Alexander (1, 2) has introduced certain topological invariants of a manifold which arise from the intersections of cycles of non-complementary dimensions, and he points out that they are not derivable from the Betti and torsion numbers, nor from the fundamental group. In the present paper we consider some topological invariants of this type on an algebraic surface, and, although we cannot define them completely, we show that they are intimately connected with the multiplications of the period matrix of the simple integrals of the first kind. We are then led to a concept which we call the “intersection group” of the surface, which is, by its definition, topologically invariant, and we show that it is also invariant under birational transformations. The proofs are based on Lefschetz's theory of cycles for an algebraic surface (4) and some simple properties of the period matrix of an algebraic curve. The results obtained here have a number of applications to the theory of ∞3 correspondences between algebraic surfaces, as we propose to show in a later paper.

The paper contains some ionization-height curves for an isothermal ionized atmosphere in which the mechanism of electron disappearance is attachment to oxygen atoms at a rate proportional to the product of the number of oxygen atoms and of electrons present. Chapman's formula for the rate of production of ions is used. The curves were obtained by means of a differential analyser.

The presence of H2 molecules reduces the temperature of the cosmical cloud to a value that is small compared with the estimate given by Eddington. The necessary conditions for the molecules to persist at the capture radius of hot stars are investigated in the present paper, and it is shown that provided that the density of the cosmical cloud is sufficiently high the molecules will not suffer appreciable dissociation, and that radiation pressure will have only a negligible effect on the hydrogen. The critical density for a typical B star appears to be about 5 × 10−21 g. per c.c.

In a recent paper, Fröhlich, Heitler and Kahn(1) discussed the deviation from the Coulomb law for the proton on the basis of the meson theory. Instead of being considered as a point charge, the proton was assumed to be surrounded by a meson field, which interacts with an external electrostatic field. They deduced the result that the Coulomb attraction between the proton and a negative point charge goes over into a strong repulsion for distances less than one-sixth of the electronic radius. This has been found to agree with the experiment by R. C. Williams on the fine structure in the spectrum of the hydrogen atom and the interpretation of this experiment by Pasternack. Later experiments(2), however, seem to indicate that the effect is much smaller than the theory predicts.

Expressions are obtained for the scattering of neutral mesons by protons according to the quantized theory. It is found that the scattering due to the g1 (point charge) interaction is much less than for charged mesons, but that the anomaly in the scattering due to the g2 (dipole) interaction remains. The bearing of this upon recent attempts to modify the theory by introducing nuclear particles with charge 2e or − e is discussed.

In order to explain the phenomena of melting, tensile strength, etc., we have to investigate the stability of crystals for finite deformations, for which deviations from Hooke's law occur. Although these deviations are in most cases of an irreversible character, it is necessary, for a systematic study, to develop mathematical methods for treating the mechanical (reversible) case of a highly strained crystal lattice, where terms of higher order than the second in the deformation energy must be taken into account.

The results of the bombardment of mercury, lead and thallium by 9 M.e.V. deuterons are reported. The following radioactive isotopes have been detected: 5·5 min., 48 min., 36 hr., 60 day mercury isotopes; 4·4 min., 10·5 hr., 44 hr., and 13 day thallium isotopes; 10·25 min., 2·75 hr., and 54 hr. lead isotopes; 18 hr. and 6·35 day bismuth isotopes. The 10·25 min. lead isotope is positron emitting, an interesting result in an element of high atomic number. Absorption measurement have been made of the radiations emitted by many of these isotopes and assignments have been made in most cases.

In conclusion we wish to thank Dr N. Feather for valuable discussions, and also for making for us a Ra E source. We are indebted to Dr Lewis for advice in setting up the thyratron scale of eight counter. This paper would be incomplete without a sincere acknowledgement of our indebtedness to the hard work of past members of this laboratory who have been mainly responsible for setting up the cyclotron.

One of us (R. S. K.) is grateful to the Royal Commissioners for the Exhibition of 1851 for the grant of an overseas scholarship which made this work possible.

The increase in the angle of reflexion from a powdered crystal varies from (1 − n) tan θ to (1 − n) (tan θ + cot θ) as its absorption increases. In either case the true lattice spacing of a cubic crystal is obtained by increasing the extrapolated lattice spacing by a fraction 1 − n of itself. The divergence of the reflected beam by refraction is small in comparison with other causes of line width.

The energy density of a cubic lattice, homogeneously deformed by a force acting in the direction of one axis, is calculated, and the equilibrium conditions and the stability conditions for any arbitrary small additional deformations are derived. A special assumption is made as to the law of force between the atoms, and the numerical calculations are performed for the face-centred lattice. In this way the strain as a function of the deformation is calculated and, from the stability conditions, the tensile strength is determined. The results are not in agreement with the experimental facts, and the possible reasons for this disagreement are discussed.

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.

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 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.

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.

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.

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.

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.