In a previous paper, under the same title, I considered the problem of how far apart two consecutive primes can be. The present paper is concerned with the opposite question. How near together can large primes lie? The published literature on this subject is scanty and, though interesting, is mainly negative in character. It appears to be very difficult to give any answer that is not trivial, or that is at all illuminating.

Some theorems in the operational calculus, taking definite integration as the fundamental operator, are proved for discrete systems. It is suggested that it is physically more satisfactory to regard the solution for a continuous system as the limit of the solutions for a set of discrete systems, rather than as the solution of a partial differential equation. The necessary and sufficient condition for the validity in this sense of the usual operational method for continuous systems appears to be that the discrete systems considered shall not tend to infinite instability; that is, that all poles of the Bromwich integrand shall always have real parts less than some fixed positive quantity. The correction for finiteness of the number of degrees of freedom is examined for a case analogous to heat conduction and found to be unimportant.

This paper gives a criterion for determining whether real, non-zero solutions of a linear differential equation of the second order have an infinite or a finite number of zeros, or, in short, are oscillatory or non-oscillatory, as the independent variable tends to infinity.

In a paper published in the Proceedings of the London Mathematical Society, (2), 40 (1936), 143, Welchman introduced the idea of fundamental scrolls from which all special scrolls may be obtained by projection, and he developed the theory of fundamental line scrolls. These fundamental line scrolls are generated by joins of pairs of corresponding points in a cyclic (1, 1) correspondence of period 2 on a canonical curve. In this paper I generalize some of Welchman's results, and then go on to consider certain fundamental scrolls which have a particular interest.

In this paper we consider the slow two-dimensional motion of viscous liquid past a sharp edge projecting into the stream, the motion being one of uniform shear apart from the disturbance caused by the projection. A special form is assumed for the boundary so that a method lately developed by N. Muschelišvili can be used in solving the biharmonic equation; a simple expression in finite terms is found for the stream function ψ1. Fig. 1 shows the section ABC of the fixed boundary of the liquid, the equation of the curve ABC is given in § 2, and ψ1 in § 3.

The derivation given by Hoyle and Lyttleton for an accretion formula proposed by them is examined. A number of arguments against its validity are put forward, especially that on the one hand their capture radius depends on the theorem that if the velocity of certain masses of gas after collision is less than the velocity of escape at the point, they will not in fact escape, while on the other hand it is clear (and is now admitted) that the gas cannot in fact move with this velocity at all. It is also shown that since, ex hypothesi, the individual molecules will all, on the average, retain their hyperbolic velocities, there is not the compelling reason for their capture that there appeared to be in Hoyle and Lyttleton's argument, where only the mean radial velocity of the centre of gravity of the mass was considered. Further, it seems improbable that the temperature of the interstellar matter can be low enough for the initial assumptions of their theory to hold.

The foregoing paper by Dr Atkinson relates to the new formula for the rate of accretion of mass by a star that has recently been proposed by us. The derivation of this formula was made by means of a dynamical argument, certain physical aspects having to be neglected.

The rate of accretion of interstellar matter by stars as proposed in a previous paper is further discussed. It is shown that this amount, while sufficient for the evolution of the majority of stars, is insufficient by a factor of the order of 10 or more to give a satisfactory description of the general evolution of massive stars and close binary systems of small mass. Consideration of the possibility of increasing the rate of accretion for such exceptional stars leads to the conclusion that this can be carried out satisfactorily only by a corresponding increase in the density of the cloud. Although we were led to this view by considering all the factors involved in accretion and showing that only a change in the density could possibly produce the required increase, it is at once clear from the accretion formula, without detailed discussion of the other quantities involved, that the density is the only factor through which effects could be introduced that do not apply to all stars quite generally. By investigating the various factors in the galaxy affecting the density, it is shown that within 100 parsecs of the galactic plane, and also in local regions, the density may rise above 10−21 g. per c.c., which gives an increase of order 100 times the normal rate for stars lying in these regions. These suggestions receive independent corroboration from investigations by Jeans relating to extra-galactic nebulae which led to average densities also of order 10−21 g. per c.c., while a further argument from geological evidence shows that the average density of material along the sun's track must be higher than 10−21 g. per c.c. It remains to be seen whether future observations will succeed in confirming these suggestions indicated by the requirements of this theory of stellar evolution.

If Tαβ is the energy-tensor, then, for any vector field ξα, the equation

follows from the energy equation

Suppose first that the physical system considered is complete, i.e. that the energytensor vanishes beyond some world tube Z. Let L be a time-like world line running inside the tube. We integrate both sides of (a) over a portion of Z, and we transform an integral over a four-dimensional region into a linear integral over L. We obtain the variational equation

in which the m's are tensors characteristic of the physical system. An essential feature of the m's is that they are symmetrical in their two last superscripts. Equation (c) has to be satisfied by every field ξα, provided that the ξ's vanish, with all their derivatives, at the ends of the integration path.

The interaction between an external electromagnetic field and a nuclear system can be expressed in terms of the multipole moments. The electric quadripole and the magnetic dipole moments of the deuteron have been calculated, taking into account the exchange forces as given by the meson theory. The cross-section of the photomagnetic effect of the deuteron has been calculated.

This work was carried out under the guidance of Dr Heitler and Dr Fröhlich. The writer wishes to express his sincerest thanks to them for suggesting the problem and many valuable comments. The writer is also indebted to Dr Kahn for discussions during the early stages of this work.

In a previous paper a new method, based on Kemmer's β-formalism, of calculating meson processes was given for the case in which the meson interacts with an electromagnetic field. This method is now extended to the nuclear interaction, so that the whole of the meson theory can be given either in tensor or in matrix form, the former being preferable when the wave aspect of the meson is important and the latter when the particle aspect is dominant.

As examples of the matrix method, derivations are given of the cross-sections for the nuclear scattering of mesons and for the production of mesons from nuclei by photons. It is pointed out that the usual non-relativistic theory of the nuclear interaction is inadequate even for very small velocities.

In a previous piece of work(1) numerous data were obtained on the dependence upon magnetizing field of the Benedicks e.m.f. in nickel wire. The work was restricted to a study of the effect of the initial increase of magnetizing field, no hysteresis loops being examined. The curves were most remarkable in that the change of e.m.f. depended upon the direction of the magnetizing field with respect to the temperature asymmetry. In the paper quoted, it was suggested that the curves should be resolved into two components, symmetrical and anti-symmetrical with respect to the field. In the present paper the former component is termed the normal component, because, as is shown later, a simple explanation of it can be given at once. We call the antisymmetrical component the anomalous effect.