The aim of this paper is to give a general theory of the ‘strength’ of Hausdorff methods of summability. These methods are defined by the linear transforms

of a sequence (sk). Here the μk form a given sequence of real or complex numbers and the Δp(μk) denote their differences of order p; i.e. Δ0(μk) = μk and

For a direct comparison of the individual attractive and repulsive terms of an intermolecular potential determined by the inductive analysis of themodynamic data with the same terms calculated by quantal methods it is desirable to carry out the analyses, in the first approximation, with an intermolecular potential of the form ø(R) = Pe−aR − A1/R6 − A2/R8. For mathematical convenience, in place of the above expression, two potential functions,

and

are considered, the first being taken to be adequate in the range of values of R between 0 and R0 (the minimum of the potential function) and the second, in the range from R0 to ∞. By dividing the problem in this way it is possible to find substitutions which permit the integration of the classical expression for the second virial coefficients (and other appropriate thermodynamic data) directly in terms of fairly simple series in | ψ0 |, R0, a and r. Finally it is pointed out that for such simple atoms or molecules as the rare gases, oxygen, nitrogen and methane r may be taken as 0·15 throughout, which considerably simplifies the application of the method to the experimental data.

Let f(t) be defined and measurable for t > 0, and suppose that it has non-vanishing moment constants

where the integrals are Cauchy-Lebesgue integrals, or possibly if f(t) is null for t > t0 but not for t < t0 (a convention to which we adhere throughout).

The expression

for the number of linearly independent line complexes of degree n in space of r dimensions was derived by Sisam as a particular case in the determination of the number of linearly independent hyperconnexes of given degrees. The purpose of this note is to generalize Sisam's result by determining the number of linearly independent algebraic forms of a special type to which we give the name k-connex.

There are many types of problem in the study of molecular structure in which integrals involving two or more centres of force arise. We may mention the calculation of molecular energy levels, the Stark effect in molecules, the van der Waals forces between molecules, and, more recently, the internal arrangement of neutrons and protons in nuclei. During the last few years the writer has been accumulating a table of these integrals, and the explicit forms for many of them are given in this paper.

1. I appreciate the careful reply which Dirac, Peierls and Pryce (hereafter referred to as DPP) have made to my criticism of the usual theory of the eigen-functions of a hydrogen atom. Their paper will be generally welcomed because there has not, I think, previously been available an authoritative and clear statement of what is assumed and the grounds for assuming it. Their defence, which is partly on lines I had not expected, has required very serious consideration. But the result of making the arguments more explicit is two-fold. The theory is perhaps more self-consistent than it had appeared to be; but on the other hand, the pressing need for amendment becomes too plain to be overlooked. I have endeavoured to show in the last twelve years that this amendment opens out very fertile extensions.

1. Definition 1. A linear set E is said to possess the property C if, to any sequence of positive numbers {ln} (n = 1, 2, …), there corresponds a set of intervals, of length not greater than l1, l2, …, which includes all the points of E.

The theory of lattice deformations is presented in a new form, using the tensor calculus. The case of central forces is worked out in detail, and the results are applied to some simple hexagonal lattices. It is shown that the Bravais hexagonal lattice is unstable but the close-packed hexagonal lattice stable. The elastic constants of this lattice are calculated.

We shall consider plane Borel sets of points, though the result and the proof hold in space of any number of dimensions and for more general sets.

The main purpose of the paper is an investigation of the stability of a certain class of Bravais lattices, namely, those with a rhombohedral cell of arbitrary angle. The potential energy is assumed to consist of two terms, each proportional to a reciprocal power of the distance. In the continuous series of lattices obtained by changing the rhombohedral angle, there are included the three cubic Bravais lattices, the simple (s), the face-centred (f) and the body-centred (b) lattices. It is shown that (f) and (b) correspond to a minimum of the potential energy, and (s) to a maximum. A method for calculating the potential energy for the intermediate rhombohedral lattices is developed, and, with the help of a certain characteristic function, it is shown by numerical calculation that the (f) lattice corresponds to the absolute minimum of potential energy, and that no extrema, other than (f), (s) and (b), exist. In the last section, the case of a compound (non-Bravais lattice) is considered, and it is shown that the equilibrium and stability conditions for the law of force assumed can be divided into one set for change of volume, and an independent set for change of shape.

We take this opportunity of expressing our sincere thanks to Prof. Born for his interest in our work, and for much valuable advice.

In the course of the development of a special method of interpreting thermo-dynamic data in terms of a generalized force field for the mutual interactions of the molecules concerned, expansions were required for integrals of the following general type, which apparently have not hitherto been discussed. We require explicit expansions of F(α, s, n), in terms of the parameter α, for the integral values 0, 1, 2, 3, … of s and n, where

1. There are several known types of expansions of Mathieu functions, i.e. mod 2π periodic solutions of Mathieu's equation ((9), chap. 19),

The simplest expansion is the Fourier series

Almost equally well known are Heine's expansion ((4), p. 414; see also (5) and (6))

and

By a well-known theorem of Kirchhoff the strain energy of an elastic solid is less in the equilibrium position than in any other position satisfying the same boundary conditions, and under the same body forces. The theorem contradicts the fact that elastic instability can occur, since two or more positions of equilibrium can then exist, and both cannot have the smaller strain energy. Numerous writers (Bryan (1), Southwell (2), Dean (3)) have explained the apparent discrepancy as due to the neglect of second-order terms in the elastic equations. This is correct so far as it goes, but it does not explain why the usual discussions of elastic instability give the right answers. The elastic constants vary somewhat with stress and in any case will be different according as a stressed or an unstressed state is taken as the standard. If any second-order terms should be included, we might expect that this variation would make a contribution comparable with those hitherto considered. Further, several theories of elasticity now exist that differ in the higher terms (Dean (3), Seth (8), Murnaghan (6)), and it may be asked whether they should lead to the same estimates of the critical loads.

1. The object of this paper is to discuss conditions of validity of the Parseval formulae for Fourier integrals:

where the transforms are defined by ordinary convergence; we shall not be concerned with the more elegant theory in which they are given by convergence in mean.

Calculations have been made of the momentum distribution and the shape of the Compton profile for gaseous methane, using the Dirac transformation theory and three distinct approximations (molecular orbital, electron-pair and self-consistent-field) to the true molecular wave functions. The bearing of the results upon the validity of the original wave functions is discussed; it is concluded that the exponents in molecular wave functions should be approximately 1·1 times greater than in corresponding atomic wave functions. Comparison is made with experimental electron scattering results.

Our thanks are due to Prof. Massey for providing us with the numerical tables of the self-consistent-field for methane in advance of publication, and to Mr E. G: Phillips of the Mathematics Department of Univeristy College of North Wales, for the loan of a calculating machine.

Formulae are found for the number of configurations of particles on two-and three-dimensional lattices when each particle (a) occupies two closest neighbour sites, and (b) consists of three groups which occupy three sites on the lattice in such a way that adjacent groups in the molecule occupy closest neighbour sites on the lattice. Bethe's method is used to determine the equilibrium conditions of the corresponding order-disorder problem, and the number of configurations is determined from these equilibrium conditions. For the case in which the molecules occupy two closest neighbour sites on the surface the determination of the number of configurations from geometrical considerations is discussed.

It is found that for molecules which occupy two closest neighbour sites the number of configurations of particles for a square lattice, a simple cubic lattice and a body-centred cubic lattice respectively are and For molecules which occupy three sites on the lattice the corresponding results are and when the molecules are perfectly flexible and and when the molecules are completely inflexible, Ns being the total number of sites in the lattice.

The author wishes to thank Dr J. K. Roberts for suggesting this work; the geometrical treatment given in § 3 was developed from manuscript notes prepared by him. The problem arose during an investigation of the vapour pressure equations of solutions in which the solute molecules are chain molecules consisting of a large number of groups, undertaken as part of the programme of fundamental research of the British Rubber Producers' Research Association, whom the author wishes to thank for the grant of a Research Scholarship.

It is shown that the theorem stated in Born's paper, and proved for the case of a linear lattice of N equal particles under certain restrictions concerning the forces between the particles, that macroscopic stability (stability for long waves) implies microscopic stability, may be extended to three dimensions for the particular case of a face-centred cubic lattice, where the effects of all neighbours, other than the first twelve neighbours, are neglected.

I take this opportunity of expressing my sincere thanks to Prof. Born for much valuable advice.

The author's general variational method is applied to the case of a particle for which second moments are important but third and higher moments are negligible. Equations of motion are obtained for the angular momentum and for the centre of mass, equations (12·35) and (12·41), with arbitrary external forces X.

The angular forces are then calculated for a charged particle with electric and magnetic moments moving in a general electromagnetic field, on the assumption that the effect of a certain part of the energy tensor, Tiii of (15·17), is negligible. This leads to the equations of angular motion, (17·13), from which it is inferred that, in order that the magnitude of the angular momentum may be integrable, the angular momentum, electric and magnetic moments must all be parallel in a frame of reference in which the particle is instantaneously at rest.

The linear forces are then calculated for the case of no electric moment, leading to the equations for linear motion (18·10). From these it is inferred that, in order that the mass may be integrable, the ratio of the magnetic moment to the angular momentum must be constant.