Absorption curves in paper have been obtained for the β-rays of actinium (B + C) and actinium C′. Some inferences regarding the shape of their energy distribution curves have been drawn, and the ranges of the β-rays of actinium B and actinium C′ fix their maximum β-ray velocities at Hρ 3430 and 6140. The potency of the method is emphasized for elements which cannot be easily investigated directly in a magnetic spectrograph.

The γ-ray ionizations of actinium B and actinium C′ have been compared under the same conditions with the β-ray ionization of actinium C′.

In conclusion, the writer wishes to thank Professor Sir Ernest Rutherford for his kind interest in the work. Thanks are also due to Dr J. Chadwick and Dr C. D. Ellis for many valuable suggestions and discussions.

The tetrahedral complex of lines in space of three dimensions may be defined as the aggregate of lines which meet four given planes in a range of points which is related to some fixed range. In the present paper we consider the more general system of lines, in space of r dimensions ([r]), which meet a certain number k of given primes (or [r−1]) in a range of points which is related to some given range. The paper is divided into three parts. In the first part, after considering successively the cases r = 2, 3, 4, we proceed by an induction argument to the general case. The section following treats the matter from an algebraical point of view, and in it we obtain in a concise form expressions for the locus (when such exists) generated by the lines. In the final section the systems of lines are shown to arise naturally as projections of known configurations in higher space.

In a recent paper, Poisson's summation formula

was proved very simply by integration by parts, subject to the conditions:

(α) for all real values of x, f(x) and f′(x) are continuous and f (x)→0, f′ (x)→0 as |x| → ∞.

(β) f(x) and f″(x) are such that the integrals, converge.

This problem was originally attacked by Stokes, and the formula obtained by Stokes for the case of steady motion (given at the end of § 1 below) is now generally used in experimental work. However, this formula throws no light on the way in which the velocity of the sphere approaches its steady value: and the only known discussion of this problem was given by Boggio (his results will be found in § 2 below). Unfortunately, when the specific gravity of the sphere exceeds ⅝ (or the constant σ, defined later, is less than 4) the solution so obtained cannot be converted into numerical results, because the constants α, β become complex numbers; and even when α, β are real, the formula is not very convenient for large values of t (as the velocity approaches the steady value).

Any one who has dealt with the convergence of multiple series soon becomes aware that simple and interesting examples are very scarce. It may therefore be worth while placing on record the following result which is not altogether trivial.

It is a familiar fact in Solid Geometry that the problem of finding a twisted cubic curve with four given tangents is poristic. In fuller statement four prescribed tangents are apparently sufficient to determine such a curve yet

(A) In general there is no twisted cubic touching four given lines,

(B) If there is one there is an infinite number.

The purpose of this note is to extend in two respects some previous work of the writer on the wave equation of a particle in a central field of force which is a Coulomb field for sufficiently large r.

The object of this paper is to make certain calculations which will be required in the estimation of the quantum defect. Hartree has defined certain solutions Gi(σ, ζ), Hi(σ, ζ) of the wave equation for an electron in a Coulomb field, viz.

In this paper we use Dirac's relativity quantum mechanics to derive the well-known Kramers-Heisenberg dispersion formula for an atom with one electron. The treatment is not limited to the case of a central field, but is quite general. An expression is also obtained for the dipole moment to which is due the incoherent scattering. The formulae obtained are similar to those obtained by O. Klein for the case of a central field. We find in this way an explicit expression for f, the number of dispersion electrons for any line of the optical spectrum (being a measure of the intensity of the line), in terms of the solutions of the four wave equations of Dirac's theory. It is further shown that the sum of the number of dispersion electrons for any state of the atom is not exactly unity, but differs from it by an amount of the order of 10−4. The result Σf = 1 has been shown by London to hold exactly for the simple wave equation as originally given by Schrödinger. It is here shown that the exact relativity treatment has a very small effect.

Hobson has given a proof of this theorem in its fullest generality. The present note gives an alternative for part of Hobson's argument. The theorem may be stated in two forms. If f(x) is a function of x, monotone when a ≤ x ≤ b, and φ(x) is integrable over the same range, then

where a ≤ X ≤ b,

(ii) the same holds with a < X < b except in some trivial cases where f(x) is constant in the open interval a < x < b. The form (ii) is not mentioned by Hobson.

The stability and the stream lines of the symmetrical and unsymmetrical double rows in a channel have already been discussed by the author. It is the object of this paper to investigate the possibility of the existence of a double row whose axis is parallel to, and not necessarily coincident with, the axis of the channel. It will be seen that, if the system of vortices is to move forward as a whole, then the only possible types of double row are those that have been discussed previously, that is, the symmetrical and unsymmetrical systems, and in both of these cases the centre lines of the row and channel are coincident. It was shown previously that the symmetrical row is always unstable and that the unsymmetrical system is stable within a particular domain of stability.

The surfaces considered in this note are Jacobians of four quadric forms (“quadrics”) in [4], where one or more of the quadrics are space-pairs. It is believed that, with one exception, they have not been noticed previously; and so little is known of non-rational surfaces in higher space that it seems worth while putting them on record.

The solution of problems in diffraction by an elementary application of Huyghens's principle is discussed. The obliquity function is investigated, using the criterion that the formula used must give the right result when integrated for the case of an undiffracted plane wave. It is shown that this is satisfied for distant points by any function which makes the integrals converge, but that to satisfy it completely a constant obliquity function is necessary. This makes a consideration of the distant boundary essential, as the integrals do not converge in this case. It is shown that a boundary distributed in a Gaussian way is completely satisfactory. The integral for the case of diffraction by a straight edge is solved exactly, leading to the usual result. Finally, the results are discussed in relation to the teaching of the subject.

I am indebted to Dr H. Jeffreys and to Mr F. P. White for their interest in this paper; also to Mr N. F. Mott, who has contributed much to the course of its development.

During my visit to Cambridge last year I was given the opportunity of carrying out in the Cavendish Laboratory some experiments with a view to studying the reflection of atoms from a solid surface. For this purpose the atoms of radioactive substances offer the advantage of extreme sensitivity and also the possibility of quantitative measurements by means of their radiations. While the main object of the experiments could not be achieved, some phenomena have been observed which seem to deserve brief mention.

The problem of the collision between an electron and an atom was first considered on the Quantum theory by Born, who has worked out in detail the case of atomic hydrogen, and has obtained formulae giving the variation of scattering with angle both for elastic and inelastic collisions. Born's solution is only approximate. The purpose of this note is to discuss the physical nature of the approximations used by him, and also to extend his results to Helium, for which experimental evidence is now available. It is found that the theoretical curve agrees with the experimental as well as the approximations used would lead one to expect. We shall confine ourselves to elastic collisions.

Miquel's Theorem, as generalised by Clifford, states that the four circles through the intersections of four straight lines, taken in triads, meet in a point; that the five such points derived from five lines, taken in sets of four, lie on a circle; that the six such circles, determined by six lines, meet in a point……. The process continues indefinitely.

While investigating the effect of a caesium film on the auto-electronic emission from a tungsten surface I noticed certain phenomena which seemed worth further investigation. Langmuir and Kingdon have shown that the emission from a tungsten filament in caesium vapour varies with the temperature of the filament in the manner shown in Fig. 1. The initial rise in the emission at low temperatures is due to the existence of a caesium layer on the filament. At higher temperatures the caesium evaporates and the emission drops until eventually the temperature becomes high enough to produce a thermionic emission from the clean tungsten surface.

In the measurement of small ionisations it is usual and desirable to employ an ionisation-chamber whose central collecting electrode serves also as an electroscope, of the gold-leaf or the quartz-fibre type. As an instrument for the measurement of quantity of electricity even the most sensitive of these is inferior, by a factor of about 100, to the combination of an ionisation-chamber and a Compton electrometer. The new type of ionisation electroscope here described combines a high voltage-sensitivity with a small capacity and has a sensitivity for quantity of electricity of the same order as the combination mentioned.