I. Lattice points in the domain | x |α + | y |α ≤ 1

Theorem 4. Let G be the star domain

where α > 0. Then, when α tends to zero,

Proof. The linear substitution

changes G into the similar domain

and so

.

Preface. In order to make this paper intelligible to the reader, I repeat the following definitions and results from my paper ‘On lattice points in star domains’, which is to appear in the Proceedings of the London Mathematical Society.

1. The problems of stress distributions in an infinite plate containing a circular hole were solved in a general manner by Bickley(1) for isotropic materials. An alternative method, using complex stress functions, was later given by Green(5) and extended so as to apply to aeolotropic materials. In the present paper Green's method is employed to determine the stress distributions that arise when the stresses are produced by isolated forces acting at points on, or near to, the edge of the circular hole, and though some of the solutions are cumbersome they are all obtained in finite form.

1. The projective transformations of F into itself … 121

2. The flecnodal curve, and the lines of F … 122

3. A geometric characterization of F, and the six different types of F in the real domain … 123

4. The τ-points and τ-planes, and a notation for the lines of F … 124

5. The incidence conditions for the lines of F … 125

6. The tetrads of the first kind … 126

7. The tetrads of the second kind … 126

8. The pairs of lines of F … 127

9. The tetrads of the third kind … 129

10. The 16-tangent quadrics of F … 130

11. The conics of the first kind … 131

12. The conics of the second kind … 132

13. No other irreducible conics lie on F … 133

14. The tangent planes of F of multiplicity greater than 3 … 136

15. On twisted curves, especially cubics and quartics, lying on F … 138

16. The T-transformations … 139

17. Construction of an infinite discontinuous group of birational transformations of F into itself … 142

18. Deduction of an infinity of unicursal curves lying on F … 143

Using the results of the preceding paper of this series (by M. Bradburn), the equation of state and the elastic constants for a monatomic face-centred cubic lattice are calculated. Central forces with a potential of the form −ar−m+br−n are assumed to act between the particles. Numerical results are obtained for five sets of values (m, n) and represented in tables and diagrams. The general features of these are discussed and compared with previous computations.

I take this opportunity of thanking Prof. Born who suggested this problem to me, for his interest in my work and his advice on many occasions. I am also indebted to Dr R. Fürth for his numerous helpful suggestions.

1. It is trivial that

for all integers n ≥ 0 implies sn = 0 (n = 0, 1, 2, …). This conclusion is no longer true if it is only known that (1·1) is true for an infinity of n. But we shall show that the truth of (1·1) for, roughly speaking, one-half of all positive integers, together with an order condition on the magnitude of sn, ensures sn = 0 for all n. This follows from

Theorem 1. Let a1 < a2 < … be positive integers, n(R) the number of a's not exceeding R, sn a sequence of complex numbers. If

(ii) sn = ο(nK)as n tends to infinity

then sn = P(n), where P(x) is a polynomial of degree less than K.

An alternative derivation of the integral equation of Waller, Heitler and Wilson is given on the basis of the stationary method of the perturbation theory. The previous theoretical treatment of the scattering of charged vector mesons by nuclear particles through the nuclear interaction is extended to pseudoscalar mesons. Numerical results calculated from the theory are given for the total and differential scattering cross-sections.

1. In what follows f(θ) is a periodic function of L2,

is the Fourier series of f(θ), and

is the nth partial sum of T(θ). We denote by n(θ) any function of θ which is measurable, is finite p.p. †, and assumes non-negative integral values only, and by n(θ, H) an n(θ) none of whose values exceeds H. We shall sometimes write N for n(θ), and N(H) for n(θ, H), to simplify the set-up of formulae.

The absolute measurement of neutron flux density by ionization methods is discussed.

The relation Eυ = JυW ρ between the energy Eυ absorbed per unit volume of a solid medium and the ionization Jυ produced in a gas-filled cavity in the medium, the average energy W expended by the ionizing particles in producing a pair of ions in the gas, and ρ the stopping power of the solid relative to the gas for these particles, is critically examined from the point of view of the measurement of the rate of absorption of neutron energy at any point in a hydrogenous medium. The accuracy attainable is assessed, in so far as it depends on our knowledge of the quantities ρ and W.

The radiation from 198Au (half-value period 2·76 ± 0·02 day) have been studied, by absorption and coincidence methods, in a wide variety of experimental conditions. For the most part the results confirm those obtained by other investigators, but, in addition, a fairly intense γ-radiation of about 65 KeV. energy has been detected. This radiation has been distinguished from the weaker fluorescent K X-radiation of mercury, emitted as the result of internal conversion of one of the γ-rays of 198Au, and it has been shown that it is not the K X-radiation of platinum. Disintegration of 198Au by K-electron capture, if it occurs at all, cannot take place in more than about 15% of all disintegrations.

1. It is familiar that the properties of the invariants and covariants of binary forms of the first four orders admit of elegant geometrical interpretations on rational normal curves and their projections. For forms of order higher than four the number of irreducible concomitants which appear in the complete system increases rapidly. It is the purpose of this note to exhibit geometrically the 23 irreducible concomitants of the binary quintic, using the rational normal curve R5 in space of five dimensions and its projection R′5 on to a prime as a foundation.

1. In this paper, a continuation of an earlier paper(1), we consider the two-dimensional motion of incompressible viscous liquid past a projection, 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 the area in the z-plane (Fig. 1) can be represented conformally on a circle in the ζ-plane by a rational function of ζ; this function contains a parameter a (0 < a ≤ 1), and by varying a the shape of the projection can be varied. Since a rational function is concerned in the conformal transformation a method lately developed by N. Muschelišvili(2) can be used in solving the biharmonic equation for the stream function, though the method actually used differs in some points of detail from that originally proposed by Muschelišvili and appears to be somewhat simpler.

1. In this paper we consider the conditions at the surface of a semi-infinite elastic body due to the action of an external force applied, in a direction at right angles to the surface, to the interior of the body.

1. The cosmical number N = . 136 . 2256 is most picturesquely described as ‘the number of protons and electrons in the universe’. This in itself would be a matter of idle curiosity. But N has a more general significance as a fundamental constant which enters into many physical formulae; it determines the ratio of the electrical to the gravitational forces between particles, the range and magnitude of the non-Coulombian forces in atomic nuclei, and the cosmical repulsion manifested in the recession of the nebulae. Its special interpretation as the number of particles in the universe arises in the following way. If we consider a distribution of hydrogen in equilibrium at zero temperature, the presence of the matter produces a curvature of space, and the curvature causes the space to close when the number of particles contained in it reaches a certain total; this total is N. We cannot say with the same confidence that the number of particles in the actual universe is precisely N, because the admission of radiation, complex nuclei, and unsteady conditions takes the problem outside the range of rigorously developed theory; but to the best of our belief these complications do not affect the total number of protons and electrons composing the matter of the universe.

It has recently been pointed out by Heitler(1) that the well-known discrepancy between the theoretical expression and the experimental results for the cross-section of scattering of charged mesons by nuclear particles can be removed by a proper consideration of the effect of radiation damping in the quantum theory. The radiation damping in quantum theory was first considered in complete detail for free electrons by Waller(2). A rigorous deduction of the integral equation set up by Waller was given by Heitler on the basis of a method developed by Góra. An alternative rigorous derivation of the integral equation has also been given by Wilson(3). Exact solutions of the integral equation for the simple scattering of mesons by nuclear particles have been found by Heitler in the non-relativistic approximation. An exact solution has not so far been given for physical problems in which the integral equations are complicated, but Wilson has given a general approximate formula for the scattering cross-section, which should be valid for all problems.

Hughes and Mann(1) have shown that, under certain conditions, the distribution curve of the energies of electrons scattered inelastically by an atom or molecule has the same shape as that of the X-ray Compton line. Hughes and his collaborators (1–5) have applied the method to an examination of a number of atoms and molecules in the gaseous state under which condition it can be assumed that each atom or molecule acts as an independent scattering centre. Coulson and the author (6–11) have applied quantum methods to derive a theoretical shape of the Compton line for certain of the molecules for which wave functions are available and have discussed the discrepancy between the experimental and the theoretical results(10, 11).