The present paper is concerned with a simple type of dynamical system, a system having only two freedoms and of “separable” type. The essential simplification introduced by this restriction is that the motions in the two coordinates can be discussed to some extent independently of each other. In Part I a method of classification of the possible orbits is explained. Four constants are needed to define the motion completely, but the general nature of the orbit depends on two constants suitably chosen, and the classification is effected by reference to a plane in which the two constants are taken as Cartesian coordinates. It is shewn that the plane of reference is divided into regions by critical curves, and that the orbits represented by points in the same region (each point representing a set of orbits, not a single orbit) are of the same general type. The various possible types of orbits, and their stability, in a special sense, are discussed. In Part II the theory is applied by way of illustration to a somewhat trivial example, the orbits for a particle under a central attraction proportional to the inverse (n + 1)th power of the distance. The most striking thing here is that the possible types of orbits are essentially the same for all values of n greater than 2. In Part III the theory is applied to the classical problem of a particle subject to an attraction of Newtonian type to two fixed centres, a problem of some interest in relation to the general problem of three bodies, and to some current questions in atomic dynamics. The problem is of course not a new one, but it seems that a satisfactory classification of all the possible orbits has not previously been given. It must be confessed that the complete classification for this problem is very laborious.

It is well established by observation that in the ordinary flow of a stream there is a slow transverse flow outwards from the middle near the bottom, and inwards near the top; and such a circulation seems to be required to explain the fact that the filament of maximum onward velocity is depressed some distance below the centre of the free surface. Qualitative explanations of the phenomenon have been offered, depending on the extra friction near the sides, but it seems that the true explanation must depend on something more fundamental. For any explanation to be satisfactory it must show uniform linear flow to be impossible; and actually it can at least be rendered highly probable that such flow can exist over a very wide range of conditions.

1. Young's criterion for the convergence of a Fourier series may be stated in the following form:

The integral

tends to zero as ω → ∞ whenever

(A) ø(t) is absolutely integrable in (0, a),

(B) ø(t) → 0 in mean as t → 0,

(C) tø(t) is of bounded variation to the immediate right of t = 0 and, as t → 0,

1. The following investigations were undertaken in connection with Cayley's problem of seven lines on a quartic surface.

1. The principal object of this note is to give a new derivation of the coincidence formula in the general correspondence of points in a rational n-space. It is supposed that the correspondence is algebraical and that there are ∞n pairs of corresponding points P, P′ (in the geometrical sense of ∞n). Denote by the symbol (i) (n − i)′ the number of pairs P, P′ such that P lies in a general i-space and at the same time P′ in a general (n − i)-space. When (P, P′) tends to a coincidence of the correspondence, the join PP′ tends to some limiting position, which may depend on the way in which (P, P′) tends to the coincidence. In general the limiting lines will lie on a conical figure—a star—with the double point as vertex. In the simplest case of an isolated double point, the star fills the whole n-space. Denote by the symbol ε(i, n) (n − i) the number of times it happens that a double point lies on a general (n − i)-space while the limiting line PP′ meets a general i-space(and lies in the n-space).

The problem here considered is a particular case of the more general problem of finding the number of [k]s, lying in a space of dimension n(n > k), which satisfy a certain number of conditions of the following type: viz. to meet a given [a0] in a point, a given, [a1] containing the [a0], in a line, etc., finally to lie in a given [ak] containing [ak−1]. The number of such conditions is such that there are just a finite number of [k]s which satisfy them all. The method which we employ is the one introduced by Schubert and commonly known as the “degeneration method” it is explained very briefly below.

1. The following proof of Bateman's expansion may be of interest since it avoids the use of Appell's hypergeometric functions of two variables, though it suffers from the defect that it presupposes a knowledge of the expansion.

1. The form of the absorption curves of homogeneous β-rays in aluminium has been investigated by means of a Geiger counter. It is found, in agreement with Schonland, that the main portion of the curve is linear, with pronounced initial and final flattenings. Similar initial flattenings were found for copper and silver, but not for gold.

2. The range of β-rays emergent at different angles has been determined, and it appears that the loss of range is due to the increased path consequent upon scattering.

3. Some evidence has been obtained of the angular distribution of plurally scattered particles, and of the existence of a most probable angle of scattering, similar to that found for α-rays by Geiger.

In classical mechanics the state of a dynamical system at any particular time can be described by the values of a set of coordinates and their conjugate momenta, thus, if the system has n degrees of freedom, by 2n numbers. In quantum mechanics, on the other hand, we have to describe a state of the system by a wave function involving a set of coordinates, thus by a function of n variables. The quantum description is, therefore, much more complicated than the classical one. Let us consider, however, an ensemble of systems in Gibbs' sense, i.e. not a large number of actual systems which could, perhaps, interact with one another, but a large number of hypothetical systems which are introduced to describe one actual system of which our knowledge is only of a statistical nature. The basis of the quantum treatment of such an ensemble has been given by Neumann. The description obtained by Neumann of an ensemble on the quantum theory is no more complicated than the corresponding classical description. Thus the quantum theory, which appears to such a disadvantage on the score of complication when applied to individual systems, recovers its own when applied to an ensemble. It is the object of the present note to examine this question more closely and to show how complete the analogy is between the quantum and classical treatments of an ensemble.

The purpose of the present paper is to illustrate the use of the Compton Quadrant Electrometer for current measurement by the rate of deflection method. There are several points in the theory and technique for this instrument which do not appear to have been mentioned in any previous papers. With the Compton Electrometer the rate of deflection method gives accurate measurements of currents of 10−14 amperes in less than a minute. The instrument is of course more difficult to use than the Dolezalek type, and it is for this reason that an attempt is made here to discuss and illustrate its use. The theory is given in Compton's original paper and there is also an important note on the subject by Cox and Grindley.

An attempt has been made in the work described in the present paper to use the method initiated by Hartree, for the numerical solution of the Schrödinger wave equation for an atom with a non-Coulomb field of force, to estimate the number of dispersion electrons (hereinafter denoted by “ƒ” for brevity), corresponding to the lines of the principal series of the optical spectrum of lithium, and also to the continuous spectrum at the head of the series. Various attempts have been made to do this for hydrogen and other atoms by an application of the Correspondence Principle, but the first successful attempt at a complete description was made by Sugiura†, who has calculated ƒ for the Lyman, Balmer, and Paschen series and the corresponding continuous spectra, by using the known analytical solutions of the wave equation for an electron in a Coulomb field. The same author has also calculated ƒ for the first two lines of the principal series of sodium, by the utilisation of an empirical field of force in the atom calculated from the observed term-values by a method based on the old quantum theory. He has estimated the contribution to Σƒ (summed for the whole series) due to the continuous spectrum by the theorem that Σƒ = 1 in the one-electron problem§. This property provides a useful check on the work when ƒ for the continuous spectrum is also calculated. In the present paper ƒ for the continuous spectrum is actually calculated and it is found that Σƒ = 1 to a good approximation.

The equations of propagation of electromagnetic waves in a stratified medium (i.e. a medium in which the refractive index is a function of one Cartesian coordinate only—in practice the height) are obtained first from Maxwell's equations for a material medium, and secondly from the treatment of the refracted wave as the sum of the incident wave and the wavelets scattered by the particles of the medium. The equations for the propagation in the presence of an external magnetic field are also derived by a simple extension of the second method.

The significance of a reflection coefficient for a layer of stratified medium is discussed and a general formula for the reflection coefficient is found in terms of any two independent solutions of the equations of propagation in a given stratified medium.

Three special cases are worked out, for waves with the electric field in the plane of incidence, viz.

(1) A finite, sharply bounded, medium which is “totally reflecting” at the given angle of incidence.

(2) Two media of different refractive index with a transition layer in which μ2 varies linearly from the value in one to the value in the other.

(3) A layer in which μ2 is a minimum at a certain height and increases linearly to 1 above and below, at the same rate.

For cases (2) and (3) curves are drawn showing the variation of reflection coefficient with thickness of the stratified layer.

Case (3) may be of some importance as a first approximation to the conditions in the Heaviside layer.