Let the eight points in space common to three arbitrary quadric surfaces be called eight associated points. These points possess the property that any one of them is uniquely determined if the other seven are given. It is interesting to give the analytical expressions for eight associated points in terms of symbols entirely free from coordinate systems. Thus we take the first eight digits 1, 2,…8 to denote the points; a group of two digits to denote the line joining these two; a group of three to denote a plane, and so on. Also a conjunction of two groups denotes the point, line or plane common to the two groups, thus (34, 678) is the point common to the line 34 and the plane 678.

In a recent paper the present writer made an attempt to reduce the ordinary optical constants of matter to a more primitive form by introducing “scattering indices.” Their use solves half the problem of optics, for general formulae connect these indices with the behaviour of matter in bulk. There remains the deduction of the scattering indices from whatever we are assuming to be the reaction to light of the actual atoms, and even in the classical theory this encounters rather formidable difficulties of convergence. As long as we are content with no great rigour it is a fairly simple matter to deduce the scattering indices for any assumed model without any of those rather difficult arguments about the polarisation of the medium which play a part in the ordinary presentation.

It is well known that in non-euclidean geometry a plane triangle has four circumcircles, and that each of these circles touches each of four other circles. The latter theorem, which is an extension of that of Feuerbach, is essentially due to Hart.

A complex or system ∞3 of conics in space of four dimensions is such that a finite number of conics pass through an arbitrary point. Linear complexes are those for which this number is unity, and are such that their curves are defined by conditions of incidence with fixed surfaces, curves and points. In this paper are discussed briefly the linear complexes defined by the condition that their curves meet an irreducible curve in four points. Denoting by a curve of order m and genus p it is found that the curves in question are The complex associated with is considered in greater detail, since it is found to have an interesting connection with the well-known Weddle quartic surface of ordinary space. In fact the conics of the system touching a space (of three dimensions) do so in the points of such a surface. The main properties of this surface can be thence deduced. In addition we discuss certain results in connection with this curve . The paper closes with certain enumerative results which were obtained in the course of the researches giving the results recorded and which we believe are worth record.

1. The plane quartic curves which pass through twelve fixed points g, of which no three lie on a straight line, no six on a conic and no ten on a cubic, form a net of quartics represented by the equation

The only really useful practical method for solving numerical algebraic equations of higher orders, possessing complex roots, is that devised by C. H. Graeffe early in the nineteenth century. When an equation with real coefficients has only one or two pairs of complex roots, the Graeffe process leads to the evaluation of these roots without great labour. If, however, the equation has a number of pairs of complex roots there is considerable difficulty in completing the solution: the moduli of the roots are found easily, but the evaluation of the arguments often leads to long and wearisome calculations. The best method that has yet been suggested for overcoming this difficulty is that by C. Runge (Praxis der Gleichungen, Sammlung Schubert). It consists in making a change in the origin of the Argand diagram by shifting it to some other point on the real axis of the original Argand plane. The new moduli and the old moduli of the complex roots can then be used as bipolar coordinates for deducing the complex roots completely: this also checks the real roots.

The theorem, due to Miquel, that the foci of the five parabolas which touch fours of five straight lines lie on a circle, when generalised projectively and dualised becomes the theorem: If six arbitrary points 1, 2, 3, 4, 5, 6 be taken and the five conics passing respectively through the five points obtained by omitting in turn 1, 2, 3, 4, 5, then there exists a conic touching two arbitrary lines through the point 6 and triangularly inscribed to these five conics. It appears, however, that the relation is symmetrical and that the conic obtained is also triangularly inscribed to the conic passing through the points 1, 2, 3, 4, 5. If the condition of touching the two arbitrary straight lines through the point 6 be omitted, we have a doubly infinite system of conics triangularly inscribed to the six conics passing through fives of six points. It does not immediately appear how this family of conics depends upon the two parameters involved, and the following direct analytical investigation of the general symmetrical figure was undertaken with a view to deciding this point.

The tensor calculus introduced in a recent paper proves a powerful method of investigating the light scattered laterally as exemplified in the blue of the sky. The late Lord Rayleigh worked out the effect on the polarisation of having non-spherical molecules, and the present note is practically a repetition of his work but with a somewhat greater generality. The method has the advantage that it does not become unmanageable for molecules of a more complicated type of asymmetry, whereas a perusal of his work suggests that it would be a very complicated matter to give the lateral scattering from a set of arbitrary orientated “optically active” molecules. As this case has at present no practical interest, I have merely indicated the procedure so that it could easily be applied when required.

The results of the present paper were developed some years ago for quite trivial purposes. It seems possible, however, to judge by a recent publication, that they may sooner or later have scientific utility. They have also a certain slight intrinsic interest.

Mystery has remained attached to the transmission of free electric rays a long way round the protuberant curvature of the earth, which has recently developed into the greatest sudden practical evolution in signalling since the telephone. The difficulty was already emphasised by the late Lord Rayleigh, as soon as the first signs of transmission across the Atlantic had been detected by the Marconi operators. The effect has been sometimes supposed to be accounted for by a hypothesis that the rays are turned downward by an upper conducting layer in the atmosphere. But conduction, as usually understood, involves dissipation, and thus loss of energy of the rays by absorption: so that a train of radiation travelling along a layer sufficiently conducting to bend the rays could not go far. In fact, by a well-known dynamical principle, if the absorption is small of the first order, the resulting decrease of velocity of the train is small of the second order and so of no account for bending rays in a varying field.

In this note we obtain a system of characteristics on which depend the main enumerative properties of complexes, or systems ∞3, of conics in space of four dimensions. The method used is applicable to a large number of problems of this nature, and we select this illustration as being analogous to congruences in ordinary space. In particular a finite number of conics pass through an arbitrary point, thereby defining the order, n, of the system. Linear complexes are those for which this order is unity. The conics of a complex satisfy eight simple conditions, in general conditions of contact with a fixed form or forms, but they may include conditions of incidence with a surface, each such counting once, with a curve, each such counting twice, or with fixed points each such counting thrice. In particular only incidence conditions can occur when the system is linear, for otherwise more than one conic would pass through certain ∞3 points of space. Points through which more than a finite number of conics pass are termed singular, as well as their loci. Directrix constructs are necessarily singular, but they do not necessarily exhaust the singular system, for the complex may possess ∞2 curves lying on a surface. In the linear case through a point of a singular curve pass ∞2 conics, and through a point of a singular surface ∞1 conics. The possibilities in the nonlinear cases are too numerous to be detailed. Similarly the system of planes of the conies may possess a singular curve, through which ∞2 of the planes pass.

The object of the present communication is to give an account of an investigation of the problem of four particles on the lines of my investigation of the problem of three particles, an account of which has been recently published by the Society.

Since the publication of a previous paper on chemical constants, some further experimental data upon the dissociation of chlorine by Wohl have appeared, from which a new and rather more satisfactory value of the chemical constant of diatomic chlorine can be calculated. Wohl concludes that Q0, the heat of dissociation at absolute zero, is – 57,000 calories. This is in good agreement with Henglein's value – 54,000 but differs from that of Trautz and Stackel, namely – 71,000, which is the value adopted in the previous paper. Q0 is necessarily an adjustable constant, and since Wold's value gives results which are a good deal more concordant with each other and with the theory, we now take Q0 = − 57,000. Wohl also uses hv0/k = 902 instead of 1093, but as the difference made by this change is very small compared with other disagreements, we retain the value 1093. The following table gives the new values of Γ (Cl2), calculated in the same way as before. The initials at the heads of the columns refer to the results of Henglein, Trautz and Wohl respectively.

The Lorentz group is denned by the equations

where the coefficients are subject to the conditions

It is an intransitive group possessing the single invariant

§ 1. The general quaternary cubic

can be expressed in various interesting canonical forms involving suitably chosen linear forms Xi, Yi, Ai. Thus, referred to the pentahedron. X1X2X3X4X5 of Sylvester the cubic becomes the sum of five cubes

with its Hessian in the form

where the coefficients ai may if necessary be taken as equal to unity and the five linear forms Xi each contain four independent parameters, making a total of twenty parameters which is the number of coefficients aijk in the given cubic C3

1. Appleton and van der Pol have shown that in a simple Triode or Dynatron generating circuit the anode potential v is related to the time t by a differential equatior of the type

where f (v) is a power series in v, and may be written

Let Λ (n) be the arithmetic function usually so denoted, which is zero unless n is a prime power pm (m ≥ 1), when it is log p. We write as usual

and

where the dash denotes that if x is an integer the last term Λ (x) of the sum is to be taken with a factor ½. We wrute further

It has been observed by various workers that, in addition to particles ejected from the various atomic levels of the atoms concerned, some are ejected by rays which correspond to the natural K X-radiation of those atoms. It has also been shown that some radioactive substances emit β particles of comparatively low energy. Therefore it was thought advisable to make an attempt to see whether this “soft” radiation has a line spectrum; for one would expect to find traces of the natural L-radiation, just as one finds evidence of the K-radiation, provided that the effect is sufficiently intense.

Beltrami's discovery that corresponding to any ruled surface there is another applicable on it having corresponding generators parallel but with the parameter of distribution of opposite signs, has brought into prominence the distinction between the applicability of two surfaces and the continuous deformation of one into the other. It is pointed out that since the parameter of distribution differs in sign, the tangent plane rotates in opposite directions as the point of contact proceeds along the two corresponding generators on the respective surfaces. One surface cannot, therefore, be deformed into the other. It is proposed in the present paper to investigate the distinction between applicability and deformability of two surfaces in general, and to connect this isolated fact with the general theory of deformation.

The object of this note is to correct an error in my paper “Extensions of a theorem of Segre's…,” the notation used being the same. The curve C4 dealt with is regarded as given by its canonical representation

and at one point in the paper we sought the locus of the lines analogous to the line A2A4 of the figure of reference for each of the ∞2 representations of this type (p. 671, small print). In the space representation of the locus there is an additional principal curve

and the order of the locus must be reduced by that of the form corresponding to the points of this conic. The locus sought is in fact none other than the cubic form, locus of chords of C4, the present system of lines being the directrix systemt†. This follows at once from the following results, which can be shown immediately using the above representation:

(1) The space joining such a line g to any tangent cuts the curve again in coincident points, and thus contains a second tangent;

(2) The line joining the points of contact of these tangents meets g, and the points give the involution