The problem of the double integral in the calculus of variations, when expressed in the parametric notation, was first fully discussed by Kobb. In this paper Kobb finds conditions for the minimising of the double integral

taken over the surface

Previous theories of the recombination of ions in gases are shown to be inapplicable to high pressures. An approximate quantitative theory which takes into account all the relevant phenomena is developed. The effect of both thermal agitation and mutual attraction in determining encounters of ions of opposite sign is considered, and the frequency of encounters which lead to recombination calculated. The criterion for an encounter to lead to recombination is that the drift of the ions towards each other due to their mutual attraction must be greater than their tendency to separate due to their Brownian movement. Excellent agreement is obtained with such experimental results as are available. Preferential recombination between an ejected electron and its parent ion is also discussed.

I am indebted to various members both of the Wills Physics Laboratory and of the Cavendish Laboratory for discussion of the problems dealt with in this paper.

1. Let F be an irreducible surface in [4] with apparent triple points, t in number; let P be an arbitrary point in the space and p1,…,pt the trisecant lines which can be drawn through P. Then, if F lies on at least ∞1 cubic primals, any one of these which passes through P will contain p1,…,pt in consequence and the linear system will be compounded of the congruence of trisecant lines of F. By a well-known theorem it follows that the grade of the system is zero. The free surface of intersection Φ of two cubic primals will be ruled, having t generators, each a trisecant of F, passing through every point on it.

1. A sketch of the first part of this paper, together with some of the second part, was first written out in October last, shortly after the publication of Mr Babbage's paper, A series of rational loci with one apparent double point. It is well known that the Del Pezzo quintic surface the only non-ruled quintic surface in [5], has one apparent double point; Babbage shows that this surface is a member of a series of loci in [2n + 1], each of which has one apparent double point; he establishes the existence of these loci from their representations on flat spaces ∑n, these representations being analogous to the plane representation of by means of cubic curves through four fixed points.

The problem of statistical investigation is the description of a population, or Kollektiv, of which a sample has been observed. At best we can only state the probability that certain parameters of this population lie within assigned limits, i.e. specify their probability density. It has been shown, e.g. by von Mises(1), that this is only possible if we know the probability distribution of the parameter before the sample is taken. Bayes' theorem is based on the assumption that all values of the parameter in the neighbourhood of that observed are equally probable a priori. It is the purpose of this paper to examine what more reasonable assumption may be made, and how it will affect the estimate based on the observed sample.

Let x be a normally distributed variable, which may without loss of generality be measured from the mean of the distribution, so that E(x) = 0, E denoting mathematical expectation. Then x satisfies the differential relation

where k2 (otherwise σ2 or (2h2)−1) is the semi-invariant of order 2. Also k1 = 0, and we know that kr = 0 for r > 2.

The present paper is devoted to a consideration of a type of surface which appeared incidentally in some recent work and which seems worth separate notice for its own intrinsic interest. The surface in question, which lies in space of four dimensions, is one that possesses a triple curve containing pinch-points and quadruple points, and as such is a particular case of a general surface in [4] with the same projective characters. The results given here are illustrative of those contained in the papers referred to, and with few exceptions the main facts which are used in the present discussion are quoted from the same source. In the section dealing with rational surfaces some novel results are obtained in connection with the degeneration of a plane curve into two portions, one of which is counted twice.

1. Among the equations arising in the theory of natural selection is the finite difference equation

where k is a constant which may have any value between 1 and − ∞ inclusive, but is often small. Its solution is discussed in the accompanying paper(1). It is a particular case of the equation

where Φ(x) is a known one-valued function of x. When k is small this may obviously be solved approximately by treating it as a differential equation

whence

To a non-singular algebraic surface in space of five dimensions there can generally (the Veronese surface, of order 4, and cones are exceptional) be drawn, from an arbitrary point, a finite number of chords. If such a surface be projected from a point into space of four dimensions, there will, therefore, in general, be a certain number of points upon the resulting surface, at which two sheets of this surface, with distinct tangent planes, have an isolated common point. Such points have been called improper double points. We consider an algebraic surface ψ, in space of four dimensions [4], with no other multiple points than such double points, which we shall call accidental double points. The chords of the surface ψ, drawn from an arbitrary point O of the space [4], form a surface, or conical sheet, of which a general generator meets the surface in two points. The locus of these points is a curve which we shall call the chord curve. This curve has an actual double point at each of the accidental double points of ψ There will also, generally, be a certain number of points of the surface which are points of contact of tangent planes of the surface passing through O (and therefore also points of contact of tangent lines through O, these tangent lines being generally tangent lines of the chord curve).

The literature about the Lorentz transformation is already so wide that some justification is needed for presenting even a small paper on the subject. It is possible to deduce the transformation from essentially different sets of relevant assumptions. The standard text-books on relativity throw little light on this question, which is probably due to the circumstances in which the Lorentz transformation first came into the field. Probably the question is important enough for the following considerations to be worthy of notice.

Equations (3) and (4) describe the changes undergone by a Mendelian population mating at random, and under intense selection.

The solution of linear differential equations by the method of Frobenius is a straightforward matter consisting merely in the substitution of a power series and obtaining the coefficients from a recurrence relation. The corresponding method when applied to difference equations is laborious and leads in general to divergent series from which convergent solutions can be obtained by a method due to Birkhoff(1). Actually a more appropriate method is to find a factorial series, but even here the direct substitution of such a series requires repeated transformations and throws little light on the structure of the equation. A method of symbolic operators was originated by Boole(2), but owing to the restricted definition of the operators it has a very limited scope. In the present paper a more general interpretation is given to the operators, and their application to a certain class of linear difference equations with rational coefficients is discussed.

In a former paper the application of Boole's operators π and ρ to linear difference equations whose coefficients are rational functions was discussed. Reference should be made to this paper for the definitions of π and ρ and for Theorems I–V. In the present paper a more general type of equation is considered. Two new theorems are proved and used to obtain the formal solution and to establish the convergence.

(1) The cataphoresis of ions and colloid particles is discussed in so far as it is affected by the ionic atmosphere.

(2) The ions in the atmosphere which carry a charge opposite in sign to that on the central particle are attracted to the central particle. However, when an electric field is applied to the liquid, they are able to drift away in virtue of their molecular energy, and the migration of the central particle is dependent on this fact. The relaxation force is the resultant of the forces between central particle and ions during this separation.

(3) Such a force draws the ions in the atmosphere after the central particle to some extent. From a consideration of the energy involved in the separation of particle and atmosphere, and of the molecular energies of the ions, we are able to calculate the number of ions which this relaxation force could draw through the liquid as though they were bound to the particle, and hence deduce the magnitude of the force in terms of the friction constant of the ions. The expression is the same as that given by Debye and Hückel.

(4) The cataphoresis equation usually employed for colloid particles takes no account of this relaxation force during migration. The corrected equation is given.

The marked disagreement between the specific heats at high temperatures of the diatomic gases as found by sound velocity measurements, and those calculated from theoretical considerations, has for long been a source of perplexity to statistical mechanists; a perplexity all the greater in view of the simple nature of the theoretical calculations which appeared to form as direct a test of Boltzmann's hypothesis and the theory of band spectra as was possible to large scale measurements. One way out of the difficulty is to attribute the differences to the experimental difficulties inherent in the sound velocity method of measurement at high temperatures(1), but it cannot be denied that there is a certain measure of agreement between the separate velocity determinations. It consequently becomes necessary to examine the theory.

1. Many experiments have shown that the natural heterogeneous β-rays from any radioactive source are absorbed according to a law which is exponential for small thicknesses of absorber but which falls below the exponential for larger thicknesses, giving, finally, a quite definite ‘kink’. The positions of these kinks have been measured by the author(1), by Feather (2) and by others. Experiments on the continuous spectra of β-rays (e.g. Gurney(3)) seem to indicate a definite high-velocity limit to the continuous spectrum; and experiments on the absorption of homogeneous β-rays (e.g. Madgwick(4)) show a definite extrapolated range for the β-rays. So one might expect the ‘kinks’ found to represent the extrapolated ranges of the β-rays of highest velocity. The author (1), using the extrapolated ranges of Madgwick(4) and Varder(5), finds this interpretation of the kinks to give values of the end-points agreeing with the direct experimental values of Gurney(6). Feather(2), on the other hand, uses the results of the kinks in certain cases to give an empirical relationship between the kinks and the end-points. Feather's relationship is purely empirical and is independent of any interpretation given to the kinks; for the purpose of using the kinks to find the end-points, Feather's relationship is sufficient, but the actual interpretation of the kinks is of interest. Feather(7) has considered the matter from the theoretical standpoint, and he finds no reason to expect any abrupt discontinuity; he interprets the apparent kink as the limit of the measurable β-ray effect.

When a mass of air, saturated with water vapour at room temperature θ1, is expanded adiabatically, the temperature falls to a lower temperature θ2. In general the resulting density of water vapour is greater than the equilibrium value at the temperature θ2 and condensation commences. This condensation causes heating of the gas and the temperature rises until finally at some temperature θ3 the density of uncondensed vapour is equal to the equilibrium vapour density at θ3. (θ1 > θ3 > θ2.) Condensation then ceases and the air rises slowly to room temperature by heat conduction through the walls of the apparatus.

By their measurements on the heating effect of the β-rays of Radium E, Ellis and Wooster(1) have proved that the continuous spectra of β-rays represent the actual β-rays of disintegration. Since the β-rays in the continuous spectrum have a wide range of energies, there arise problems which can best be understood by considering a particular series of disintegrations, which we may regard as typical; the most suitable series to consider is the following:

The main problem that arises is to reconcile the facts that the β-rays from Thorium B and Thorium C are emitted with a wide range of energies, while before, in Thorium A, and after, in Thorium C and Thorium C′, we have bodies which emit α-rays which are homogeneous, or, at most, consist of a few homogeneous groups [Rosenblum(2)].

The method of applying the scattering correction in accurate γ ray absorption measurements is discussed, and it is shown that the scattering correction only diminishes, as the solid angle subtended by the chamber at the source is reduced, if the absorber diameter is reduced correspondingly. The method of calculation of the scattering correction in cases in which the absorber is placed very close to the chamber is explained, and the result of the calculation for one particular chamber is shown to be in satisfactory agreement with the experimental values. The scattering correction allows the ionization function of chambers to be determined and an (ionization function)/(wave-length) curve is shown for a steel thick-walled high-pressure ionization chamber.