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 Before it was written the author had the advantage of consulting an unpublished manuscript by Professor Comessatti on the classification of surfaces in S4; in this work the regularity of surfaces of order eight is considered and, in part,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.

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.

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.

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.

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”).

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.

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.

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. In a recent paper I established new conditions for a form φ of order n, homogeneous in r + 1 variables, to be expressible as the sum of nth powers of linear forms in these variables; and for this expression, if it exists, to be unique. These conditions, I further showed, may be stated as general theorems regarding the secant spaces of manifolds Mr in higher space, namely:

Necessary and sufficient conditions that through a general point of a space N, of h (r + 1) − 1 dimensions, there passes (i) no, (ii) a unique (h − 1)-dimensional space containing h points of a manifold Mr lying in N are that

(i) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr meets Mr in a curve, so that Mr cannot be so projected upon a linear space of r dimensions;

(ii) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr does not meet Mr again, so that Mr can be so projected, birationally, upon a linear space of r dimensions..

This paper gives experimental results for the scattering of 4 and 2 volt electrons in nitrogen, which are compared with Stier's theoretical curves. An approximate theory of the scattering of electrons in nitrogen is given which is applicable when the wave length of the electron is small compared to the internuclear distance. The results are compared with experiment.

The purpose of this paper is to provide a point of vantage from which to attack combinatorial problems in what may be termed modern, synthetic, or abstract algebra. In this spirit, a research has been made into the consequences and applications of seven or eight axioms, only one [V] of which is itself new.

In a recent paper we have discussed certain general principles underlying the determination of the most efficient tests of statistical hypotheses, but the method of approach did not involve any detailed consideration of the question of a priori probability. We propose now to consider more fully the bearing of the earlier results on this question and in particular to discuss what statements of value to the statistician in reaching his final judgment can be made from an analysis of observed data, which would not be modified by any change in the probabilities a priori. In dealing with the problem of statistical estimation, R. A. Fisher has shown how, under certain conditions, what may be described as rules of behaviour can be employed which will lead to results independent of these probabilities; in this connection he has discussed the important conception of what he terms fiducial limits. But the testing of statistical hypotheses cannot be treated as a problem in estimation, and it is necessary to discuss afresh in what sense tests can be employed which are independent of a priori probability laws.

A three-phase star-connected alternating-current motor of simplest type, connected to busbars maintained at sinusoidal potentials e1, e2 and e3 with respect to the star-point and connected to a shaft load which exerts a reactive torque f on the rotor, is dynamically specified by the following functions:

where the are the armature currents in the three phases, i is the current in the amortisseur circuit, Q the current in the field circuit, θ the rotor position angle, T the kinetic energy, V the potential energy, S the Rayleigh dissipation and U the activity of the external forces on the machine. The self-inductances of the armature circuits and the mutual inductances between them are assumed to be constant.

The problem of discriminating between the different real forms of the rational twisted quartic was first attempted by Adler in 1882. His catalogue is incomplete and some of his results are inaccurate, as has been pointed out by Richmond †. The work was however superseded by a paper in 1891 by Rohn‡ who considered the curve in relation to its allied Steiner quartic surface; his results may be compared with those given in the final section of this paper. Substantially the same results were obtained by Vietoris§ who mapped the curve on a conic by a somewhat different method from that used by Adler.

Some experiments are described in which the efficiency of ionisation of potassium at a hot molybdenum surface was investigated. The degree of ionisation observed is consistent with the most recent measurements of the electron work function of molybdenum.

1. The surface here discussed arises most naturally in the study of a certain cubic primal in space of five dimensions. Let G be a cubic primal in five-dimensional space, containing two planes M1, M2 which do not intersect. From any point of G can be drawn one transversal to M1, M2; this does not meet G again, and meets a fixed prime p in one point. Thus G can be birationally projected upon p, and is rational.