1. The series

is the simplest and most familiar power series whose radius of convergence is zero. It is natural to regard it as a development of the function G(z) defined, when

by

For

say;and

or

according as x is positive or negative. Thus the series (1·1) is an asymptotic series for G(z) in the sense of Poincaré.

The object of this note is to obtain necessary and sufficient conditions for the existence of

on the assumption that ø is integrable in any finite positive interval not including zero (0 < α ≤ x ≤ β < ∞). On reference to the literature on the topic, I find no exhaustive analysis of this problem anywhere.

A linear vector m-space Rm defined along a curve (C) in a Vn and lying in the complete osculating space of (C) will be called a characteristic Rm of (C) if it is auto-parallel along the curve and makes constant angles with its tangent and the principal normals. Curves admitting a characteristic R1 have been studied by Hayden under the name of generalized helices and generalized by me§. In this paper we give a complete determination of the curves with a characteristic R2. Curves whose curvatures are proportional to a set of constants, which have been considered by Syptak for the particular case when Vn is an Rn, form one of the classes of curves of this type. As a consequence, the existence of the curves admitting a characteristic Rm (m > 2) is partly established, but the problem has not been completely solved. At the end we prove two theorems in connexion with two particular types of characteristic Rm's.

In a previous paper a general solution was given for problems of stress distributions in a plate containing circular holes of varying sizes arranged in any manner. This work was a generalization of special methods used by various writers for particular arrangements of holes. The types of stress distributions were, however, confined to those which produce zero force resultants on each hole and the solutions were therefore independent of the elastic constants. Bickley has studied distributions of stress round one circular hole in an infinite plate when the force resultants on the hole are no longer zero, and a few other problems of this type have been dealt with by other writers.

A short survey of Born's theory of the thermodynamics and melting of crystals is given. It is shown that Lindemann's and Grüneisen's law for the normal melting temperature can be deduced from this theory, and that the dependence of the melting temperature on pressure, and of the compressibility and the elastic constants on pressure and temperature, as predicted by the theory, are in good agreement with experiment. Several connexions between breaking and melting, suggested by the fundamental ideas on melting and stability of crystals, are discussed and verified. Finally a relation between the heat of melting and the heat of sublimation is deduced and compared with experiment.

A general discussion is given of the momentum wave functions of simple molecules in which only single bonds occur; the momentum functions are obtained by transformation of the space wave functions, for which two distinct approximations are possible. In each case it appears that the presence of the bond decreases the mean component of the velocity in the direction of the bond, increases the mean component of velocity perpendicular to the bond, and increases the mean momentum averaged over all directions. The analysis is illustrated by consideration and H2; in the latter case the strong dependence upon the mutual orientation of the two electronic momenta is emphasized.

The momentum distribution has been calculated for an electron in one of the hybridized orbitals of a carbon atom, and is shown as a function of the degree of hybridization. The mean momentum shows a steady increase from pure s to pure p binding. Similar calculations have been made for the momentum distribution in the C—H bond, and it is shown that this also depends upon the degree of s-p mixing. The mean momentum for the bond is less than for an isolated hybridized carbon orbital, but behaves similarly to such an orbital when the degree of hybridization is varied. The differences are about within the limit of experimental determination.

The discussion of momentum distribution is extended to cover electrons in double bonds and conjugated bonds of fractional order. The mean radial distribution function is obtained, and it shows that the mean momentum of the electrons is less than if they were rigidly attached to their own particular nuclei. Polar diagrams show that, if electronic mobility is high in any direction in a molecule, the momentum is generally small in that direction.

In developing the theory of adsorption taking into account the interaction between adsorbed particles it has been usual to use a physical model in which it is assumed that there is a fixed interaction energy between particles adsorbed on neighbouring sites on the surface. In this paper the differences between the behaviour of this model and that of actual surfaces are discussed by considering a one-dimensional film in which the potential energy of a single adsorbed particle varies continuously and periodically with its position on the surface and in which there is a repulsive force between adsorbed particles which varies with the distance between them according to an inverse power law. For such a physical model the variation of the heat of adsorption with the fraction of the sites occupied is considered in detail and it is shown in particular that there is much less difference between the behaviour of mobile and immobile films than is indicated by the earlier model in which a fixed interaction energy is assumed. These results are considered in connexion with the interpretation of experiments on the adsorption of hydrogen on tungsten.

The geometrical constraints on the forms of quaternary alloy equilibrium diagrams have been derived.