The introduction of radio aids to navigation of the hyperbolic type, such as Decca, Gee and Loran, resulted in the need for special charts. These charts, which are usually of the scale and projection appropriate to the occasion, have overprinted on them red, green and purple lattice lines, representing the families of confocal hyperbolae drawn with respect to the Master and Slaves of the radio system. The production of these lattice charts was extensively developed during the war, both in Great Britain and the U.S.A. To provide the mariner with satisfactory marine charts for use with the English Decca Chain, the Hydrographer of the Navy has latticed all the principal coastal charts of England and Wales, and the Danish Hydrographer has similarly latticed the Danish coastal charts.

In applying the general theory of intermolecular forces to the complicated problem of ionic solvation, the minimum requirements are an exact knowledge of (1) the force laws governing the interaction of two isolated solvent molecules as a function of their separation and the various angles of mutual inclination, (2) the force laws governing the interaction of an ion with one solvent molecule, also as a function of their distance apart and of their mutual inclinations, (3) the average disposition of solvent molecules in the pure liquid and (4) the disposition of the solvent molecules in the force field of the ion, particularly in its immediate vicinity. None of these theoretical requirements is available for systems of practical interest. The established laws of intermolecular force are those relating to spherically symmetrical fields, applied to gases (J. E. Lennard-Jones; see (1)), in which the molecular arrangements are random, and to crystals (M. Born (2)), in which the arrangements are orderly. In seeking to extend these laws to liquids and solutions, we are hampered at the start by our ignorance of directional forces in general and of the degree of disorder prevailing in condensed fluid systems. What is attempted here is a simplification of the problem of applying intermolecular force theory to ionic hydration by the introduction of assumptions which can claim some experimental warrant. How far the introduction of these assumptions may swamp other factors must remain an open question. At any rate, the attempt removes many anomalies, and allows of the absolute calculation of ionic dimensions in aqueous solution from intermolecular force constants determined without reference to the condensed states of matter.

The name ‘cardinal function’ was given to

by E. T. Whittaker (1), who considered it as a ‘smooth’ approximation to a function f(x), having the same values as f(x) at the points a + rw (r = 0, ± 1, ± 2, …). It has since been extensively studied (2), mainly from the point of view of interpolation theory. Hardy (3), however, observed that the functions νr(t) defined by

form a normal orthogonal set on the interval (−∞, ∞), for r = 0, ± 1, ± 2, …. This fact suggests a discussion of the cardinal series from the point of view of mean-square approximation.

1. Introduction. A detailed paper describing the theory and calculation of equivalent head winds on air routes has been prepared by J. S. Sawyer and is being published as a Meteorological Report. The present paper is a simplified account which draws largely on Mr. Sawyer's paper.

When regular air services over a route are being planned, the wind at the chosen level of flight is an important factor. From the long term point of view, aircraft require to be designed so as to operate economically against adverse winds, if not on all occasions at least on all but a small percentage of occasions; for example, British Overseas Airways Corporation usually specify an 85% regularity of operation as a minimum. Similarly, for short term planning, wind information is necessary for the computations regarding the averages and extremes of fuel consumption, payload, time of flight, and so on. The aircraft operator consequently requires to know the frequency distribution of winds on any given route at any time of the year.

The initial collision (see pp. 571–2). The instantaneous velocity change of the hammer (from V0 to V) when it first strikes the liquid film should be calculated from energy and not momentum considerations, since the hammer has momentum in the vertical direction whilst the liquid is expressed in a horizontal direction and its total momentum is, by symmetry, zero at every instant. The kinetic energy dE of an annulus of liquid of radius r will be ½mc2, where m = 2πrhρdr, and c, the radial velocity of flow, is ½rV/h, since the liquid starts moving in plug flow. Integrating from r = 0 to r = R, we find that the total kinetic energy E imparted to the liquid film is . Assuming that extraneous energy losses, including any energy imparted to the anvil, are negligible, we may equate this to the energy loss of the hammer . Hence

To a first approximation this yields

The term at the right-hand side of the denominator is one-half that given in the original derivation (equation (20)), so that the instantaneous decrease in the velocity of the hammer is even less marked. For the given case where M = 400g., R = 1 cm., ρ = 1.6 and h = 5 × 10−2 cm., the velocity decrease is less than 2%.

Vol. 44 (1948), pp. 566–80.

It seems probable that the magnetic compass, the mariner's compass as it is often called, first made its appearance at some time between the tenth and twelfth centuries A.D. Attempts have been made to prove that the compass was in existence earlier, and that some of the early voyages could not possibly have been performed without its help; but there is no evidence to support this contention and a certain amount to show that other methods were used.

Again, it has been suggested that some obscure remarks by certain classical writers could be explained if the author were assumed to be describing the use of a compass, whilst having only the haziest notion of what it was. Among such is the legend of Abaris, the Hyperborean priest, who visited Greece, some say from Britain, many centuries B.C., and had with him a mysterious arrow which showed him the way. If we cut out the fancies of some writers who credited him with the possession of a kind of aircraft (capable of supersonic speeds), it is not difficult to imagine that the arrow may have been a magnetized needle. The difficulty is in understanding how no one down the centuries which followed came to write of the compass. There is, of course, little doubt that much scientific knowledge was possessed by the ancient priestly cults and the knowledge of the compass may have been kept secret for their own ends. If the great library of Alexandria had not been destroyed we might have had the answer to the riddle; but perhaps there is none and the whole tale of Abaris is a fabrication.

In the discussion on Blind Estuary Pilotage held by the Institute on 18 June 1948, the writer (see Journal, Vol. 2, No. 1, p. 60) put forward the suggestion that the well-known principle of radar shadowing could be used to provide accurate leading patterns for ships navigating straight channels or harbour entrances. Two possible patterns of this sort are illustrated in Figs. 1 and 2.

The pattern shown in Fig. 1 uses six ‘targets’ and two ‘screens’ and would be suitable for any channel wider than about 200 yards. The channel illustrated is meant to be about 200 yards wide, and the configuration of the pattern—though not the dimensions of the targets and screens—is drawn to the same scale.

It is well known that a solution of the wave equation

for which u = ∂u/∂t = 0 initially outside a surface S0, vanishes at time t in the exterior of a surface St parallel to, and at normal distance ct from S0, so that the wave fronts of disturbances represented by the solutions of the wave equation obey the laws of geometrical optics. Analogous results hold for the solutions of any linear hyperbolic second-order partial differential equation with boundary value conditions of the ‘Cauchy’ type. But the wave fronts of solutions of problems in which some of the boundary conditions are of the type representing reflexion do not seem to have been treated, and in particular the case of diffraction, when there is a ‘shadow’, does not seem to have been considered at all.

P. F. Everitt (Journal, Vol. 2, No. 1, p. 49) has discussed the problem of precomputing sights but gives the impression that the A.N.T.s cannot be used for this purpose; this is not the case, provided of course that the accuracy of 1 or 2 miles obtainable from A.N.T.s is acceptable. My experience of the method described below is confined to marine use with Hughes' Tables, but the same limitation of being confined to integral degrees of latitude and hour angle applies, so the method is clearly applicable in the air. Provided that the observation is actually obtained, the extra work involved in computing the sight in advance is trifling, and there are often appreciable advantages.

The simplest exact classical theory, of a particle possessing charge and dipole moment proportional to the spin leads to a wave equation with an explicit spin interaction with the field. This wave equation is used in this paper to calculate the scattering probability coefficient in the Compton effect by two methods. The first method uses the unquantized radiation field; the second uses Dirac's scheme of field quantization. The scattering probability coefficient is given in the general case in terms of the charge e, the constant of proportionality in the dipole moment C, and the quantities k, l specifying the spin. Some special results are considered for particles of spin ½ and 1. Formulae are given for the scattered intensity as a function of the angle of scattering, and the total scattering cross-section as a function of the energy of the unpolarized incident radiation. Graphs are given for these formulae when the magnetic part of the dipole moment takes the empirical values in the case of the electron, proton and neutron. In the case of the meson, normal and abnormal values of the dipole moment are used.

The essential elements in any application of statistical theory are

(1) the set of hypothetical physical states θ which may give rise to the observed phenomena,

(2) the set of all possible observations or samples x,

(3) the set of actions α which may be taken when x is known.

The whole procedure in the common applications can be divided into two parts:

(A) the estimation of θ from the known observation x,

(B) action on the basis of this estimation.

1. The method introduced by Dirac (1) in his classical theory of the interaction of an electron with an electromagnetic field has since been extended to the interaction of point particles with more general wave fields, such as occur in the neutral meson theory. Bhabha (2, 3) considered the scattering of neutral mesons by nucleons and gave arguments to show that typically quantum-mechanical effects should be unimportant if the energy of the incident meson is less than the rest energy of a nucleon, so that a classical theory with damping is appropriate to such problems. Now only the charged mesons are observable, and therefore a discussion of the classical theory of the interaction of a nucleon with a charge-bearing field is desirable. Apart from a paper by Fierz (4) which omits the dipole-like coupling terms, there is little published work on this subject.

In Ewald's theory of crystal optics, based on classical principles, an optical wave through a crystal lattice of polarizable atoms (idealized as isotropic oscillators) is a self-sustaining system of vibrations so constituted that, on the one hand, the electric moment of each atom is caused to oscillate by the electromagnetic field and, on the other hand, the electromagnetic field is itself the resultant field due to the superposition of the dipole waves produced by the lattice atoms. Born extended the theory to the case of movable lattice ions and showed that in an optical wave the electromagnetic field is so coupled to the lattice vibrations that each lattice ion vibrates in phase with the local field which, as in Ewald's case, is itself produced by the vibrating ions. In Born's original theory, the motion of the lattice particles had to be treated by classical mechanics. It is shown that the results of the quantum-mechanical treatment of the lattice motion agrees exactly with the classical theory. Not only is the induced current at each lattice point in phase with the local field, but the magnitude of the current is also identical with the classical value, completely independent of whichever vibrational state the crystal might be in.

This paper describes a number of proposals concerning the correction of the errors of magnetic compasses. The first proposal concerns the correction, by means of special soft iron correctors, of the deviating effects of both vertically and horizontally induced magnetization.

A scheme for removing constant error without having to use algebraic addition or even having to record the existing deviations is next described.

A scheme is then suggested whereby, provided no constant deviation is present, a compass may be corrected for the effects of permanent magnetism in four operations, each of which consists of the complete removal of deviation on a particular heading by a particular corrector.

Possibilities of putting compasses below decks without seriously affecting their performance are also discussed; these particular suggestions cannot apply to merchant ships in which magnetic cargo is being transported. Suggestions are also made of correcting the compass by modifying the disposition of the magnetic material in the bridge structure instead of by adding a corrector to the compass binnacle.

Finally, a scheme is introduced to maintain an aeroplane compass mounting parallel to the surface of the Earth during a level turn by using the centrifugal force due to the turn.

This paper puts forward five separate suggestions concerned with the correction and siting of magnetic compasses. The suggestions are unrelated to each other except in the fact that they have all resulted from viewing the problems from a deliberately unconventional standpoint. Some of the solutions proposed may be of interest to those concerned with magnetic compasses in ships, aircraft and vehicles.

1. The particular set of symmetric polynomials known as S-functions has recently been shown to be of importance in a variety of algebraic problems. Many of the applications of these functions depend upon the properties of the operation which Littlewood terms ‘new multiplication’, by which, from two given S-functions {μ}, {ν} of respective degrees m and n, is constructed a symmetric function {μ} ⊗ {ν} of degree mn. Littlewood has devoted a considerable part of his paper (2) to explaining various methods by which this function can be expressed in terms of S-functions of degree mn. None of these methods is really simple (in the sense that the rule for expressing the ordinary product {μ} {ν} in terms of S-functions of degree m + n is simple); and, indeed, it would be unreasonable to expect any simple rule of general validity for writing the resulting expression down since, for instance, a knowledge of the explicit expression for {μ} ⊗ {n} would yield immediately a knowledge of all the linearly independent concomitants, of degree n, of an algebraic form of type {μ} in an arbitrary number of variables. Nevertheless, the evaluation of {μ} ⊗ {ν} in particular cases is often necessary. Of the methods suggested by Littlewood for performing this evaluation the one which seems to be normally the simplest (his ‘third method’) involves a process which is not shown to be free from ambiguity, and which can actually be shown by examples to give, in certain cases, alternative solutions, the choice between which must be made by other considerations. It is therefore perhaps worth while putting on record a quite different method of procedure, which, apart from any intrinsic interest which it may possess, seems to be quite practicable for most of the actual evaluations performed by Littlewood.

In this paper we consider the theory of point-curve correspondences on a single surface F, i.e. correspondences in which to each point of F corresponds an algebraic curve of F. The results previously obtained for induced and extended correspondences between two surfaces require some modification here, as we can consider the self-correspondences induced on a curve C of F, and the related theory of extended correspondences, which is now complicated by the existence of the identical correspondence on C. We also develop a theory of correspondences with (non-zero) valency, and show that for a surface whose Riemann matrix is pure and without complex multiplication (and hence for surfaces of ‘general moduli’), all correspondences are valency correspondences, this result being exactly analogous to the well-known theorem for curves. We also consider generalized valency correspondences which are an extension to surfaces of the concept of correspondences of multiple valency on a curve. The analogy between our theory and the known results for curves does break down in one important respect. It is not true that every surface possesses valency correspondences even of the generalized kind, the existence of such correspondences involving restrictions on the intersection group of the surface.