In three other papers in this number of the Proceedings will be found direct experimental evidence that in a β-ray disintegration the γ-rays are emitted after the electron has been ejected from the nucleus. In this paper we discuss this result in more detail and especially its bearing on the general picture of the β-ray disintegration.

It is now well known that radium B is an isotope of lead of atomic number 82 with a mass 214, and consequently, if the atoms of radium B are bombarded by an external source of electrons, the spectrum excited in it should be identical with that of lead atomic number 82. A very interesting question arises with regard to the L radiation emitted by a source of radium B during its spontaneous transformation. At the moment of the expulsion of the disintegration electron from radium B, the internal atomic structure of radium B corresponds to an element of number 82, but an instant later, when the electron has escaped from the nucleus, the charge on the latter is 83 and there must follow a reorganisation of the external electrons. Under these conditions, we cannot be certain whether the L spectrum of radium B should correspond to an element of number 82 or 83. Since the excitation of the L spectrum is for the most part due to the action of the rays from the nucleus, the spectrum should correspond to number 82 if the emission of the γ-ray precedes the escape of the disintegration electron and number 83 if it is subsequent to this process.

This paper is a study of a new method of enumeration of the partitions of multipartite numbers.

Incidentally an algebraic function, which is derived from the repetitional exponents of partitions of unipartite numbers, presents itself. The generating function which enumerates the partitions of unipartite numbers is expressible in terms of these functions and finds in such expression its fullest connection with the divisors of numbers. There are also similarly derived functions connected directly with bipartite, tripartite, etc. numbers. It has not been necessary to study these for the purposes of this paper.

The tensor calculus introduced in a recent paper proves a powerful method of investigating the light scattered laterally as exemplified in the blue of the sky. The late Lord Rayleigh worked out the effect on the polarisation of having non-spherical molecules, and the present note is practically a repetition of his work but with a somewhat greater generality. The method has the advantage that it does not become unmanageable for molecules of a more complicated type of asymmetry, whereas a perusal of his work suggests that it would be a very complicated matter to give the lateral scattering from a set of arbitrary orientated “optically active” molecules. As this case has at present no practical interest, I have merely indicated the procedure so that it could easily be applied when required.

The theorem of Wallace that if four arbitrary lines he taken in a plane the circumcircles of the four triangles formed by them in threes meet in a point—in which for circles we may equally well take conies drawn through two arbitrary points I, J of the plane —may be proved very easily from three dimensions. Through the four given lines draw four arbitrary planes, forming a tetrahedron X, Y, Z, T. A unique cubic curve can be drawn through I, J and the four vertices. Projecting this curve from the point T upon the original plane, we get a conic which passes through I, J and the vertices of the triangle formed by the lines in which the plane meets the planes TYZ, TZX, TXY, that is, by three.of the given lines. This conic, which is one of the conics of the theorem, also passes through the third point K, other than I, J, in which the original plane meets the cubic curve. Projecting in turn from the other three vertices of the tetrahedron, we get the other three conics of the theorem, which also pass through K.

The problem of the temperature distribution in transformer or other laminated cores in which heat is generated at a uniform rate, does not appear to have been treated by exact mathematical methods. A good deal of work of a semi-empirical nature has been done by Vidmar, Kuchler and others, the object of which has been to enable designers of transformers to obtain an estimate of the maximum temperature likely to prevail in the core.

In this note we obtain a system of characteristics on which depend the main enumerative properties of complexes, or systems ∞3, of conics in space of four dimensions. The method used is applicable to a large number of problems of this nature, and we select this illustration as being analogous to congruences in ordinary space. In particular a finite number of conics pass through an arbitrary point, thereby defining the order, n, of the system. Linear complexes are those for which this order is unity. The conics of a complex satisfy eight simple conditions, in general conditions of contact with a fixed form or forms, but they may include conditions of incidence with a surface, each such counting once, with a curve, each such counting twice, or with fixed points each such counting thrice. In particular only incidence conditions can occur when the system is linear, for otherwise more than one conic would pass through certain ∞3 points of space. Points through which more than a finite number of conics pass are termed singular, as well as their loci. Directrix constructs are necessarily singular, but they do not necessarily exhaust the singular system, for the complex may possess ∞2 curves lying on a surface. In the linear case through a point of a singular curve pass ∞2 conics, and through a point of a singular surface ∞1 conics. The possibilities in the nonlinear cases are too numerous to be detailed. Similarly the system of planes of the conies may possess a singular curve, through which ∞2 of the planes pass.

Some previous work on the theory of the electrical counter is extended by a detailed consideration of the ionic process involved. It is shown that with a positive point the total number of ions produced should be proportional to the initial ionisation. Evidence is adduced to show that with a negative point extinction of the discharge is probably brought about by a local increase of pressure, accompanied by an electrical wind along the sides of the point. The latter would bring about the discharge of a definite quantity of electricity, independent of the initial ionisation, provided the initial pressure was not too low. It is shown that these considerations account for a number of other phenomena connected with discharges from small electrodes, and that the quantities involved are of the right order of magnitude.

The experimental work in connection with this paper has been carried out in the Cavendish Laboratory. The author desires to thank Professor Sir E. Rutherford for his continual interest in its progress, and further to thank Professor Appleton, and Dr Chadwick for much helpful discussion of several of the points involved.

§ 1. The six Plücker coordinates of a straight line in three dimensional space satisfy an identical quadratic relation

which immediately shows that a one-one correspondence may be set up between lines in three dimensional space, λ, and points on a quadric manifold of four dimensions in five dimensional space, S5. For these six numbers pij may be considered to be six homogeneous coordinates of such a point.

The stability of a charged raindrop has been discussed mathematically by Lord Rayleigh. The case of an uncharged drop in a uniform electric field is perhaps of more meteorological importance but a mathematical discussion of the conditions for stability turns out to be very much more difficult in this case, owing to the fact that the drop ceases to be spherical before it bursts. Moreover it does not seem possible to express its geometrical shape by means of any simple mathematical expressions. On the other hand, by using a soap bubble instead of a water drop it was found possible to carry out experiments under well-defined conditions in this case, whereas experiments with Rayleigh's charged drop would be difficult.

The following is a simple example of what Poincaré calls probabilité du continu.

Two points are marked at random on a unit line and it is assumed that all positions of either point are equally probable. What is the average or probable positive distance between the two points?

(1) An apparatus has been devised in which the composition of the vapour phase passing over the catalyst can be varied at will, or two vapours can be used alternately as reactants.

(2) By using ethyl alcohol, and amyl alcohol from fusel oil alternately as reactants, the poisoning of a very active copper gauze catalyst has been followed and the temperature coefficient of the dehydrogenation of ethyl alcohol by the copper has been measured during the poisoning.

(3) The activity of the catalyst, poisoned by amyl alcohol, fell in accordance with the simple logarithmic law

(4) Confirmation was obtained that the rates of dehydrogenation of ethyl and amyl alcohols were equal.

(5) The temperature coefficient of dehydrogenation on the catalyst was unchanged throughout the poisoning, though the reaction velocity was reduced 30 times.

(6) The temperature coefficient was also shown to be unchanged during sintering, the reaction-velocity having been reduced 10 times.

(7) Equations are derived for the rate of poisoning of an exponential distribution of centres of activity when the poison attacks the more active centres selectively.

(8) The relation deduced

cannot usually be distinguished (except with special relations between the constants) from the simple logarithmic law of poisoning.

(9) Thus the exponential distribution normally poisons like a surface composed of homogeneous centres, the temperature coefficient being unchanged by the poisoning.

(10) The equation

describes the reaction-velocity as a function of time, and temperature, and gives adequate explanation of previous experimental results on the forms of the heating and cooling curves with “poisonous” alcohols.

(11) Frenkel has shown , where τ0 is the vibration frequency of the molecule perpendicular to the surface, and μ0 is the heat of desorption of one gram molecule. Thus if the heat of desorption is not small compared with the energy necessary to cause chemical disruption of the molecule, the previous equation becomes

the form of the equation being unaltered.

The velocities of electrons emerging through an aperture in an anode to which they were accelerated have been investigated.

The velocity distribution of electrons passing through holes ranging from 0·13 mm. to 3·24 mm. in diameter in a thin copper sheet is shown in Fig. 2. In this case the accelerating field between filament and anode was 204 volts.

Corresponding results for a capillary tube 0·40 mm. in diameter and 16 mm. long are shown in Fig. 3, where the curves are plotted for different accelerating potentials. Experiments were also made with shorter lengths, down to 5 mm., and with a tube 1·25 mm. in diameter. All the tubes investigated gave the same distribution of velocities of the electrons in the emergent beam, within the limits of error of the experiments.

In conclusion the authors wish to express their thanks to Sir Ernest Rutherford for his interest and advice during the course of the work.

In the Transactions of the American Math. Society, vol. I, 1900, and vol. IV, 1904, Mr E. Kasner treats exhaustively the (2, 2) double binary form and discusses the theory connected with double binary forms and multiple binary forms in general terms. He shows the relations between the systems of multiple binary forms with digredient variables and the forms with cogredient variables. Hitherto nothing seems to have been written on systems of triple binary forms (with regard to higher forms see a paper on the (1, 1, 1, 1) form by C. Segre); so here I propose to discuss the complete system of a (1, 1, 1) binary form which consists of six forms connected by one syzygy. When two of the variables are the same we naturally get the (2, 1) form and when the three are the same We get the (3) or cubic binary form.

The purpose of this note is to calculate, on the basis of the assumptions which underlie the theory of the stopping of α-particles by matter, the influence of space quantization on this phenomenon. The results arrived at may be capable of an experimental test and will serve to illuminate these assumptions and to indicate their limitations.

The Lorentz group is denned by the equations

where the coefficients are subject to the conditions

It is an intransitive group possessing the single invariant