1. It has been shown by Pieri, and independently by James, that the trisecant planes of a quartic curve in [4] which meet a line meet also another quartic curve intersecting the former in six points. James generalises the theorem and shows that trisecant planes of a quartic curve C, which meet a quartic curve C1 having six points in common with C, also meet another quartic curve C2, and that the relation between the three curves is symmetrical. The object of this note is to give a simple proof of this theorem and to discuss the representation by which this proof is effected. The method used is analogous to that of Pieri; an interesting differential method is adopted by C. Segre who shows that the foci of the first and second orders, of any linear system of ∞2 planes in [4] of which two planes pass through a point, determine a conic and five points respectively, in any plane of the system: the trisecant planes of C meeting C1 give a particular case of such a system.

In a recent paper I discussed plane congruences of order two in [4] and obtained congruences of types (2, 6)1, (2, 6)2, (2, 5), (2, 4) and (2, 3). The method employed was due to Segre, who showed that a plane congruence of order two in [4] has in general a curve locus of singular points which is met by each plane in five points. Then, if we can find a curve in [4], composite or not, with an ∞2 system of quadrisecant planes of which two pass through an arbitrary point, the planes must all meet a residual curve, and we shall have obtained a congruence of the second order and a fifth incidence theorem.

The linear subspaces [k] of a space [n] have been represented by Grassmann by the points of a rational locus V(n−k)(k+1), of dimension (n−k)(k+1), in ; the representation is effected by taking as independent homogeneous coordinates of a [k] in [n] the determinants of order k+1 which belong to the matrix, of k+1 rows and n+1 columns, formed by the co-ordinates of any k+1 linearly independent points of the [k].

Nicod, using Sheffer's operation “│” as a primitive (undefined) idea, obtained a drastic reduction in the number of primitive propositions (postulates) used by Whitehead and Russell in their theory of deduction for “elementary” propositions. In this reduction Nicod accepts Sheffer's replacement of the primitive ideas “∼” and “v” and the primitive propositions *1·7 and *1·71 of the Principia by a single primitive idea and a single primitive proposition, and, by adding two ingeniously devised postulates of his own, derives from the three postulates the Principia's six primitive propositions *1·1, *1·2, *1·3, *1·4, *1·5, *1·6. But the author does not consider the consistency or the independence of his three postulates; does not discuss the Principia's remaining primitives—the ideas “φx” and “⊢. φx” and the propositions *1·11 and *1·72; and fails to prove that his postulates are derivable from those of the Principia.

An approximate calculation of the electromagnetic field of a vertical dipole at the surface of a conducting earth, for small angles of the radius vector with the horizontal, was given by Sommerfield; the case of large angles with the horizontal has been studied by a number of writers. It is proposed here to develop formulae for the vertical dipole by a method which takes into account the singularities of the integrand of a certain integral more accurately than is done by Sommerfield; the analysis is developed especially for small horizontal angles and small numerical distances.

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

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 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.

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.

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.

Measurements have been made of the positive ion currents emitted by tungsten at temperatures between 3000°K. and 3200°K. The results are in general accordance with the values calculated on the basis of the Saha equation from the rate of evaporation of neutral atoms, the electron work-function and the ionisation potential of tungsten. The “work-function” associated with the ionic evaporation appears to lie between 10 and 11 electron-volts.

Some general considerations regarding the design of hot cathode X-ray tubes for producing an accurately constant and repeatable beam of X-rays are first given. The construction of a tube in accordance with these considerations is then described. This tube has been used to show that except with small tube-currents the quality of the vacuum in such a tube has little influence on the steadiness of the beam if the electrical input to the tube can be satisfactorily controlled.

It is shown that the X-radiation associated with the fluorescence often observed in X-ray tubes has usually but not always a negligible effect on the steadiness of the beam.

Some factors limiting tube current when the filament is enclosed in a box containing a focussing slit are mentioned, and the mechanism of formation of the focus is discussed with reference to its size. The use of an auxiliary electrode to increase the tube current and improve the focussing is also discussed.

Here the notion of density is made more precise, by definition in terms of the fundamental orthoscheme. (The fact that reciprocal polytopes have the same density is a direct consequence of the precise definition.) A new criterion for regular polyhedra is given, superior to Petrie's in that it applies to degenerate, as well as to finite, polyhedra. The cases are examined in which the actual vertex figures of one polytope bound another; and this idea is developed into the process called alternation, which provides an alternative method for calculating density in four dimensions.

After a review of the magnetic and spectroscopic properties of compounds of the rare earth and of transitional elements of the first group, the types of bands in spectra of the latter are considered in detail. Reasons are advanced for the view that these are of “molecular” origin in ferric salts, but that the spectra of some transitional compounds contain lines which cannot be regarded in this way. Such lines are shown to be ionic, or quasi-ionic, and accord well with the suggestion of inter-combination transitions such as 4F—2G.

The theoretical intensity conditions, as a function of multiplet intervals, are in agreement with the observed appearance of intercombination transitions in Cr, and possibly also in Mn and Co. Similar considerations apply to the rare-earth spectra. As an example we have shown that the strong absorption in Pr and Nd compounds, and the much weaker lines found in Sa and Eu, are in perfect agreement with the theoretical prediction of intercombinations.

It is pointed out that there is some confusion concerning the meaning of the sign of a mutual inductance. The question is examined from first principles and it is shown that since the sign of a mutual inductance depends upon conventions as to sign of current directions, these, as well as the sign of the mutual inductance, must be given in order to specify what is really required—the relation between the directions of winding of the primary and secondary coils. It is also shown that in many cases it is possible to replace mutual inductance by a quantity which is numerically equal to the mutual inductance but which is independent of current directions. The two methods applied to a Wheatstone network yield consistent results.

With the aid of experimental results on the absorption of homogeneous β-rays in aluminium, a method is developed for calculating the absorption curve of β-rays forming a continuous spectrum. When applied to certain known spectral distributions reasonable agreement is obtained with the experimental absorption curves of these heterogeneous β-rays. Distribution curves with momentum of the β-rays from actinium C″, uranium X2, thorium C and thorium C″ are found, which lead to similar agreement.

The distribution curves with energy and with momentum for eight β-ray bodies and a table of average energies emitted in thése spectra are shown.