Nous ne considérerons, en général, que les sous-ensembles de l'intervalle [0, 1] et nous désignerons par et respectivement l'ensemble des nombres rationnels et l'ensemble des nombres irrationnels de cet intervalle.

Nous allons employer pour les fractions continues la notation

Étant donnée une suite de nombres naturels {zn}, nous dirons que le nombre correspond à la suite {zn} et vice versa, et nous écrironns z = v({zn}). Pareillement, nous dirons qu'un ensemble correspond à une famille Φ de suites de nombres naturels si l'on a E = v(Φ).

1. Any number x between 0 and 1 may be expressed uniquely in the form

where xr is a non-negative integer less than r (r = 2,3,…). We consider the set E of numbers x for which

We establish an inequality connecting the dimensional number of the set E with certain constants of the series

in particular we show that, when ξτ = rθ, the dimensional number of E is θ. We are concerned with the measure of Hausdorff.

The reflexion of a train of simple harmonic waves by a convex paraboloid of revolution, and by a parabolic cylinder, has been discussed by Lamb. In the present paper these results are extended to the reflexion of plane waves of arbitrary form. It is found that on the introduction of suitable variables the equation of sound propagation transforms (in each case) into a simpler equation whose general integral can be obtained by quadratures. Two unknown functions are introduced during the integration, which have to be determined from the boundary conditions. This involves in both cases the solution of a Volterra integral equation, which is effected numerically by calculation of the first terms in the series development of the resolving kernel. An interesting feature of the solutions obtained is that when a suitable time scale is introduced (for a sharp-fronted pulse the time must be counted from the onset of the wave), the reflected wave experienced is the same at all points on any paraboloid (or parabolic cylinder) confocal with the reflector.

1. Introduction. It is a classical result in hydrodynamics that a solid moving in an infinite liquid under no forces is capable of steady translational motion in any one of three mutually perpendicular directions. In the general case such a motion is only possible in three directions, though of course particular solids are capable of steady motion without rotation in an infinity of directions.

We consider the motion of a particle in a plane field of force. We take rectangular cartesian coordinates (x, y) in the plane, and denote by V(x, y) the potential of the field, and by h the (constant) energy of the motion. If A and B, whose coordinates are (x0, y0) and (x1, y1), are any two points on an orbit, the orbit is characterized by the property that

taken along the orbit is stationary as compared with the integral taken along a neighbouring curve joining the same points. Here s denotes the length of the are measured from A to B. This is one form, sometimes called Jacobi's form, of the principle of least action. The value of the integral, taken along the orbit, and expressed in terms of the coordinates of the termini and the constant of energy,

is the action function for the field V.

It has been shown by M. Born (1) that the equation of state and other thermo-dynamical properties of crystals, for example the melting temperature and its dependence on pressure, can be derived by a logical development of the lattice theory of crystals. I was able to show in a recent paper (6) that the results of a very large number of experiments of various kinds on the mechanical and thermal behaviour of solids are in satisfactory agreement with this theory. In the present paper I wish to show that new experiments of Bridgman (7) on the compression of solids under very high pressure can also be used for a comparison with the same theory, and that the results of this investigation are in favour of the theory. Some considerations on the range of stability of a lattice under a high uniform negative pressure are added, which support the idea that there is a close relation between the theory of stability of crystal lattices and the theory of melting.

The paper describes the results obtained from a study of the bombardment of gold with deuterons of energy up to 9·1 M.e.V. The reactions found to take place in gold are the formation of 2·7 day Au198 by a (d-p) process and of 32 hr. Hg198* in a metastable state by a (d-n) process. The β-ray and γ-ray energies associated with each radioactivity have been measured by absorption methods. Excitation functions for the two reactions have been determined and the results have been compared with those to be expected on theoretical grounds.

The author takes this opportunity of expressing his gratitude to the past and present members of the Cavendish Cyclotron Laboratory for assistance in running the cyclotron. He wishes to thank the Royal Commission for the Exhibition of 1851 for the award of a Science Scholarship.