1. The study of queues has been of interest to mathematicians and engineers for the past forty years, and a very extensive literature on the subject exists. The applications of the theory are many and varied, from Erlang's original work on telephone engineering, to present-day studies in the design of airports. The number of papers which deal with the theoretical side of the subject is, however, small, and it therefore seems desirable to attempt to develop a general theory which will cover the diverse practical requirements, so that the unity in the applications will become apparent. This paper gives such a development in the case where there is a single queue and a single server attending to it; the theory of multiple queues or many servers seems, except under simplifying assumptions which do not always correspond to reality, to be a problem of considerable difficulty. Previous work on the subject has mainly been confined to a special case where the customers arrive at random, as, for example, in a recent paper by Kendall(5);* the present theory makes no such assumption and allows the customers to join the queue in other ways, though the theory simplifies when the more restrictive assumption is made. The centre of interest in the present theory is the waiting times of the customers.

1. The method of converging factors, for hastening the convergence of slowly convergentseries and improving the accuracy of asymptotic expansions, was introduced by J. R. Airey and is well known to computers (see Airey(1) and Rosser(2)). The principle is as follows. It is required to compute a quantity which is expressed as an infinite series

The series may be either convergent or asymptotic and divergent.

(i) In the penultimate line of § 1, the reference should be to example (iii)

(ii) The right-hand side of equation (31) should be divided by 3.

(iii) Two lines below equation (33), v should be defined as the highest common factor of n, u1, u2,…

(iv) Similarly in the penultimate line of the paper (before the References) the words ‘and of n’ should be added.

A method is given for obtaining sequential tests in the presence of nuisance parameters. It is assumed that a jointly sufficient set of estimators exists for the unknown parameters. A number of special tests are described and their properties discussed.

This paper describes two methods of determining whether a sextant observation of the Moon should be of the upper or lower limb. The first method requires the use of a celestial globe or planisphere; the second is mathematical and requires a certain amount of computation, but is exact.

The line which separates the dark portion of the Moon's disk from the illuminated portion is known as the terminator, and the straight line joining the ends of the terminator is that used in this problem. This line is always perpendicular to a line from the Moon to the Sun (the horns being always turned away from the Sun), so that its position can be predicted from the geometrical relations of the Earth, Sun and Moon. This enables the limb to observe to be determined.

In recent years, the most striking development in the philosophy of navigation has surely been the change in attitude towards errors. Ten years ago an error was regarded as something that just did not happen in the best navigational circles; today we accept it as the normal concomitant of observation, and we study to reduce it.

This change in philosophy is really nothing more than theory following in the footsteps of the practical man, for the practical man has always regarded a position plotted on a chart with a certain amount of healthy scepticism. It may therefore be of value to discuss the whole business of errors from the point of view of the practical man, avoiding complicated mathematics and dealing in demonstrations rather than in rigid proofs. Navigation has been described as the business of conducting a ship on its way across the sea or an aircraft on its way across the sky. The word conducting implies safety. Indeed, safety is the keyword in navigation.

This paper is an account of the methods that have been used with the EDSAC for the solution of algebraic equations. Three repetitive or iterative methods are examined: Bernoulli's method, the root-squaring method, and the Newton-Raphson method. Experience with the EDSAC has shown that, as in hand computing, quadratically convergent methods are to be preferred to those less rapidly convergent. In particular, the Newton-Raphson method has proved the most useful. Several examples are given in the appendix.

It is proved that the matrix algebra for any relativistic wave equation of half-odd integral spin can be factorized as the direct product of a Dirac algebra and another, called a ξ-algebra. The structure and representation of ξ-algebras are studied in detail. The factorization simplifies calculations with particles of spin > ½, because the ξ-algebra contains only one-sixteenth as many elements as the original matrix algebra.

During the Antarctic summer season of 1949 the Norwegian sealer Norsel, which had been chartered by the expedition, landed the Norwegian-British-Swedish Antarctic Expedition in Queen Maud Land in the Antarctic. In January 1951 the ship took down relief personnel and stores and for this trip the Admiralty, at the invitation of the planning committee, appointed the author as an observer. The decision to appoint a member of the navigation branch was made partly so that the techniques of ship handling and navigation in ice could be studied; and since the captain of Norsel, Captain Jakobsen, was extremely experienced in every aspect of ice navigation, a great deal of valuable information was acquired. The Institute had also suggested a number of astronomical observations that might usefully be made in polar areas and the Admiralty gave its approval for the observer to make them.

1. The Hahn-Banach theorem on the extension of linear functionals holds in real and complex Banach spaces, but it is well known that it is not in general true in a normed linear space over a field with a non-Archimedean valuation. Sufficient conditions for its truth in such a space have been given, however, by Monna and by Cohen‡. In the present paper, we show that a necessary condition for the property is that the space be totally non-Archimedean in the sense of Monna, and establish a necessary and sufficient condition on the field for the theorem to hold in every totally non-Archimedean space over the field. This result is obtained as a special case of a more general theorem concerning linear operators, which is analogous to a theorem of Nachbin ((6), Theorem 1) concerning operators in real Banach spaces.

In the last line on p. 615, read ‘360’ instead of ‘372’, omit the parenthesis which follows and, on p. 616, substitute ‘343’ for ‘348’. We are indebted to Dr M. E. Rose for drawing our attention to this error.

The Institute was planned to bring together not merely the professional navigators, but all who could contribute to the exchange and advance of knowledge bearing on navigation. The founders were wise when they gave effect to the thought that the small-boat sailor and the sailplane aviator could contribute to the progress of our science and art, and could greatly strengthen the hand of the professional provider and the professional user of the processes and material of navigation. The reasons for exchanging and advancing our knowledge are as diverse and yet as closely interacting and mutually reinforcing as are the individual interests of our members.

In this paper I prove the following results.

Theorem 1. Let f(x, y) = ax2 + bxy + cy2be an indefinite form. Then there exist real (x0, y0) such that

for all (x, y) ≡ (x0, y0) (mod 1). If f represents 0 there are indenumerably many such incongruent (x0, y0).

After taking a latitude at noon and obtaining a position, a course and distance to a position nearer the destination is usually required. A course and distance from an estimated noon position may be worked prior to noon and then corrected at noon by the observed latitude and morning position line, by means of a plotting chart, giving reliable results quickly and accurately.

In Fig. 1, BA and CA represent the course and distance from B to A and C to A respectively. BD and EC are arcs of concentric circles with centre A; and AC = AE and AB = AD.

If BD and CD are small in comparison with AB, arcs BD and EC may be considered straight lines and the figure BECD a rectangle. If the rectangle can be solved, the distance AC can be deduced from AB, i.e. AC = AB - DC or AC = AB + DC if C lies the other side of BD.