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.

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.

The test of significance of the difference between two means was extended by means of Fisher's methods of analysis of variance to test jointly the differences between several means, the test being essentially the same as the test of significance of two variances. But though it is true in many cases that we are chiefly concerned with two independent estimates of variance, and the method of analysis of variance is available, the problem of jointly testing the variances obtained from several independent samples does not fit obviously into this scheme.

The surfaces whose prime-sections are hyperelliptic curves of genus p have been classified by G. Castelnuovo. If p > 1, they are the surfaces which contain a (rational) pencil of conics, which traces the on the prime-sections. Thus, if we exclude ruled surfaces, they are rational surfaces. The supernormal surfaces are of order 4p + 4 and lie in space [3p + 5]. The minimum directrix curve to the pencil of conics—that is, the curve of minimum order which meets each conic in one point—may be of any order k, where 0 ≤ k ≤ p + 1. The prime-sections of these surfaces are conveniently represented on the normal rational ruled surfaces, either by quadric sections, or by quadric sections residual to a generator, according as k is even or odd.

This paper illustrates the problems of general linear propagation mainly by analysis of the conditions necessary for effective resonance, which are now of great technical importance. Another theory is also broached, that of the relation of the arterial pulse to the blood-flow, first explored with adequate hydraulic instincts by Young in 1808.

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.

By a finite lattice† is meant any finite class having the following properties:

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.

In a paper by Mr Lyons and myself an elementary proof of the following theorem was given:

Given three lines a, b, c in space, if a′ is the line of shortest distance of b and c, b′ that of c and a, and c′ that of a and b; and if x is the line of shortest distance of a and a′, y that of b and b′, and z that of c and c′, then x, y, z have a common line of shortest distance.

Two perpendicular intersecting straight lines will be said to be normal to each other. Then the Petersen-Morley theorem is: Given arbitrarily three lines a, b, c, let a′ be the unique normal to b and c, b′ that to c and a, and c′ that to a and b; then the lines x, y, z, normal respectively to the pairs of lines a and a′, b and b′, c and c′, have a common normal.

In recent seismological work several questions have arisen concerning the construction of tables from data containing a certain amount of accidental error, and it seems that some of the methods may be of more general interest. The first is the problem of smoothing. A method suggested by Dr L. J. Comrie was as follows. Suppose that we have values of y for five equally spaced values of x, and that we wish to make the second differences vary as smoothly as possible. We try to find a cubic polynomial such that the sum of the squares of the deviations of y from it will be a minimum. It is found that the polynomial is less than the observed value of y at the centre of the range by of the central fourth difference; or approximately by of the fourth difference. I have since found that the method had previously been suggested by Sir G. H. Darwin. It gives a great improvement in the steadiness of the differences and thereby makes interpolation much easier.

Three main stages may be marked in the development of the theory of the optical properties of metals. First, there is Drude's original theory, based on Maxwell's equations; in this theory the current density j at any point in a metal is supposed to be equal to the product of the electric vector of the light and of the conductivity of the metal. The theory yields the well-known Hagen-Rubens formula for the reflecting power, which appears to be in agreement with experiment for very long wave-lengths (λ > 10μ), but leads to completely incorrect results in the optical region. Various investigators ‡ have therefore modified the theory to take account of the finite mass of the electron; the formulae obtained pass over into the Drude formulae for sufficiently long wave-lengths. Finally a quantum theory of the phenomenon has been given by Kronig§, the electrons being treated as moving in a periodic field due to the crystal lattice in the manner originated by Bloch; this theory, in its turn, becomes identical with the modified classical theory if the periodic lattice is neglected.

It has long been realised that there are difficulties in understanding the continuous β-ray spectra emitted by certain radioactive bodies, if the conservation of energy, as applied to nuclear processes, is to be retained. In the case of radium E, for example, the energy of the disintegration β particles varies from quite low values to about a million volts, the mean being nearly 400,000 volts.

1. Many insulating crystals, such as rock salt, after exposure to high energy radiation, e.g. X-rays, β-rays, ultra-violet light or γ rays, acquire, as a result of this treatment, two new properties: (i) a new absorption band, usually situated in the visible spectrum, well separated from the continuous absorption in the far ultra-violet, and, when intense, giving visible coloration to the crystal in ordinary light; (ii) the power of showing what has been termed the inner photo-electric effect, i.e. when subjected to an electric field and then illuminated with light in the region of the new absorption band, the “activated” crystal gives an instantaneous response in the form of a small electronic conduction; in the dark or for light outside this band it is still non-conducting (Fig. 1).

Let SsTt denote a space of s + t dimensions with the property that the square of the distance between any two points

is of the form

where s of the ε's are equal to + 1 and t of them to — 1.

Several papers have recently been published which include proofs of the following:

Theorem 1. Those trisecant planes of a rational normal quartic curve Cu which meet a second rational normal quartic curve Cv having six points in common with Cu also meet a third rational normal quartic curve Cw through these same six points. The three suffixes may be permuted in any way, the three curves forming a symmetrical set.

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.