In this paper we are concerned with curves (C) of the following types:

where k1, k2, …, kh−1, kh = 0 (h ≤ n) are the curvatures of (C) relative to the space Vn in which (C) lie. Hayden proved that a curve in a Vn is an (A)2m, h = 2m + 1, if and only if it admits an auto-parallel vector along it which lies in the osculating space of the curve and makes constant angles with the tangent and the principal normals. Independently, Sypták∥ stated without proof that a curve in an Rn is a (B)n if and only if it admits a certain number of fixed R2's having the same angle properties; he also gave to such a curve a set of canonical equations from which many interesting properties follow as immediate consequences.

The formation of 62Cu, 106Ag and 120Sb by deuteron bombardment of copper, silver and antimony respectively is reported. The energy-yield curves for these reactions were determined and it is concluded that in each case the process involved is taking place, in all probability, through the formation of a ‘compound nucleus’ of the Bohr type.

It is assumed that the viscosity of a liquid is due to the transfer of momentum by the irregular movement of the holes (see part I of this paper). A formula for the absolute value of the viscosity and its dependence on the temperature can be derived by the theory and is in agreement with its experiments.

The theory of holes in liquids, suggested in a previous paper, is developed by means of classical statistical mechanics, and it is shown that the principal thermodynamic properties of the liquid state can be derived in this way and that they are in numerical agreement with the experiments.

1. In the classical theory there is no difficulty in treating the effect of radiation damping on the scattering of light by a free electron in so far as it is a result of the conservation of energy. In the non-relativistic approximation the equation of motion of a free electron under the influence of a light wave is

with the periodic solution

The total energy radiated per second is then

and the total cross-section(1) is

Formula (1) differs from the Thomson formula by the factor 1/(1 +κ2). This factor becomes appreciable for energies ħν ≥ 137mc2.

The theory of holes, developed in part I of this paper, leads to the existence of a metastable state of a liquid. It is identified with the superheated state, the main properties of which can be accounted for in this way. A theory of the supersaturation of gases in liquids is given on the same line.

The notion of fractional dimensions is one which is now well known. The object of the present paper is the investigation of the dimensional numbers of sets of points which, when expressed as continued fractions, obey some simple restriction as to their partial quotients. The sets considered are naturally of linear measure zero. Those properties of the partial quotients which hold for almost all continued fractions make up the subject called by Khintchine ‘the measure theory of continued fractions’.

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.

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.

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.

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.

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.

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).

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.