1. In his book, A Theory of Time and Space, the writer showed that, contrary to the generally accepted view, the ideas of Congruence can be built up from purely ordinal considerations, provided that we do not confine ourselves to linear order, but admit what, for convenience, he has called Conical Order. This Conical Order is built up from the purely abstract asymmetrical relations of before and after and does not pre-suppose the existence of the cones which are used to illustrate it.

Given four lines in a plane, we know that the circumcircles of the four triangles formed by them meet in a point. Kühne's theorem states the analogous property for Euclidean hyperspace, viz., that if from n + 2 primes in space of n dimensions we obtain n + 2 simplexes by omitting each prime in turn, the hyperspheres circumscribing the simplexes meet in a point when n is even, but not when n is odd. Mr Grace, in a recent reference to the property, gives a short and interesting proof by help of the rational curves which osculate the given primes and the prime at infinity. It is the purpose of the following note to obtain the result by considering the points of intersection of certain cubic loci.

By a twisted quartic of the first kind is intended the curve in space of three dimensions defined by two quadratic equations, i.e. the complete intersection of two quadric surfaces. If this intersection consist of the aggregate of two or more curves of lower order than the fourth, it will be called a degenerate quartic. In general, a twisted quartic of the first kind will be referred to for brevity as a quartic.

If a bar is vibrating under water it will set up circulating currents which possess kinetic energy and so in effect add to its inertia: consequently the frequency under water will be less than the frequency in air. The effect of the water is precisely the same as if the density of the bar had been increased: the apparent increase of mass is usually called the “added mass” of the bar and the total apparent mass the “virtual mass”, and these terms will be used subsequently.

The potential of an electric charge in a gravitational field has been discussed very recently by Professor Whittaker and by E. T. Copson. The latter gives the solution

for a point centre at “distance” a from a mass of gravitational radius α. The coordinates used are such that the metric is given by

In various communications published in the Journal of the Chemical Society the author has discussed the effect of different nuclei on the absorption spectra of various compounds. The problem has been further investigated with a number of substances derived from aniline, o-and m-cresols, and α-and β-naphthylamines, and this communication is to give an account of the results. A condensed Cd-spark was used as the source of radiant energy, and alcoholic solutions of the substances were examined in the usual way.

In two recent papers the writer has given an account of a practical method of finding the characteristic values and functions of Schrödinger's wave equations for a given non-Coulomb central field. For terms of optical spectra the method is effectively the following. We take the wave equation in the form

and require the values of ɛ for the solutions which are zero at the origin and at r = ∞. We consider the result of integrating this equation outwards from P = 0 at r = 0 to a radius r0 at which the deviation from a Coulomb field is negligible, and inwards from P = 0 at r = ∞ to the same radius, with a given value of ɛ; the characteristic values are those values for which these two solutions join smoothly on to one another, i.e. for which they have the same value of η = −P′/P at this radius. For a given ɛ, the solution zero at the origin depends on the particular atom; the solution zero at infinity can be expressed in a form independent of any particular atom.

Sommerfeld has recently made a very important advance in the theory of electric conduction by employing the conception of electron waves and the new mechanics. His theory has been most successful in explaining a large number of phenomena. It treats them, however, from the macroscopic point of view and does not examine the atomic processes involved. The present note, intended to be merely tentative, seeks to attack the latter aspect of the problem.

A simple explanation of Pauli's principle was first given with the wave mechanics. Its interpretation in the new theory was that the wave functions of Schrödinger were antisymmetrical in all the electrons concerned. Thus when the interactions of the electrons may be neglected, the wave function (for a system of n electrons) can never be of the form

in nature, but only of the form

Hence σ, τ,…ω must be all distinct.

The electron theory of metals revived by Sommerfeld assumes that an electron moves in a metal as though this were an equipotential medium. Considering the nuclei fixed and regularly spaced we obtain a potential periodic in space coordinates. To study the effect of such fields we may simplify the problem so as to contain only one periodic term for each coordinate in its expression for potential. This problem can be reduced further to a one-dimensional one, of which the simplest example is the motion of an electron in a field with potential cos x or sin x. Darwin has shown that a suitable combination or packet of elementary de Broglie waves is capable of moving coherently in several instances. The motion of such a packet is found to be equivalent to that of a particle in classical dynamics with the Heissenberg uncertainty relation. The wave packet is used here for the motion of the electron in a periodic field. The result obtained is equivalent to that of classical dynamics. The wave packet again moves as a particle with an uncertainty relation.

The following is a preliminary account of experiments made in an attempt to obtain information about the relation between the energy of a positive ion and the effect produced on a surface which it bombards. The energy relations which hold in collisions between rapidly moving positive ions and the molecules of a gas or of a solid are of extreme importance in the theory of gaseous discharges at low pressures, in particular of the initiation of the self-sustained discharge.