I remember that on V-J Day in 1945 a heroic band of Common-wealth delegates, greatly reinforced by the cooperative presence of colleagues from the United States, decided that they would not rise in celebration of victory but would continue their work, which included an estimate of the probable air-traffic density at the world's great airports in or about this year 1950. We made the best estimate we could of the numbers in which aircraft would in this present year be likely to be moving in and out of Heathrow (as it was then called), of Amsterdam, Brussels and Paris. Looking back through the mist of five years I detect a band of enthusiastic and starry-eyed optimists.

In this year of 1950 the British aircraft industry has an absolute and undisputed lead in the design and manufacture of civil turbo-jet and turbo-propeller aircraft. This time again I look forward, but I warn myself that we are another group of starry-eyed optimists if we think that another five years hence Britain will hold that undisputed lead, or will be comfortably established in a position of monopoly in the sale of jet-propelled civil aircraft.

Because of the greater speed of modern aircraft and the increasing complexity of air-traffic systems, air navigation is changing and breaking away more surely from its marine counterpart. This divergence means that navigational equipment designed for marine navigation is becoming less and less suitable for use in the air, and the special needs of air navigation are slowly gaining recognition. This paper concerns air navigation charts; its aim is to show the shortcomings of the charts at present in use, and to suggest a specification for a new style of chart that will meet modern requirements.

At the beginning it is well to define exactly what is meant by the word ‘Chart’. In I.C.A.O. terminology it is widely defined and includes amongst other things maps; but maps are a special case and their use is limited to visual flight. In this paper, therefore, the word is used in its older sense of a sheet upon which the actual navigation is performed.

The triple velocity correlation, in turbulence produced by inserting a square-mesh grid near the beginning of the working section of a wind tunnel, has been measured for mesh Reynolds numbers of RM = 5300, 21,200 and 42,400 (RM = UM/ν, where U is the mean wind speed in the working section of the tunnel and M is the centre to centre spacing of the rods making up the grid; ν is the kinematic viscosity of air). At the lowest Reynolds number the correlation has been measured at distances downstream of the grid varying from 20 to 120M. This range covers practically all of the initial period of the decay of turbulence, where the turbulent intensity varies as t−1.

1. Introductory. The present paper is mainly concerned with some forms for the reciprocal of the square matrix u of which the typical element is

and k is not an integer within the range ± (n − 1) inclusive but is otherwise arbitrary; thus

Introduction. If ξ is a real number we denote by ∥ ξ ∥ the difference between ξ and the nearest integer, i.e.

It is well known (e.g. Koksma (3), I, Satz 4) that if θ1, θ2, …, θn are any real numbers, the inequality

has infinitely many integer solutions q > 0. In particular, if α is any real number, the inequality

has infinitely many solutions.

1. A slow two-dimensional steady motion of liquid caused by a pressure gradient in a semi-infinite channel is considered. The medium is bounded by two parallel semi-infinite planes represented in Fig. 1 by the straight lines AB, DE. The stream-function ψ is a biharmonic function of x, y which exactly satisfies the condition that AB, DE must be stream-lines, but the condition that there must be no velocity of slip on these boundaries is satisfied only approximately, and the calculated velocity of slip gives a measure of the accuracy of the solution.

The early history of the Trinity House has been lost, partly in the mists of time, but mostly in the loss of records surrendered during the Civil War. The house itself has been destroyed by fire on three occasions, the first in the Great Fire of London, 1666; then in 1714 when a fire destroyed the Custom House, most of Water Lane where the Trinity House was situated and a part of Tower Street; and again in 1940 when fire, caused by incendiary bombs, destroyed nearly all the possessions of the corporation, some of which they had owned for upwards of 300 years. Such early records as they possessed, however, and their plate, which were stored elsewhere, have survived.

The corporation received its first charter, a copy of which is in their possession, from King Henry VIII in 1514, and the probability is that a Guild or Fellowship of some sort had been in existence for some time before, possibly under the same name, as a hall and almshouses were standing in Deptford in that year.

1. It is well known that, if two n × n matrices A, B commute, then there is a non-singular matrix P such that P−1AP, P−2BP are both triangular (i.e. have all their subdiagonal elements zero). This result has been generalized, and, in particular, has been shown to hold even if the commutator K = AB − BA is not zero, provided that K is properly nilpotent in the polynomial ring generated by A, B.

In a short paper recently published in these Proceedings, E. J. F. Primrose determines amongst other things a projective invariant N of a space cubic K and a linear complex L, N being of the third degree in the coefficients cik of the complex L. The purpose of the following remarks is to establish the connexion between this invariant N and the well-known projective comitants of the cubic K.

It was shown by Sas (1) that, if K is a plane convex body, then it is possible to inscribe in K a convex n-gon occupying no less a fraction of its area than the regular n-gon occupies in its circumscribing circle. It is the object of this note to establish the n-dimensional analogue of Sas's result, giving incidentally an independent proof of the plane case. The proof is a simple application of the Steiner method of symmetrization.

In § 1 a Markoff chain is defined, and a theorem of Kolmogoroff relating to its asymptotic behaviour is stated. Its stable distributions are examined in § 2 and some further results are obtained. These are then applied in §§ 3, 4 to a certain generalization of the cascade process, regarded as a Markoff chain with a special kind of matrix.

Nature has provided few bright stars and these are unevenly distributed in the sky. Even for air navigation it is necessary to use stars fainter than the first magnitude. There is then a choice which rapidly widens as the limiting magnitude is decreased. Brightness is not the sole criterion; distributipn is also of importance. In choosing a number of stars for use in astronomical navigation, no two people can be expected to give precisely the same weight to small variations of brightness and distribution; there thus arise different selections for similar purposes. The navigator, however, expects that the star he knows and observes will be catered for in such lists; he cannot be expected to remember that a particular star (no fainter and perhaps even brighter than neighbouring stars) is in one list but not in others. Any rule which seeks to include all stars brighter than a given limiting magnitude results in too long a list and an uneven distribution. There is thus a strong case for the selection of a list of stars, to which permanent names and numbers can be attached, which can be used for all purposes of astronomical navigation.

An investigation by G. L. Clark is found to provide exceptionally instructive illustrations of the calculation of the variously defined ‘densities’ and ‘masses’ occurring in general relativity theory and of their physical significance. In particular, the effects of motion and stress upon gravitational mass are elucidated. An explicit example shows how kinetic energy is included in the gravitational mass.

1. Introduction. Following recent developments in the theory of stochastic processes, a beginning has been made by various authors with the associated problems of sampling and statistical inference as these are concerned with dependent and ordered observations, they are distinct from and usually more difficult than the corresponding problems for independent and unordered observations.

It is advantageous in automatic computers to employ methods of integration which do not require preceding function values to be known. From a general theory given by Kutta, one such process is chosen giving fourth-order accuracy and requiring the minimum number of storage registers. It is developed into a form which gives the highest attainable accuracy and can be carried out by comparatively few instructions. The errors are studied and a simple example is given.