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).

The object of this paper is to give proofs of formulae, previously obtained by Schubert (for lines) and Palatini (for planes) by degeneration methods, by a direct application of the coincidence formulae of Pieri and Severi which are referred to below. In the case of planes the essential part of the proof is the deduction of the relation (10) below, from which the required formulae follow directly. The formulae in question have as their object the expression of a product of two incidence conditions as the sum of a number of simple ones.

The formulae of this paper are chiefly concerned with multiple secants and tangents to surfaces in spaces of dimension four to seven; they are obtained by the functional method of Cayley and Severi, which was first applied to scrolls by C. G. F. James. The majority of the results have already been given for scrolls in James's third paper; but whereas the functional equations in this case involve only two variables, those of the present paper contain four, and the work is considerably more complicated. It is also to be noted that, though the formulae for scrolls might perhaps have been expected to follow as particular examples from the corresponding formulae for surfaces, this is not always the case. The reason for the discrepancy is not always clear, although in some instances theoretical reasons for disagreement can be assigned.

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.

The theorem given by Professor Mordell in the preceding paper is capable of a slight generalisation. I first prove that the series

where α, β, γ, … are positive, converges only when κ > Σ (a′/α), the summation referring to those exponents a′ of a, b, c, … for which a ≥ 0; or, if there are no such exponents, the series converges only when κ > a/α, k > b/β, …

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.

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.

It is well known, having been pointed out in the first instance by Plücker, that a plane quartic curve consisting of four unifolia, in Zeuthen's sense, i.e. four even circuits each containing one concave portion or bay, has all its twenty-eight bitangents, and all their fifty-six points of contact, real; since each bay has a bitangent whose points of contact bound it, and each pair of circuits have four common tangents. It has not, so far as I know, been remarked that the algorithm of Hesse and Cayley, in which each of the bitangents is indicated by a pair out of the symbols 1 2 3 4 5 6 7 8, can be applied in a very simple way to such a curve.

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.

The work of Heitler and London has shown that the first order perturbation energies of two atoms taken as a function of the separation between the nuclei gives a fairly good description of the behaviour of the atoms with regard to possible molecular formation. For example, the dissociation energy of the hydrogen molecule calculated in this way is of the same order of magnitude as the accepted observed value, being in fact about 25% too small. The theory has been generalised by Heitler, and also by London and others, to the case where each atom contains many electrons, but detailed calculations do not appear to have been made for systems more complex than that of two hydrogen atoms.

R. H. Fowler and L. Nordheim have presented a theory which seems to offer a satisfactory explanation of the phenomenon commonly known as the auto-electronic discharge, in which electrons are extracted from cold metallic cathodes by intense electric fields. Their theory considers the passage of electrons through a potential “hill” of the type shown in Fig. 1, where V is the potential energy of an electron at a point distant x units from the surface.

The experiments described below were under-taken to follow up the observation, made independently by Dr Chadwick and one of us (M. L. O.), that the disintegration by sputtering of a plane cathode in a gas discharge was not uniform but showed a maximum at a small distance from the edge. By running the discharge for many hours under steady conditions it was possible to remove a narrow ring of metal at a distance of about a millimetre from the edge of a thin platinum cathode, the middle portion and the extreme edge remaining whole. A photograph of a cylindrical cathode taken by Kaye shows the same effect very clearly.

The present paper contains an account of some work done in an attempt to produce high speed electrons, using an indirect method suggested by Sir Ernest Rutherford. Although it was not found possible to make the method work, yet a description of it may be of interest on account of the importance attached to any method of obtaining fast electrons.

Generally, when an alternating voltage is applied to a cumulative grid rectifier no grid current flows at mean grid potential. The behaviour of a rectifier working under these conditions is examined. An expression is derived for calculating rectified current for any applied voltage. As this equation is rather cumbersome to apply, a very simply empirical formula is given which is applicable for any value of applied potential whatever.

An expression is derived for the power absorbed by the rectifier. It is shown that as the applied voltage increases, the apparent resistance of the rectifier decreases and approaches half the value of the grid leak resistance.

Further, it is shown that rectified current depends on the peak value of the applied potential and that it is almost independent of ordinary wave-form variations, even when the applied voltage is small.

By slightly modifying the expression for rectified current we find that, in an amplifier, the grid current is a measure of V and, in an oscillator, the grid current is a measure of the output. The only condition is that V > 2b in both cases.

In conclusion I wish to express my indebtedness to Professor C. E. Inglis for placing at my disposal the facilities of the Cambridge University Engineering Laboratory, and to Mr E. B. Moullin for the interest he has taken in the work.

This paper is concerned with the electric and magnetic field close to conductors which carry an alternating current. Hertz's famous solution of Maxwell's equations gave the field at a great distance from a magnetic or electric dipole whose moment was alternating harmonically: but from its very character it could not describe the field at points which were insufficiently remote from the origin for the source to appear as a dipole. The great development of radio communication has made important certain problems about the field in the vicinity of wires which are carrying an alternating current. Thus imagine that an aerial of some thousand feet in height and very many acres in extent has been erected. Before the designer of this vast and costly structure can calculate the rate of energy output, he must suppose himself so far removed that the aerial has dwindled to a mere speck in the aether. Such a process must seem both unsatisfactory and unsatisfying to those who take the responsibility of obtaining radio communication. It is already known that the field at any point can be expressed formally when the distribution of current and charge in the source is postulated: it is shown in this paper that the field at all points very near to the source can be expressed by simple formulae. Of course it is not an exact solution for the field due to a conductor of specified shape, because the postulated distribution of current and charge is incorrect.

This paper considers the stability of a valve amplifier which has an oscillatory circuit in the grid and in the anode circuit. An exact condition for stability is obtained for circuits which have no resistance, and it is shown that when both circuits have the same natural frequency instability is then impossible if the anode inductance exceeds μ times the grid circuit inductance. This condition is believed to be new; in practice it is unnecessarily severe, but it is believed that stability should be sought by increasing the anode inductance and the grid circuit capacity. The stability of circuits which have resistance is too cumbersome to express generally, but it is discussed on broad principles; the stability and amplification of multistage amplifiers are also considered briefly. The following papers should be compared with the present one.

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.

A preliminary investigation is described in which a type of gold leaf source suitable for use with an expansion chamber is developed. Such a source is necessary for the detailed examination of β particles of energy less than about 50,000 electron volts by the Wilson method. Some results obtained with sources of radium D and thorium B + C of this type are presented in outline.

The writer would like here to acknowledge the help and encouragement he derived from many discussions with Professor Sir Ernest Rutherford and with Dr J. Chadwick during the course of the investigation, and further to express his thanks to Mr G. R. Crowe for assistance in the preparation of some of the sources.