The equations of propagation of electromagnetic waves, simple harmonic in time, in an optically anisotropic stratified medium are obtained from the treatment of the refracted wave as the resultant of the incident wave and wavelets scattered by the elements of volume of the medium, and are reduced to a simple form.

The primitive property of the medium, from which the other optical properties are derived, is the scattering tensor, relating the induced dipole moment per unit volume to the applied electric field.

The relation between the dielectric tensor (corresponding to the dielectric constant of an isotropic medium) and the scattering tensor is obtained.

A medium consisting of classical oscillators in an external magnetic field is then considered, the scattering tensor and dielectric tensor are evaluated for such a medium, and finally a formula for the refractive index is obtained.

For an ionised medium the formula differs from that obtained by Goldstein; the difference is due to the inclusion in the present treatment of a term omitted by Goldstein; the significance of this term is discussed, and its inclusion justified.

Taking this term into account makes an important difference to the properties of the medium for long waves; an example is given.

1. In a recent paper Landau has shown that, when electrons are moving freely in a magnetic field, they exhibit, in addition to the paramagnetic effect of their spin, a diamagnetic effect due to their motion. This result is rather unexpected, since it is quite contrary to the classical case. There it might appear as though the circles described by the electrons must produce a magnetic moment, but the error was long ago pointed out by Bohr. The motion of the electrons must be confined to some region by means of a boundary wall, and the electrons near the wall describe a succession of circular arcs, repeatedly bouncing on the wall, and slowly creeping round it in the direction opposite to that of the uninterrupted circles; when the moment of these electrons is taken into account, it exactly cancels out that due to the free circles. In Landau's work it is of course necessary to consider the boundary, but he shows how allowance is to be made for it by an appropriate process. The complete justification is rather subtle, and so it may be worth considering a special case, admitting of exact solution, which takes the boundary into account, and so makes it possible to follow more closely the analogy between the classical and quantum problems. With regard to the general case with a boundary wall of any type, we shall only observe that the different results arise, because in the wave problem ψ must vanish at the bounding “potential wall,” and so will be small near it; this upsets the balance of the electric current near the wall, and yields the magnetic moment.

Configurations of points in higher space have been known and studied for some time. In what follows we shall classify those configurations of points lying on the hyper-sphere in [4] which may be said to possess the group property. In this classification we shall seek to enumerate only those configurations which are essentially different from this point of view, in order to bring out any geometrical differences which might throw light on the fundamental problems of the theory of linear groups.

Engineering is not, as some are mistakenly apt to imagine, a comparatively recent introduction into Cambridge curricula. Professor William Farish, B.D. of Magdalene, Jacksonian Professor and our first President, contributed the first paper in Vol. I of our Transactions, “On Isometrical Perspective,” a method of which he appears to be the inventor. The reason for the invention he explains. He was evidently teaching engineering—or applied mechanics—so he wanted numerous models of machines and “would have found it difficult to procure a warehouse large enough to contain them.” So he obtained a number of interchangeable parts: loose brass wheels all gearing together, pulleys, axles, bars and frames from which different models could be built up, in fact he invented the forerunner of Meccano and all its competitors. His assistants needed drawings showing them how to assemble the parts for these different models, and he devised this very simple and readily comprehensible method for making such drawings.

1. Let ℱ be a family of functions X (t) defined in (0, 2π). The functionals to be considered are of a particular class, namely those which define a correspondence between each function X of ℱ and a number which we denote by F(X). It is convenient to refer to F(X) as the functional.

Almost every species is, to a first approximation, in genetic equilibrium; that is to say no very drastic changes are occurring rapidly in its composition. It is a necessary condition for equilibrium that all new genes which arise at all frequently by mutation should be disadvantageous, otherwise they will spread through the population. Now each of two or more genes may be disadvantageous, but all together may be advantageous. An example of such balance has been given by Gonsalez(1). He found that, in purple-eyed Drosophila melanogaster, arc wing or axillary speck (each due to a recessive gene) shortened life, but the two together lengthened it.

This paper deals with the calculation of the inter-atomic force at large distances for the cases of hydrogen and helium, and is based on the method used in calculating the polarizability of helium.

No satisfactory formula has so far been derived theoretically for the photoelectric absorption of X-rays and γ-rays. The empirical law

has hitherto been generally accepted as giving approximately the variation of the photoelectric absorption coefficient per electron, with atomic number Z and wave length λ for X-rays of wave length greater than 100 X.U., and the validity of this law has often been assumed for γ-rays also.

1. Mordell has recently proved that if A, B, C, D, P, Q are any real constants such that Δ = AD − BC ≠ 0, and if ΔBC ≠ 0, then the inequalities

are satisfied by integer values of x and y.

The importance of experimental investigation of the intensity of scattering of X-rays by crystals has long been realized, since it affords valuable information regarding the charge distribution in the atoms of the crystal. Recently Debye has pointed out the importance of similar investigations of the scattering by gases. Here diffraction effects may be observed due to regularities in the structure of the gas molecules as distinct from crystal diffraction, where the regularities are due to the orderly arrangement of crystal atoms. This provides a means of obtaining important information as to the structure of molecules, and Debye and his collaborators have obtained interesting experimental results in this field. Debye has also introduced an approximate theory of the results obtained, but owing to the great complexity of the molecules considered it is at present impossible to proceed any further in the theoretical discussion. However, the case of molecular hydrogen may be worked out for short X-rays, since the charge distribution in the ground state of the molecule is fairly well known. It is the purpose of this paper to calculate the intensities of short X-rays scattered from hydrogen gas. The case of other diatomic molecules will not be very dissimilar.

Let (X1, …, Xn) be the coordinates of the centre of a unit cube C, in n-dimensional space, whose (n − 1)-dimensional faces are parallel to the axes of coordinates. Further let the X's be integers. Let εi = ± 1 for r (≤ n) values of i, 1 ≤ i ≤ n, and let εi = 0 for the remaining n − r values of i. Then (X1 + ε1, …, Xn + εn) gives the centre of a cube C′, which touches the cube C along an (n − r)-dimensional edge or face. The cube C and the cubes C′, for all possible arrangements of the ε's, which are subject to the above conditions, form a symmetrical arrangement of cubes. This paper discusses the possibility of completely filling space by means of the packing together of such sets of cubes.

The thyratron valve is essentially a triode which contains a small quantity of some inert gas, usually mercury vapour. If a suitable voltage be applied between anode and filament, with sufficient negative grid bias, the valve behaves like an ordinary vacuum tube; but, as the grid is made more positive, at a certain critical voltage an arc strikes between the anode and the filament, with enormous increase in the anode current. The grid then exercises no further control over the anode current, which must be limited by external resistance so as not to exceed the saturation emission current from the filament. If the voltage across the tube exceeds the “disintegration” potential, of some 20–25 volts, the cathode will be disintegrated by positive ion bombardment. Limitation of the anode current to the emission of the filament ensures that the voltage, while higher than the ionization potential of mercury, is considerably below the disintegration potential.

1. The method generally employed in investigating the angular scattering of electrons in gases consists essentially of firing a directed beam of electrons into a “field-free” enclosure, containing the gas to be investigated at a pressure of a few hundredths or thousandths of a millimetre. The electrons in the beam will be scattered, on colliding with the gas molecules, through various angles. Those which are scattered between θ and θ + dθ are selected by means of a pair of slits which can be set at any desired angle to the electron beam. The electrons selected by these slits are then resolved by means of retarding potentials or electric or magnetic fields into those which have been elastically scattered and those which, on being scattered by the molecule, have excited or ionised it. By plotting the final current received by the Faraday cylinder against θ, we can obtain the angular scattering curve for the elastically or inelastically reflected electrons.

In an investigation which I hope to publish shortly, I think I have been able to improve my theory of the constant hc/2πe2 and to bring it at last into a precise form. No alteration is made in the value 137 obtained in the work already published. The recent advance has been mainly due to the fresh light thrown on the foundations of wave-mechanics by Dr Dirac's book. With a fuller understanding of the “theory of 137” it has been possible to discern opportunities for extension in several directions, and it is with these developments that the present paper deals. They are still in a rudimentary state; but since the theory appears to give correctly either accurate or approximate values of the masses of the electron, the helium atom, and the cosmos in terms of the mass of the proton, it would seem to be on the right lines. Moreover the principle of “ignoration of degrees of freedom” on which the numerical predictions depend is strongly suggested by the theory of the constant 137. If my view is right the only arbitrary constant of nature is the number of particles in the universe—if the number is arbitrary.

Some applications of Fourier series in the analytic theory of numbers.

Page 589, equation (3–10), after “k>0” insert “and 0<R(s)< 1,” and for “2nπi/k” read “2nπix/k.”

Page 589, equation (3·11), for

Add also ‘The evaluation of the integrals given in (3·11) is obvious when 0 < R (s) < 1, and then holds also for 0 < R (s) < 2 by the theory of analytic continuation.”

1. A description of a simple method of demonstrating electric lines of force around a charged body is given.

2. The instability of liquid surfaces in an electric field is used to obtain an idea of the order of magnitude of the charge developed on tourmaline on cooling through about 140° C.

3. A new method of measuring the electric moment of tourmaline is described.

1. It is known, from the theory of the Riemann theta-functions, that the canonical series of a general curve of genus p has 2P−1 (2P − 1) sets which consist of p − 1. points each counted twice. Taking as projective model of the curve the canonical curve of order 2p − 2 in space of p− 1 dimensions, whose canonical series is given by the intersection of primes, we have the number of contact primes of the curve. The 28 bitangents of a plane quartic curve, the canonical curve of genus 3, have been studied in detail since the days of Plücker. The number, 120, of tritangent planes of the sextic curve of intersection of a quadric and a cubic surface, the canonical curve of genus 4, has been obtained directly by correspondence arguments by Enriques. Enriques also remarks that the general formula 2P−1 (2p −1) is a special case of the formula of de Jonquières, which was proved, by correspondence methods, by Torelli§.

We now consider some properties of “almost all” series

where the numbers vary independently in the interval (0, 1). What we need is a definition of measure in the space of sequences or, what is the same thing, in the space

of infinitely many dimensions. Such a definition has been given by Steinhaus. He defines a correspondence between the cube (7.2) and the interval (0,1) in the following way:

In three earlier papers the author has developed the theory of the operational wave equation, which was originally suggested by Professor Eddington as the most extensive generalisation possible of the linear wave equation devised by Dirac. The wave function,ψ, plays a very minor rô1e in the development of this theory and, in reality, it is introduced simply to provide an operand which shall be patient of the action of the wave operators, A1, A2, A3, A4. The object of this paper is to show that the wave function may be entirely eliminated from the theory, which then takes the form of a “ matrix mechanics,” i.e. a set of relations between matrices representing the coordinates, the momenta and the spin operators.