Let α, β, γ, δ be real numbers with Δ = |αδ −βγ| > 0, and let ξ, η denote the linear forms

In a recent note the writer has examined the varieties whose generic curve sections are canonical curves of genus p, of general character, and whose surface sections contain only complete intersections with primals; following Fano's classification, we call these varieties of the first species. Such varieties are all rational provided that r > 3 and p > 6. In the present paper we consider their representations on linear spaces for the case r = 4, from which, in conjunction with the previous results, we conclude that fourfolds of the first species exist if, and only if, p ≤ 10; this agrees with the conjecture made by Fano in the case r = 3. It will be seen that the representation of these varieties on [4] provides interesting illustrations of Semple's formulae for composite surfaces in higher space.

A. S. Besicovitch has defined the dimension of a point-set X in n-dimensional Euclidean space in terms of its exterior Hausdorff measure as follows (2). Let (δ, X) be any enumerable class of sets whose point-set union contains X and whose members are each of diameter less than δ. Let (δ, X) denote the class of all (δ, X).

Before considering in any detail the importance of precise navigation to air traffic control, or attempting to define the word ‘precise’ in this context, it seems desirable to go back to fundamentals and consider certain essential characteristics of a volume of traffic flowing between individual pairs of airports in a system comprising a large number of such airports, randomly distributed and heavily loaded.

The dip of the horizon resulting from the observer's height of eye can be calculated from the formula Dip = 1′·06√H, where H is the height of eye in feet, or by Dip = 1′·93√h, where h is the height in metres. In this paper the latter form will be used so that comparison can be made with studies in international publications.

Das Schwarzsche Lemma der klassischen Funktionentheorie einer Veränderlichen kann als vollständige Beschreibung der inneren Abbildungen des Einheitskreises (mit Fixpunkt O) aufgefaßt werden. Ist nämlich f(z) eine in |z| ≤ 1 reguläre Funktion mit |f(z)| ≤ 1 und f(0) = 0, so muß |f′(0)| ≤ 1 sein, und es stellt diese Ungleichung bekanntlich den ersten Schritt zur Lösung des Koeffizientenproblems der im Einheitskreis gleichmäßig beschränkten Funktionen dar. Die Aussage

kann andererseits so gedeutet werden, daß jede innere Abbildung des Einheitskreises mit Fixpunkt O auch innere Abbildung jedes konzentrischen Kreises |z| ≤ |ρ| < 1 ist.

In a previous paper(1) a number of problems in probability were considered arising out of the finite resolving time of a recording apparatus. It was shown that, although the probability distributions themselves are complicated, their Laplace transforms are relatively simple. Recently Feather(2) has discussed a number of additional problems arising in this connexion, including the number of coincidences, and the number of k-clusters. The method developed in (1) is readily applicable to these problems, and it will be shown how complete probability distributions can be derived. An alternative type of recorder is sometimes considered (3) which remains dead as long as events succeed each other at intervals less than τ. The mathematical problem here is somewhat different, but it can again be effectively dealt with by the use of Laplace transforms.

This paper is an application of O. G. Sutton's theory of eddy diffusion to a case for which it was not originally developed; the theory is used to predict the dispersion of a cloud of falling droplets released from a point source at great height above the surface of the earth. Some experimental and theoretical values are given for the widths of ground area covered by liquid drops released from sources situated at heights between 1000 and 5000 ft., and agreement between theory and experiment is seen to be quite reasonable. This paper therefore provides further support for the use of O. G. Sutton's theory in problems of this type. The theory has practical applications, e.g. in the spraying of crops from aeroplanes.

In a paper published in these Proceedings I proved that there are only a finite number of quadratic fields in which Euclid's Algorithm (E.A.) holds. Recently Davenport has found a new proof of this theorem based on the theory of the minima of the product of linear inhomogeneous forms.

There has always been, since the very early days of aviation, some form of control exercised over air traffic, an attempt to curtail random activities. The earliest example is perhaps that of the somewhat unfortunate individual who was obliged to stand for long periods in a highly dangerous position at the end of a paraffin flare path. This example shows how soon it was realized that where a collection of individual units are all trying to get to the same place, some of them possibly at the same time, some co-ordination and orderliness must be introduced, and that this could only be achieved by some third party. It was soon seen that certain difficulties prohibited individual countries from making their own decisions on these matters and that, like the high seas, aviation requires a common ‘highway code’. With the ending of the first world war, and the beginnings of international aviation, came the formation of the International Commission for Air Navigation, which had amongst its objects the agreement between contracting states on standardization of rules of the air, and of visual and aural signals. It might well be said that, out of I.C.A.N., air traffic control was born in the period 1919–20.

In Fart 1 the oscillations are examined which may develop in supersonic flow through the divergent part of a nozzle. Hooker's theory is found to be confirmed by observations on a steam nozzle.

In Part 2 it is shown that Taylor's approximate theory of flow between circular arcs can also analyse the throat conditions when the velocity of approach to the throat is supersonic. Limiting symmetrical flow occurs when the velocity at the centre of the throat just exceeds the local velocity of sound, no matter what the radii of the arcs and the ratio of the specific heats of the gas may be. The unique asymmetrical case is merely the reverse of Taylor's solution for the flow when the velocity of approach is subsonic.

A numerical comparison has been made of the velocities across the throat as calculated by Taylor's method and by Fox and Southwell's iterative process when the velocity of approach is subsonic. The agreement is good for limiting symmetrical flow but not so satisfactory for the unique asymmetrical case.

The technical processes of navigation are much the same in all craft, but there are certain requirements and problems which are peculiar to ocean racing. Compared with big ship navigation, the principal differences are:

(1) The ship's speed is irregular and is not always exactly known; it is also much lower so that the tidal streams and other sets play a much more important role.

(2) The course steered is often a constantly varying one, particularly when the vessel is on the wind, or before the wind.

(3) Leeway is a far larger factor.

(4) The violence of the motion and the lack of space greatly increase the difficulty of plotting and calculating.

(5) The methods of fixing the ship are generally less exact and less readily available.

When simultaneous position lines are unobtainable a running fix or transferred position line is generally used. This method depends for its accuracy on the true assessing of course and speed. If, instead of transferring one of two position lines, three are obtainable, only the course need be assessed and used to obtain a position. It is assumed that the speed is constant, or nearly so—a reasonable assumption.

1.1. It is often necessary in nuclear physics research to know the distribution in amplitude of a series of electrical impulses. An instrument which reveals this distribution directly is called a ‘pulse-amplitude analyser’, or, in the jargon of the late war, when the problem of building such an instrument was first seriously considered, a ‘kick sorter’. Many kick sorters have been described or discussed ((1)—(16)), and many new types are at present under development.

In this paper, a family of exact solutions of the problem of two-dimensional flow of a compressible perfect fluid about a cylinder is found, the solutions being generalized from those for the flow of an incompressible fluid about an elliptic cylinder of arbitrary eccentricity and angle of attack. The circulation is taken to be zero and the speed of the fluid at infinity subsonic. This analysis is an application of the general theory given by T. M. Cherry (1, 2); it was done to exhibit the details of the analysis for a flow other than that corresponding to the low-speed flow past a circular cylinder.

This paper describes a microwave direction finding equipment and gives the results of flight tests carried out to investigate its possible use for air navigation.

The system operates in the 3.2 cm. wavelength band. The aircraft equipment consists of a receiver which sweeps rapidly over a small frequency band, a directional aerial which is scanned in bearing and a cathode ray tube display. The ground beacons are low power c.w. oscillators with an omnidirectional aerial and operate at separate fixed wavelengths to facilitate identification.