The purpose of this paper is to provide a point of vantage from which to attack combinatorial problems in what may be termed modern, synthetic, or abstract algebra. In this spirit, a research has been made into the consequences and applications of seven or eight axioms, only one [V] of which is itself new.

1. In a recent paper I established new conditions for a form φ of order n, homogeneous in r + 1 variables, to be expressible as the sum of nth powers of linear forms in these variables; and for this expression, if it exists, to be unique. These conditions, I further showed, may be stated as general theorems regarding the secant spaces of manifolds Mr in higher space, namely:

Necessary and sufficient conditions that through a general point of a space N, of h (r + 1) − 1 dimensions, there passes (i) no, (ii) a unique (h − 1)-dimensional space containing h points of a manifold Mr lying in N are that

(i) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr meets Mr in a curve, so that Mr cannot be so projected upon a linear space of r dimensions;

(ii) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr does not meet Mr again, so that Mr can be so projected, birationally, upon a linear space of r dimensions..

The problem of discriminating between the different real forms of the rational twisted quartic was first attempted by Adler in 1882. His catalogue is incomplete and some of his results are inaccurate, as has been pointed out by Richmond †. The work was however superseded by a paper in 1891 by Rohn‡ who considered the curve in relation to its allied Steiner quartic surface; his results may be compared with those given in the final section of this paper. Substantially the same results were obtained by Vietoris§ who mapped the curve on a conic by a somewhat different method from that used by Adler.

1. The surface here discussed arises most naturally in the study of a certain cubic primal in space of five dimensions. Let G be a cubic primal in five-dimensional space, containing two planes M1, M2 which do not intersect. From any point of G can be drawn one transversal to M1, M2; this does not meet G again, and meets a fixed prime p in one point. Thus G can be birationally projected upon p, and is rational.

1. In this note we investigate, from a new point of view, some properties of cubic primals in [4]. These are enunciated by Fano as follows.

In a recent publication the problem of determining the abundance of two interacting species of animals was investigated, and the fundamental equations were established.

In a recent paper we have discussed certain general principles underlying the determination of the most efficient tests of statistical hypotheses, but the method of approach did not involve any detailed consideration of the question of a priori probability. We propose now to consider more fully the bearing of the earlier results on this question and in particular to discuss what statements of value to the statistician in reaching his final judgment can be made from an analysis of observed data, which would not be modified by any change in the probabilities a priori. In dealing with the problem of statistical estimation, R. A. Fisher has shown how, under certain conditions, what may be described as rules of behaviour can be employed which will lead to results independent of these probabilities; in this connection he has discussed the important conception of what he terms fiducial limits. But the testing of statistical hypotheses cannot be treated as a problem in estimation, and it is necessary to discuss afresh in what sense tests can be employed which are independent of a priori probability laws.

This paper gives experimental results for the scattering of 4 and 2 volt electrons in nitrogen, which are compared with Stier's theoretical curves. An approximate theory of the scattering of electrons in nitrogen is given which is applicable when the wave length of the electron is small compared to the internuclear distance. The results are compared with experiment.

Some experiments are described in which the efficiency of ionisation of potassium at a hot molybdenum surface was investigated. The degree of ionisation observed is consistent with the most recent measurements of the electron work function of molybdenum.

A three-phase star-connected alternating-current motor of simplest type, connected to busbars maintained at sinusoidal potentials e1, e2 and e3 with respect to the star-point and connected to a shaft load which exerts a reactive torque f on the rotor, is dynamically specified by the following functions:

where the are the armature currents in the three phases, i is the current in the amortisseur circuit, Q the current in the field circuit, θ the rotor position angle, T the kinetic energy, V the potential energy, S the Rayleigh dissipation and U the activity of the external forces on the machine. The self-inductances of the armature circuits and the mutual inductances between them are assumed to be constant.