The present note was inspired by the desire to see whether the exposition of the theory of the Segre cubic primal of ten nodes was simplified by using only five coordinates instead of the six redundant coordinates introduced by Stéphanos and Castelnuovo. But the simple remark that the ten nodes may be separated into two simplexes which are polars of one another in regard to a quadric—which may or may not be novel—suggests the comparison of the theory of five associated lines in space of four dimensions with familiar properties of eight associated points in three dimensions; especially as it appears (§ 8) that the ten nodes do not form an associated set. With the repetition, for the sake of clearness, of several results which are familiar in general terms, there seems enough novelty to make the note of some utility. The form found for the equation of the Segre primal in five coordinates (§ 5) seems also noticeable.

Let a1, a2, … be a sequence of variables. Write

We use the notation Bn = Bn (a1, …, an) to indicate the variables on which Bn depends.

Let d(m) denote the number of divisors of the integer m. Chowla has conjectured that the integers for which d(m + 1) > d(m) have density ½. In this paper I prove and generalize this conjecture. I prove in § 1 a corresponding result for a general class of functions f(m), and in § 2 the result for d(m) which is not included among the f(m). I employ the method used in my paper: “On the density of some sequences of numbers.”

The Abel sum of the series can be written in the form

where

This suggests that we should define the “Abel limit”, as u → ∞, of any function A(u) as being given by the expression (1) whenever this exists. We shall, however, in this paper, restrict ourselves to functions A(u) which are bounded in any finite interval. Since we are concerned only with the behaviour of A(u) as u → ∞, this does not involve any serious loss of generality, while we avoid difficulties arising from the divergence of integrals at finite points. We note that the expression (1) can be written in the form

where

A1(u) may conveniently be described as the “Abel transform” of A(u).

When a probability distribution is specified by more than one parameter, the statistical information in a sample on an unknown α will usually depend on the values of the other unknowns. For large samples where the joint distribution of the estimates tends to normality, and the efficiency of the estimation to a maximum, little theoretical difficulty exists. In a discussion(1) of the more general theory of small samples, I have stressed the importance of properties of sufficiency.

It is proved that if there exists a sufficient statistic for the estimation of an unknown parameter of a population, the frequency function of the population must be of a certain type.

It is shown that some modification of previous theory of the intrinsic accuracy of statistics is necessary when the range of the population sampled is a function of the parameter to be estimated.

Finally, the theory is extended to sufficient sets of statistics, i.e. sets of statistics which together contain all the information provided by a sample about an unknown parameter.

In the study of random events and associated fluctuations such as occur in the shot effect, a theorem first stated and discussed by Dr N. R. Campbell can often be employed. It applies on any occasion when there occur at random a number of events whose effects are additive. Let us suppose that a single event occurring at time tr causes at time t an effect f(t − tr) in some part of the observed system, and that the effects of different events are additive, so that the total effect or output is ϑ(t), given by

We may suppose that the same events cause another set of effects g(t − tr) with output ϑ(t), where

Both the functions are assumed to be bounded and integrable in the Riemann sense, as are all the functions studied in physics.

In this paper we consider the slow two-dimensional motion of viscous liquid past a sharp edge projecting into and normal to the undisturbed direction of the stream. The liquid is supposed bounded by rigid planes represented by ABCDE in Fig. 1, and, apart from the disturbance caused by the projection, is assumed to be in uniform shearing motion. The stream function is then a bi-harmonic function that must vanish together with its normal derivative at all points of the boundary, and must be proportional to y2 at a great distance from the projection.

It is shown that the eigenvalues of the angular momentum of the electromagnetic field containing a number of charged particles, apart from spin angular momentum, are only the integral multiples of ħ. This is shown by using cylindrical polar variables, and taking a particular choice of the vector potential in which the radial component is zero, defined explicitly in terms of the magnetic field strengths. By expanding in terms of the Fourier functions einθ, the angular momentum is separated out into terms independent of one another, each taking on only integral values in units of ħ.

The arguments all apply equally well to a modified field theory such as that of Born and Infeld.

In a recent paper Dirac has shown that by passing from the ordinary Euclidean space to a four-dimensional conformal space, some of the equations of physics can be written in a tensor form, the indices of which take on six values. Those equations which can be written in this form are then invariant under conformal transformations of the Euclidean space. Among the equations of physics which have this more general invariance are the Maxwell equations, as was proved by a direct transformation a long time ago by Cunningham, and Bateman, so that Dirac's paper provides an alternative and more general proof of this result. Certain errors ∥ in Dirac's paper, however, necessitate a reformulation of the proof. Before we do this in § 2, we briefly recapitulate in § 1 some of the general results derived there. In § 3 we investigate further the conformal invariance of the wave equation for an electron in the presence of a general electromagnetic field.

The disintegration of nitrogen by slow neutrons has been studied in photographic emulsions of different sensitivity, which enable an unambiguous distinction to be made between the emission of α-particles and protons. Evidence has been obtained that the disintegration takes place according to the reaction

with a cross-section of about 10−24 cm.2

A repetition has been made, with the introduction of electrical counting methods, of the experiment of Bothe and Geiger on the simultaneous production of a scattered quantum and of a recoil electron in the scattering of X-rays. Evidence has been obtained that such coincidences do actually occur, in agreement with the results of the original experiment, and with the Compton theory of scattering.

A source of ions consisting of a low voltage arc, yielding homogeneous ion currents of the order of ½ milliampere at low potentials is described. With such a source, atomic disintegrations have been observed with ion beam energies of less than 8kV. Approximate values of the absolute yield for protons from the deuteron on deuterium disintegration, and α-particles from the protons on lithium disintegration are obtained.

The dissociation of hydrogen by a hot tungsten filament has been studied under conditions such that all the atomic hydrogen produced is effectively removed by reaction with molybdenum or tungsten oxide. The rate of production of atomic hydrogen is many times greater than was inferred from earlier work. With the tungsten at constant temperature the rate of dissociation is proportional to the square root of the pressure. A formula is given for the rate of production of hydrogen atoms per sq. cm. of the tungsten per second.

The fact that the rate of production of atomic hydrogen at a tungsten surface at a given temperature is proportional to the square root of the hydrogen pressure means either that the important process is the evaporation of atoms from an adsorbed film which over the whole range of experimental conditions is sparsely occupied, or that the production of atoms is in the main due to a process in which a hydrogen molecule strikes a bare tungsten atom in the surface, one atom being adsorbed and the other evaporating and the surface being almost completely covered over the whole range of experimental conditions. Either process leads to a temperature variation in the rate of atom production in agreement with experiment. A definite decision between the two processes cannot yet be made.

The results of experiments on the diffusion of hydrogen through metals from a pressure p on one side to a vacuum on the other show that at high pressures the amount diffusing varies linearly with p½ but that at low pressures it varies more rapidly. The difficulty usually encountered when diffusion from an adsorbed layer into the solid is considered theoretically is that the theory indicates that saturation should be reached. In this paper it is shown that this difficulty is due to the omission of an important process at the surface and that by including this process the experimental results can be explained.

In automatic temperature-controlling apparatus a temperature-sensitive element, commonly called a thermostat, serves to increase or decrease a supply of heat to the body or oven whose temperature is to be prevented from changing. The thermostat and heating elements are commonly both electrical, the thermostat comprising a “master coil” whose resistance changes with temperature, the heating element consisting of a “slave coil” carrying the heating current. The temperature of the master coil is made to control the current in the slave coil, usually by the interposition of some form of relay with the appropriate translating and amplifying apparatus. Since every form of relay exhibits “backlash”—the critical values of the operative signal at make and at break are not exactly equal—such a system necessarily oscillates or “hunts” through a range at least as great as the backlash of the relay; and since further there is thermal separation between the master coil and the slave coil, the hunting has a range exceeding the backlash of the relay. It is a common experience with such thermostatic apparatus that, owing to this action, continued improvement towards constancy of temperature is not attainable by increasing the delicacy with which the temperature of the master coil controls the current in the slave coil.

Recent reports contain tables of a parameter K required in calculating the performance of an airscrew by a new method.

The method of calculating K (due to Goldstein) is unsuitable for large values of the pitch especially near the tip of the airscrew.

In the case of infinite pitch we fall back, for a two-bladed airscrew, on the problem of a rotating lamina in two dimensions.

The solution for a cross lamina (corresponding to a four-bladed propeller) is given below and the tables of K for four blades are completed.

A formula for the limit of K/Kp at the airscrew tip is given for a propeller with any number of blades, where Kp is an approximate value of K due to Prandtl.

K for any number of blades is given in the form of an infinite series. The case of three blades is discussed in detail.