En prenant connaissance d'un mémoire récent de M. G. Doetsch(2), mon attention a été ramenée sur quelques remarques de nature élémentaire que la lecture de la note de M. Hardy, citée ci-dessus, m'avait suggérées et qui me paraissent apporter quelque clarté sur l'origine des résultats obtenus par ces auteurs relativement à la composition des noyaux de Fourier. Des remarques analogues sont contenues dans un article de M. Hermann Kober qui paraitra prochainement(3) et dont M. Hardy m'a obligeamment communiqué les épreuves. La voie que j'ai suivie diffère de celle de M. Kober; je l'expose brièvement ci-dessous.

1. Prof. Fekete, in a letter to the author, wrote: “…Satz von Myrberg über den Zusammenhang zwischen dem transfiniten Durchmesser und dem logarithmischen Mass zugehörig zum Exponenten 1 + ε, ε > 0 (vgl. Satz 10 in seiner Acta Arbeit).…Ich würde einen elementaren Beweis für den letzteren Satz sehr begrüssen.” The object of this note is to supply such a proof but, since the concepts involved may not all be matters of common knowledge, I shall begin by outlining the necessary definitions, the relevant known results, the nature of the theorem and its relation to what is known.

The problem of residual intersections is one of considerable importance in algebraic geometry. In its most general form the problem may be stated as follows. On a given variety V of d dimensions two varieties A and B, of respective dimensions k and k′, pass through a variety C whose dimension r is not less than r′ = k + k′ – d. It is required to determine the variety D of dimension r′ which forms the residual intersection of A and B. The classical paper on this subject is that of Severi*. He considers the case in which V is a linear space, and obtains a large variety of enumerative results connecting the characters of the residual intersection with those of the given loci.

This note is concerned with the relations between the invariants Ωi (i = 0,1,…, d – 1) of the canonical system on an algebraic Vd. Here Ω0 is the grade of the canonical system, and Ωi is the virtual arithmetic genus of the Vi of intersection of d – i canonical Vd–1's.

1. It often happens that we have a series of observed data for different values of the argument and with known standard errors, and wish to remove the random errors as far as possible before interpolation. In many cases previous considerations suggest a form for the true value of the function; then the best method is to determine the adjustable parameters in this function by least squares. If the number required is not initially known, as for a polynomial where we do not know how many terms to retain, the number can be determined by finding out at what stage the introduction of a new parameter is not supported by the observations*. In many other cases, again, existing theory does not suggest a form for the solution, but the observations themselves suggest one when the departures from some simple function are found to be much less than the whole range of variation and to be consistent with the standard errors. The same method can then be used. There are, however, further cases where no simple function is suggested either by previous theory or by the data themselves. Even in these the presence of errors in the data is expected. If ε is the actual error of any observed value and σ the standard error, the expectation of Σε2/σ2 is equal to the number of observed values. Part, at least, of any irregularity in the data, such as is revealed by the divided differences, can therefore be attributed to random error, and we are entitled to try to reduce it.

1. A well-known theorem on probability may be stated as follows. Let λdt be the chance that a certain event will happen in time dt, so that λT is the average number of occurrences of the event in an interval T. If we take a large number of repetitions of the interval T the actual numbers of occurrences will fluctuate about the mean λT, being say λT + x′, λT + x″,…. Then the theorem states that

The ejection of an α-particle by a radioactive substance is an event of the type contemplated and the theorem was applied to this physical problem by von Schweidler in 1905. Much similar work, both theoretical and experimental, has been done at intervals since then*.

1. The elastic stability of a long flat plate when acted on by a shearing force along its edges has been discussed by Southwell and Skan*, and Leggett† has solved the shearing problem when the plate has a small constant curvature. The present paper deals with the elastic stability of a long corrugated plate which is acted on by a shearing force along its generators, the work being applicable to plates with any given number of bays forming the corrugations. It is supposed that the plate is thin and that the depth d of a bay is a small multiple of the semithickness h; little progress has been made without some such assumption as the second, which enables us to reduce the fundamental equations to a soluble form.

The solution of problems in two-dimensional potential theory depends, generally speaking, on the discovery of a conformal transformation suitable for the particular region involved, and the degree of success in the applicability of the method seems to be seriously limited by the complexity of the transformation. Only one serious attempt seems to have been made, by W. G. Bickley*, to formulate a general theory coordinating the different types of boundary value problem for general and inclusive classes of curves, but Bickley's analysis is somewhat involved and not very suitable for certain types of problem. In a re-examination of the subject on the basis of certain suggestions made to me by Prof. Livens, I have discovered what appears to be a much simpler and equally general method of formulating directly the solution for a comprehensive class of problem dealing with simple closed cylinders with curved or rectilinear boundaries, and including all those cases for which solutions are known. In its theoretical aspects the method seems to be perfectly general, but to put it into practice it is necessary to determine explicitly the particular form for the conformal transformation of the space outside the cross-section of the cylinder to the space outside a circle with the points at infinity coinciding.

1. Introduction. The importance of the frequency spectrum of the lattice in problems concerning the properties of the crystalline state has often been emphasized. In particular, all the properties of a crystal which depend on the heat motion of the constituent particles require for their explanation a knowledge of the actual form of the vibrational spectrum.

The statistical treatment of order-disorder transformations, as developed by Bethe and Peierls, has been extended to alloys having superlattices of the AB and AB3 types but compositions which differ slightly from the ideal compositions required for the formation of perfect superlattices. If the composition of the alloy is specified by the concentration c of one component, which we have chosen to be the A component, then the results show that for alloys of the AB type the critical temperature should have a maximum value when c = ½, that is, for the ideal composition. This is in agreement with experiment. For alloys of the AB3 type it is found that the maximum value should occur for some value of c > ¼. This result disagrees with that obtained experimentally, but both results agree, at least qualitatively, with those obtained by the theory of Bragg and Williams.

In a recent paper, the solubility of hydrogen in palladium was discussed. The statistical theory employed was essentially the same as that given by Fowler† for critical adsorption, the only modification necessary being to take into account the dissociation of the hydrogen molecules on absorption. In the present paper Peierls' theory‡ is applied to the same problem in order to enable a comparison to be made between the theories. According to both theories we must postulate that the absorbed (adsorbed) particles go into a definite number of potential energy holes and that the heat of absorption increases as the number of holes filled increases. Fowler uses the Bragg and Williams type of approximation to determine the energy of the particles in the holes; while Peierls supposes that there is an interaction energy V for each pair of neighbouring adsorbed particles and that all those arrangements of the particles in the holes having the same number of nearest neighbours have the same energy. The calculation of the dependence of the energy on the arrangements is approximated to by a method due to Bethe.

Bethe's method, used in the problem of order-disorder transitions in alloys, is applied formally to the problem of molecular rotations in solids. To apply this method, we assume that the solid is entirely homogeneous, and that the state of rotation is the same throughout the solid. Without this assumption, the application of this method is impossible.

A particular form of the mutual potential energy between two neighbouring molecules has been chosen, and classical statistics is employed throughout. The calculations are made entirely after the manner of Bethe, and the similarities of the two problems are pointed out. The result is that there is a critical temperature, and also a discontinuity in the specific heat of the magnitude of some ten times the Boltzmann constant per molecule, arising from the sudden setting in of the rotations as the temperature is increased beyond the critical point. Agreement with the experiments is bad, indicating that a more profound theory is necessary.

In an investigation of the disintegration of by protons, resulting in the emission of three α-particles, Dee and Gilbert* found that the observed distributions of ranges and angles of these particles could be explained by assuming that one α-particle is first ejected, leaving a short-lived, excited nucleus which then decomposes into two more α-particles. The half-life of this excited beryllium nucleus was estimated to be about 10–21 sec.

During the course of a cloud chamber investigation of the occurrence of rare events accompanying the passage of fast β-particles through matter, it was found* that existing methods of introducing high-energy electrons into the chamber were unsatisfactory. Electrons of sufficiently high energy could only be obtained from sources which emit, in addition, a large amount of γ-radiation. Absorption of this γ-radiation in the walls and gas of the chamber resulted in the ejection of numbers of stray electrons, whose presence rendered the observation of the fast β-ray tracks a matter of some difficulty.

A high-speed recording counter circuit is described which has proved very reliable. The basic circuit element employs two triodes in a symmetrical circuit which has two stable conditions. The circuit is triggered alternately from one stable state to the other by the applied impulses acting through a circuit comprising two “Westector” cuprous oxide rectifiers and an inductance.

It is easily possible to make a counter operate accurately at a speed of 20,000 random impulses a minute, provided that the associated recording meter operates in less than 1/100 sec.

A “watch-dog” thyratron recording circuit is also described. This is useful where the slow speed of the mechanical recording meter would introduce loss.

A method of studying magnetic properties in alternating fields and its application to the intermediate state of a superconducting sphere are described. It is shown that the behaviour of the sphere in alternating fields of frequencies between 10 and 100 is different from that in very slowly varying fields, the difference becoming less for lower frequencies and for smaller radius. When the specimen is in the intermediate state, energy losses are produced in it by the alternating field, showing that the induced currents are out of phase with the alternating field. The induced currents do not, however, flow in the same way as in a homogeneous isotropic conductor, and this is probably on account of an anisotropic structure of the specimen when in the intermediate state. This structure depends strongly on the amplitude of the alternating field in such a way that the effective average conductivity is increased with decreasing amplitude, and this amplitude effect depends on the size of the specimen. Evidence is given for a time lag in the magnetocaloric effect, and this time lag decreases with decrease of the size of the specimen, being of order sec. for a sphere of about 1 cm. diameter.

In conclusion I wish to thank Prof. Lord Rutherford and Dr Cockcroft for their interest in this work; Dr Peierls for suggesting the experiment and for much valuable advice and assistance; Mr Shire for advice on some technical points; and various members of the staff of the Royal Society Mond Laboratory for help in the measurements.

Some applications of an alternating field method to various problems of superconductivity are described. It is shown how the method can be used to measure the resistance and self inductance of a metal ring, and this was applied to show how the resistance of a superconducting tin ring is restored by a magnetic field. The method is also convenient for a rapid determination of the order of magnitude of the specific resistance of a substance, and to see whether or not it becomes superconducting; several elements were examined in this way, but no new superconductors were found.

It is concluded that the “overshoot phenomenon” in superconductivity described by F. B. Silsbee was due to his method of mounting the specimen.