one-one and continuous. The study of this ‘transformation theory’, initiated by Poincaré, has been developed chiefly by G. D. Birkhoff(l,2). His results have been applied to problems of ‘non-linear mechanics’ by N. Levinson(3).

The study of differential properties of functions of two real variables in the light of the modern theory of functions of real variables was started by H. Rademacher in 1919. Although much work has since been done on the subject, and a great many general results have been obtained, a number of questions have remained open, some of which will be discussed in the present work.

The Institute has now completed two years of its existence. The papers which have been read before it during these two years have covered a wide range of subjects and have served to emphasize the many ramifications of the science of navigation. Because of the high speed of modern aircraft, air navigation presents more problems and of greater variety than surface navigation, but even on the problems of surface navigation there has been ample scope for a wide range of discussion. The Institute has taken a prominent part in the discussion of the proposals for the revision of the Abridged Nautical Almanac. It might with some reason have been supposed that there was nothing more to be said on the methods of reducing astro-sights and determining position at sea. The problem is perfectly straightforward and there is a limit to the number of different ways in which the spherical triangle can be solved. But the essential basic data can be presented in a variety of ways, while there are many possible methods of presenting tables for the solution of the spherical triangle. The decision to use Greenwich hour angle instead of right ascension in the Abridged Nautical Almanac has followed its adoption in the Air Almanac; the revised Almanac will have an entirely different format from the present, while the methods of reducing sights must be correspondingly modified.

Practical experience in radar-assisted marine navigation has shown that the closer one is to the general land areas, navigational seamarks, &c, the easier position fixing from radar information becomes, although some particularly congested and confined waters provide exceptions to this rule. There are usually large numbers of more suitable targets available at the closer ranges, and increased ease of identification results from objects being closer to the observer. The latter effect, of course, arises from initially knowing one's position more accurately, and this in turn is based on the shorter ranges at which bearings and/or ranges are being taken.

A method is given for the calculation of the surface waves of small amplitude generated on deep water by a normal velocity distribution of period 2π/σ prescribed over a submerged circular cylinder. The method of solution involves a system of linear equations in an infinite number of unknowns; this system always possesses a solution. The unknowns may be obtained as power series in a parameter Ka, convergent for sufficiently small values of the parameter. When the parameter is not small, the equations can be solved by infinite determinants. It is shown that the reflexion coefficient of waves incident on a fixed circular cylinder vanishes, as was first shown by Dean. The pulsations of a submerged cylinder are discussed when the normal velocity is the same at all points of the cylinder at any given time.

In connexion with moment problems, S. Verblunsky proved the following two theorems:

Theorem I. (a) If f(x) is integrable in (−∞, +∞) and satisfies 0 ≤f(x) ≤ 1, then there exists a function σ(x), bounded and non-decreasing in (−∞, +∞), such that, for ζ = ξ + iη and η ≠ 0,

This paper describes tests of an approach procedure in which aircraft approach the airport on a radial track, circle the airport on an orbit and finally approach to land on another radial track in line with the runway. Ninety-six approaches were made with a representative selection of airline pilots in an aircraft fitted with the Australian Distance Measuring and Multiple Track Range Equipment.

The flight tests provide information as to the accuracy with which pilots can follow a set track, both for track accuracy and turning, when provided with adequate navigation aids. The average time for the particular procedure used was 18 minutes, and 90% of all times were within ±1 minute of the average. Maximum departures from the correct track occurred during orbiting where the average track was 0·2 mile inside the nominal orbit, and in 90% of approaches the departure did not exceed ±0·8 mile. Detailed analysis of the turns is also given in which it is shown that, on the average, turns occupy ¾ minute instead of the ½ minute required for a correctly executed Rate 1 turn.