The problem of the two bodies has been treated on the new mechanics by Dirac, Pauli, and Schrödinger, who have independently derived the Balmer terms. The present paper is an attempt at a more complete solution. In particular, formulae are derived for the line intensities of the hydrogen spectrum, for the photoelectric effect and its inverse, and for the continuous absorption spectrum in the ultraviolet and in the X-ray regions. Also the probabilities of transition, deflection and capture are computed for the collision of an electron and an ion. Numerical values are only obtained, however, for the simplest line intensities. It is hoped to treat the problem in greater detail.

If it is assumed that the series electron of an atom polarises the core, then it has been shown by Born and Heisenberg that the polarisability α of the core in a given state may be calculated from the corresponding term value by means of the approximate formulae, where q is the quantum defect, δν is the difference between the term and the corresponding hydrogen term, R is the Rydberg constant in cm.−1, α1 is the radius of first hydrogen orbit, n is the principal quantum number, k is the azimuthal quantum number, For terms with small quantum defect either of the formulae (1) and may be used, but for terms with large quantum defect (1) gives a higher degree of accuracy.

During the last year the homogeneous reactions of haemoglobin, and the heterogeneous reactions of the red blood corpuscle, have continued to furnish us with abundant opportunity for the application of our method for measuring the velocity of very rapid chemical reactions*. The recent types of apparatus, which we have devised, have not departed in essentials from the general principle previously employed, but a number of modifications and extensions have been introduced to meet with greater efficiency the special conditions which were imposed upon us. Since these developments would seem to be of much wider application than the special purposes for which they were in the first place designed, we think that it may be useful to workers in other fields to give a brief summary of their general features here. In the present paper we shall accordingly describe the methods we have adopted for 1. Very fast reactions. 2. Very slow reactions. 3. Very dilute solutions. 4. Small quantities of fluid. 5. Reactions following one another in rapid succession. 6. Reactions involving short-lived transient compounds. 7. The detection of concentration gradients in heterogeneous reacting solutions. Special apparatus has been devised to deal with each of these cases which will be briefly described in the following sections.

The particular form of Taylor's theorem considered states that: If ƒ(x)is n times differentiable at a thenwhere.

This correspondence has been established by Mr H. W. Turnbull, Proc. Camb. Phil. Soc. vol. XXII. (1925), pp. 694–9, a construction which leaves uncorrelated the lines of a special linear complex and the points of the quadric lying in one tangent fourfold; these have to be correlated by means of an independent construction.

In the famous paper in which he developed his theory of radiation, Einstein investigated two problems. He investigated first the problem of what distribution of radiant energy in the steady state would be set up by an enclosed assembly of atoms in thermodynamic equilibrium radiating and absorbing according to the assumptions of his theory; he showed that the distribution so set up was given by Planck's law. He investigated secondly the velocities set up amongst the atoms in consequence of the random variations in direction of the emissions and absorptions; provided that emissions are assumed to be directed, so giving rise to recoil momentum, he showed that the mean square velocity-component in a given direction, , is equal to its equipartition value, given by .

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 velocities initially, 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.