Rosseland proposed a theory of the production of characteristic X-rays. He calculated the number of atoms from which on the average an electron with definite energy will remove an electron from a particular core orbit, but neglected the velocity of the core electron in its orbit and the increased velocity of the impinging electron due to the atomic field. Taking these into account considerably alters his formula.

New factors arise in a species by the process of mutation. The frequency of mutation is generally small, but it seems probable that it can sometimes be increased by changes in the environment (1,2). On the whole mutants recessive to the normal type occur more commonly than dominants. The frequency of a given type of mutation varies, but for some factors in Drosophila it must be less than 10−6, and is much less in some human cases. We shall first consider initial conditions, when only a few of the new type exist as the result of a single mutation; and then the course of events in a population where the new factor is present in such numbers as to be in no danger of extinction by mere bad luck. In the first section the treatment of Fisher (3) is followed.

Thomas's differential equation for the average field inside a heavy atom is analogous to Emden's equation for the polytropic equilibrium of a star. Emden's result that the total gravitational potential energy of a star is calculable once the differential equation has been solved is adapted to give the total electrostatic energy, and hence the total energy of binding, of an atom built on Thomas's model. This should be equal to the sum of the successive ionisation potentials. The total energy is found to be proportional to N, where N is the atomic number. The values found agree with Hartree's calculations of the successive ionisation potentials of certain atoms.

An account is given here of the measurements of the mass absorption coefficients of iron, nickel, copper and aluminium over a wave-length range 0·705 Å.U. to 1.932 Å.U.

There are in the Fifth Book of Euclid's Elements twelve propositions, the object of which is to prove the equality of two ratios.

In this paper the results of some measurements of the Hall effect in single crystals of iron are described. This work was undertaken to determine whether the Hall coefficient varied with the direction in the crystal for which it was measured. Such a variation had been observed in other magneto-electric phenomena—the change of resistance in longitudinal* and transverse† magnetic fields—and had thrown some light on the nature of these effects. It was thought possible, therefore, that the study of the Hall effect in single crystals might yield some fresh light on this unexplained phenomenon.

A Fourier integral is said to be of finite type if its generating function vanishes for all sufficiently large values of ¦x¦. Because the coefficient functions are defined by integrals over a finite range, the behaviour of such a Fourier integral usually resembles closely that of the corresponding series.

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.

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.

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. 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.

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

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.

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†.

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.

§ 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 †.