1. The problem of the partition of numbers, first investigated in detail by Hardy and Ramanujan (1), has in recent years assumed importance on account of its application by Bohr and Kalckar (2) in evaluating the density of energy levels in heavy nuclei. A ‘physical approach’ to the partition theory has been made by Auluck and Kothari (3), who have studied the properties of quantal statistical assemblies corresponding to the partition functions familiar in the theory of numbers. The thermodynamical approach to the partition theory, apart from its intrinsic interest, draws attention to aspects and generalizations of the partition problem that would, otherwise, perhaps go unnoticed. Thus we are led to consider restricted partitions such as: partitions where the summands are repeated not more than a specified number of times; partitions where the summands are all different; partitions into summands which must not be less than a specified value; partitions into a prescribed number of summands, and so on. The generalization that seemed to us to be the most interesting is the extension of the partition concept to include partitions into non-integral powers of integers.

Dirac has suggested that the quantization of electric charge could be explained by the existence of magnetic monopoles. In view of this hypothesis, this paper investigates what theoretically would be the behaviour of such monopoles in a Wilson cloud chamber. The treatment, which for simplicity is basically classical, closely follows Bohr's work on the decrease of velocity and ionization properties of α- and β-particles, and expressions are derived for the rate of decrease of energy and the number of ion-pairs produced per centimetre by a monopole passing through a gas. These expressions are then discussed with particular reference to the case of heavy particles, and the main differences between them and the corresponding expressions for α-particles both as to range and ionization are indicated; these differences can be summarized by saying that monopoles have much shorter paths, but create many more ion-pairs per centimetre than α-particles. Also, the very sharp increase in the ionization at the end of the path of an electric particle is missing, the ionization for the monopole decreasing to a small amount near the end of the path.

The purpose of this note is to indicate the relation between (a) the problem of evaporation from a finite area of rectangular shape into a turbulent air stream and of the boundary layer on a flat plate of finite width, and (b) the problem of the boundary layer on a semi-infinite plate and evaporation from an infinite quadrant.

1. This paper deals with the differential equation

(dots denoting derivatives with respect to t), where for large x the ‘restoring force’ term g(x) has the sign of x and the ‘damping factor’ kf(x) is positive on the average. It will be shown that every solution of (1) ultimately (for sufficiently large t) satisfies

with B independent of k. The conditions on f(x), g(x) and p(t) (stated in §§ 2, 3) are rather milder than those assumed by Cartwright and Littlewood (1, 2) and Newman (3) in proving similar results.

It is now generally recognized that, under instrument conditions, traffic congestion at busy airports is already bad and any further increase in traffic density would be unmanageable without improvements in methods of air traffic control. A study of the problems of air traffic control by full-scale tests is prohibitively expensive and a practical alternative is required. This paper describes a series of investigations into air traffic control problems by simulation methods.

Laboratory equipment suitable for such experiments is described briefly and has been shown to give a reliable quantitative assessment of the effect of changes in a control system. A number of possible control systems were studied in detail.

1. Let x1, x2, …, xn, … be a set of independent variables each with a uniform probability distribution in 0 ≤ x ≤ 1. If 0 ≤ α < β ≤ 1 we denote by FN (α, β) the number of x1, …, xN which satisfy α < x ≤ β,

and put RN(α, β) = FN(α, β) − N(β − α).

Direction finding is the oldest, simplest, and probably most reliable radio aid yet offered to the mariner. In some ways it may be likened to a magnetic compass and, if its range is very much smaller, it has the inestimable advantage over a compass of giving bearings on a very large number of stations instead of one only. It is the only navigational aid, other than radar, in which the performance is controlled by the shipborne installation and is not subject to technical failures which may occur on land-based transmitters.

It is clearly not possible, in the space of one paper, even to touch on all aspects of interplanetary navigation, and some whole fields must be ruled out from the start. This paper will not be concerned with any discussion of the space-flight as such; there is no point in considering navigational problems at all unless one can assume (at least for the sake of argument) that there is to be some practicable way of lifting a ship out of the Earth's gravitational field, and then of accelerating it out of the Earth's orbit; but even granting this major premise a considerable amount of selection is still essential.

Many stochastic problems arise in physics where we have to deal with a stochastic variable representing the number of particles distributed in a continuous infinity of states characterized by a parameter E, and this distribution varies with another parameter t (which may be continuous or discrete; if t represents time or thickness it is of course continuous). This variation occurs because of transitions characteristic of the stochastic process under consideration. If the E-space were discrete and the states represented by E1, E2, …, then it would be possible to define a function

representing the probability that there are ν1 particles in E1, ν2 particles in E2, …, at t. The variation of π with t is governed by the transitions defined for the process; ν1, ν2, … are thus stochastic variables, and it is possible to study the moments or the distribution function of the sum of such stochastic variables

with the help of the π function which yields also the correlation between the stochastic variables νi.

It is well known that the operators mainly employed in quantum theory are hermitian; it is less well known amongst physicists that they are required, in addition, to be self-adjoint. This is essential for the validity of the result known in quantum theory as the representation theorem and in the mathematical theory as the resolution of the identity. The purpose of this paper is to show that the self-adjoint operators can be characterized by a condition which is nearer to having a physical significance than those given in the literature.

Boundary-layer equations for the unsteady flow near an effectively infinite flat plate set into motion in its own plane are subjected to von Mises's transformation. Solutions are obtained for the flows in which gravity is neglected, the Prandtl number σ is arbitrary, and the plate has a constant temperature and a velocity that is either uniform or, with dissipation neglected, non-uniform. Explicit solutions are obtained for the case in which the viscosity μr varies directly as the absolute temperature Tr. Solutions are also obtained for the diffusion of a plane vortex sheet in a gas, and for the boundary layer near a uniformly accelerated plate of constant temperature when gravity is not neglected. For the non-uniform motion of a heat-insulated plate, dissipation not being negligible, a solution is obtained when σ is 1 and μr ∝ Tr. The relative importance of free convection due to gravity and forced convection due to viscosity is discussed, and a solution is obtained, with μr ∝ Tr, for the free convection current set up near a plate that is at rest in a gas at a temperature different from that of the plate, dissipation being neglected.

Recent papers on the future of astronomical navigation in the air indicate that the present trend of development is directed towards surmounting two principal obstacles: the time lag between the shots and the final fix, and inaccuracies due to accelerations of the bubble. Even in aircraft at speeds between 200–300 m.p.h. these two factors tend to discourage the use of astro-navigation even where, as is particularly the case in lower latitudes, radio aids are not available.

1. The aim of this note is to prove the

Theorem. Let

where the λnare real and

and let

Then

A similar result holds for infinite seriesconverging uniformly in [−T, T].

It is well known that, if a trigonometric series

converges to zero in 0 ≤ x < 2π then all the coefficients are zero. To generalize this property of the series, sets of uniqueness have been defined. A point-set E in 0 ≤ x < 2π is a set of uniqueness if every series (0·1), converging to zero in [0, 2π) − E, has zero coefficients. Otherwise E is a set of multiplicity. For example, every enumerable set is a set of uniqueness. An account of the theory may be found in Zygmund (2), chapter 11, pp. 267 et seq.

The paper reviews some of the major problems which have been encountered in the development of airborne direction finding equipment, and describes, in rather more detail, the latest electrical and mechanical improvements which have been evolved to provide the modern, fast moving aircraft with a more rapid or instantaneous position indication.

Various methods of utilizing the information provided by the direction finder are discussed and questions of calibration and ultimate accuracy considered.