The fact that the Mathieu equation, and, as was proved in the preceding paper, the more general Hill equation, can admit of but one solution of period π or 2π, suggests the problem of determining how far the second solution deviates from periodicity when the first solution has either the period π or the period 2π

The equation to be considered is of the type

where p (x) is continuous for all real values of x, even, and periodic. It is no restriction to suppose that the period is π, and this assumption will be made, so that the equation is virtually Hill's equation.

1. The relation

in which we suppose each of a, b, c, a2 − c2 to be different from zero, leads, from a value θ, to two values of φ, which we may denote by θ1 and θ−1. Each of these, put in place of θ, leads, beside the value θ of φ, to another value of φ, say, respectively, θ2 and θ−2. If θ2 or θ−2 be put for θ, the same relation leads to two values of φ, say, θ1 and θ3 or θ−1 and θ−3, respectively. And so on. It may happen that θn is the same as θ, in which case also θ−n is the same as θ; this we may express by saying that the relation is closed, or that there is closure, after n links. It is the object of the present note to express in reduced form, in terms of a, b, c, the condition that this may be so. Evidently, if n = pq, the condition of closure after n links is satisfied when the condition for closure after p links (or also after q links) is satisfied. But there is a condition for closure after n links which is not satisfied for any less number of links; this is the condition which we call the reduced or proper condition for closure after n links, and it is this which we seek to express.

In a paper by Slater a new theory of radiation is discussed and the following explanation of the breadths of spectral lines is given: “An atom in the ith. state has a probability Pi of suffering in unit time a transition. Thus there is a probability Pi that the vibrations of each of the oscillators will simultaneously cease. But we shall assume that, in addition to this probability Pi of ceasing its oscillation altogether, each oscillator has also an independent probability Pj of suffering an interruption in which it ceases its oscillation as if it were leaving the state, but immediately begins again as if it were entering the same state” (i.e. with an arbitrary phase difference). “This term Pj is the same as the probability that an atom in the jth state will leave that state.” The total probability of interruption of vibration is thus (Pi + Pj) which is symmetrical with regard to the two end states. This makes the breadths of absorption and emission lines equal and so satisfies Kirchhoff's law.

The experiments described in this paper were carried out in the University of Sydney during November 1924 and, owing to the short time which was available, are of a purely preliminary character. The following account is given as they are not likely to be resumed in the immediate future.

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.

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.

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.

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.

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.

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.

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.

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 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

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