The paper deals with new solutions of the differential equations of the two-centre problem which are expressed in terms of the confluent hypergeometric function. By means of this function a solution of the λ-equation (1) is obtained which enables the separation constant A to be found for small values of the parameter p in a rapidly convergent series, which for the special case of m = 0 is still more rapidly convergent. The solution for the μ-equation (2) in the general case is of the same character as regards rapidity of convergence as solutions previously obtained, but when m = 0 it again possesses a higher rate of convergence as compared with solutions given by other investigators.

Theoretical calculations on the mean radial momentum distribution of the electrons are made for the molecules CH4, C2H6, C2H4 and C2H2; from these distributions the profiles of the Compton line are deduced. It is assumed that each electron acts as a single scattering centre so that the mean radial distribution for the molecule is just the sum of the various “partial distributions” for each electron separately. The electrons are supposed to be paired together in the formation of localized bonds, and the contributions from each type of bond have to be separately determined. When superposed, these give the momentum distribution for the whole molecule. Such distributions are similar in shape, but the peak value of the momentum curves moves to higher values of p as the C—C bond becomes more saturated.

A comparison of the various partial distributions for C2H4 shows that the C—H bond is an important factor in the momentum distribution of such molecules. In other hydrocarbon molecules, we may presume that the proportion of C—H bonds will be very significant in determining the breadth of the momentum distribution curve, and hence of the width of the Compton line; in fact the half-width of the Compton line decreases both with an increase in the number of the C—H bonds, and with increased saturation of the C—C bonds.

In the only case where experimental results are available, CH4, the discrepancy between theory and experiment amounts to about 30%. Reasons for this discrepancy are discussed; in part the discrepancy may be attributed to the approximations used in the wave functions; but reasons are given for supposing that in part also it arises from an incorrect interpretation of the experimental results.

Let M be a bounded and closed set of points in the complex z plane; d(M), a set-function which is of great importance in potential and function theory, may then be defined as follows. n points z1, z2, …, zn in M are so chosen that the product of the mutual distances

has the greatest possible value Then it can be proved that

exists. Thus the set-function d(M), named by Fekete the transfinite diameter of M, is defined.

The problem considered in this paper is that of finding the least possible h = h(x) such that a given arithmetic function a(n) should keep its average order in the interval x, x + h, i.e. that we have

and

as x → ∞.

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.

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.

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.