The suffixes used in logic to indicate differences of type may be regarded either as belonging to the formalism itself, or as being part of the machinery for deciding which rows of symbols (without suffixes) are to be admitted as significant. The two different attitudes do not necessarily lead to different formalisms, but when types are regarded as only one way of regulating the calculus it is natural to consider other possible ways, in particular the direct characterization of the significant formulae. Direct criteria for stratification were given by Quine, in his ‘New Foundations for Mathematical Logic’ (7). In the corresponding typed form of this theory ordinary integers are adequate as type-suffixes, and the direct description is correspondingly simple, but in other theories, including that recently proposed by Church(4), a partially ordered set of types must be used. In the present paper criteria, equivalent to the existence of a correct typing, are given for a general class of formalisms, which includes Church's system, several systems proposed by Quine, and (with some slight modifications, given in the last paragraph) Principia Mathematica. (The discussion has been given this general form rather with a view to clarity than to comprehensiveness.)

If a particle counter is subject to the radiation from a pure radioactive source of sensibly constant activity, it is axiomatic to suppose that the series of events constituted of the traversals of the counter by ionizing particles is a random series in relation to distribution in time. There are various reasons why the related series of events formed by the occasions on which the counter responds to these ionizing particles is not a random one—and there may also be cases in which the primary source does not, in fact, give rise to particles traversing the counter in a random sequence. The counter responses will necessarily depart from strict randomness in time because of the finite recovery time of the counter; they may also depart from strict randomness through a not uncommon counter defect, the occurrence of ‘spurious’ discharges correlated in some way with the ‘true’ discharges brought about as a result of the ionization produced by the particles. On the other hand, the original ionizing events will not have the effect of a random sequence, even with a pure radioactive source, if secondary radiations are emitted—even in a fraction of the disintegrations—with a time delay appreciable in relation to the counter resolving time, and, clearly, they may depart considerably from true randomness in time if the source is not a pure source, and, in particular, if there is produced in the source material a ‘daughter’ radioelement of lifetime comparable with this resolving time, or with the mean time between counter discharges.

In 1939 I published a paper in the American Journal of Chemical Physics (Born(1)) in which I tried to develop the thermodynamics of a crystal lattice in the domain of classical (Boltzmann) statistics. Definite formulae and numerical tables for the temperature dependence of the elastic constants up to the melting point were obtained; nevertheless the work was unsatisfactory not only because the approximations used were rough and their accuracy not known, but because a fundamental difficulty with respect to lattice stability turned up. This led to a series of investigations by my collaborators and myself, published in these Proceedings under the title ‘On the stability of crystal lattices’ (quoted here as S I to SIX), by which the difficulty mentioned has been removed. It is now possible to return to the original problem, to which the present series of papers is devoted. In this introduction I wish to recapitulate the whole situation and to explain the plan of the following papers which will be published by my collaborators.

In previous papers by Misra(1) and by Born and Misra(2) lattice sums of the type required in discussing the stability of a cubic crystal of the Bravais type in which the forces are central have been calculated. In the investigation of the thermodynamic properties of crystals a more general type of lattice sum occurs, which involves the phases of the waves. In the present paper a method of calculating these sums is developed and tables are computed.

The purpose of this paper is the determination as accurately as possible of the equation of state for the simplest stable crystal, the cubic face-centred lattice, and the comparison of the result with those obtained by rough approximations(1) in order to get an estimate of the reliability of the latter if applied to more involved problems where exact calculations are impossible.

The problem of the energy loss by radiation of an electron in a Coulomb field has been solved, using relativistic equations and the Born approximation, by Bethe and Heitler, and, using exact non-relativistic equations, by Sommerfeld; the first of these solutions is valid only for relatively high and the second for relatively low energies. In this paper an exact solution of the problem is given using the relativistic Coulomb wave functions, but depending on a final stage of numerical computation. The method is an extension of that used to determine cross-sections for pair production.

A number of errors have unfortunately been made in the last section of the above paper.

(1) ‘log’ should be deleted from equation (44).

(2) The simplification of equation (45) which was presented neglects some important terms, with the result that the later equations give only the limiting values of osmotic pressure and vapour pressure at infinite dilution. The section following equation (45) must therefore be ignored.