It is difficult to get satisfactory noon observations when the zenith distance is very small, because the Sun is rapidly changing in azimuth, the sextant must be swung through a wide arc of the horizon, and the Sun only dwells at the highest altitude for a very short period. To avoid these difficulties it is suggested that, instead of trying to measure the altitude of the Sun, one should measure its distance from a point on the horizon in a known azimuth. To make this measurement the Sun is brought down to the required point on the horizon, tilting the sextant as necessary; we shall distinguish this type of observation from the normal one, made with the sextant vertical, by calling the angle so obtained the horizon distance. At normal altitudes this method, however sound in theory, breaks down because one cannot measure the azimuth with sufficient accuracy. However, at very high altitudes the rate of change of horizon distance with change of azimuth of the point from which the distance is measured becomes so small that it is not necessary to know the azimuth with such precision. In fact, at a zenith distance of 1°, the largest we shall consider here, the greatest error caused by an error of 1° in the azimuth is l″, and for smaller zenith distances the maximum error is proportionately less.

The three basic problems of navigation are determination of position, definition of direction, and steering the direction required. In polar regions, all these problems are affected either by the high latitude, or by the polar climate, or by both. In this paper the various problems which arise are reviewed, possible solutions are discussed, and it is indicated where further research and experiment might be desirable.

Two of the strongest emission lines in the spectra of the planetary nebulae were identified by Boyce, Menzel and Payne (1) as due to forbidden transitions in the ground configuration of doubly ionized neon. The transitions are of wave-length 3967 and of wave-length 3869 in the configuration Ne III 1s22s22p4. These lines are excited by inelastic collisions of electrons with the neon ions. In order to investigate these transitions in detail a knowledge of the collision cross-sections is required. As a first step in this direction, wave functions have been calculated for the 3P state by the method of the self-consistent field, including the effect of exchange. In later papers these wave functions will be used in the calculation of collision cross-sections, in a study of the departures from the Landé rule in the triplets of the p2 and p4 configurations and, it is hoped, in the calculation of line strengths for certain permitted lines of astrophysical importance in the spectrum of Ne II.

In this paper, the theoretical double and triple velocity correlation functions, f(r), g(r) and h(r), which correspond to Heisenberg's spectrum of isotropic turbulence, are obtained numerically for two Reynolds numbers. One set of these correlations is for the limiting case of infinite Reynolds number. In addition, a method is developed for deriving the approximate form of the double correlations for any Reynolds number, which is not too small, from the corresponding correlations for infinite Reynolds number. These theoretical correlations are then compared with the results of experiment.

1. Introduction. Necessary and sufficient conditions are established for a real quadratic form to be positive definite on a linear manifold, in a real vector space, explicit in terms of the dual Grassmann coordinates for the manifold.

1. Introduction. In a recent contribution to these Proceedings Alladi Rama-krishnan (23) has discussed a number of problems which arise when the development of a cascade shower of cosmic rays is considered from the standpoint of the theory of stochastic processes. As has often been remarked (see, for example, the important study by Niels Arley (1)), there is a close formal analogy between such physical phenomena and the growth of biological populations. In particular, if the distance of penetration (t) is identified with the time, and the energy (E) of a particle in the shower is replaced by the age (x) of an individual in the population, there emerges an obvious analogy between stochastic fluctuations in the energy spectrum of the shower and similar fluctuations in the age distribution of the population.

1. In his classical investigation of the motion of a sphere moving (without spin) in a fluid at rest at infinity, Stokes showed (using his method of ‘stream functions') that the sphere experiences a thrust − 6πμau, where a is the radius of the sphere, and μ, u have their usual meanings.

The concept of the Green's vibrational function given in an earlier paper by the author is used to obtain a general expression for the disturbance from a point source. The potential due to transient sources of sound moving with subsonic and supersonic velocities is derived from this. It is found that the Doppler effect for a supersonic source differs from that for a subsonic source. In the former case it is found that two frequencies are heard simultaneously from a source emitting a note of one frequency.

The theory is applied to determine some solutions of the two dimensional equation of supersonic, irrotational compressible flow, corresponding to the flow around an aerofoil taking into consideration the entropy changes at the shock wave.

Language holds many pitfalls for the unwary. A housewife picks up her iron, and so does the golfer: a butcher uses the steel and so does the soldier: a government nationalizes iron and steel. No wonder the foreigner makes ludicrous mistakes. And we in our turn, coming across an Italian book entitled Il Compasso da Navigare might be pardoned for supposing its subject-matter to be the navigating compass. This is not so. It is a learned and comprehensive treatise under the title belonging to the oldest surviving Sailing Directions for the Mediterranean Sea, and as the author, Bacchisio R. Motzo, clearly proves, such Directions were called a Compasso, and by a natural transference the same name was given to the sailing chart which was drawn to accompany the Directions. What we call a mariner's compass, or simply a compass, the Italians term Compasso-bussola or more usually just bussola. When English sailors began to use written Directions or ‘Pilots’ they called them Rutters (from the French Routier), while an alternative Italian name for a Compasso was Portolano, which also meant the accompanying chart.

1. Introduction. In a recent paper Clark (1) has dealt with the problem of the rotating disk, the material of which is such that the waves of dilatation in this particular material travel with the velocity of light. The material of the disk is supposed to be under an isotropic stress p when in a strained state, and the relation between stress p and the dilatation Δ is found to be connected by an expression

where a = density in the unstrained state, and Δ is given by

Ui (i = 1, 2, 3) are the components of the strain.

An extension is proposed to the theory of Verwey and Overbeek on the force between two colloidal particles in a dilute sol. The effect of the Coulomb interaction of the ions in the dispersion medium is considered. It is shown that this contributes an additional repulsion term to the mutual free energy of the two particles. This correction term behaves as an inverse power and is the predominant effect at large separations, although it is a second-order effect at small separations. For parallel plates, this power is the fourth, and for two spheres, the sixth. A tentative calculation shows that, at least for thin plates and low electrolyte concentrations, our new repulsion will outweigh the attractive London-van der Waals force considered by Verwey and Overbeek, and more recently by Casimir and Polder, when the plates are far apart.

This paper discusses the possibilities of using diagrammatic solutions for accurate astronomical navigation and gives a detailed description of two methods of doing so that have been put into effect in Germany. These methods, the ARG1 and the Astronomischer Rechenatlas, have previously been described in German publications and referred to in numerous reports and memoranda written in this country and the United States. So far as is known no authoritative description of these two aids to navigation has yet been published in English, and it is appropriate that this, the first, should have been written by Dr. Freiesleben, who played a prominent part in their development.—Ed.

The development of radio aids to navigation has not made astronomical navigation redundant, but it has given more force to earlier demands for the simplification of methods of reduction. No longer is it acceptable to have a large variety of alternative methods, and it is probable that attempts will be made to select a single method that is both simple and universally applicable.

A general theory of the interaction of two charged identical colloidal particles of arbitrary shape is developed. An expression for the Helmholtz free energy of the electric double layers is obtained by the methods of statistical mechanics. The condition that there is equilibrium between the ions adsorbed on the surfaces of the colloidal particles and those dissolved in the dispersion medium is accounted for by requiring that the free energy of the whole system be a minimum with respect to variation of the ionic density on the surfaces.

The theory presented here is a further development of the work of Verwey and Overbeek. The conclusions of this paper are that in dilute sols, to which the present investigations are restricted, the results of these authors require extension in two directions. First, there is a correction to the mutual energy of two particles, due to the Coulomb interaction of the ions in the bulk of the solution. Secondly, no special assumption concerning the relation between the surface potential (or charge) and interparticle separation need be introduced. The equations set up to determine the free energy of interaction at the same time yield the ‘adsorption isotherm’ for the ion type which is common to the solution and the surface of the particle.

1. Introduction. In the course of an important memoir on integral functions of finite order†, G. Valiron discusses ‘fonctions orientées’, that is, functions with zeros an such that arg an tends to a limit as n → ∞. He obtains results that include the following two theorems, in which Vρ(x) denotes a function of the form , where α1, …, αν are positive integers, and n(r) has its usual significance in the theory of integral functions:.

In the paper under the above title I have given an example of a homoeomorphism of a plane under which the positive half-orbit† of a certain point is everywhere dense on the plane, and I have made a statement that the same is true for every point of the plane, except the origin.

This paper describes two methods by which lines of position may be obtained by observation of stars (in pairs) without the use of instrumental scale readings either for altitude or for azimuth. These lines of position represent arcs of great circles of the Earth. An instantaneous observation is proposed.

Great circles determined by star positions form, on the celestial sphere, a lattice which is not deformed by precession. Whether or not he be at rest relative to the rotating Earth, the observer's zenith traverses this lattice, which thus may serve in celestial navigation precisely as does a well-marked terrain in contact flying over land, or as do ranges and markers in coastal and harbour navigation. Observation of the zenith-transit of an element of this lattice is akin to the process of noting the moment when an aircraft crosses a coastal line or other visible terrestrial mark or to the recognition of when a vessel near shore is on a certain range. Observation of this type is as rapid as the navigator may be able to make it since it is the observation of the occurrence of an instantaneous situation.