1. Die unendliche Folge der Partialsummen {sn} der Fourierschen Reihe einer integrierbaren und nach 2π periodischen Funktion f(ξ) der reellen Variabeln ξ ist an einer Stelle ξ = x, wo die Funktion stetig ist, (C, 1) limitierbarzu f(x) = s als Grenzwert, d.h. es ist

Suppose that k(x) is of bounded variation in (− ∞, ∞) and that s(x) is bounded and continuous in the same range. Let

The object of this paper is to complete the proof of the

Theorem. Let P(√d) be the quadratic field of discriminant d > 0. Then Euclid's algorithm does not hold in P(√d) if d is sufficiently large.

It is well known that solutions of partial linear differential equations of the second order and of elliptic type are uniquely determined by their boundary data, and that they assume their maximum and minimum values on the boundary. The usual proofs make use of the principle of superposition and are therefore not applicable to non-linear problems. But recently Pryce has proved the uniqueness theorem for the non-linear equations of minimal surfaces and of Born's electrostatics. These equations are the Euler equations of the variational problem

k = + 1 corresponds to the case of minimal surfaces in n + 1 dimensions; k = − b−2, n = 3 corresponds to Born's electrostatics. Pryce's procedure depends essentially on the notion of conjugate variables in the calculus of variations for multiple integrals and can therefore be extended to a wide class of differential equations arising from variational problems (for several functions of several variables) as we show in § 3.

1. Taylor has shown (1) that two characteristic lengthsλ and λη may be defined for turbulent fluid motion. The length λ, which is connected with the dissipation of energy, is, for isotropic turbulence, given by

where is the mean rate of dissipation of energy per unit volume and represents the mean square value of any component of velocity. The length λη can be defined in terms of thus

where For isotropic turbulence Taylor assumed that

where B is a constant. Since the turbulence is isotropic,

and so, from (1), (2), (3) and (4) we have

The probability of an electron having an energy greater than a certain fraction of its initial energy after traversing a certain thickness of matter is calculated for different values of the initial energy. Whereas in previous calculations the energy losses by radiation and by ionization have been treated separately, and the results added, the two types of losses are here combined and their effect on each other included. The results may be compared with those of experiment for values of the initial energy up to a few times the initial energy up to a few times the critical energy of the substance traversed. The theoretical values obtained are found to give better agreement with the results of experiment than do those obtained by the less accurate method.

Solutions of Fock's equation for the self-consistent field with exchange have been carried out for the negative ions Na−, K−. The results indicate that these negative ions can certainly exist; and calculations of the energy values, and estimates of the errors of these values, indicate that these ions are probably stable and not liable to autoionization.

The wave functions for these negative ions are required in calculating atomic cross-sections for the capture of electrons by the neutral atoms.

By solving the fundamental integral equation an expression is obtained for the electrical resistance which takes into account, to the first approximation, the mutual influence of the impurities and the lattice vibrations. It is found that deviations from Matthiessen's rule are to be expected and that these deviations are surprisingly large. The formulae derived only indicate the trend of the resistance curve, but this trend is not confirmed by experiment. Whereas the theory indicates that the mutual influence of the impurities and the lattice vibrations should decrease the electrical resistance, the experimental results of Grüneisen on copper show that, when deviations from Matthiesen's rule occur, the resistance is increased.

According to quantum mechanics there exists in any non-metallic crystal a band of allowed electronic energy levels which are unoccupied when the crystal is in its state of lowest energy. We call this band the conduction band; the crystal can conduct electricity if electrons are raised into the conduction level from lower levels. According to the theory of semi-conductors given by Wilson, there exist in these substances lattice imperfections at which an electron can exist in a bound stationary state below the conduction band, electrons being raised from these levels into the conduction band by the thermal agitation of the surrounding atoms.

In a recent letter to Nature F. London(3) has called attention to a discontinuity in the derivative dCV/dT of the specific heat with respect to temperature which arises in a perfect Einstein-Bose gas at low temperatures. When discussing the properties of a perfect gas satisfying the Bose statistics, Einstein(1) remarked that at low temperatures something resembling a condensed phase should appear. The temperature at which this condensation should begin is given by the equation

where n is the number of particles of mass m per unit volume and the other symbols have their usual significance.

In order to be able to identify the type to which an adsorbed film belongs from its experimentally determined behaviour, it is necessary to know theoretically how various simple and general types, which are likely to occur frequently, behave. Films formed on a rectangular solid lattice with another atom at the centre of each rectangle are considered. Each adsorbed molecule is assumed sufficiently large to exclude the possibility of adsorption on the four neighbouring sites and, owing to the rectangular shape of the lattice, reacts appreciably with two and not with four neighbouring adsorbed molecules. The statistics and kinetics of the film are considered. It is shown that the variation of heat of adsorption with fraction of sites occupied is quite different from that of other types of film which have so far been treated. Immobile films of the same type are briefly discussed.

The adsorption of gaseous molecules in a monolayer on solid surfaces has been studied theoretically by several workers, namely Fowler, Peierls and Wang. They have shown that the adsorption isotherm exhibits critical phenomena if there is assumed to be an attractive force between the neighbouring adsorbed molecules, and this has provided an explanation of the critical condensation phenomena observed in deposition experiments, for example those of Cockcroft on the deposition of cadmium on copper. The actual critical phenomena observed consist in the deposition of many layers which are formed if the first layer has fairly started. It is of interest, therefore, to consider if there are other ways in which critical phenomena can arise than through interactions in the first layer. The general problem of adsorption of an arbitrary number of layers is too complicated to study because of the mathematical difficulty in constructing the partition function. We therefore confine ourselves to the consideration of adsorption in two layers only.

A detailed account is given of an absorption method, using the β-radiation of radium E as a standard of comparison, by which the maximum energy and some indication of the form of the energy spectrum of the β-particles from any other radioelement may be obtained. The method is applicable alike to weak sources and to sources of finite thickness and is designed to yield the largest amount of information under such limitations. Experimental results with sources of UX2, Na24, Ag106 and P32 are treated by the method described. A modified empirical range-energy relation is given.

It is shown that a long-lived radioactive cobalt isotope, which had been previously obtained by Livingood and Seaborg by bombardment of cobalt with slow neutrons, can be produced by irradiation of nickel with fast neutrons.

The absorption curve for cosmic-ray shower particles has been observed in lead and iron with a counter arrangement. Since the observed curves are in good agreement with the absorption curve of cosmic ray electrons calculated by Heitler, it has been concluded that nearly all the showers are of the cascade type.

Observations of Schmeisser and Bothe which have previously been interpreted as giving experimental evidence for the occurrence of penetrating particles in showers are not in contradiction with the observations given here, so that they also must be interpreted in terms of energetic electron cascades.

The rate of sputtering of magnesium in oxygen was observed with an apparatus which permitted the field immediately in front of the target to be reversed. Change in the sign of the field did not affect the rate of sputtering which remained abnormally low. It is therefore concluded that, while the low rate is in some way due to the presence of an oxide film, it is not due to charged particles being returned to the cathode by the electric field, as had been suggested by other workers.

The magnetic anisotropy of crystals of calcite and sodium nitrate does not alter between room temperature and liquid nitrogen temperature. By using such diamagnetic crystals, it is easy to compare strong magnetic fields to 1 part in 1000. The effect is large enough for the measurement of fields of the order of 100 gauss, but in small fields the accuracy is limited by ferromagnetic contamination.