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’.

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.

1. Introduction and summary. A chain of N links is allowed to assume a random configuration in space. The extent of the chain in any direction is defined as the shortest distance between a pair of planes perpendicular to that direction, such that the chain is contained entirely between them. In the present paper the probability distribution of the extent is discussed, starting with a chain in one dimension for which formulae are derived for the probability and mean extent for all values of N. The limiting forms for large N are then considered. The results are extended to the case of a chain in three dimensions, and it is shown that the extents in two directions at right angles tend to be independently distributed when N is large. It is assumed that the links are infinitely thin, so that a point in space may be occupied by the chain any number of times.

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.

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.

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.

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.

A general method is given of calculating the effect of damping on the collision cross-sections for problems involving free electrons and mesons. The result is expressed in the form of an integral equation which can only be solved if certain simplifying assumptions are made. It is shown that radiation damping has a negligible effect on the scattering of light by free electrons, so that the Klein-Nishina formula is unchanged by the inclusion of damping effects. The effect of damping for mesons, on the other hand, is extremely large and reduces the cross-sections considerably. The main problems considered are the nuclear scattering of mesons and the energy radiated by mesons during collisions. The revised cross-sections are much more reasonable than those calculated previously, but on account of the inadequacy of the data no detailed comparison with the experimental results is possible.

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.