For material stratified in parallel planes in local thermodynamical equilibrium we have Milne's well-known result that

where πF is the constant net integrated flux of radiation, τ is the optical depth and B is the “ergiebigkeit” which is related to the temperature T by the relation

σ being the Stefan-Boltzmann constant. If we define the effective temperature by the relation

we have from (1)

where T1 is the temperature ar τ = 1. In this note we establish a similar result for extended photospheres where the curvature of the outer layers is properly taken into account.

This problem, in the particular case of the hydrogen molecular ion, is so well known that a detailed reference to the large number of papers dealing with it is unnecessary. The investigations to which chief reference is made will be found at the end of this paper.

The molecular-orbital method has been applied to a study of in its ground state and excited levels, and the relative importance of the perturbation and variational methods has been considered in some detail, as well as the effect of certain integrals which, in discussions of molecular structure, have often been neglected. It appears that the ion should exist in stable equilateral form with a nuclear distance about 0·85 Å., and that all excited levels are unstable.

Reasons are given for supposing that the molecule H3 is linear and not triangular.

It is well known that an electron possesses a magnetic moment. Frenkel has further deduced by relativistic arguments that the electron must also possess a real electric moment. Dirac has deduced, from his linear wave equation, a relation which indicates formally the existence of an imaginary electric moment as well as the real magnetic moment. It is shown that Frenkel's argument is not sound, since we now know that a relativistic state of affairs may not admit of tensorial representation, and that Dirac's reasoning is not sufficiently precise. When it is replaced by precise reasoning the magnetic moment remains as before, but the terms representing the electric moment disappear from the result. The electric moment therefore does not exist.

It has been shown by Chadwick and Lee(1), using a high-pressure ionization chamber, that if one neutrino is emitted by each disintegrating Ra E nucleus, then the neutrinos do not produce more than one pair of ions in 150 km. of air at N.T.P. Calculations based on the wave mechanics show that the ionization due to a neutrino having a magnetic moment of one Bohr magneton would be very easily detectable(2), whereas it has peen estimated that if the neutrino has no magnetic moment at all its encounters with nuclei will be as scarce as one in 1016 km. of water(3). I have investigated the matter again, using two Geiger-Müller counters, instead of an ionization chamber. The counters have the advantage of giving a discharge for a single pair of ions(4). The two counters of 15×5cm. were connected in parallel, and filled with air at 76mm. pressure. They were shielded on all sides by 45 mm. of lead. A source of 7 mg. of Ra (D, E, F) was placed at a distance of 11 cm. from the centres of the two counters, so that the total solid angle subtended by the counters at the source was 0·10, expressed as a fraction of the complete sphere. Assuming that both the disintegrating atoms of Ra D and Ra E give one neutrino, then the number of neutrinos crossing the counters per minute is 3·1 × 109. The average effective length of the path in the counter, reduced to N.T.P., is 0·3 cm., so in one min. the total range in air of all the neutrinos crossing the counters is 9·3 × 108 cm.

The commonest form of Langmuir's adsorption isotherm is

where θ is the fraction of the surface of the solid covered by adsorbed molecules, p the gas pressure in equilibrium with the adsorbed layer and A = A (T) a function of the temperature alone. This formula is usually derived by a kinetic argument which balances the rates of deposition and re-evaporation. It is perhaps not without interest to show that formula (1) and similar formulae can be obtained directly by the usual statistical methods which evaluate all the properties of the equilibrium state of any assembly. The ordinary derivation is apt to obscure the essentially thermodynamic character of (1) and to lead one to think that its form depends on the precise mechanisms of deposition and re-evaporation, whereas in fact it depends only on the whole set of states, adsorbed and free, accessible to the molecules in question. By suitable use of the usual technique for handling assemblies obeying the Fermi-Dirac statistics the saturation effect can be naturally incorporated in the theory ab initio.

In a previous paper the present writer gave a solution of the problem of determining the circulation around a thin elliptic cylinder in a steady stream of slightly viscous fluid when the major axis of the cylinder is inclined at a small angle to the direction of the flow at infinity. The present note gives a largely qualitative analysis of the manner in which this circulation is built up when the motion of the fluid or cylinder is started instantaneously from rest with a given velocity.

In a recent paper I developed a method for calculating the probability of the creation of an electron pair in the collision of two charged particles moving with a relative velocity very near that of light. I showed there that under certain conditions it is legitimate to treat one of the colliding particles, say the heavier one of charge Z2 and rest mass M2, which we shall call the particle 2, as fixed at the origin of coordinates, and the other, of charge Z1 and rest mass M1, which we shall call the particle 1, as moving classically along a straight line with uniform velocity V in the direction of the z-axis, passing the other particle 2 at a minimum distance of approach (impact parameter) b. I developed expressions (given by (18) to (22) of A) giving the probability of the transition of an electron from an initial state of negative energy E0 and momentum p0 lying in an element of momentum space dp0 to a final state of energy E and momentum p lying in an element of momentum space dp, under the combined perturbing influence of the two colliding particles when they pass at a minimum distance b. The initial state of the electron, left vacant after the transition, appears as a positron of momentum p+ = − p0. To get the differential effective cross-section for the creation of the above pair, we must integrate this probability over all values of the impact parameter b, and this integration can be performed easily as shown in A. The final result can be written as a sum of a finite number of doubly infinite integrals (A, (24)). The purpose of this paper is to carry through the evaluation of these integrals for certain special cases, and to consider the effects of screening. The results have already been communicated in A.

Buri's method of solution of the equations of turbulent motion in a boundary layer has been used extensively in a previous paper. Since the results of that investigation have not been compared with experiment, nor has any other application of Buri's method been checked experimentally, it seems advisable to apply this method to a problem for which the experimental results are known.

1. The purpose of this paper is to examine in detail the types of radiation emitted by a quantum mechanical system (representing a nucleus) when the radiating particle changes its azimuthal quantum number l by two, one, or nought. We assume that the radiating particle is spinless* and that it moves in a field possessing central symmetry, so that the particle may be described by a wave function ψn,l,m which factorizes into radial and angular parts,

We assume secondly that the nucleus has a finite radius aN of the order of 10−12 cm., that is to say that π(r) = 0 for r > aN.

The number of ions produced by a neutral particle of small mass with a magnetic moment equal to n Bohr magnetons has been calculated (I–III) and found to be 103n2 per km. path in air at N.T.P., being practically independent of the mass and energy of the particle (IV). A comparatively large fraction of the secondary electrons produced have high energy and may therefore be counted if they are produced in the wall of the Geiger counter instead of in the counter itself. This fact should facilitate the experimental detection of the ionization considerably (V).

A direct method of obtaining the limitations of crystal symmetry on physical phenomena is described and discussed in greater detail for the case of diamagnetic magnetostriction. Formulae are obtained for the variation of the longitudinal and transverse effects with crystal orientation in the case of bismuth.

Recently Kohler, on the basis of some experiments by Verleger, has questioned the validity of the usual assumption that the linear Hall effect is entirely perpendicular to the current. The available experimental data are however contradictory, so a simple magneto-resistance experiment with a bismuth crystal was made, which suggested that the usual assumption was, after all, valid. The disymmetry of the Hall effect is discussed, and some of Kohler's results are generalized.

We have examined the two-dimensional gliding of a semi-infinite plate on the surface of a stream of finite depth. We have calculated the lift L in terms of the ratio D/H, where H is the depth of the stream at infinity up-stream, and D is the height of the trailing edge of the plate above the surface of the stream.

In particular we have found that the trailing edge cannot be at a height of more than 0·07H approximately, above the up-stream fluid surface.

In conclusion, the writer would like to thank Prof. G. I. Taylor, F.R.S., for suggesting this problem and for his continued interest in it.

Bell and Wolfenden have published a theory of diplogen concentration by electrolysis, which, to a considerable extent, explained the empirical result, that the efficiency of diplogen separation is constant. This theory depends on the assumption that a quantity, γ, introduced by Gurney is constant. In a later paper Bell writes that this assumption would “probably not be justified in a strict examination of the problem”, but he gives no quantitative estimate.

The temperature dependence of the photo-electric effect with the image field is discussed, the result obtained agreeing with Fowler's T2 law near the threshold. Fowler's law is discussed, and shown to rest upon a mistake in the normalization of the free wave functions.

The two methods of analysing experimental results due to Fowler and Du Bridge are also discussed, and it is shown that in general the latter should give the more reliable results.

The author wishes to repeat his thanks to Prof. R. H. Fowler and to Mr A. H. Wilson for their constant interest in the work.

The probability of excitation of helium by electron impact to the more important doubly excited levels is calculated, and comparison made with the experimental results of Priestley and Whiddington who have observed two lines due to double excitation. The calculated relative intensities and angular distributions are found to be in qualitative agreement with experiment, and reasons are given for the lack of quantitative agreement.

Experiments on the β-ray spectrum of thorium C + C″ by the Wilson cloud chamber method were made. From a certain number of photographs taken, the upper limit of the continuous spectrum was found at 9250 Hρ and no β-particles of higher energy were observed. This value for the upper limit agrees fairly well with the result obtained by Henderson's recent experiments by the method of Geiger-Müller counter and focusing magnet.

The idea of “free” electrons is purely classical, but it is convenient to introduce it also in the quantum theory of metals. In order to calculate the number of free electrons N F from a quantum mechanical model it is necessary to give an appropriate definition.

In previous communications* the author has discussed the recombination of gaseous ions, subsequently extending the theory to include large ions† and also to apply to the coagulation of colloids and of smokes‡. The predictions of the original theory were in excellent agreement with experiment, but discrepancies appeared when it was generalized. The author is now indebted to Dr F. J. W. Whipple for the discovery of a numerical error in the first paper, the correction of which leads on the whole to a much more satisfactory agreement with experiment.