The four papers which follow were read to a meeting of the Institute held at the Royal Geographical Society on 19 May. They provide an account of the four systems of radio position fixing that are most widely used at sea, and give both a description of the experience gained and a general survey of what has been achieved since the international meetings on radio aids to marine navigation were held in 1946 and 1947. Certain requirements for marine radio position fixing systems were formulated at these meetings and a measure of agreement was reached on the types of navigational aid which should be developed. This review of the four principal systems, and of aspects of some of them which may not be widely appreciated, may perhaps give an opportunity for assessing how much progress has been made over the last few years and of the problems that remain to be solved.

Before the war the only radio navigational aids in current use by shipping were direction finding systems, operating in the medium frequency band of the radio spectrum.

During the war, not only were many new types of position fixing systems evolved, but radar techniques were also developed, thus making available a wide range of new facilities of potential use both to ships of many different categories and to port and harbour authorities for the supervision of marine traffic.

The practice of precomputing astronomical sights in the air, so reducing the delay between taking the observations and plotting the position lines on the chart, appears to be gaining in popularity. The observer selects a future instant in time, and calculates the altitude and azimuth of a suitable body or bodies, for an assumed position near the D.R. position at that time. Then, if the observation is taken at the time chosen, the intercept is obtained directly by comparing the sextant reading with the precomputed altitude, corrections having previously been made for systematic errors (e.g. dome refraction, personal error, &c.)

The zeros of solutions of the general second-order homogeneous linear differential equation are shown to satisfy a certain non-linear differential equation. The method here proposed for their determination is the numerical integration of this differential equation. It has the advantage of being independent of tabulated values of the actual functions whose zeros are being sought. As an example of the application of the method the Bessel functions Jn(x), Yn(x) are considered. Numerical techniques for integrating the differential equation for the zeros of these Bessel functions are described in detail.

The type of Markoff process which is considered in this paper corresponds to a system capable of n states, the time being regarded as a continuously varying parameter. At any instant t the probability distribution is represented by the vector

Several recent papers have been concerned with a group G, of order 28.36.5.7, which can be represented as a collineation group in five dimensions. This collineation group is generated by harmonic inversions (projections) which leave fixed a point (the vertex) and a prime (said to be conjugate to the vertex). There are 126 projections in G and the set of 126 vertices form a configuration which is described in a paper (Hamill (3)) in which the operations of G are expressed as products of at most six projections. The group leaves invariant six algebraically independent primals; these have been determined (Todd (6)), and the equations of the simplest are given. The simplest invariant is a sextic primal, and some properties of this have been recorded in an earlier paper (Hartley (4)).

The aim of this article is to discuss a paradox of importance in the quantum theory of measurement. The paradox was first propounded by Einstein, Podolsky and Rosen in 1935(1), and was discussed subsequently by Bohr(2), Furry(3), Schrödinger(4, 5), as well as in texts (6, 7). We begin by formulating the paradox.

The determination of a ship's position by means of radio observations from fixed shore transmitters has now been practised for a considerable period of time, but up to the outbreak of the last war the accuracy obtained from the conventional direction finding systems then in use was only sufficient to provide a general estimate of position suitable for off-shore navigation in waters where the ship's position need be known only to within a mile or two. It was not until, under the pressure of wartime requirements, the modern range of radio aids to marine navigation were produced that means were provided by which the mariner was enabled to determine his position with an accuracy comparable with that of the best visual fixing but with the great advantage that these observations could be made in all conditions of visibility and weather. At the end of the last war there were three systems of outstanding merit available: Gee, Loran and Decca. All three systems, as is well known, employ a combination of master and slave stations, the transmissions of which provide radio position lines capable of accurate calculation and plotting on navigational charts.

The publication of the paper Astro-fix by Computation by Oliver C. Collins in the April number of the Journal serves as a reminder that the observation for the determination of the latitude from two altitudes, usually known as the double altitude problem, is one of some celebrity in the history of navigation.

The following notes on the subject may be of some interest although they are far from complete, having been compiled from such books as happened to be readily available, and can only be considered as an introduction to the subject.

1. The tabulated values of the Legendre polynomials suggest that the right-hand minimum of Pn(x) changes monotonically as n increases. Let xr, n be the value of x which gives the rth extreme value to the left of 1 of Pn(x). Then we can show that

where jr is the rth pösitive zero of J1(z), and that after some term the sequence Pn(xr, n) is monotonic with the moduli of the terms decreasing. We cannot, however, show that the sequence is monotonic from the place at which its terms become significant.

The high concentration of aircraft near aerodromes, with consequent collision risk is well known. It is at present necessary, in poor visibility, to guarantee the indispensable commodity of safety by the sacrifice of schedule regularity and fuel economy. So long as this is the case, the problem of terminal navigation and control must be regarded as one still demanding a satisfactory solution.

The four papers printed here were read to a meeting of the Institute held at the Royal Geographical Society on 17 February 1950. They present, each from a detailed experience of particular requirements, an enquiry into one of the major problems of flight operation at the moment. It is clear, both from the papers and from the nature of the problem itself, that no solution can be considered apart from its navigational aspects, and that the whole conception of control is inevitably bound up with the question of position. The problem is therefore of its essence a navigational one.