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.

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.

It has recently been pointed out by Heitler(1) that the well-known discrepancy between the theoretical expression and the experimental results for the cross-section of scattering of charged mesons by nuclear particles can be removed by a proper consideration of the effect of radiation damping in the quantum theory. The radiation damping in quantum theory was first considered in complete detail for free electrons by Waller(2). A rigorous deduction of the integral equation set up by Waller was given by Heitler on the basis of a method developed by Góra. An alternative rigorous derivation of the integral equation has also been given by Wilson(3). Exact solutions of the integral equation for the simple scattering of mesons by nuclear particles have been found by Heitler in the non-relativistic approximation. An exact solution has not so far been given for physical problems in which the integral equations are complicated, but Wilson has given a general approximate formula for the scattering cross-section, which should be valid for all problems.

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.

Hughes and Mann(1) have shown that, under certain conditions, the distribution curve of the energies of electrons scattered inelastically by an atom or molecule has the same shape as that of the X-ray Compton line. Hughes and his collaborators (1–5) have applied the method to an examination of a number of atoms and molecules in the gaseous state under which condition it can be assumed that each atom or molecule acts as an independent scattering centre. Coulson and the author (6–11) have applied quantum methods to derive a theoretical shape of the Compton line for certain of the molecules for which wave functions are available and have discussed the discrepancy between the experimental and the theoretical results(10, 11).

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)