The following note was suggested by an interesting paper written by F. P. White, where many references are given. It refers to a theorem given by W. F. Meyer, by whom the proof is indicated as possible by generalization of an intricate analytical proof given by him for a simple case. His result is that if on the rational curve of order r, in space [r], say the curve cr [r], there be an involution ∞k, of sets of m points, expressed, suppose, by an equation

then the spaces [r − 1], formed from r points of any one of the polyhedra of m points, are an aggregate ∞k of primes of this space [r], which is of class (m − k, m − r), the notation (p, q) meaning the binomial coefficient p! / q!(p − q)!. By Meyer, the conditions k < r ≤ m are supposed to be satisfied. But there is a theorem for r < k ≤ m − 1, relating to selected [r − 1], formed from r points of any one of the polyhedra of m points. The general theorem may be formulated thus: In a space [r], the equation of any prime may be expressed by λu + μv +... + ρw = 0, where u = 0, v = 0, …w = 0 are any r + 1 given independent primes, and λ, μ, …ρ are coefficients which may be described as prime coordinates of the [r].

It is a familiar fact that the arithmetic genus pa and the arithmetic linear genus ω of a general surface are linear functions of its four projective characters; and we find by direct calculation that a similar property holds for the numerical invariants of a general threefold. The question thus arises, whether this result can be established a priori for any algebraic variety Vk of general type, since in that case we should have a simple means of determining its numerical invariants. It has been shown by Severi that, subject to a certain assumption, the arithmetic genus pk of Vk is a function of its projective characters, while it is known that, for k ≤ 4, pk coincides with the arithmetic genus Pa obtained by the second definition (§ 5). In the present paper we obtain, by using Severi's postulate, expressions for the arithmetic genera of a V3 and a V4 in terms of their projective characters. We obtain also the characters of their virtual canonical systems and hence derive formulae for the relative invariants Ωi. For this purpose we replace certain projective characters of Vk by others which are more easily computed and better adapted to a simple notation.

1. While the general theory of surfaces has received a great deal of attention in recent years, there remain a number of difficulties in the application of the results of that theory to the study of particular surfaces. I propose to discuss here certain details which, from their nature, do not seem to be amenable to very general treatment. I shall therefore consider a number of examples with a view to illustrating the way in which certain classes of surfaces may be expected to behave.

Let a1, a2, …, an be a set of n positive integers. Then it is easily seen that the set of positive integers not divisible by any aν has a density, i.e. that if Nn(z) is the number of such integers not exceeding z, then z−1Nn(z) tends to a limit when z → ∞; and that

where

and where [u1, …, uμ] denotes the least positive common multiple of the positive integers u1, …, uμ.

The object of this paper is to evaluate an infinite integral involving Bessel functions and parabolic cylinder functions. The following two lemmas are required:

Lemma 1.

provided that R(m) > 0.

A definition of “closer” and “closest” as applied to estimates of statistical parameters is given, and it is shown that we can sometimes prove that estimates properly derived from sufficient statistics are the closest possible.

The scaling of a gamma distribution, and the location and scaling of an exponential distribution and of a rectangular distribution are discussed in detail, and the closest estimates of the parameters obtained.

The differential analyser has been used to evaluate solutions of the equation

with boundary conditions y = y′ = 0 at x = 0, y′ → 1 as x → ∞, which occurs in Falkner and Skan's approximate treatment of the laminar boundary layer. A numerical iterative method has been used to improve the accuracy of the solutions, and the results show that the accuracy of the machine solutions is about 1 in 1000, or rather better.

It is shown that the conditions are insufficient to specify a unique solution for negative values of β a discussion of this situation is given, and it is shown that for the application to be made of the solution the appropriate condition is that y′ → 1 from below, and as rapidly as possible, as x → ∞. The condition that y′ → 1 from below can be satisfied only for values of β0, greater than a limiting value β0, whose value is approximately − 0·199, and which is related to the point at which the laminar boundary layer breaks away from the boundary.

The ratios of the inter-multiplet separations for the lowest states of O, O+ and O++ obtained by Slater's method depart considerably from the observed values. In this method it is assumed that matrix elements of the Hamiltonian involving two different configurations are negligible, so that each state can be described by a single configuration, whereas these matrix elements are probably appreciable, and a better approximation is obtained by use of a wave function corresponding to a superposition of more than one configuration. The effect of this superposition of configurations has been called “configuration interaction”, and the general theory of it is discussed in Condon and Shortley's Theory of atomic spectra, Chap. xv. It is shown that it occurs only between terms of the same L and S, and of the same parity (∑l even or odd). Few quantitative applications, however, have yet been made. A calculation by Bacher for Mg shows that the effect can be considerable although the states are quite widely separated.

A method of obtaining an approximate solution of Schrödinger's equation with a non-central field of force is given for the case when a solution of a similar equation involving the angular coordinates only is known.

In conclusion I should like to thank Prof. J. E. Lennard-Jones, F.R.S., for suggesting the problem and for his continued interest in it, and also the Department of Scientific and Industrial Research for a grant.

The object of this paper is to compare some of the wave functions which have been suggested for the normal helium atom, including one calculated in the first section of the paper, as regards energy, magnetic susceptibility and electric polarizability. Since this atom provides the simplest two-electron problem, such a comparison is of special interest and may indicate the direction in which improvements might be made in the calculation of wave functions for more complicated atoms.

The magnetization curves of short tin, lead and tantalum cylinders were measured by the force method, and it was shown that although for tin there was no appreciable hysteresis in the transverse position, there was a very marked hysteresis in the longitudinal position, of the same form at different temperatures but depending on the sharpness of the cylinder rims. Lead and tantalum showed additional temperature dependent features which could be attributed to impurities.

The hysteresis due to shape in the longitudinal cylinder is analogous to the hysteresis of alloy ellipsoids, in that the magnetization for a given field always lies within a certain boundary curve and varies “classically” within this boundary. This analogy suggests that the mechanism of the shape hysteresis for a pure superconductor may be similar to the sponge mechanism proposed by Mendelssohn for alloys, but it is also possible that the hysteresis is due to the formation of macroscopic superconducting rings. The considerations which fix the boundary curve are probably similar to those in the superconducting ring, and in this connection some new measurements on a lead ring are reported. The paper concludes with a discussion of the time lag between magnetization and field, and it is suggested that this is not a primary phenomenon.

Konopinski and Uhlenbeck have shown that the empirical β-ray spectra seem to require a modification of Fermi's original expression for the interaction leading to β decay‡. Their expression includes a derivative of the neutrino wave function. The purpose of the present paper is to investigate the most general type of such an interaction term and the possibility of introducing second-order derivatives. We have to deal with the transition§

neutron + neutrino → proton + electron

or proton + anti-neutrino → neutron + positron.

The whole system has the equations of motion

where ψ1 and ψ2 are the wave functions of the initial and final states in configuration space, H1 and H2 are the Hamiltonians of the particles in the initial and final states, and A is a small interaction operator.