In two previous notes I have given the various types of non-singular surfaces with sectional genera 4 and 5. In this paper the methods previously adopted are applied to the non-singular surfaces of sectional genus 6. For purposes of classification these may be divided into the four classes:

(1) Regular, non-rational.

(2) Rational.

(3) Irregular, non-referable.

(4) Referable.

In a recent paper, Fowler discusses the adsorption isotherm for a monatomic layer, assuming an interaction between neighbouring atoms in this layer. In this paper we do not intend to give a contribution to the physical problem, but merely to show how the statistical problem involved can be solved by a method due to Bethe.

The paper first scrutinizes thoroughly the variety of compositions which lead to the same quantum-mechanical mixture (as opposed to state or pure state). With respect to a given mixture every state has a definite probability (or mixing fraction) between 0 and 1 (including the limits), which is calculated from the mixtures Statistical Operator and the wave function of the state in question.

A well-known example of mixtures occurs when a system consists of two separated parts. If the wave function of the whole system is known, either part is in the situation of a mixture, which is decomposed into definite constituents by a definite measuring programme to be carried out on the other part. All the conceivable decompositions (into linearly independent constituents) of the first system are just realized by all the possible measuring programmes that can be carried out on the second one. In general every state of the first system can be given a finite chance by a suitable choice of the programme.

It is suggested that these conclusions, unavoidable within the present theory but repugnant to some physicists including the author, are caused by applying non-relativistic quantum mechanics beyond its legitimate range. An alternative possibility is indicated.

In 1907 Enriques and Severi published an extensive and fascinating account of hyperelliptic surfaces. In general a hyperelliptic surface is that expressed by the necessary relation connecting three meromorphic functions of two variables which have four columns of periods. Such functions arise naturally by associating the two variables, in accordance with Jacobi's inversion problem for hyperelliptic integrals of genus 2, with a pair of points of a hyperelliptic curve. When the primitive periods of the functions are those arising for the curve, and the set of three functions chosen is representative, in the sense that only one pair of (incongruent) values of the variables arises for given values of the functions, the surface is called by Enriques and Severi a Jacobian surface; but, if several sets of (incongruent) values of the variables arise for given values of the functions, say r sets, the surface is said to be of rank r. For example, when the three functions are all even, to each set of values of these there belong not only the values u, v of the variables, but also the values −u, − v, and r is thus even, being 2 at least, as in the case of the Kummer surface. In the paper referred to, many cases in which r > 1, corresponding to particular hyperelliptic curves possessing involutions of order r, are worked out. In general the method followed consists in arguing, from the character of the associated group of order r, to the character and equation of the hyperelliptic surface Φ of rank r; and from this the Jacobian surface F is inferred upon which there exists an involution of sets of r points, the surface Φ being the representation of this involution. The argumentation is always beautiful, but often not very brief. The hyperelliptic surfaces for which the primitive periods of the functions are not those of a hyperelliptic curve are also shown in the paper to arise from involutions on the Jacobian surface; with these I am not here concerned.

Second order focusing can be obtained with an apparatus, which contains a circular magnetic field and differs little in its dimensions from the apparatus at present in use. The lay-out is shown to scale in Fig. 4: the essential data are collected at the end of Section V, and indications of the accuracy to be expected are given at the end of Section III.

The very small water drops which are considered in this note are as a rule met with in the form of clouds with a high numerical density of drops. The interior of such a cloud will never be in temperature equilibrium with the walls of a containing vessel, and under these conditions measurements of the rate of fall of the top of a cloud are not measures of the Stokes free fall of a single drop. Optical methods of measurement assume, therefore, a particular importance. The angular diameter of the diffraction halo has long been used as a measure of drop size in clouds; the applicability of this method for small drops will be considered. The criterion of “Rayleigh scattering” has also been employed as showing that the diameter of the drops in question was very much smaller than the wave length of light. A third method which may be applied to drops of diameter of the order of one wave-length will be discussed.

The magnetic susceptibility of antimony both parallel and perpendicular to the trigonal axis is independent of field down to 4° K. The numerical value of the susceptibility parallel to the trigonal axis decreases with increasing temperature, similarly to that of bismuth, but perpendicular to the trigonal axis there is no temperature dependence. The results at higher temperatures are compared with earlier measurements and the comparison suggests that the susceptibility of antimony, like that of bismuth, is very sensitive to addition of foreign elements.

When relative motion of a viscous incompressible fluid of constant density and an immersed solid body is started impulsively from rest, the initial motion of the fluid is irrotational, without circulation. This is shown by observation, and may be seen in many of the published photographs of fluid flow. The theoretical proof is exactly the same as that given, for inviscid fluids, in treatises on hydrodynamics; for it may be assumed that the viscous stresses remain finite. The fluid in contact with the solid body is, however, at rest relative to the boundary, whilst the adjacent layer of fluid is slipping past the boundary with a velocity determined from the theory of the velocity potential. There is thus initially a surface of slip, or vortex sheet, in the fluid, coincident with the surface of the solid body. In other words, there is a “boundary layer” of zero thickness. The vorticity in the sheet diffuses from the boundary further into the fluid, and is convected by the stream. The boundary layer grows in thickness. (The same results follow from a consideration of the equations for the vorticity components in a viscous incompressible fluid, or of the equation for the circulation in a circuit moving with the fluid.)

An unsuccessful attempt has been made to detect artificial radioactivity in aluminium after bombardment with electrons at 300 kV. During these experiments a transitory activity produced by cleaning the walls was observed in a Geiger counter. It is suggested that this might be due directly or indirectly to a progressive oxidation of the inner surface of the walls.

In his chapter on correspondences between algebraic curves Prof. Baker has raised a problem concerning the possibility, when we are given the equations of Hurwitz for a correspondence between two algebraic curves, of obtaining therefrom a reduction of the everywhere finite integrals on either curve into complementary regular defective systems of integrals. The problem is stated as an unproved theorem, an exact formulation of which is given below. The object of the present note is to give a proof of this theorem on the lines of Prof. Baker's chapter.

The steady state theory of sound propagation in exponential loud-speaker horns is well known. The problem of transients, however, has not been treated hitherto, so in this paper it is proposed to consider what happens when the air particles at the throat of the horn are suddenly set in motion. To avoid difficulties which arise in treating sound propagation in the horn when the linear dimensions of the cross-section are comparable with the wave-length of the sound, the following

artifice is postulated. The axis of the horn is linear and the air column is divided up into a large number of frictionless conduits by very thin rigid partitions as indicated in Fig. 1. The cross-section of each conduit expands exponentially from the throat towards the mouth of the horn. The major linear dimension of the section of a conduit at the mouth is small in comparison with the wave-length of the highest frequency to be reproduced.

The groups of rotations that transform the regular polygons and polyhedra into themselves have. been studied for many years. Lately, increasing interest has been shown in the “extended” groups, which include reflections (and other congruent transformations of negative determinant). Todd has proved that every such group can be defined abstractly in the form

This group is denoted by [k1, k2, …, kn−1], and is the complete (extended) group of symmetries of either of the reciprocal n.-dimensional polytopes {k1, k2,…, kn−1}, {kn−1, kn−2,…, k1}. There is a sense in which these statements hold for arbitrarily large values of the k's. But here we are concerned only with the cases where the groups and the polytopes are finite. The finite groups are

[k] is simply isomorphic with the dihedral group of order 2k (e.g. [2], the Vierergruppe). [3, 3,…, 3] with n − 1 threes, or briefly [3n−1], is simply isomorphic with the symmetric group of order (n + 1)!.

1. There is a lemma given by Severi which is of importance because it is used by him in his proof that the number of finite Picard integrals belonging to an algebraic surface is equal to the irregularity of the surface; it is also used by Castelnuovo † in his proof of the same result. The lemma is: If upon an algebraic curve there is an irreducible algebraic series, ∞1, of sets of s points, of index r; and if the sets of sr points which consist of all the r sets which contain a given point (this point taken r times) move in a linear series as this point varies, then any one of the r sets of s points separately moves in a linear series. The proof given by Severi was held satisfactory by Castelnuovot, but I find difficulty in stating it with precision; and Castelnuovo gives an entirely different proof, founded on an enumerative formula due to Schubert (loc. cit. p. 341); this is also the course adopted by Enriques-Chisini.

1. In a former paper we solved the following problem (Problem I). Given k + 1 real numbers co, …, ck, to find necessary and sufficient conditions that there shall exist a function f(x) in (0, 1) which satisfies the conditions

In spite of a large amount of general theory, it remains a surprising fact that the number of known irregular surfaces in space of three dimensions is quite small. Moreover, although the evaluation of the arithmetical genus pα is comparatively direct, it is often a matter of considerable difficulty to calculate the geometrical genus pg. This paper deals with a number of surfaces for which the equations of the canonical system can be written down and their number counted. An extension to primals in higher space is also suggested for the simplest case.

1. The gliding of bodies on the surface of water has received mathematical treatment by Wagner for both two- and three-dimensional cases. In this note, attention is confined to the two-dimensional gliding of a plane plate on a stream of infinite depth. Wagner's method is approximate and only applies when the angle of incidence of the gliding plate to the stream is small. He expresses such quantities as the total lift on the plate, not in terms of the total length of the plate, but in terms of the length of the pressure surface, which is the distance between the trailing edge of the plate and the point of stagnation. Further, his solution requires that the direction of the jet or spray which is formed should be parallel to the length of the plate. If Lis the total normal lift on the plate, β the small angle of incidence of the plate to the stream, V the velocity of the plate, δ the thickness of the spray and c the length of the pressure surface, then Wagner gives

1. As is well known, the electrical resistance of a metal is very greatly in-creased by the addition of a second metal with which it forms a solid solution. The increase Δρ in the resistivity due to the addition of a small percentage of the second metal is in general independent of the temperature (Matthiessen's rule), though there are oertain exceptions (e.g. Cr in Au). The quaritum-mechanical explanation of both these facts was first given by Nordheim, and may be expressed as follows: the electrical conductivity of any metal may be written in the form

where τ is the “time of relaxation”, equal to half the time between collisions, and N is the effective number of free electrons per unit volume: hence, for the resistivity, we have

It is a well-known fact that the non-conservation of energy during β decay may be explained by assuming the existence of light neutral particles, neutrinos, which are ejected from the nucleus at every act of β emission and which enable all the conservation laws to be fulfilled. A study of the energy distribution of recoil atoms during β decay might furnish some data for testing this hypothesis. However, in order to determine the energy distribution of recoil atoms consider-able experimental difficulties have to be overcome. The energy of recoil atoms of ordinary radioactive substances is of the order of magnitude of 1 volt, i.e. of the same order of magnitude as the adsorption energy of atoms on a surface. Thus, on escaping from the'surface on which they are adsorbed, they will lose an amount of energy of the same order of magnitude as that which they possess, and their velocity distribution will therefore be completely distorted. This difficulty may be met by employing an artificial, light radioactive substance, whose recoil atoms may have energies some ten times greater than 1 volt. For this reason active carbon C, obtained by bombarding boron with deuterons according to the reaction 10B+2H→11C+n, was employed as a radioactive substance. Moreover, the fact that the emission of positrons from 11C is not accompanied by the emission of γ-rays has the advantage of simplifying the interpretation of the results.

It is well known that the eigen-functions corresponding to the free electrons in a one-dimensional lattice are of the form

where C (x) is a periodic function with period c (the lattice constant), and Kis a continuous variable.