The theorem that if two triangles be inscribed in a conic their six sides touch another conic is, of course, to be found in all the text-books; it is apparently due in the first place to Brianchon. The further remark, that if three triangles be inscribed in a conic the three conics obtained from them in pairs have a common tangent, is to be found in Taylor's Ancient and Modern Geometry of Conics; it was made independently by Wakeford.

The object of these notes is twofold. First, to connect the theory of the possible shapes of Jacobi's ellipsoidal equilibrium figure, for a given mass of homogeneous rotating liquid, with the theory of its stability and other properties given by Prof. H. F. Baker in his paper “On the stability of rotating liquid ellipsoids” in Proc. Camb. Phil. Soc. vol. 20, pp. 190–97. And secondly, to indicate two inconclusive reasonings on the subject in Thomson and Tait's Natural Philosophy (2nd edition), arts. 778 and 778′, and to complete the proofs of the results in question.

In the recent experiments on the Zeeman effect in strong magnetic fields made by Mr H. W. B. Skinner and myself for obtaining an intense source of light for a small fraction of a second, we have been discharging a condenser battery consisting of 32 Leyden jars connected in parallel with a capacity of one-tenth of a microfarad. The discharge was made through a small spark gap by means of an oil immersed switch (the details of this arrangement are to be found in the above-mentioned paper). When the discharge was produced a curious phenomenon was sometimes observed. After the battery had been charged to its maximum tension, and the switch had been put into operation, only a small spark occurred in the spark gap, and the main part of the discharge went over the top of the insulators of one of the Leyden jars.

In this paper the method of infinitesimal transformations of coordinates, used by Weyl to determine conditions that a function of the tensors gik and φi, and certain of their derivatives, should be a scalar density, is applied (with certain modifications so as to give tensor relations) to functions of and . It is known that in order that such a function should be a scalar density it must be a homogeneous function, of degree ½n, of , and this must of course be deducible from the equations found by the infinitesimal transformations. In view of the part which these equations may play, as “equations of energy,” etc., in purely affine field theories, it seems desirable that the connection should be explicitly shown, and this is done in § 3.

1. It has been shown that the action of alpha particles on paraffin is independent of the state of aggregation of the substance.

2. Defects in the customary procedure for the decomposition liquids and solids by the emanation tube method are discussed.

3. A lecture demonstration of the chemical effect of alpha particles is described.

The particular form of Taylor's theorem considered states that: If ƒ(x)is n times differentiable at a thenwhere.

Statistical mechanics is concerned primarily with what are known as “normal properties” of assemblies. The underlying idea is that of the generalised phase-space. The configuration of an assembly is determined (on classical mechanics) by a certain number of pairs of Hamiltonian canonical coordinates p, q, which are the coordinates of the phase-space referred to. Liouville's theorem leads us to take the element of volume dτ=Πdp dq as giving the correct element of a priori probability. Any isolated assembly is confined to a surface in the phase-space, for its energy at least is constant; when there are no other uniform integrals of the equations of motion, the actual probability of a given aggregate of states of the proper energy, i.e., of a given portion of the surface, varies as the volume, in the neighbourhood of points of this portion, included between two neighbouring surfaces of constant energies E, E + dE; it therefore varies as the integral of (∂E/∂n)−1 taken over the portion. If I be the measure of the total phase-space available, interpreted in this way, and i that of the portion in which some particular condition is satisfied, then i/I is the probability of that condition being satisfied.

In this paper the results of some measurements of the Hall effect in single crystals of iron are described. This work was undertaken to determine whether the Hall coefficient varied with the direction in the crystal for which it was measured. Such a variation had been observed in other magneto-electric phenomena—the change of resistance in longitudinal* and transverse† magnetic fields—and had thrown some light on the nature of these effects. It was thought possible, therefore, that the study of the Hall effect in single crystals might yield some fresh light on this unexplained phenomenon.

Two alternative forms of the CO2 molecule have been suggested by various authors who have discussed the band spectrum data. The specific heat curves based on these models are considered here. It is found that neither is quite satisfactory over the whole range of temperature and we discuss the difficulties for the low temperature and high temperature portions separately. In order to get agreement for low temperatures we find it necessary to introduce a further hypothesis about the molecular model which also seems to explain one or two outstanding difficulties in interpreting the fine structure of the bands. This assumption does not make any difference at higher temperatures where we show the error in one of the curves to be of the order we should expect to be accounted for by a centrifugal stretching of the molecule.

Much attention has been focused on the excitation of atomic spectra by electrons whose velocities do not greatly exceed the critical value, and it is now possible to form a rough idea of the behaviour of the “excitation function,” or probability that an electron of given velocity shall excite on impact, when the energy of the electron is only slightly greater than that required for excitation. But the form of this excitation function for much higher velocities is still in considerable doubt.

The equation to be considered is of the type

where p (x) is continuous for all real values of x, even, and periodic. It is no restriction to suppose that the period is π, and this assumption will be made, so that the equation is virtually Hill's equation.

The methods of geometrical optics provide an approximate solution of the equation of wave propagation, if the following condition, given by De Broglie, is satisfied. If V is the phase velocity, λ the wave-length, l the direction of greatest increase of V, and θ the angle between l and V, then the condition is that

that is, the relative change of V over a distance of the order of θ must be small†.

This correspondence has been established by Mr H. W. Turnbull, Proc. Camb. Phil. Soc. vol. XXII. (1925), pp. 694–9, a construction which leaves uncorrelated the lines of a special linear complex and the points of the quadric lying in one tangent fourfold; these have to be correlated by means of an independent construction.

There are two ways of investigating the velocity distribution of an electronic emission—the retarding potential and the magnetic deflection methods. These have been applied to the case of the photo-electrons by Millikan and Ramsauer respectively. On one important point the results disagreed; Millikan found a definite maximum velocity, while Ramsauer obtained an asymptotic falling off of the number of electrons with increasing velocities. In a critical discussion of the two methods Klemperer has shown that they agree when photoelectric activity of the collecting electrode in the first, and when electronic reflection at the walls and slits in the second, are eliminated.

We may summarize our conclusions as follows. If the rotating doublets have quite different angular velocities initially, then they repel each other with a force (R) given by

where

and

ø1 and ψ1 being the (constant) angular velocities of the two doublets.

If the doublets have the same angular velocities initially, and the same moments of inertia, then over a certain range of r we have

where

and ω is the common value of the angular velocities of the doublets. When the doublets correspond to hydrogen atoms in their principal quantum orbits, the range of distance becomes 5 Å. to 50 Å. and the formula for R reduces to

This is a law of force of the type found empirically by Lennard-Jones for helium, neon, and argon. The attractive term in this formula is larger than the attractive terms found by Lennard-Jones. The repulsive term, however, which leads to a “diameter” of 3·31 Å., is in very satisfactory agreement with the repulsive terms found by Lennard-Jones.

Measurements of the saturation currents produced between two parallel plates by a source of radium C of carefully tested purity have been performed by an accurate galvanometer method. From the results a transformation constant

has been assigned to radium C corresponding to a half-value period of 19·72±·04 minutes.

Similar measurements with radium active deposit have enabled a value to be estimated for the period of radium B between 26·7 and 26·8 minutes, which is the limit of accuracy of the experimental data.

In the network shown diagrammatically in Fig. 1, A0A3, A3A5,… are resistances of values a1, a3,… joined in series with one another, and A3B3, A5B5,…are resistances of values 1/a2, 1/a4,…: the points B0B3,… are all on a cable of negligible resistance. The members A0A3, A3A5,… will be called the series members of the network, and the members A3B3, A5B5,… will be called the shunt members of the network: the points A0B0, will be called the input terminals of the network. If a potential difference is maintained between the input terminals, currents will flow in the members of the network.

A method of using a valve for amplifying ionisation currents 100,000 times is described, which avoids the instabilities usually associated with such apparatus. The necessary conditions to be satisfied, while extremely simple, are based on theoretical conditions which are discussed in full. Used in conjunction with a galvanometer of sensitivity 200 mm. per micro-ampere, the system behaves in a similar manner to a low capacity quadrant electrometer of sensitivity 6350 mm. per volt, shunted by a leak of 360 megohms, the value of the latter being slightly greater for negative currents than for positive. In addition, it has the advantages of portability, compactness, and ease and rapidity of erection and operation, being specially suitable for lecture demonstrations.

The method of compensation can also be applied to other valve circuits, resulting in a much steadier zero.

The absorption by matter of energy from a beam of X-rays follows laws which are now well known. The first stage is the ejection from the absorbing atom of a high speed electron. This electron, in turn, produces pairs of ions from some of the molecules through which it passes until its energy is all spent. The process is essentially a discontinuous one in space, and the proportion of atoms affected at a given time is always exceedingly minute, even with an intense beam of radiation. With a beam of average intensity an individual atom would suffer ionisation, on an average, about once in a million years.

The glow discharge between cold aluminium electrodes in air, oxygen, nitrogen and hydrogen has been analyzed by Langmuir's method, for pressures between 0·1 and 0·4 mm. Hg, current densities of from 0·02 to 0·2 mA./sq.cm., and applied potentials between 300 and 700 volts. An annular exploring electrode has been used. It has been found that whilst practically the whole fall of potential is localized across the cathode dark space at the lower pressures, a fall of as much as 40 volts can exist across the remainder of the discharge at the higher pressures. Reversal of the electric field has been found in the negative glow, and in certain cases in the Faraday dark space, when conditions are favourable for passage of an electron current by diffusion against the field. In several instances the negative glow was at a higher potential than the anode. Two groups of electrons occur in the negative glow, together with a single fast group at the anode boundary of the cathode dark space, and a single slow group in the Faraday dark space.