It is recognized that the main hope for advance in a rational theory of aeroplane propulsion lies in extensive use of the general principle of conservation of resultant momentum, even more than on considerations of energy; for energy can be dissipated by friction, while the relation of momentum to force is absolute.

This is a sequel to recent communications to the Society, on “The General (m, n) Correspondence” and “The (2, 1) Correspondence.” For convenience the reference numbers here adopted will be consecutive with what has preceded, so that the present work opens with § 6 as a direct consequence of previous results in earlier sections.

§ 1. “Stretches” joining Points in a Metrical Space.

If x, y, z are any points in a metrical space, then by definition

Regarding x and y as fixed, we may define the set of points z for which

as the “stretch joining the points xy” or more shortly the stretch xy*. Then for points z not on the stretch,

The stretch xy always contains the points x and y themselves, which we may call the “end-points” of the stretch. In particular cases, it may contain no other points: e.g. with the function δ0(x, y) = 1 when xǂy, each stretch contains only its end-points †.

In his remarkable memoir ‘On certain arithmetical functions’* Ramanujan considers, among other functions of much interest, the. function τ(n) defined by

This function is important in the theory of the representation of a number as a sum of 24 squares. In fact

where r24 (n) is the number of representations;

where σs(n) is the sum of the 8th powers of the divisors of n, and the sum of those of its odd divisors; and

The theorem which we prove here seems obvious enough when stated, but it appears to have been overlooked by the numerous writers who have discussed the subject, and the proof is less immediate than might be expected.

Statistical mechanics is concerned primarily with what are known as “normal properties” of assemblies. The underlying idea is that of the generalised phase-space. The configuration of an assembly is determined (on classical mechanics) by a certain number of pairs of Hamiltonian canonical coordinates p, q, which are the coordinates of the phase-space referred to. Liouville's theorem leads us to take the element of volume dτ=Πdp dq as giving the correct element of a priori probability. Any isolated assembly is confined to a surface in the phase-space, for its energy at least is constant; when there are no other uniform integrals of the equations of motion, the actual probability of a given aggregate of states of the proper energy, i.e., of a given portion of the surface, varies as the volume, in the neighbourhood of points of this portion, included between two neighbouring surfaces of constant energies E, E + dE; it therefore varies as the integral of (∂E/∂n)−1 taken over the portion. If I be the measure of the total phase-space available, interpreted in this way, and i that of the portion in which some particular condition is satisfied, then i/I is the probability of that condition being satisfied.

The methods of geometrical optics provide an approximate solution of the equation of wave propagation, if the following condition, given by De Broglie, is satisfied. If V is the phase velocity, λ the wave-length, l the direction of greatest increase of V, and θ the angle between l and V, then the condition is that

that is, the relative change of V over a distance of the order of θ must be small†.

§ 1. Sir John Herschel gave the condition which must be satisfied in order that a symmetrical optical system, free from spherical aberration for two conjugate axial points, may also be free from spherical aberration for two neighbouring and conjugate points upon the axis of the system; but Herschel's condition applies only to first order aberration, i.e. to aberration depending upon the cube of the inclination of the ray to the axis. Abbe shewed, later, that this condition could be included in a wider result, viz. that the spherical aberration, supposed zero, is stationary for axial variations provided that the incident and emergent rays for two conjugate axial points, associated with modified magnification m, satisfy the relation

where θ and θ′ are their initial and final inclinations to the axis; and by ‘modified’ magnification is meant the ratio of the reduced sizes of the image and object.

Henderson proposed a theory of the stopping of swift α-particles by matter. He treated the electrons in the atoms as free and at rest and ignored all collisions of the α-particle with them except those in which the electron would on that assumption gain sufficient energy to leave the atom. Fowler has shown that Henderson's theory gives stopping-powers only of about 60% of those observed. Fowler also made a calculation of the stopping-power of hydrogen by combining the effects of close collisions treating the electron as free and slight collisions as perturbations of the electron's motion in a circular orbit. He obtained much better agreement. This method is, of course, the natural extension of Bohr's original theory to that model.

Considerable advances have been made in recent years in our knowledge of the γ-ray emission of radioactive bodies. It has been established quite definitely that series of definite frequencies are emitted, forming a characteristic line spectrum of the nucleus, and methods of measuring the frequencies have been devised so that at present the γ-ray spectra of most of the radioactive bodies are well known.

It is well known that the calm which commonly comes on a cold clear night is due to the cooling effect of the ground on the lowest layers of the atmosphere. It frequently happens that the wind at a height of a few hundred feet does not die down at all at night; in fact observations taken at the top of the Eiffel tower show that there is frequently a slight increase in wind at that height. The cooling of the air near the ground makes the lower layers heavier than the higher layers and thus tends to prevent the formation of turbulence at the earth's surface. Since it is only through the medium of turbulence that the higher layers are able to drag the lowest layers over the ground the surface air stops moving.

An α-particle passing through a gas loses velocity because it gives up energy to the gaseous atoms. If we can calculate the average energy transferred to an atom, the stopping power of the gas follows immediately. The earlier theories, by Thomson and Darwin, took account of only the close collisions, in which the α-particle actually passes through the atom. Bohr, however, in 1913, took into account the transfers of energy to atoms at distances from the track considerably larger than atomic dimensions. His calculation was purely classical, and dealt with an atomic, model in which the electrons were capable of simple harmonic motions. The result was in very satisfactory agreement with experiment. For atoms containing each a single electron, with mass μ, charge ε, and natural period ω, Bohr's formula is

where − dT/dx is the rate of loss of energy by an α-particle whose charge is E, and velocity v; N is the number of atoms per c.c.; and γ = 1·123.