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.

The general theory of surfaces has been developed without much attention to the simpler aspects of the subject and, in particular, to the problem of determining some elementary criteria of regularity or referability; there is, in fact, only one well-known criterion of this kind, due to Castelnuovo and Enriques. (Theorem 2 of the present work.) It seems desirable, therefore, to establish some tests which require no more knowledge of a surface than its order, sectional genus and normal space. A number of such tests is given in the present paper. Before it was written the author had the advantage of consulting an unpublished manuscript by Professor Comessatti on the classification of surfaces in S4; in this work the regularity of surfaces of order eight is considered and, in part, those of order nine; the four-dimensional case of Theorem 11 and a particular case of Theorem 6 and Theorem 14 are given. Theorems 4 and 5 are also proved by Comessatti. For permission to publish these results and for the help derived from them the author's grateful thanks are due.

1. Saks calls an integral complete if it satisfies two conditions, of which one, the condition (C), is as follows.

If f(x) is defined in (a, b) and is integrable in each interval (a + ε, b − η), where a < a + ε < b − η < b, and

exists, then f(x) is integrable in (a, b) and the integral over (a, b) is equal to the above limit.

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 aim of the following exposition is to give some idea of the scope of modern algebra in the light of the theory of ideals. It consists for the most part of definitions, examples, and the statement of theorems without proof; but the selection is severely restricted even in idealtheory. Thus no reference is made to the highly important matter of homomorphic, or multiply isomorphic, correspondence and its consequences.

1. Elsewhere I have expressed the opinion that a linearly measurable plane set of points, of upper density ½, has zero projection (i.e. projection of measure zero) on almost all directions. Though unable to prove this I have there managed to obtain a partial result. I would not attempt to guess to what extent this is true of a general irregular linearly measurable plane set; these sets are not amenable to the methods used in the case of those of upper density ½, the latter having a remarkable and simple structure. It is a quite trivial matter to construct an irregular set, of upper density 1, with positive projection on some one direction; and from it we can deduce another such set with positive projections on a denumerable every-where-dense set of directions. But, working on a suggestion due to Mr Besicovitch, I have obtained an irregular linearly measurable plane set, of upper density 1, with positive projection on a set P of directions, where the members of P in any sector have cardinal t, though P has measure zero. It may be that the set of points has positive projection on directions other than those of P, but there are no obvious indications of any such directions.

When any metal film is deposited on the surface of a massive copper oxide slab and illuminated with visible light, a current flows in such a direction that the electrons pass from the Cu2O to the metal in greater numbers than from the metal to the Cu2O. This is the reverse direction to the easier flow of electrons under an applied voltage when the contact is allowed to act as a rectifier. This effect is sometimes called the “Vorderwandeffekt” to distinguish it from the similar group of phenomena which occur at the natural interface between massive copper and Cu2O when the surface of the copper has been oxidized in situ (“Hinterwandeffekt”).

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).

An account is given of attempts to detect radiation in thunder clouds by means of photographic plates attached to balloons. Eleven balloons were sent up and the plates attached to ten were recovered and examined.

The result was negative in every case.

A small ionisation chamber with quartz condenser intended for detecting radiation in thunder clouds is described.

These observations were made at the Solar Physics Observatory, Cambridge, and I have to thank the Director for giving me every facility.

I am also greatly indebted to Professor C. T. R. Wilson for his continued advice and interest.

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.

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.

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.