In a recent paper the present writer made an attempt to reduce the ordinary optical constants of matter to a more primitive form by introducing “scattering indices.” Their use solves half the problem of optics, for general formulae connect these indices with the behaviour of matter in bulk. There remains the deduction of the scattering indices from whatever we are assuming to be the reaction to light of the actual atoms, and even in the classical theory this encounters rather formidable difficulties of convergence. As long as we are content with no great rigour it is a fairly simple matter to deduce the scattering indices for any assumed model without any of those rather difficult arguments about the polarisation of the medium which play a part in the ordinary presentation.

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.

It has been observed by various workers that, in addition to particles ejected from the various atomic levels of the atoms concerned, some are ejected by rays which correspond to the natural K X-radiation of those atoms. It has also been shown that some radioactive substances emit β particles of comparatively low energy. Therefore it was thought advisable to make an attempt to see whether this “soft” radiation has a line spectrum; for one would expect to find traces of the natural L-radiation, just as one finds evidence of the K-radiation, provided that the effect is sufficiently intense.

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.

In a previous paper the values for some lines of the β-ray spectrum of the natural L radiation of radium B have been given. The purpose of the present paper is to show that many of these lines can be accounted for by assuming that rays corresponding to L X-rays, but generated within the atom, act on various atomic levels. Before proceeding to do so, however, it will be advisable to give a fuller account of the experimental methods.

It is well known that the β-ray type of disintegration is usually accompanied by the emission of γ-rays, and it is a matter of great importance to decide which of these two phenomena occurs first, that is, whether the γ-rays are emitted before the nucleus disintegrates or afterwards. It is not practicable to attempt a direct solution of this problem, but since the emission of the disintegration electron results in a change of the nuclear charge of + 1 it would be sufficient to determine the atomic number of the radioactive body at the moment of emission of the γ-rays. For instance, an atom of the β-ray body radium B has an atomic number 82, but after disintegration when it becomes radium C it has an atomic number 83, so that if the γ-rays were known to be emitted from a body of atomic number 82 it would mean that they were emitted before the disintegration, whereas if the atomic number 83 were found it would show that they were emitted afterwards.

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.

In a recent series of papers analytical methods have been introduced which allow of a mathematically simple treatment of the theorems of statistical mechanics for the usual assemblies of isolated or effectively isolated systems. By this we mean that the individual component systems may be treated for energy content as if they were never interfered with. It is only then that energy can be assigned to systems rather than to the assembly as a whole, and it is on this partition of energy among the systems that the analysis is based. When this independence, for example between separate atoms, breaks down as in a molecule, and still more in a crystal, we can take the whole complex to be a system. The analysis will still apply, and if we can formulate the dynamical motions of the complex system, we can still make progress. The essential step for any system is to construct its partition function. Examples of such constructions for molecules and crystals will be found in the papers quoted, and are of course otherwise well known.

The following paper is in two parts.

In Part 1 it is shown that Kronecker's theorem can be extended in the form

If is greater than ηfor all sets of integers l1, l2, … l3 less in absolute value than K/σ and not all zero, l also being an integer, then, for any x1, x2 … x3, an integer q less than l/(ησ8) can be found such that qv1–x1, qv2–x2…qv3–x4 all differ from integers by less than σ;. K, L depend only on s.

It is an immediate corollary that if

is greator than for all sets of integers l1, l2… l3, l less in absolute value than K/σ and not all zero while F(v1, v2… v3), l less in absolute value than K/σ and not all zero while F(v1, v2… v3, v) is periodic period 1, in v1, v2 … v3,v, then T can be found between 0 and such that

Where K, L depend on s, and N on s and the bounds of

Landé has indicated a connection between the “relativity” doublets of X-ray spectra, and the doublets and triplets of optical spectra, and so has obtained a formula for the separation of optical doublets and triplets. This formula can only be expected to hold if the orbit of the series electron penetrates into the core, so it may be possible to obtain evidence on the question whether an orbit penetrates from the doublet or triplet separation of the corresponding term.

From this evidence it is concluded that, except for lithium-like atoms, p terms of all known spectra correspond to penetrating orbits; this disagrees with Bohr's assignment of quantum numbers for those terms of the spectra of neutral atoms of the Cu and Zn sub-groups. For d terms the evidence is not so definite, but it seems possible that for Cs I and Tl I the d terms correspond to penetrating orbits, in disagreement with Bohr's assignment. It is shown that the Landé formula seems to hold approximately for separations in multiplet spectra when the terms belong to a sequence of the Rydberg type.

Ideas suggested by Heisenberg in a recent paper would involve some modifications of the interpretation of the magnitudes of doublet separations, but at least the conclusions as to the p orbits penetrating still stand.

1. The axioms necessary for the construction of a proof of Pappus' Theorem, regarded as a theorem in the geometry of projective space of three dimensions, fall into three groups:

I. Axioms of Incidence;

II. Axioms of Order, giving the properties of the relation “between”, and establishing the order type of the projective line as cyclical and dense in itself;

III. An Axiom of Continuity.

It is customary in treatises on projective geometry to adopt in Group III the Axiom of Dedekind, which states that if the points of a segment are divided into two classes, L and R, which have each at least one member, and are such that no member of L lies between two points of R, nor vice versa, then there is a point of the segment, not an end-point, which is neither between two points of L nor between two points of R. This axiom, however, assumes considerably more than is necessary for the proof of the theorem.

Thirty years ago, in a paper on continued fractions, Stieltjes published a definition of the integral which bears his name. His replacement of the variable of integration x by a more general “base function” φ(x)—a change which throws so much light upon other theories of integration—received at first little attention, but has later sprung into greater prominence; so much so that Professor Hildebrandt, in summarizing these various theories in a paper to the American Mathematical Society, makes the statement that “it [the Stieltjes Integral] seems destined to play the central rôle in the integrational and summational processes of the future.” Yet even now the integral and the allied theory of differentiation with respect to a function have been subjected to little detailed analysis, and the possibilities of extension have been only touched upon. It is the object of this present paper to establish certain results which are of some value in themselves and which prepare the way for an attack upon the integral.

It has long been realized that a Stieltjes integral can be defined and developed on the same lines as a Lebesgue integral, using in place of the measure of a set E the variation of an increasing function ø (x) upon E; but only the more obvious—and less useful—properties of such an integral seem to have been stated. The original integral defined by Stieltjes possesses the remarkable Integration by Parts property that if either of exists, so does the other, and their sum is equal to

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.

In the year 1663 John Wallis, Savilian Professor of Geometry in the University of Oxford, delivered a lecture in which he claimed that he had proved Euclid's Postulate of Parallels according to the strictest laws of demonstration after Euclid's manner.