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.

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.

We are concerned in this paper with linear integro-differential equations of the form*

and

where kr(y) are given functions with bounded variation in any finite interval, g(x) is known, and f(x) is to be determined.

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. 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. We first compare the λ-ic characters of two polynomials with respect to each other and the prime p, the polynomials being irreducible with respect to p. This is done by relating each to the resultant of the polynomials. We then show in § 6 that the ordinary quadratic character of one prime with respect to another is also the resultant of two polynomials.

1. A method has lately been developed by N. Muschelišvili (1) for the solution of problems of the slow two-dimensional motion of viscous liquid and of the corresponding problems of plane stress and plane strain, in cases in which the area in the x, y-plane that is concerned can be represented conformally on the interior of the circle |ζ| = 1 in the ζ-plane by a relation of the form z = x + iy = r(ζ), where r(ζ) is a rational function of ζ. In most problems in which the method has been used the function r(ζ) has been a simple one, but it is of importance to consider a rational function of as general a form as possible since, given any relation z = f(ζ), it will usually be possible to find a rational function that approximates to f(ζ) throughout the circle |ζ| = 1 and for a close approximation a complicated function r(ζ) will in general be required.

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

.

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. 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

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.

1. Problems in elasticity which are concerned with isotropic rectangular plates have attracted the attention of many writers both from the theoretical and practical points of view. When the boundary conditions are of the simply supported type the solution of the problems is usually simple, although when double Fourier series are used the validity of such solutions is not very clearly shown in most cases. Satisfactory exact solutions for many classical problems in which the edges of the rectangular plate are clamped have only been obtained in recent years, but approximate strain energy methods often gave results which were useful for practical purposes.

1. In this paper an analysis is given of the distribution of stress in a semi-infinite elastic medium due to the action of an external force applied to the interior of the medium. It will be assumed throughout that the force acts in a direction perpendicular to that of the boundary of the solid; the analysis is similar when the line of action of the force is parallel to the boundary and is therefore not given here. The equations of plane strain parallel to the x-y plane are employed; physically this is equivalent to assuming that there is no component of the displacement vector in a direction normal to the x-y plane or, what is the same thing, that the external force is applied along are infinite line parallel to the axis of z.

In an improved method of strip-theory calculation of the performance of an airscrew a factor K occurs in the formula

which, for an airscrew with N blades, connects the total interference velocity w1 and the circulation Γ at radius r. In this formula ø is the inclination to the plane of rotation of the airscrew of the velocity of the air relative to the airscrew blade; for infinite pitch ø=½π For large but finite pitch the calculation of K is difficult; in computing tables of K it is therefore useful to have the values for infinite pitch, which can be calculated by a much simpler theory than that needed for the general case†.

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.

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.

It has been found possible to generalize Ising's (4) treatment of the linear chain with nearest-neighbour interaction, and his main result, that such interaction cannot lead to cooperative effects, has been shown to hold for very many, if not all, types of lattice in three dimensions. Reasons are given for presuming a similar result for the second nearest-neighbour model. It is found, however, that even a small interaction of the long-range type can lead to cooperative effects at high temperatures, provided that the nearest-neighbour interaction is sufficiently strong. It is therefore claimed that explicit account must be taken of both types of interaction in any acceptable theory, and a possible method of estimating their relative importance is indicated.

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.

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. I proved in 1908(1) that if A = Σam and B = Σbn are convergent, and

for large m and n, then

is convergent (necessarily to AB); and this theorem has been extended in a number of directions both by other writers and by myself. Thus we may replace (1), when am and bn are real, by

we may use conditions unsymmetrical in am and bn; we may put the same problem for the product of any number of series; and we may consider modes of ‘Dirichlet multiplication’ based on sequences λm and μn, reducing to Cauchy's when λm = m and μn = n. The appropriate references will be found in(2).