1. Having given the terms sn of a sequence

then Hölder's means are defined by

Cesàro's means are defined by

and a third kind, σ, of mean which will be used in this paper is defined by

1. The problem of finding the number of space cubic curves which pass through p given points and have 6 – p given lines as chords has been solved by several different methods. The similar problem, in space of four dimensions, of the number of rational quartic curves which pass through p given points and have 7 – p given trisecant planes has been solved for p > 1 by F. P. White and for p = 1 by J. A. Todd. In a recent paper I discussed a similar problem for elliptic quartic curves. My present object is to apply the method of that paper to the problems mentioned above and to obtain two other sets of numbers which I believe to be new. They are (1) the number of rational normal quartic curves which have three assigned chords, pass through p assigned points, and have 3 – p assigned trisecant planes; (2) the number of curves of intersection of three quadrics in [4] which have a suitable number of assigned points and chords. The evaluation of Schubert symbols which occur in the work is done by means of a formula due to Giambelli‖. This formula is stated in a more simple form in a note at the end of this paper (7).

The object of this paper is to find the characteristic functions and the characteristic numbers of the partial differential equation

valid in a domain G, and where on Γ, the boundary of G. The method employed is to transform two quadratic forms to their common self-conjugate “triangle“ of reference. The solution to the problem is given by this method in a simple manner, without the use of the integral equation theory, or the use of minimal sequences.

In a paper recently published I found it necessary to extend those investigations of Lord Rayleigh which deal with the instability of the surface common to two streams of fluid. The Rayleigh investigation is equivalent to a first order approximation to the flow and concludes that the initial form of the disturbance of the common surface is maintained but that its amplitude increases exponentially with the time, the form of the surface at time t being, with particular values inserted,

where a is the initial amplitude, 2π/k the wave-length, U the velocity of the upper stream, – U the velocity of the lower stream, and ρ the density of both fluids. A second order approximation to the flow for the above particular case shows that the original disturbance does not grow symmetrically, the ordinate of the common surface being given by

(A slightly amended form of this solution is given byequation III(8).) The solution suggests the initial stages in the formation of vortices. The problem was further discussed by a numerical step-by-step method and showed the formation of vortices quite clearly.

1. In the Lehrbuch der abzählenden Methoden Zeuthen considers the problem of determining the nature and multiplicity of the tangent lines and planes to a degenerate surface in ordinary space. The surface φ in question consists of n arbitrary planes; these meet in pairs, forming double lines, and in threes, forming triple points. The tangent cone to φ from an arbitrary point consists of plane pencils, each counted twice, as follows from the known theory of degenerate plane curves. According to Zeuthen the tangent planes are of two types:

(i) those passing through one of the triple points; there are thus such systems, and each is to be counted six times;

(ii) those passing through certain points S lying on the double lines; these depend upon the passage to the limit, it being assumed that φ is obtained as the limiting form of some nondegenerate surface. Two such points lie on each double line, so that there are n(n−1) such systems of planes.

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.

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.

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).

(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.

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.

The application of the quantum theory to the collision of electrons with atoms has now been worked out in some detail. For fast collisions the calculations present no difficulty, since Born's method may be applied and the probability of excitation of a given level of a simple atom can be worked out quite accurately. The corresponding problem for the case of slow collisions is much more difficult and requires a knowledge of the exact solution of the equation representing the motion of electrons in the static field of the atom considered. Using such solutions obtained by Faxén and Holtsmark's method, and allowing also for electron exchange, Massey and Mohr have been able to calculate excitation probabilities for low voltage impact in hydrogen and helium.

In an alternating current motor (of simplest type) the internal E.M.F. e, the terminal tension e, the useful electrical power transformed p and the rotor phase angle φ are connected by the known formula

in which

where r is the resistance and l the self-inductance of the armature and ω the angular velocity of the rotor.

An account is given of a simple experiment designed to illustrate quantitatively the phenomena of coupled oscillations. Two similar small magnets are suspended in the earth's magnetic field at a suitable distance apart so that there is appreciable magnetic interaction between the two oscillatory systems. Under the conditions employed, the equations of motion reduce to a simple form, and the experiment may be used as a method of measuring the intensity of the horizontal component of the earth's magnetic field.

The upper portion of the β-ray spectrum of thorium C″ has been investigated by measuring the tracks produced in a Wilson chamber. No trace of a “tail“ is found, no tracks of Hρ greater than 10,800 being observed in some 600 to 800 disintegrations. The end-point is placed at 9400 Hρ, somewhat higher than the value obtained by other workers.

A high-tension supply suitable for all types of Geiger counter must supply up to 5000 volts, free from A.C. ripple and other electrical disturbances, and it must be possible to rely on the voltage remaining constant to within one or two parts in a thousand over considerable intervals of time. With the apparatus here described these requirements have been fulfilled sufficiently well for satisfactory operation of Geiger-Müller tube-counters from ordinary A.C. mains, and since the method of keeping the voltage constant does not appear to have previously been used for this purpose, it seemed desirable to record it briefly.

The spontaneous disappearance of an electron and proton in the form of radiation has often been suggested as a possible source of energy to maintain the emission of radiation from hot stars. It can be calculated that the energy of a single quantum formed in this way should be about 1000 million electron-volts. In order to account for the absorption of the ultra-penetrating radiation observed by Regener in deep lakes, and interpreted on the basis of the Klein-Nishina formula, a quantum of still higher energy is required. Sir James Jeans has recently calculated that the transformation of a helium atom into a single quantum would provide a radiation of about the required penetrating power.

Previous work on the existence and period of radium C′ is discussed with reference to an experiment of Jacobsen which provides evidence that a γ ray transformation of period comparable with that of radium C′ precedes the expulsion of α particles. It is shown that, from Jacobsen's results, part of the γ radiation from a source of recoil atoms should originate in the space surrounding the source.

A rough calculation is made which shows that the γ rays above the source should be detectable by ordinary methods, and a description is given of an ionisation method capable of detecting the effect. The γ rays predicted by Jacobsen's experiment were tested for by using specially prepared sources of radium C. Phenomena associated with α recoil were also investigated for sources of radium (B + C) and thorium (B + C).

No evidence of a γ ray emission from the space above any of the sources was obtained. The negative result indicates that the interval between the departure of the disintegration particle and the emission of the γ ray quantum is considerably less than 10−5 second.

The interaction between gas atoms and solid surfaces has been studied experimentally for many years. The first detailed study of the nature of this interaction was perhaps due to Knudsen, who in his classical researches introduced the idea of the accommodation coefficients for energy or momentum exchanges between gas atoms and a solid surface, when the gas atoms and the solid are at different temperatures or possess different mass motions. Knudsen and other investigators have given numerical values for these accommodation coefficients for various gases and solid surfaces, which seem to indicate that the accommodation coefficients are never small and are often of the order unity. This means that the gas molecules before reflection accommodate themselves almost completely to equilibrium with the temperature or motion of the wall. Before the recent work of Roberts referred to below, which has inspired this paper, no special precautions had been taken in the preparation of the wall surface, and as we now know the walls used by all previous investigators must have been completely covered with at least a mono-molecular film of gas. Thus the old observations of the accommodation coefficient do not determine it under precise conditions, and find in fact a value many times larger than that found by Roberts for the energy exchanges between the gas and a clean tungsten surface. For helium and metal with a dirty surface the value of the accommodation coefficient is some 6 times as large as the true value for the clean surface. The older values of the coefficients were so large that there were apparently grave difficulties in the way of any simple theory, but this is so no longer.