This paper is an attempt to apply the methods of the theory of algebras to the more general problem of the structure of semi-groups, i.e. systems with one composition, satisfying the associative law.

Two theorems by the late Prof. F. Morley have aroused special interest, one being the chain of circle properties discovered by De Longchamps and independently by Pesci, Grace and by Morley himself, and the other the remarkable property that if the angles of any triangle are trisected, then the trisecting lines meet in pairs to form an equilateral triangle. In a previous paper‡ the present writer has shown that the circles making up the chain may be derived from a rational normal Cn in [n] by the following process.

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 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.

The interaction between an external electromagnetic field and a nucleus, including its exchange meson field, has recently been investigated by several authors (1). The interaction between a vector potential A and a nucleus has been found to be expressible in terms of the electric and magnetic multipole moments of the latter. It is the object of this note to discuss the corresponding interaction with a scalar potential V, and its connexion with previous results.

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.

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.

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 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 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 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.

A further study has been made of the deuteron bombardment of silver. The following radioactive isotopes are formed: 2·4 min., 26 min. and 225 day along with the silver fraction; and 6·7 hr. and 1 year along with the cadmium fraction. Results of absorption measurements of the radiations emitted are given. The energy-yield curves have been determined for the 2·4 min. and the 26 min. Ag isotopes and also for the 6·7 hr. and 1 year Cd isotopes. The formation of the 26 min. body from silver has been explained on the basis of a new type of disintegration, namely the (d-p, 2n) reaction.