La théorie que nous nous proposons de présenter ici provient d'une synthèse de plusieurs théories particuliéres.

In a paper ‘The projective generation of curves and surfaces in space of four dimensions’, F. P. White considers the representation of prime sections of the general Bordiga surface F6 in [4] by quartic curves in the plane through ten fundamental points. We shall make use of several results from the same section of that paper, and, for convenience, list them as follows. From a point P′ of F6 can be drawn an ∞1 of trisecant lines lying on a cubic conical sheet. The curve of intersection of these trisecants and F6 is a curve C8, of order eight and with a double point at P′, which meets each of the ten lines on F6 in two points. In the plane, the representation of this curve is a curve C7, of order seven and with eleven double points, namely, the ten fundamental points and the point P which corresponds to P′. C7 is hyperelliptic, containing an involution of points Q, R corresponding to the points Q′, R′ in which a trisecant from P′ meets F6. The envelope of joints of corresponding points Q, R in the plane is a conic; this conic will continually be referred to as Σ.

In an interesting recent paper Schrödinger(5) contributes much new information on the properties of the hypercomplex algebra used in meson theory(1, 2, 4, 6, 7) which is defined by the relations(3)

The object of this note is to show that the equation of the quadric in n dimensions can be transformed to a ‘focus-directrix’ form which is exactly similar to those for two and three dimensions: this form is extended throughout by the introduction of a constant k. A short discussion of confocal quadrics is appended.

The paper considers the application of Dirac's classical theory of radiating electrons to consider the straight line motion of an electron towards a fixed proton and the straight line motion of two oppositely charged electrons. The paper also gives a discussion of the probable solutions in two- and three-dimensional motions of an electron moving round a fixed proton. In all these cases there appears to be no solution which would permit a collision between the two particles.

I take this opportunity to express my deepest gratitude to Prof. Dirac for his patient guidance and supervision, and also to Christ's College for a scholarship.

Note added. In the problem of the rectilinear motion considered in section (2), if we take the charges on the particles to be of like signs, then we find that the equations of motion have a solution which corresponds to a collision. In place of equation (4) we have

It can be easily verified that under suitable initial conditions there is a solution with x→0 and y→∞. Near x=0 the solution is approximately . Thus a collision is possible for like charges and not possible for unlike charges. The result, in both cases, is the opposite of what we should expect from elementary physical considerations.

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.

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.

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.

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.

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.)