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 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.

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

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

The object of this note is to direct attention to an entirely false inference which was drawn from the analysis given in my previous paper with the same title.

The theorem that, if four arbitrary lines be taken in a plane, the four circles about the triangles formed by threes of these lines, meet in a point, can be generalised to space of any even number of dimensions, as was recognised by Mr J. H. Grace in 1897. In the plane case the centres of the four circles lie on another circle passing through the point of concurrence of these four; it has been sought to prove that the centres of the n + 2 spheres, similarly arising in space of an even number, n, of dimensions, also lie on a sphere†.

1. Mr J. P. Gabbatt has discussed in the most recent Part of the Proceedingsof this Society the Pedal locus of a simplex in hyperspace. It is, however, possible to regard the pedal property of the circumcircle somewhat differently and so to seek other extensions. Given a circle, any three points on it are vertices of an inscribed triangle, and the feet of the perpendiculars on the sides from any fourth point of the circle are collinear. Is there any curve in space on which an analogous property holds for any five points, viz. that the feet of the perpendiculars from any one upon the faces of the tetrahedron formed by the other four are coplanar?

It will be shown that curves of order n exist in Euclidean space of n dimensions on which any n + 2 points have such a property; but that the curves cannot be real if n is odd.

1. Geodesic curves, even of simple surfaces such as quadrics, are almost always transcendental: occasionally they are algebraic: still more rarely do they belong to simple and familiar types. The form of the differential equation referred to proves conclusively that even such integrals as are expressible by elementary functions must be quite exceptional.

The problem of finding the number of r-dimensional regions which are situated in a space Sn of n dimensions, and which satisfy a suitable number of conditions of certain assigned types (called “ground-conditions”) has been investigated by Schubert. A special class of such problems arises when the r-dimensional region is merely required to intersect k regions Pλ of rλ dimensions (λ = 1, 2 … k) situated in general position in Sn where for the finiteness of the sought number we must have

In a previous paper it has been shown that the corrosion of zinc, cadmium, iron and lead is connected with electric currents set up through unequal distribution of oxygen; the currents have been measured and compared with the weight of metal corroded away. The “unaerated” areas are anodic, and thus the attack tends to become concentrated upon the points relatively inaccessible to oxygen. When a drop of salt water is placed on an iron sheet, the central part of the drop suffers anodic corrosion, giving rise to ferrous salts, whilst the marginal ring over which oxygen has ready access to the metal becomes cathodic, alkali being produced there. Where the ferrous salt and alkali come into contact, ferrous hydroxide is precipitated, which oxidises to ferric hydroxide; after a few hours we get a membrane of brown ferric hydroxide extending right over the drop, but surrounded by a ring of clear alkaline liquid. All the actual corrosion of the metal is confined to the area within the membrane; the metal below the external ring of clear liquid, to which oxygen has access, suffers no attack.

1. There seems to be little doubt that the detailed mechanism of ionisation or excitation by electron impact will not admit of description, even to a first approximation, by the system of classical mechanics. However the experimental work of Franck and Horton and their collaborators, has shown that, within the limits of the error of experiment, an electron is able to impart the whole or part of its kinetic energy to the atom it excites. When its initial energy is greater than the excitation energy, the electron, instead of being brought to rest, retains the excess as kinetic energy.

I have to correct the proof of the assertion on p. 471 of my paper that if f (t) is integrable over (0, a) and ν ≥ – ½, then

The instrument to be described is a modification of an electrostatic oscillograph originally invented by Prof. Taylor Jones. In the original instrument a thin phosphor-bronze wire is stretched parallel to a flat metal plate towards which it is attracted when a difference of potential is established between them. A small mirror is fastened by one edge to the middle of the wire, and the other edge of the mirror is pivoted by bringing up to it a flat cork against which it presses. The motion of the wire is recorded by reflecting a beam of light from the mirror on to a rotating drum.

Radiologists have often had cause to note certain apparent anomalies in the behaviour of their apparatus. It is quite well known that different X-ray tubes, excited by different kinds of high tension apparatus, yield X-radiation of markedly varying quality and quantity even under conditions which, as measured by spark gap and milliammeter (the usual measuring instruments of the radiologist), are apparently identical. These anomalies seemed to offer an interesting field for investigation, and one which might not be without value on the practical side.

The two letters now communicated are from G. G. Stokes to W. Thomson, of dates Dec. 12–13, 1848, three years after Faraday's great magneto-optic discovery. They formulated already the permissible types for general equations of propagation, virtually on the basis of the very modern criterion of covariance,—relative to all changes of the spatial frame of reference in the case of active fluids, but having regard to the fixed direction of the extraneous magnetic field in the Faraday case. Their form was elucidated in each case by correlation with a remarkable and significant type of rotational stress in a propagating medium.