1. In a recent paper Landau has shown that, when electrons are moving freely in a magnetic field, they exhibit, in addition to the paramagnetic effect of their spin, a diamagnetic effect due to their motion. This result is rather unexpected, since it is quite contrary to the classical case. There it might appear as though the circles described by the electrons must produce a magnetic moment, but the error was long ago pointed out by Bohr. The motion of the electrons must be confined to some region by means of a boundary wall, and the electrons near the wall describe a succession of circular arcs, repeatedly bouncing on the wall, and slowly creeping round it in the direction opposite to that of the uninterrupted circles; when the moment of these electrons is taken into account, it exactly cancels out that due to the free circles. In Landau's work it is of course necessary to consider the boundary, but he shows how allowance is to be made for it by an appropriate process. The complete justification is rather subtle, and so it may be worth considering a special case, admitting of exact solution, which takes the boundary into account, and so makes it possible to follow more closely the analogy between the classical and quantum problems. With regard to the general case with a boundary wall of any type, we shall only observe that the different results arise, because in the wave problem ψ must vanish at the bounding “potential wall,” and so will be small near it; this upsets the balance of the electric current near the wall, and yields the magnetic moment.
A study is made of the actual trajectories of fluid particles in certain motions of classical hydrodynamics. When a solid body moves through an incompressible fluid, it induces a drift in the fluid, such that the final positions of the particles are further on than those from which they started. The drift-volume enclosed between the initial and final positions is equal to the volume corresponding to hydrodynamic mass, that is, the mass of fluid to be added to that of the solid in calculating its kinetic energy. This result is proved quite generally. The work involves integrals which are not absolutely convergent, and these are discussed in relation to the general mechanics of fluids. When the trajectories are considered of the fluid surrounding a rotating body, it is shown that the fluid particles slowly drift round the body, even though the motion is irrotational and without circulation. There seems to be in some respects a closer resemblance between the behaviour of the idealized hydrodynamic fluid and a real fluid than might be expected from the well-known discrepancies between them.
Einstein’s equations for the orbits round an attracting point mass, here called the sun, are examined so as to see whether there are orbits which end in the sun, as there are in the corresponding case of electrical attraction when relativity is allowed for. With the measure of the radius as usually taken, it is shown that no hyperbolic orbit can have perihelion inside r = 3 m , and an elliptic orbit cannot have perihelion inside r = 4 m . Particles going inside these distances will be captured. Circular orbits are possible for any greater radius. If 3 m < r < 4 m the orbit is unstable; with one disturbance it falls into the sun, with the opposite it escapes in a spiral to infinity. If 4 m < r < 6 m , it is also unstable, either falling into the sun, or moving out to some aphelion at a greater radius before returning to its circle. Only if r > 6 m is the orbit stable. A study is made of the travel of light rays. No light ray from infinity can escape capture unless its initial asymptotic distance is greater than 3√3 m . A field of stars surrounds the sun, and is viewed in a telescope pointed at the sun from a distance. If the field as seen is mapped as though in a plane through the sun, each star, in addition to its direct image, will show a series of faint ‘ghosts’ on both sides of the sun. The ghosts all lie just outside the distance 3√3 m . A few technical details are given about the orbits of the captured particles.
on the Reflexion of Equation (26) is suitable for use in computations for love temperatures. The expanded form for %(0) analogous to (24) is suitable for use with high temperatures and is more immediately intelligible than (26). This form is %(O)=27rvII(a) ~2 (2JII(¢))"-'-n~-I n ! x ~a + n+l " n+2 + n+3 .(27} and the corresponding form of the second virial coefficient is n=l ?t ! f(d-a)d ~ 2(d-~)-"d (d-~)83~ (28) × L" ~-~-+-i n+2 ÷ n+3 5" xoii. The Reflexlon of ,Y-Ra~/s from Imperfect Crystals.
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This is a sequel to a paper of 3 years ago, which studied the orbits of ‘comets’ near a ‘sun’ regarded as a point source of gravitation according to general relativity. That paper expressed the forms of the orbits in terms of elliptic functions, but its method was not so well adapted to a study of the time in those orbits. In the first half of the present work these orbits and their associated times are described in a simple form, the results being expressed in terms of integrals of elementary functions, which can be easily worked out either by quadratures or by approximation. One result of the earlier paper was the proof that no orbit can have perihelion inside r = 3 m , and in the later part of the present work a method is proposed in order to study this region, since no comet can return from it. It is supposed that flashes are emitted both from a distant observatory and from a comet, each signalling the ticks of his clock according to the time it is keeping. These are observed by the other and compared with the time on its own clock. The method serves to describe occurrences between r = 3 m and the ‘barrier’ at r = 2 m , and it points to some unexpected results in the matter of the comet passing the barrier, which call for explanation.
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