Let f(x, y) be a binary cubic form with real coefficients and determinant D ≠ 0. In a recent paper, Mordell has proved that there exist integral values of x, y, not both zero, for which

These inequalities are best possible, since they cannot be satisfied with the sign of strict inequality when f(x, y) is equivalent to

for the case D < 0, or to

for the case D > 0.

Suppose that n runs through all integral values, that (øn) is a system of normal orthogonal functions for the interval (−∞, ∞), and that ψn is the Fourier transform of øn. Then, by Parseval's theorem for Fourier transforms,

and (ψn) is also a normal orthogonal system.

Let G(θ) denote the limit points of the fractional parts of θn (n = 1, 2, 3, …), and T the set of numbers θ for which G(θ) contains an infinity of points. In an earlier note it was proved that if θ is a rational fraction greater than unity then θ belongs to T. The results of this note concern the limit points of the powers of algebraic irrationals.

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 → ∞.

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 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.

The momentum distribution for the electron in the hydrogen molecular ion has been calculated for various wave functions, including the one used by James with which he obtained such a good value for the binding energy. The method adopted for this particular wave function is outlined and the results show appreciable change with improvement in the wave function. In conclusion there are discussed the implications of the present calculations on similar work on the H2 molecule.

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.

Experiments on the bombardment of platinum with 9 M.e.V. deuterons are reported. Radioactive isotopes of platinum of periods 32 min., 18 hr. and 2·8 days, and isotopes of gold of periods 2·6 and 5·6 days are reported. The 2·6 day gold isotope is formed directly by deuteron bombardment as well as by decay of the 32 min. platinum isotope. The absorption of the β-rays in aluminium and of the γ-rays in lead has been measured, and assignments have been made for most of the isotopes reported. Excitation function measurements have been made for the 32 min. and 18 hr. platinum isotopes, and the form of the energy-yield curve is the same for both these elements. The form of the curve seems to favour the Oppenheimer-Phillips process for the formation of these radioelements by the reaction.

The rates of sputtering in hydrogen and deuterium have been compared using ions drawn from a low voltage arc accelerated through a known potential. The rates are equal within a few per cent.

One of us (K.I.R.) is indebted to the Industrial Research Council of Eire for a maintenance grant.

The deposits formed by allowing sputtered material to fall on a glass plate, through a fine grating of wires placed close to the plate, are studied in detail. Evidence obtained showing the presence of material in the shadow of the wires and other less direct evidence gives strong support for the view that the sputtered material is able to move about on the surface. It is also shown that when the grating is very fine a considerable fraction of the incident material is not condensed.