1. The steady motion of an incompressible viscous fluid, due to an infinite rotating plane lamina, has been considered by Kármán. If r, θ, z are cylindrical polar coordinates, the plane lamina is taken to be z = 0; it is rotating with constant angular velocity ω about the axis r = 0. We consider the motion of the fluid on the side of the plane for which z is positive; the fluid is infinite in extent and z = 0 is the only boundary. If u, v, w are the components of the velocity of the fluid in the directions of r, θ and z increasing, respectively, and p is the pressure, then Kármán shows that the equations of motion and continuity are satisfied by taking

Professor Oystein Ore of Yale University has kindly called my attention to the close relation between my paper and some earlier researches of Dedekind. I should like to correlate, as far as possible, my results with his.

A method for studying the adsorption of hydrogen on a clean tungsten surface has been developed which depends on the fact that the accommodation coefficient of neon is different for a bare surface and for a surface with an adsorbed film on it.

It has been shown that on a bare tungsten surface, saturation occurs at a partial pressure of hydrogen of 4.10−4 mm. of mercury and probably much lower, so that the adsorption is of the nature of chemi-sorption.

Important deductions can be drawn concerning the process of activation if any for this chemi-sorption of hydrogen on a bare tungsten surface.

The accommodation coefficient of neon with a tungsten surface covered with a hydrogen film of the type investigated is 0·17 at 295°K, and 0·32 at 79°K.

The experiments described were carried out during my tenure of a Moseley Studentship at the Cavendish Laboratory, and I should like to thank Lord Rutherford for his interest in the work and the Council of the Royal Society for their continued support. I should also like to thank Professor Rideal for his interest in the results.

In the first part of this paper I investigated the nature of the isolated singular points which can appear on an algebraic surface without affecting the conditions of adjunction and the arithmetic genus of the surface. It appeared that such points have neighbourhoods analysable into connected chains or trees of curves, each rational and of virtual grade − 2, and arranged either in a single chain (giving a binode, or a conic node if there is only one curve) or in three chains of, say, n, p, q curves respectively, one end curve of each chain meeting one which we may call the central curve of the tree. The values of n, p, q are not arbitrary, but satisfy either p = q = 1, or n ≤ 4, p = 2, q = 1; the unodes given by the former case we call U, those given by the latter case U, with a suffix indicating the reduction in class in each case. I pointed out the similarity between these results and Coxeter's enumeration of groups generated by reflexions, there being a one-one correspondence between these singularities and Coxeter's groups, with the restriction that the only groups with which we are concerned are those in which any two primes of symmetry are inclined at either π/2 or π/3. In fact, the curves which make up a complete neighbourhood correspond to the bounding primes of a fundamental region, two curves which do not meet corresponding to mutually perpendicular, primes, and two which do to primes inclined at π/3.

A method is described for measuring the heat of adsorption of hydrogen on a clean tungsten filament. The value obtained is 2·8 × 104 calories per mol.

The columnar theory developed by Jaffé to account for the recombination of ions in alpha particle tracks is extended to beta rays by taking account of the clusters of secondary ionisation. Reasonable agreement is obtained with experiment. Recombination in proton tracks produced in hydrogen by neutrons is shown to be in agreement with the columnar theory, but in the case of nitrogen nuclear tracks in nitrogen the recombination is only a hundredth of that predicted by the theory. An explanation of this effect is advanced, and it is suggested that recombination is likely to be abnormally small for all heavy nuclei of velocities not exceeding 5 × 108 cm. per sec.

An experimental determination of the coefficient of recombination of ions in nitrogen and hydrogen at pressures of 20, 40 and 90 atmospheres is reported.

My thanks are due to Dr Chadwick for interest in this work, and to Dr Gray and Dr Tarrant for advice on the experimental technique of high pressure ionisation measurements. I am indebted also to the Department of Scientific and Industrial Research for a maintenance grant.

In connection with his work on singularities of surfaces, Du Val asked me to enumerate certain subgroups in the symmetry groups of the “pure Archimedean” polytopes n21 (n < 5), namely those subgroups which are generated by reflections. For the sake of completeness, I have enumerated such subgroups of all the discrete groups generated by reflections (including the symmetry groups of the regular polytopes). The work involved being somewhat intricate, several slips would have been overlooked but for the information that Du Val was able to supply from the (apparently remote) theory of surfaces.

Tests of the simple type of cloud-chamber previously described by C. T. R. Wilson(1), using various ionising sources, indicate that satisfactory photographs can be obtained over a range of final pressures in the chamber extending from atmospheric down to 7 cm. of mercury.

In the second part of this paper I pointed out how the presence of any singular point or points of the kind considered in the first part, on a rational surface, corresponds to a subgroup of the group of symmetry of the polytope which represents the properties of the system of base points in the plane representation of the surface. The subgroup is generated by reflexions, and may be the direct product of one or more factors (all the primes of symmetry in one factor being perpendicular to all those in any other factor). Each factor corresponds to a singular point on the surface, namely (in Coxeter's notation), a factor [ ] to a conic node C2, a factor [3n] to a binode Bn+2 (n ≥ 1), a factor [3n,1,1] to a unode Un+5, (n ≥ 1), and finally a factor [3n,2,1] to a unode The possible subgroups in the finite groups that arise have been enumerated by Coxeter; and we shall find that every subgroup generated by reflexions in the group of symmetry of the polytope in ε dimensions which represents ε base points corresponds to a possible configuration of the base points, in which just those rational curves of grade − 2 are actual which correspond to primes of symmetry belonging to the subgroup; without exception, for ε ≤ 6; with one exception—the subgroup [ ]7—for ε = 7; and with three exceptions—the subgroups [ ]7, [ ]8, and [31,1,1 × [ ]4—for ε = 8. Since moreover the system |k| of cubics passing simply through all the base points is in all these cases an actually existing system, for which all the rational curves of grade − 2 are fundamental, its projective model (or in the case ε = 8, in which |k| is only a pencil, the projective model of the system |2k|) provides a rational surface on which all the sets of curves corresponding to the subgroups in question actually appear as singular points.

Recent experiments on α-particles and protons have demanded an accurate knowledge of the range-velocity relation of these particles, more particularly in the case of protons, since in most experimental work their energy can only be determined indirectly by measuring their range and by deducing the velocity from a range-velocity curve.

Observations are described in which a cloud chamber was used to detect a penetrating radiation apparently produced in thunderclouds. Comparisons are made between the results obtained by Schonland and Viljoen in South Africa, there being a similarity in the magnitude of the effect observed. An analysis of the directions of arrival of the particles shows that the most favoured direction is from the north at a large angle to the direction of the magnetic field of the earth. The results obtained, though not conclusive, encourage a continuation of the experiments.

Owing to its physical and chemical properties being greatly different from those of any of the liquids which have hitherto been used in the Wilson cloud chamber, mercury has been used in the experiments described in this paper and the condensation phenomena of its vapour at different temperatures observed. Before constructing the apparatus it was considered necessary to get from theoretical considerations some idea about the magnitude of the critical supersaturation for mercury vapour in equilibrium with a drop carrying unit charge. Assuming that J. J. Thomson's formula.

where s is the supersaturation of mercury vapour in equilibrium with a drop of mercury of radius a, charge e, density σ and surface tension T, the value of which is assumed here to be independent of the radius of the drop, K the specific inductive capacity of the dielectric surrounding the drop, and R the gas constant for one gramme of weight, all at temperature θ, can be applied to the present problem, this critical supersaturation sm is given by the formula

The problem of the enumeration of the different arrangements of n letters in an n × n Latin square, that is, in a square in which each letter appears once in every row and once in every column, was first discussed by Euler(1). A complete algebraic solution has been given by MacMahon(3) in two forms, both of which involve the action of differential operators on an expanded operand. If MacMahon's algebraic apparatus be actually put into operation, it will be found that different terms are written down, corresponding to all the different ways in which each row of the square could conceivably be filled up, that those arrangements which conflict with the conditions of the Latin square are ultimately obliterated, and those which conform to these conditions survive the final operation and each contribute unity to the result. The manipulation of the algebraic expressions, therefore, is considerably more laborious than the direct enumeration of the possible squares by a systematic and exhaustive series of trials. It is probably this circumstance which has introduced inaccuracies into the numbers of 5 × 5 and 6 × 6 Latin squares published in the literature.

A considerable amount of work has recently been done on the application of wave-mechanics to the theoretical study of chemical reactions. This has consisted chiefly in calculating activation energies and strengths of various bonds by consideration of electronic states in molecules. Some work has also been done on actual reaction mechanisms. It is evident from the latter that, owing to the large masses of the particles concerned, the quantum theory and the classical treatment will give different results only for reactions involving hydrogen or diplogen. Previous attempts to deal with such reactions have consisted simply of calculating the permeability G(W) of a barrier of height equal to the activation energy for protons of energy W. The reaction rate is then assumed to be given by

When I was invited by Christ's College to deliver this year the Liversidge Lecture, it was indicated to me kindly but firmly that I was to talk about heavy hydrogen. I appreciate most deeply the honour you have done me by this invitation, especially by coupling my name (as it were) with this distinguished American newcomer to physical and chemical science. I cannot claim myself to be much more than an interested spectator of the rapid progress which has already been made in the study of this new substance, and the facts that I shall comment on (and the many others which time will force me to neglect) have been established by the collaboration of many workers—some of them our colleagues here, who would be much better qualified to speak at first hand about heavy hydrogen than I. Professor Liversidge laid it down in his deed of gift that his lecturer shall not deal with generalities or give a mere review of his subject, or give an instructional lecture suitable for undergraduates, but shall primarily try to encourage research and to stimulate himself and his audience to think, and to acquire new knowledge. In trying to fulfil his wish I shall not therefore catalogue all the new interesting properties of heavy hydrogen; this will mean, I fear, that I can make little or no further reference to its extremely exciting properties as an atomic projectile in disintegrating lithium and other nuclei, or to the equally exciting prospect that heavy water may have entirely distinct biological properties from ordinary water, being even lethal to many organisms, or to the wonderful possibilities of the new complex organic chemistry.

An apparatus for the production of beams of fast positive ions is described. Measurements of the cross-section for capture of an electron by a positive ion from a neutral gas atom are given on the case of He+ ions in helium and protons in helium and hydrogen. These are compared with the theory given by Massey and Smith, and as far as possible with previous experiments.