The extension of Mellin's inversion formula expressed by the equations has been considered by Fowler who shows that some form of Stieltjes integral is essential to Poincaré's proof of the necessity of the quantum hypothesis. Fowler confines his discussion to a restricted type of function φ (y) which is sufficient for the physical problem. It will be proved here that the formulae hold with a general Stieltjes integral in the first equation.

A Fourier integral will be described as of finite type if the range of integration of the integrals by means of which the coefficients in the Fourier integral are defined is a finite interval instead of the usual (−∞ ∞). Thus is of finite type (p, q), such that where ƒ(x) is the generating function of the series.

§ 1. If the kinetic potential for the relative motion of two masses is written with an added constant as

a close connexion with the relativity quadratic appears. The latter is in fact

where

a modification of the primary form

which shows an unaltered determinant. The condition in respect to the determinant, suggested, I believe, by Schwarzschild, is one which to me appears to give the most significant form to the results. From the dynamical standpoint we may regard it as imposing a counterpoise in the inertia coefficients to the modification introduced by the potential; or from a geometrical point of view we may regard it as minimizing the departure from the normal use of coordinates. An illuminating example of the loss of meaning that accompanies transformation in which this condition is disregarded is furnished by the isotropic form which is sometimes given to Einstein's quadratic.

1. During the last sixty years the principal questions presented by the higher singularities of plane algebraic curves have been completely solved, and definite results obtained. The two most successful lines of research have been by expansions and quadratic transformation. By each method it has been shown that a higher singularity may be looked upon as containing concealed or “latent” multiple points or lines in addition to those immediately recognized; and from each, with the help of small quantities, has been constructed a topological explanation of these latent multiple elements, which are accounted for as situated in the immediate vicinity of the point and line base of the singularity. Further, by each method it has been proved that as regards the numerical relations known as Plvicker's equations a singularity produces the same effect as a definite number of nodes, cusps, bitangents, and stationary tangents.

The absorption by matter of energy from a beam of X-rays follows laws which are now well known. The first stage is the ejection from the absorbing atom of a high speed electron. This electron, in turn, produces pairs of ions from some of the molecules through which it passes until its energy is all spent. The process is essentially a discontinuous one in space, and the proportion of atoms affected at a given time is always exceedingly minute, even with an intense beam of radiation. With a beam of average intensity an individual atom would suffer ionisation, on an average, about once in a million years.

We may summarize our conclusions as follows. If the rotating doublets have quite different angular velocitiesinitially, then they repel each other with a force (R) given by

where

and

ø1 and ψ1 being the (constant) angular velocities of the two doublets.

If the doublets have the same angular velocities initially, and the same moments of inertia, then over a certain range of r we have

where

and ω is the common value of the angular velocities of the doublets. When the doublets correspond to hydrogen atoms in their principal quantum orbits, the range of distance becomes 5 Å. to 50 Å. and the formula for R reduces to

This is a law of force of the type found empirically by Lennard-Jones for helium, neon, and argon. The attractive term in this formula is larger than the attractive terms found by Lennard-Jones. The repulsive term, however, which leads to a “diameter” of 3·31 Å., is in very satisfactory agreement with the repulsive terms found by Lennard-Jones.

This communication is a sequel to a former paper on “The General (m, n) Correspondence” read before this Society in March, 1926*. It contained the general exposition (§ 1) and the theory of complete and closed sets (§ 2). For convenience these reference numbers will be here adopted, so that the present work, which opens with § 3, will be understood as a direct application of the previous sections.

In this paper the method of infinitesimal transformations of coordinates, used by Weyl to determine conditions that a function of the tensors gik and φi, and certain of their derivatives, should be a scalar density, is applied (with certain modifications so as to give tensor relations) to functions of and . It is known that in order that such a function should be a scalar density it must be a homogeneous function, of degree ½n, of , and this must of course be deducible from the equations found by the infinitesimal transformations. In view of the part which these equations may play, as “equations of energy,” etc., in purely affine field theories, it seems desirable that the connection should be explicitly shown, and this is done in § 3.

The relevant results from the writer's previous paper on the relation between spectra of atoms of different atomic structure are summarised.

For non-penetrating orbits no new theoretical results are obtained, and there are few known spectra (other than those of lithium-like or sodium-like atoms already treated in the first paper) on which to test the relations previously obtained. Values of the polarisability for the A1+ and Si++ ions are calculated from terms of Al I and Si II respectively corresponding to non-penetrating orbits, and are shown to be very much greater than the values of the polarisability of the neon-like ions Al+++ and Si++++.

The main new results are those for penetrating orbits. Assuming a central field of force, it is shown that the quantum defect q for such an orbit can be expressed as the sum of contributions from the electrons in groups of core orbits of different principal quantum number n, and further that if for a given atom in different states of ionisation corresponding orbits of the series electron are compared, the contribution to q from a set of core orbits of given n is very nearly independent of the degree of ionisation so long as the number of electrons in core orbits in the group remains the same. It follows that, if q is the quantum defect for a term of the spectrum of an atom core charge C, the core of which contains SM orbits of principal quantum number M and none of higher quantum number, and q″ is the quantum defect for the corresponding term in the spectrum of the atom of the same element with core charge C + sM, which differs from the atom of core charge C only in lacking the core orbits of principal quantum number M, then q −; q″ is approximately the contribution from the core orbits of principal quantum number M to the quantum defect for the term of the atom core charge C.

Further, it is shown that if corresponding terms of different atoms of the same electronic structure are compared, then for large values of C the contribution to q from any group of core orbits should tend asymptotically to be proportional to the average time mean radius of these orbits, and its reciprocal should tend asymptotically to be linear in C.

Somewhat similar relations are obtained for the quantity Q = dq/d (ν/R) which measures the variation of quantum defect within a sequence.

These theoretical results, and in particular the result that l/(q − q″) should tend asymptotically to be proportional to C, are compared with the values of q deduced from the terms of such observed spectra of aluminium-like and copper-like atoms as are available, and it is found that though the theoretical relations are only established as asymptotically true for large C, there is a considerable measure of agreement with spectra for small values of C, which are the only ones which can be observed.

The dynamical problem of the “diatomic molecule” is solved on the new mechanics. The terms of the rotational energy are , where ; the weights of the corresponding states are 2m; the frequencies differ a little from the classical ones. Finally the intensities are slightly different from those computed by Kemble; the main term agrees with that of Fowler, but the positive branch is only slightly stronger than the negative. The central line vanishes. The intensities are valid only for the fundamental band.

§ 1. My primary aim here is to give some account of relativity in connexion with axial rotation, and in particular to deal with the disk problem. The justification of the use of a special quadratic form for this purpose suggests the propriety of some introductory remarks of a more general character.

1. Soit donnée l'équation de Mathieu

a et q étant des paramètres réels.

1. It is pointed out that the signal strength of a wireless wave at a distant point depends on:

(a) the electrical constants of the ground,

(b) the curvature of the earth,

(c) the existence of an “atmospheric” ray coming downwards from the Heaviside layer.

2. We can eliminate the effect of (b) and (c), and so obtain direct evidence about the electrical constants of the ground, by making measurements near the transmitter.

3. Measurements on wave lengths of 300 m. and upwards give information about the resistivity of the ground, and on shorter wave lengths (15 m.) give the dielectric constant of the ground.

4. Attenuation measurements have been made over short distances for wave lengths of 1600 m. and 360 m.

5. The results are compared with those calculated from Sommerfeld's theory and show close correlation for distances beyond 10 wave lengths, but show deviations from the theory for shorter distances.

They give as values for the resistivity of the ground

p = 1·8 × 1013 E.M.U.

for the Daventry signals, and

p = 0·6 × 1013 E.M.U.

for the London signals.

The present work is the outcome of a study of the special type of (3, 2) correspondence on a circle, namely that between a point and the extremities of its pedal line. This led to considering correspondences in general, and to the formulation of concepts which are believed to be new, for example the Canonical Forms, § 1 (2), and Multipliers, § 1 (3). The case of the (n, 1) correspondence has already been given by the author.

It is well known that charcoal possesses catalytic properties. It accelerates the decomposition of hydrogen peroxide, the removal of bromine from ββ-dibromo propionic acid and enables certain oxidations to take place in air at ordinary temperatures.

The theory of catalytic action recently advanced qualitatively by Pease, Taylor, Armstrong and Hilditch, Constable, which the author has endeavoured to treat quantitatively, is extended in this paper to the explanation of the effect of pressure and inert diluents on catalytic action on saturated surfaces. The results are shown to be in agreement with experiment, and further inferences may be made concerning the relative mean lives of the reactant and the diluent on the surface.