The method of the previous paper is extended to determine the surface states of a simple cubic crystal on the approximation of tight binding. It is also applied to the case of a semi-infinite linear chain when the atomic s- and p-states are to be regarded as degenerate, the existence of surface states being again predicted.

Multivariate generalizations. In multivariate statistical analysis, common terms such as variances and correlation coefficients have received certain generalizations. Wilks (7) has called the determinant |V|, where V is the matrix of variances and covariances between several variates, a generalized variance; certain ratios of such determinants have been called by Hotelling(5) vector correlation coefficients and vector alienation coefficients. While these determinantal functions have properties which justify to some extent this kind of generalization, it sometimes seems more reasonable to leave any generalized parameters, or corresponding sample statistics, in the form of matrices of elementary quantities. This is stressed by the formal analogy which then often exists between the generalized and the elementary formulae.

The so-called Lindelöf principle is nothing more than a transformation and systematic application of the simple Schwarz lemma. Nevertheless, it is a very powerful tool for the solution of many questions in the theory of functions. It is based on the conception of “subordinate” functions.

In the first part of the paper a slow two-dimensional motion of viscous fluid is considered which approximates to a motion of uniform shear past an infinite fixed plane, and differs from this motion because there is a gap in the plane (Fig. 1). A simple expression in finite terms is found for the stream function.

The molecule of a dissociating gas is treated as a dynamical system with n normal modes of vibration with incommensurable frequencies. The assumption that the molecule dissociates when one coordinate attains a critical high value leads to a unimolecular velocity constant expressed as an integral; analytical expressions are derived for bounds to its value. Calculations in a simple case indicate that the velocity constant is approximately of the form

This is in general agreement with experiment if the constant W is identified with the corresponding empirical energy constant.

The quantum theory of the heavy electron field is applied to calculate approximately the cross-sections for “photoelectric” absorption of heavy electrons by a bound nuclear particle and for emission of a heavy electron by a free nuclear particle on collision with a nucleus. The absorption cross-section per nuclear particle is of the order 10−26 cm.2 for heavy electrons with energies up to 108 e.V., but probably decreases rapidly at higher energies. Although the energy loss of a proton, of energy 108 e.V. or higher, due to heavy electron emission is found to be greater than that due to radiation, the cross-section is very small (⋍ 10−29 cm.2) and the phenomenon is unlikely to be observed in a cloud chamber. In the present state of theory it is impossible to decide whether radiative emission of heavy electrons by nuclear particles is capable of contributing appreciably to the heavy electrons observed at sea-level but, if so, the cross-section for the process must be very much greater at high energies than in the non-relativistic energy region which we have investigated.

The possibility, pointed out in a previous paper, of superlattice formation in an adsorbed layer when the adsorbed atoms tend to repulse each other is developed in detail. Both Bragg and Williams's approximation and Bethe's approximation are used, but restricted to superlattices of the type AB. In a range of θ, the fraction of surface covered, a superlattice is found to be possible. Bragg and Williams's approximation shows further that the state with the lowest free energy is the one with a superlattice when the latter is possible. Rough kinetic expressions are also given. The equations derived from the principle necessary to preserve equilibrium are found to reduce to those of detailed balancing, and they also agree with the formula obtained statistically. As expected, all the corresponding results obtained from the two methods become the same when the number of nearest neighbours of a site approaches infinity, provided that the product of this number and the interaction potential of two adsorbed atoms occupying two neighbouring sites remains finite.

The only significant result is the large value of the slope of isotherms (the rate of change of the pressure with respect to the fraction of adsorption) when there is a nearly complete superlattice.

Experiments are described which show the effect of current on the transition of cylindrical superconductors to the normal state produced by increasing an external magnetic field. The transition curves were determined with an accuracy of 0·1%. Both longitudinal and transverse fields were used.

The following results were found.

(a) Transverse field

(i) The field strength at which the first traces of resistance appeared was independent of current strength for reasonably small currents (up to about 100 mA. for 0·010 cm. wire).

(ii) With increase of current the transition curve at first changed shape, becoming more and more nearly linear, and then was displaced parallel to itself toward smaller values of the applied field by an amount in accordance with Silsbee's hypothesis.

(iii) The change in shape of the transition curve was accurately reproducible and was identical for specimens of the same material and diameter.

(iv) The results for specimens of different diameter could be accurately correlated if the measured values of the resistance were plotted against the current divided by the specimen diameter raised to the power (l + R/Rn), i.e. I/(diam.)(1+R/Rn), where R/Rn is the ratio of the measured resistance to the resistance just above the transition point.

(b) Longitudinal field

(i) With increase of current, however slight, the transition curve was displaced parallel to itself, toward smaller values of the applied field, without showing any change in shape.

(ii) The displacement of critical field for a given change of current was completely independent of the diameter of the specimen.

(iii) The observed shift to smaller fields was much greater than could be accounted for by the additional field produced by the current at the surface of the wire.

(iv) If we denote the (extrapolated) value of the critical field for zero current by (Hc) and the difference between this and the observed value for a current (I) by (ΔHc) then I and Hc are connected by the empirical relation

where A and n are constants depending on the nature of the superconducting metal.

when Hc is expressed in gauss and I in milliamperes.

Approximate expressions are obtained, when the distance R between the nuclei is very large, for that portion of the wave function in the two-centre problem which depends on the hyperbolic coordinate μ. From these expressions the number and the approximate position of the nodes in μ can be deduced and hence the rules can be found by which that state of the combined atom at R = 0 can be determined which corresponds to a given state of the atom when the nuclei are completely separated. These rules are also applicable to the cases where the two atoms which can be formed when the nuclei are completely separated have the same energy. The converse problem of finding what state of the completely separated atom corresponds to a given state of the combined atom at R = 0 can also be solved by the use of the rules.

The fine structure of the Al L23 and Zn M23-absorption edges has been determined using the experimental methods described in an earlier paper(1).

A table containing numerical data for the Al and Zn edges, together with revised data for the edges of Li, Mg, Ni and Cu is included.

It has been found by one of the authors (1) in collaboration with Dr H. Jones that a flow of heat in liquid He ii is accompanied by what seems to be a transfer of momentum. The effect can be seen when the channel through which the heat and liquid flow consists of a smooth-walled glass capillary, such as shown in Fig. 1a. Due to the high thermal conductivity of He ii, a considerable part of the heat put into the reservoir is carried down through the capillary to the bath. When a steady heat flow exists, a flow of liquid takes place in the opposite direction, and the level of the liquid in the reservoir is seen to be higher than that in the bath. Smooth capillaries, however, produce a rise in level of only 1 or 2 cm. at most, since the viscosity of the liquid is small and hydrostatic pressure pulls the accumulated liquid in the reservoir back through the capillary. When the heat flow is large, violent surging is observed in the reservoir, but there is no further rise in level.

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.

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.