The method of applying the scattering correction in accurate γ ray absorption measurements is discussed, and it is shown that the scattering correction only diminishes, as the solid angle subtended by the chamber at the source is reduced, if the absorber diameter is reduced correspondingly. The method of calculation of the scattering correction in cases in which the absorber is placed very close to the chamber is explained, and the result of the calculation for one particular chamber is shown to be in satisfactory agreement with the experimental values. The scattering correction allows the ionization function of chambers to be determined and an (ionization function)/(wave-length) curve is shown for a steel thick-walled high-pressure ionization chamber.

Let x be a normally distributed variable, which may without loss of generality be measured from the mean of the distribution, so that E(x) = 0, E denoting mathematical expectation. Then x satisfies the differential relation

where k2 (otherwise σ2 or (2h2)−1) is the semi-invariant of order 2. Also k1 = 0, and we know that kr = 0 for r > 2.

After a review of the magnetic and spectroscopic properties of compounds of the rare earth and of transitional elements of the first group, the types of bands in spectra of the latter are considered in detail. Reasons are advanced for the view that these are of “molecular” origin in ferric salts, but that the spectra of some transitional compounds contain lines which cannot be regarded in this way. Such lines are shown to be ionic, or quasi-ionic, and accord well with the suggestion of inter-combination transitions such as 4F—2G.

The theoretical intensity conditions, as a function of multiplet intervals, are in agreement with the observed appearance of intercombination transitions in Cr, and possibly also in Mn and Co. Similar considerations apply to the rare-earth spectra. As an example we have shown that the strong absorption in Pr and Nd compounds, and the much weaker lines found in Sa and Eu, are in perfect agreement with the theoretical prediction of intercombinations.

With the aid of experimental results on the absorption of homogeneous β-rays in aluminium, a method is developed for calculating the absorption curve of β-rays forming a continuous spectrum. When applied to certain known spectral distributions reasonable agreement is obtained with the experimental absorption curves of these heterogeneous β-rays. Distribution curves with momentum of the β-rays from actinium C″, uranium X2, thorium C and thorium C″ are found, which lead to similar agreement.

The distribution curves with energy and with momentum for eight β-ray bodies and a table of average energies emitted in thése spectra are shown.

Here the notion of density is made more precise, by definition in terms of the fundamental orthoscheme. (The fact that reciprocal polytopes have the same density is a direct consequence of the precise definition.) A new criterion for regular polyhedra is given, superior to Petrie's in that it applies to degenerate, as well as to finite, polyhedra. The cases are examined in which the actual vertex figures of one polytope bound another; and this idea is developed into the process called alternation, which provides an alternative method for calculating density in four dimensions.

Nicod, using Sheffer's operation “│” as a primitive (undefined) idea, obtained a drastic reduction in the number of primitive propositions (postulates) used by Whitehead and Russell in their theory of deduction for “elementary” propositions. In this reduction Nicod accepts Sheffer's replacement of the primitive ideas “∼” and “v” and the primitive propositions *1·7 and *1·71 of the Principia by a single primitive idea and a single primitive proposition, and, by adding two ingeniously devised postulates of his own, derives from the three postulates the Principia's six primitive propositions *1·1, *1·2, *1·3, *1·4, *1·5, *1·6. But the author does not consider the consistency or the independence of his three postulates; does not discuss the Principia's remaining primitives—the ideas “φx” and “⊢. φx” and the propositions *1·11 and *1·72; and fails to prove that his postulates are derivable from those of the Principia.

It is pointed out that there is some confusion concerning the meaning of the sign of a mutual inductance. The question is examined from first principles and it is shown that since the sign of a mutual inductance depends upon conventions as to sign of current directions, these, as well as the sign of the mutual inductance, must be given in order to specify what is really required—the relation between the directions of winding of the primary and secondary coils. It is also shown that in many cases it is possible to replace mutual inductance by a quantity which is numerically equal to the mutual inductance but which is independent of current directions. The two methods applied to a Wheatstone network yield consistent results.

An approximate calculation of the electromagnetic field of a vertical dipole at the surface of a conducting earth, for small angles of the radius vector with the horizontal, was given by Sommerfield; the case of large angles with the horizontal has been studied by a number of writers. It is proposed here to develop formulae for the vertical dipole by a method which takes into account the singularities of the integrand of a certain integral more accurately than is done by Sommerfield; the analysis is developed especially for small horizontal angles and small numerical distances.

In a former paper the application of Boole's operators π and ρ to linear difference equations whose coefficients are rational functions was discussed. Reference should be made to this paper for the definitions of π and ρ and for Theorems I–V. In the present paper a more general type of equation is considered. Two new theorems are proved and used to obtain the formal solution and to establish the convergence.

The literature about the Lorentz transformation is already so wide that some justification is needed for presenting even a small paper on the subject. It is possible to deduce the transformation from essentially different sets of relevant assumptions. The standard text-books on relativity throw little light on this question, which is probably due to the circumstances in which the Lorentz transformation first came into the field. Probably the question is important enough for the following considerations to be worthy of notice.

By their measurements on the heating effect of the β-rays of Radium E, Ellis and Wooster(1) have proved that the continuous spectra of β-rays represent the actual β-rays of disintegration. Since the β-rays in the continuous spectrum have a wide range of energies, there arise problems which can best be understood by considering a particular series of disintegrations, which we may regard as typical; the most suitable series to consider is the following:

The main problem that arises is to reconcile the facts that the β-rays from Thorium B and Thorium C are emitted with a wide range of energies, while before, in Thorium A, and after, in Thorium C and Thorium C′, we have bodies which emit α-rays which are homogeneous, or, at most, consist of a few homogeneous groups [Rosenblum(2)].

1. The temperature variation of the paramagnetic susceptibility of most of the solids follows the generalised Curie law

as found first by Kamerlingh Onnes and Weiss. This gives a linear relation between 1/ψ and T. The value of C is a measure of the atomic magnetic moment, and if this moment is expressed in terms of the Weiss magneton-number p, then

where CM is the value of C when ψ refers to a gram-molecule. The experimental results for a paramagnetic substance are usually expressed in terms of p and θ. Weiss (1), Foex (2), Cabrera and others have found that in some substances there are discontinuities in the slope of the 1/ψ, T curve. Thus Weiss (3) finds that magnetite above its Curie point shows several sudden changes in the 1/ψ, T curve, which he has interpreted as corresponding to the magneton-numbers p = 4, 5, 6, 8,10. Similarly, for cupric chloride (anhydride) Weiss (4) gives p= 9.2 and 10, for the temperature ranges — 140° to 20° and 20° to 500° respectively. Nickel sulphate (5) has also been found to possess a transition point at about — 113°, where the p value on cooling changes from 14.6 to 18.2.

(1)A beam of positive ions of mercury produced from an arc in mercury vapour and fired upon a surface of nickel produces an emission of electrons.

(2)For a fresh untreated nickel target the electron emission is of the order of 1·5 per cent. for ion energies less than about 600 electron-volts, and rises to about 15–20 per cent. at 2000 electron-volts.

(3)After degassing thoroughly at a red heat the secondary electron emission is found to fall to about half the value observed with a dirty target. Continued bombardment with mercury ions leads to a progressive decrease in the emission, which approaches a final steady state after some hours. In this steady state the ratio is about 2·3 per cent. at 2000 electron-volts. After degassing afresh, the same process is repeated.

(4)Tests were made to assure that the emission is in reality one of electrons from the struck target, and that it is due to the impact of the positive ions.

It is well known that the planes which meet four given lines in a space of four dimensions meet a fifth line, determined by the first four. Also the trisecant planes of a rational quartic curve in [4] which meet a line meet another rational quartic curve. These two theorems are of the same type and may conveniently be called “fifth incidence theorems”.

In the above paper it is pointed out that before we may use the usual wave equation for the propagation of current along a straight wire in free space it is necessary that (a) the inductance L and capacity C per unit length of the wire are independent of the current distribution and (b) that the product LC = 1/c2. This has been shown to be the case.

The inductance and capacity of an element of a conductor are defined generally in the way suggested by Moullin and it is shown that the radiation resistance term, which is a function of the current distribution, is small compared with the corresponding terms involving inductance and capacity. The usual wave equation is derived from Maxwell's equations for the case of a tubular conductor in which the current is a function of the distance z along the conductor only. If the curl of the current is not equal to zero this will not represent the true state of affairs.

It is pointed out in the earlier part of the paper that a complete solution of the problem, taking into account the radiation resistance, and the skin effect, involves the solution of an integral equation which is given.

The field about a tubular current, with a current distribution J0 sin kz along it, is examined. It is found that, for a long wire, the electric force is perpendicular to the surface at every point since we neglect the contribution from the ends. The magnetic force is in circles about the axis as would be expected.

A small component parallel to the current, and in phase with it, is introduced by the charges upon the ends of the tubular conductor, and this component determines the radiation resistance. From this it follows that the radiation resistance could be increased by placing a capacity on the ends—thus increasing the charges there.