I wish to thank Dr. Clemence for pointing out1 the flaw in the interpretation drawn in my paper that the error in observed altitude due to tilt of the sextant is a maximum when the altitudes are in the neighbourhood of 45°. The purpose of a correction table is either to supply the correction or to indicate the most favourable or unfavourable circumstances. I agree that my table does not fulfil this purpose; the argument β is unknown. Moreover the table gives corrections for all combinations of altitude and β this is incorrect, as some values of β are impossible for some altitudes, and a square form of table is therefore not permissible.

1. The forces acting on the two-dimensional aerofoil in a bounded uniform stream have been found for a variety of cases, and in the present paper an attempt is made to extend the theory to include linear shear-flow. The special case of the symmetrical Joukowsky aerofoil in unbounded shear-flow has been solved by Tsien (6) using real-variable theory. A more satisfactory method is using complex-variable technique indicated in § 3, and is applicable to a more general shaped aerofoil. The effect on the lift and moment when a plane boundary is present is then considered. When the aerofoil is not too near the boundary, the lift and moment can be expanded in powers of the ratio of a typical length in the aerofoil to the height of the aerofoil above the boundary, by following exactly the technique used by Green (1) in a recent paper dealing with the same problem but with a uniform fluid flow. In the last section the limiting case of the flat plate touching the boundary with its trailing edge is discussed.

In all sciences, the experimenter is constantly seeking to improve both the accuracy of his observations and the speed with which they can be obtained. He wishes, further, to use the minimum of basic equipment, and to obtain the maximum simplicity in collecting his results. Unfortunately, these factors frequently act in opposite senses: an expensive apparatus, for example, may be required to collect accurate data quickly. In certain fields, it is even worth while deliberately to sacrifice accuracy in order to gain advantage both in speed and simplicity. Thus, in astronomical navigation, it may be possible to simplify sight reduction, to cut down on the bulk of tables carried in the air, and to reduce the time taken for observation and reduction of a sight. The penalty will be a loss in accuracy, but, provided this is not reduced beyond a certain level, it may well be worth while. A review of the requirements for air navigation, and an examination of the problems of position finding by the stars against these requirements may lead to some conclusions as to whether an advantage is to be obtained by simplifying the methods.

Astro-navigation cannot be regarded at present as one of the principal aids to air navigation. It suffers, by comparison with radio aids, in that it is slow in use and much more subject to interruption by bad weather. The average navigator needs at least 15 minutes to take and plot a two-star fix with present methods. This is quite a long time in the air, though not prohibitive with present-day aircraft, and in consequence astro-navigation is now used extensively only on long flights over sea or desert where radio aids are lacking.

The use of artificially pulsed sources of particles in experiments in nuclear physics has already given rise to powerful methods of investigating short-period phenomena and of studying the effects of ‘slow’ neutrons of different energies. On the other hand, the use of such sources introduces complications in counting experiments which are not present when steady sources are employed and it is possible to assume that the time distribution of ionizing events in any counter is a strictly random one. Again, sources which are effectively steady in respect of one counting arrangement are not necessarily so in respect of all. Ordinary cyclotron sources (i.e. cyclotron sources which are not ‘artificially’ pulsed) belong to this class; they have justifiably been considered as steady in the majority of experiments hitherto performed, but the flux of primary particles reaching the target is definitely periodic in time (with the period of the R.F. Voltage applied to the dees), thus a cyclotron will appear as ‘naturally’ pulsed whenever it is used with a counting system for which the effective resolving time is sufficiently short (say < 10−7 sec.).

The properties of (1) consistency and (2) asymptotic normality of maximum likelihood estimates of one unknown parameter, for independent observations, have been discussed rigorously by Cramér ((1), p. 500) and Doob (2). These properties were first stated by Fisher (3).

If astronomical navigation can be made to give its results in as simple a manner and as quickly as do the modern electronic systems, then clearly it can provide an aid whose usefulness will remain undisputed. The aim, then, should be to develop a system of astro-navigation which will give an immediate and continuous reading of, for instance, latitude and longitude; and it seems quite possible at the moment to envisage the nature of such a system. Its realization is perhaps another matter; and this may take time that perhaps the development of other systems will render ill-spent.

The ideas put forward in this paper are only concerned with the fundamentals of what, in the writer's opinion, is a possible solution to the problem; they are also not all new.

Throughout the history of marine navigation it has been recognized by seamen that soundings of the sea bed are a most natural and valuable aid to safe passages. Yet it is probably true to say that the methods of obtaining such soundings and for recording and using the information so obtained remained relatively primitive for longer than any of the other arts or systems used by seamen.

Records can be found which indicate that the earliest navigators used poles or lines and plummets to measure the depths of water on the coasts and in the rivers. An example of the use of soundings by Phoenician seamen is recorded by Strabo, who relates that the captain of a Phoenician ship on a voyage to the Casseterides for tin thought himself too closely followed by a Roman ship and deliberately sought a shoal which he knew of on which to run his vessel to avoid capture and the loss of his precious secrets. Then Hanno, the famous Carthaginian admiral of about 500 B.C., at his furthest point south on the West Coast of Africa, outlined the contours of the River Niger, and the Greek admiral Nearchus at the same time made his voyage from the Indus to the Euphrates by soundings in the rivers and on the coast.

A considerable volume of knowledge is now available on random fluctuations (noise) as regards the behaviour in amplitude. Familiar names in this field are those of Uhlenbeck and Ornstein(7), Fürth(1) and Rice (4), although very many others have made valuable contributions. A particular class of problem, of considerable practical importance, exists when the frequency spectrum is limited to a relatively narrow range. The resulting noise has then the character of a more or less regular oscillation modulated randomly in amplitude and phase. In this case, if we write the fluctuation in the form

(where R(t) and θ(t) are variables changing slowly in comparison with sinω0t), it is clear that the magnitude of the envelope R(t) and the phase θ(t) are now the significant quantities. Rice (4), among others, has made a study of the statistical properties of R, deriving in particular the correlation function R(t) R(t + τ) in terms of the characteristics of the (power) spectrum ω(f). Fürth and the writer (2) have extended this work and carried out a collateral experimental investigation.

In order to determine the mathematical form of the distribution of gene frequencies in a population, R. A. Fisher (1, 2) applied two different but complementary methods. First, differential equations were set up for certain cases and solved. An extension of these results arose from the expression by Wright (10,11,12) of the conditions as an integral equation. But strictly, neither integral nor differential equations can represent the conditions exactly, since integration is substituted for summation, and differentials for minimum steps (1/2n) in gene frequency. The second method employed by Fisher was exact, even though the solutions involved new functions. The properties of generating functions were utilized to yield functional equations, and the special case of rare mutations was dealt with in detail. In the present paper this powerful method is extended to the general case where mutation rates are not necessarily small.

This paper deals with some details of the application of the Lorentz-Dirac equations of motion of an electron to two simple cases, (a) with no incident field, and (b) with an incident pulse of radiation. In case (a), the field-energy distribution in the self-accelerating motion of the electron when the electron has built up a velocity close to the velocity of light is considered. Numerical and graphical methods are used to form a picture showing how the field energy tends to be concentrated in a disk for this state of motion. A discontinuity of the advanced field variables of the self-accelerating electron is also examined. In case (b), the general solution of the equations of motion for the velocity of the electron is worked out accurately. In the physical solution, this brings out the effects of radiation damping when the pulse has great intensity. Some aspects of the non-physical solutions are pointed out.

A number of important Markoff processes, with a continuous time parameter, can be represented approximately by a discrete process, interesting in its own right, of the following type. A class of individuals gives rise seasonally (in January say) to a number of new individuals (children), the probabilities of an individual having 0, 1, 2, … children being p0, p1, p2, …. These probabilities are the same for all individuals and are independent. The individuals formed each January are regarded as a new generation, and only this generation is capable of reproducing in the next January. Let

so that F(x) is the probability generating function (p.g.f.) of the number of children of an individual. Clearly the series for F(x) is absolutely convergent when |x| < |1.