1. Let (a, b) denote an unbounded interval. Given the numbers s0, …, sn−1, consider the problem of finding what conditions they must satisfy in order that there shall be a function f(x), 0 ≤ f ≤ 1, such that

1. The only double binary forms for which the complete systems of invariants and covariants are known appear to be the (1, 1), (2, 1) and (2, 2) forms, for which the complete systems were determined sixty years ago by Peano. In the present paper we determine the complete system of the binary (3, 1) form and establish its irreducibility. The system proves to contain twenty forms, and is thus scarcely more complicated than that of the (2, 2) form which includes eighteen concomitants, and much simpler than that of the (2, 1, 1) form derived by Gilham. It is hoped in a subsequent note to interpret the system geometrically.

1. In this paper we shall show that the complete system of concomitants of a binary (4, 1) form in two sets of binary variables (x0, x1), (y0, y1) consists of 63 forms. The methods employed are those of a previous paper.

1. A paper with the same title was published with Professor H. F. Baker in 1920; that paper was severely compact and would seem to deserve elucidation, more detail, and some revision. We use a symbolism which, while leaving the proofs geometrical, readily converts itself into ordinary analysis, as we shall show in § 12.

If (a1, b1, c1), (a2, b2, c2), (a3, b3, c3) be three sets of numbers subject to the six relations such as and with a positive determinant, then the four tetrahedra, M, M1, M2, M3, whose angular points have coordinates given by the rows of the four matrices

are such that every two are mutually inscribed. The sixteen points so arising are the nodes of a Kummer surface having for one form of its equation

In this paper we consider the properties of a point-curve correspondence between two (generally distinct) surfaces F and G, paying special attention to the idea of ‘induced’ and ‘extended’ correspondences. We also outline the theory of point-curve corre spondences between a curve and a surface, which is not only of interest in itself, but is of importance because such correspondences arise, as do correspondences between curves, as induced correspondences of point-curve correspondences between two surfaces. The results depend chiefly on Lefschetz's theory of correspondences between algebraic curves(1) and the properties of intersections of cycles on an algebraic surface given by the author (2), as well as on the preceding paper of this series (3). Use is also made of a paper by Severi (4) on the self-correspondences on a variable curve of an algebraic surface. The theory of correspondences on a single surface will be further developed in the next paper on this subject.

1. The general theory underlying sampling inspection methods introduced during the war under the name of sequential analysis, applicable in cases where the sampling units can be taken serially, has mainly been developed by Wald and is now published (5). The original purpose of the present investigation was to exhibit the main structure of the distributional theory relating to the size of sample required by noting its relationship with the classical ‘random-walk’ problem. Thus in Part I of this paper are derived the distribution and characteristic functions of the absorption time for one-dimensional random-walk theory with constant drift and absorbing barriers. Attention is confined to the asymptotic case of numerous independent displacements, for which it is well known that the unrestricted total displacement becomes Gaussian; the motion may also be treated as continuous. Since these results were obtained Wald has independently given an alternative and more general discussion of this problem((4); see also Tweedie (3)), but the extension here of the method of images used by Chandrasekhar (1) still appears of interest, and has advantages over alternative direct methods of solution for the distribution function involving Fourier expansion (cf. 2).

Rayleigh's method of deducing the probability distribution of the amplitude of the sum of n equal vibrations of random phase is generalized to the case when the amplitude of each vibration is a definite function of its phase. The same method is applied to the shot effect and it enables the distribution of random noise to be obtained. Campbell's Theorem and its generalizations can then be deduced from this.

In developing the theory of radiative transfer in expanding atmospheres (Chandrasekhar, 1945 a, b) the author has recently encountered certain novel types of boundary-value problems in hyperbolic equations which appear to merit consideration for their own sake. As related to the equation

the boundary-value problems which occur are of the following general type

i.e. along AB in Fig. 1. Along AD (x = 0 and 0 ≤ y ≤ l2) and BC (x = l1, 0 ≤ y ≤ l2) we are further given that

and

where φ(y) and ψ(y) are two known functions. The problem is to solve equation (1) in the rectangular strip ABCD satisfying the stated boundary conditions.

In a recent paper (1) an analysis was given of the distribution of stress in a semi-infinite elastic medium deformed by the pressure of a rigid body on part of the plane boundary, the remainder of the plane being free. In that form of the problem—the so-called ‘Boussinesq problem’—the normal displacement of a point within the pressed area was prescribed and the distribution of pressure over that area determined. In this paper the corresponding analysis is given for the case in which the pressed area and the distribution of pressure over it are both prescribed and the normal displacement of a point on the free surface is determined.

1. The properties of partitions of numbers extensively investigated by Hardy and Ramanujan (1) have proved to be of outstanding mathematical interest. The first physical application known to us of the Hardy-Ramanujan asymptotic expression for the number of possible ways any integer can be written as the sum of smaller positive integers is due to Bohr and Kalckar (2) for estimating the density of energy levels for a heavy nucleus. The present paper is concerned with the study of thermodynamical assemblies corresponding to the partition functions familiar in the theory of numbers. Such a discussion is not only of intrinsic interest, but it also leads to some properties of partition functions, which, we believe, have not been explicitly noticed before. Here we shall only consider an assembly of identical (Bose-Einstein, and Fermi-Dirac) linear simple-harmonic oscillators. The discussion will be extended to assemblies of non-interacting particles in a subsequent paper.

A set of relativistic classical motions of a radiating electron in an electromagnetic field are derived from the principle of conservation of energy, momentum and angular momentum. It is shown that these equations lead to results more in harmony with the usual scheme of mechanics than do the Lorentz-Dirac equations. When applied to discuss the motion of the electron of the hydrogen atom, these equations permit the electron falling into the nucleus, whereas the Lorentz-Dirac equations do not allow this. When applied to consider the motion of an electron which is disturbed by a pulse of radiation, the solution is in a more symmetrical form. For scattering of light of frequency ν the expression for the scattering cross-section is found to be the same as the classical Thomson formula for small ν, and to vary as ν−4 for large ν.

The paper deals with the motion of a Gaussian wave-packet in a parabolic potential field, contrasting the quantum-mechanical behaviour of the packet with that which follows from pure classical mechanics.

The behaviour is entirely classical only if the initial phase of the space-wave-function is linear in the space coordinate; in which case the wave-packet is initially a ‘best-possible’ packet according to Heisenberg's uncertainty relationship.

The paper concludes with a brief reference to the similar problem for a free particle.

Note added in Proof, 16 February 1946. Through the courtesy of Mr M. S. Bartlett, we have been privileged to see the manuscripts of some work by Mr J. E. Moyal and himself, in which the motion of a probability wave-packet is discussed in more general terms than our own: and in which it appears that the motion of our wave-packet, which we have described as non-classical, is indeed actually classical provided that a certain correlation exists between momentum and position in the p, q-distribution function. With this knowledge we now find that classical and quantal predictions agree precisely provided that the initial classical probability function is given by

this is a particular case of their more general results.

The statistical treatment which was given previously to determine the variation of the heat of adsorption of polar molecules is extended to show how the variation of the dipole moment with the fraction of the surface covered can be taken into account. The equations have been used to determine the variation of the heat of adsorption of ammonia on a non-conducting surface. The contributions to the heat of adsorption due to the van der Waals and to the electrostatic forces are of the same order of magnitude and of opposite signs so that the resultant variation in the heat of adsorption is very much less than would be expected from a consideration of forces of one kind only. The contributions to the heat of adsorption, which are made by the electrostatic forces when the dipole moment is constant and equal to M0, when it is constant and equal to M1, and when its variation with the fraction of the surface covered is taken into account, are compared for the whole range of values of θ. It is shown that the mutual depolarizing action of the molecules reduces the magnitude of the total variation in the heat of adsorption by a considerable amount. The heat curve which is calculated from the equations given by the statistical analysis is compared with that which is obtained when it is assumed that the particles form a random distribution over the surface, and the effect of the clustering of the adsorbed particles on the surface on the variation of the heat of adsorption is determined.

The number of configurations of mixtures of dimer and single molecules, of trimer and single molecules and of trimer and dimer molecules is examined by using the Bethe technique when there may be some vacant sites. This condition provides a check on the internal consistency of the Bethe method through the integrability of the resulting partial differential equations.

Exchange energy between electrons moving in a conductor is assumed to be such that the mean contribution per electron is proportional to the product of the total current within a defined range and the speed of the electron in the direction of the total current.

The ideal resistance of a conductor can be neutralized by such an exchange energy at sufficiently low temperatures.

The surface of the conductor, and internal surfaces of discontinuity if sufficiently marked, are regarded as sites for two-dimensional resonance states in equilibrium with the three-dimensional resonance states in the metal.

The superconducting transition occurs when, at the critical temperature, the neutralization of the ideal resistance of the surface by-passes the residual resistance of the body of the metal.

At the same critical temperature, equilibrium conditions ensure that exactly the right number of electrons enter the surface states to give the whole metal an ideal diamagnetic susceptibility.

These equilibrium conditions are disturbed by a magnetic field, and the transition temperature is shown to depend on the magnetic field in the observed manner.

The intermediate state is explained as one in which the number of electrons in the superconducting surface states is constrained by a non-uniform magnetic field to remain at an intermediate value which is inadequate to produce ideal diamagnetism.

By regarding either the outer surface or internal surfaces of discontinuity as the only regions becoming perfectly conducting, the details concerning restoration of resistance by a current are well explained, as also are the observed size effects in thin films, and the phenomena of ‘non-ideal’ superconductors among the alloys.

Finally, the observed distribution of superconductors in the periodic table receives a satisfactory qualitative explanation.

A simple model of independent curled-up molecules is examined theoretically. The critical temperature is found to be of the right order of magnitude but does not increase sufficiently rapidly with the size of the molecule. The Onnes constant and the critical density show the discrepancies expected from the theoretical equation of state, and the former also has an incorrect dependence on molecular weight. In the critical region, therefore, the paraffins cannot be regarded as an assembly of independent molecules, curled up in the way expected at high temperatures and low densities.