The principal object of the present paper is to give a set of sufficient conditions, of considerable generality, under which a plane curve assigned to pass through certain points with given multiplicities shall be irreducible. The case considered is that in which the assigned multiple points of the curve are of such generality that the conditions presented to the curve at these points are independent. We shall shew that subject to certain explicit restrictions on the points, which ensure, for example, that no three points shall be collinear whose prescribed multiplicities in aggregate exceed the order of the curve, the curve determined by the points will be irreducible if its freedom is not negative, save in the case when it degenerates into a repeated elliptic curve. A familiar example of this last case arises in the problem of constructing a sextic with nine assigned nodes; in general the curve in question consists of the unique cubic through the nine points, counted twice.

1. Considerable advantage has resulted from the postulation of unreal elements in projective geometry. In the first place these unreal elements were defined in terms of points represented by complex coordinates, and their use in purely geometrical reasoning had become well established before any serious attempt was made to justify this use, independently of algebraic considerations, by providing real representations of the unreal elements. The first successful attempt was that of von Staudt, who represented an unreal element by an elliptic involution associated with an order. In this system an ordered set of four real points is required to specify an unreal point. The system is comparatively simple to deal with in a single real plane, more complicated in a single real [3], and rapidly increases in complexity as the number of dimensions of the real field is increased.

The considerations set forth in the present paper are valid for the most part in parallel form for integration in space of any number of dimensions, and in the widest measure for any modification of the process that may be devised naturally: integration of functions not necessarily one-valued, with respect to functions of sets, or functions of ∑, not necessarily unchanged by subdivision, in fields as wide or as restricted as may be convenient, as long as we maintain the possibility of forming subdivisions of arbitrarily small “norm,” and of splitting up such a subdivision arbitrarily into two parts, each of which may be subdivided independently in any manner, to give a subdivision of the whole. These conditions are essential for the proper working of the theory, and in particular for the validity in more dimensions of Hyslop's method referred to in 5.

The principles and general properties indicated in the first part of the present theory are those which belong most strictly to the province of the “algebra of many-valued quantities,” by their practically exclusive reliance on notions of equality, union, and passage to the limit in this domain.

1. In a paper published recently in these Proceedings, I have proved the formulae

where R(m)>0;

where R(m) > −½; and

where R(m) >− 1. In each case n is unrestricted.

An abstract continuous group, G, is an abstract space whose points have a continuous associative multiplication law, with division. The object of this note is to sketch a proof, which will appear in full elsewhere, that if the space G is locally Cartesian and compact, and the group G is Abelian (commutative), then G is the ordinary closed translation group in n dimensions: i.e. the space is that obtained from the “cube”

1. The perturbations of a focal elliptic orbit are as easily found by Quaternions as the orbit itself.

1. This note is a sequel to a previous one. Our first object is to rectify some mistakes in I. For these corrections I am indebted to the kindness of Prof. J. R. Wilton. In the enunciation of Theorem A (I, p. 159) the equation

should be altered to

The mistake occurs at the end of I (p.169) where it is erroneously assumed that

the correct being

1. A locus Vn, of dimension n, in [2n + 1], for example a curve in [3] or a surface in [5], has ∞ 2n chords, of which a finite number pass through a general point of the space.

1.1. In a linear space Sr, of r dimensions, we may consider involutory transformations determined by linear systems of primals whose freedom is r and whose grade, that is, the number of free intersections of r primals of the system, is two; we shall be concerned only with complete systems, systems of all the primals (of given order) satisfying certain fundamental linear conditions, for example, containing a point, or a curve, or touching a plane at a given point. The primals of such a system Φ that pass through a point P form a system ∞r−1; any r linearly independent primals of this subsidiary system will meet in one other point Q outside the base elements, and Q is common to every primal of Φ through P; further, all the primals of Φ that contain Q must contain P. We thus have an involutory transformation of the space Sr.

The rate of recombination of ions in the copper arc is studied by means of the probe, the post-arc ion density being obtained directly as a function of time. Experimental results agreeing with the theory are obtained, and a coefficient of recombination, α = 5·7 × 10−9, is found. If the temperature of the gas in the path of the arc immediately after extinction is 2500°C., a comparison between the above value and that accepted for normal conditions shows that α varies approximately as the 2·5 power of the temperature, which is in agreement with the existing data.

Experience has shown that simple absorption measurements lead to quite trustworthy estimates of the maximum energies represented in the continuous spectra of many β ray bodies. For the interpretation of these experiments the range-energy relation for homogeneous particles is apparently all that is necessary. In the present paper a detailed analysis is given of the various factors which conspire to this useful result. Analytical simplifications are possible when the energy distribution in question extends beyond about 7 × 105 electron volts energy. Under this limitation it is shown that the empirical result follows from the known form of the absorption curve for homogeneous particles and the most general considerations regarding the characteristics of such measuring instruments as are regularly employed. From the analysis in one case, however, it appears that, despite numerical agreement between the results of different methods of investigation, caution is still necessary in any statement concerning the absolute nature of the high energy limits of continuous β ray spectra.

A development of the “echo” method of observation of the height of the Kennelly-Heaviside layer is described. Short wave-trains of radiation lasting sec. are transmitted at regular sec. intervals, and observed stroboscopically, together with their echoes, by means of a cathode ray oscillograph. Details of the transmitting and receiving apparatus are given. The rapid fluctuation in amplitude of the echoes is easily studied by this method.

The advent of quantum mechanics has provided a means of treating the interaction of atoms and the formation of molecules and has had very many successes in this field. In particular the phenomenon of the saturation of valence forces appears as a natural consequence of the theory. Nevertheless there are very many difficulties still to be overcome.

In a previous paper the nature of the phenomena occurring when electrons are ejected from metal surfaces by the impact of metastable atoms and positive ions was discussed. It was shown that electrons may be ejected by excited atoms at considerable distances from the surface, and by considering the case of an excited hydrogen atom at the centre of a spherical cavity in the metal the order of magnitude of the effects to be expected and the velocity distribution of the ejected electrons was determined. In this paper the theory will be extended in two directions.

1. The wave function of an atom containing many electrons has not yet been solved completely, even that of helium being as yet unknown. In the absence of a direct solution of the Schrödinger equation for the electrons in an atom, various attempts have been made to devise approximate methods of solution in particular cases. The particular case of helium, being the easiest, has received considerable attention and a number of approximate wave functions appropriate to the normal state have been constructed. These functions usually contain empirical constants which are adjusted to make the energy of the system a minimum. Zener has attempted the more ambitious programme of finding the wave functions of all the atoms in the first period of the Periodic Table (Lithium to Neon), and has made interesting discoveries as to the way in which the wave functions differ from atom to atom. This work also is based on the variation of parameters.

A portable apparatus is described which is capable of measuring directly, by means of a loop aerial, the magnetic field in an electromagnetic wave. Accurate measurements are possible of magnetic fields corresponding to field strengths of 0·2 millivolts per metre. Special means of providing small known calibrating E. M. F. S are described. The apparatus can be used to measure signals over the range 6 microvolts to 300 millivolts. Used in conjunction with a small portable vertical aerial, field strengths down to 2 microvolts per metre can be measured.