purely the ratio of the inertia of an electron to its charge and did not involve any "mass of a particle." The experiments of Kaufmann (Compt. Rend. Oct. 1902) have since proved that the mass associated with the electron in cathode rays is entirely electromagnetic. Several subsequent estimations have been made of a by various authors on the assumption that the inertia of the electron is purely electric. Kaufmann gives 10-13 cms., while Abraham in his comprehensive paper (Ann. d. Ph. [4] x. 1903) writes

The equation by which these values have been calculated is that of J. J. Thomson (to a constant près)

I= 2e2/3ac2,

where is the ratio of the two units of electricity.

If this inertia belongs to a cube of æther of edge 2a the density p of the æther is

Now for the æther rigidity will be determined exactly as for a metal at absolute zero. The rigidity No2, where o is the surface-density of the laminar distributions of electricity equivalent to the neutrons. So, just as in section 1, we write

The density and rigidity of the æther being found, we get the velocity of light by the formula

V=(N/p)=c(π/2)2 = 1·25c.

(31)

With p=1÷4a3/3 we would get V=cπ/12}='90c. (32) The coefficient of c ought to be 1, but numerical difficulties occur in connexion with the arbitrary adoption of spheres and cubes as standard shapes. The important point is that by exactly the same process as was applied to matter a rigidity has been found for the æther which, along with the density, gives the velocity of light. It should be noticed that N and p both vary as a 4, and so a disappears in V. With e of the order 3× 10-19 and e/I=6× 101, we have a of the order 2×10-13. Hence from (29) and (30) we get the values

These huge values for the density and rigidity of the æther result from the small value of a being raised to the power -4. They may be taken as upper limits, for an ather made of

electrons in contact. If the electrons in the æther, instead of being in contact, formed doublets at the centre of massless spheres made of æther, the density and rigidity of the whole æther due to the doublets would be reduced by a factor obtained by raising the ratio of a to the radius of the æther sphere to the power 4. We have of late, however, been familiarized again with the conception of a very dense and proportionally rigid æther (for example, Reynolds, 'Scientific Papers,' vol. iii., finds for his granular medium a density 101). At present we are concerned with working out the consequences of the electron theory. On being applied to the æther it leads to the above density and rigidity, which cannot be dismissed for their absurdity merely because of their magnitude. Other lines of inquiry will have to furnish the data necessary for decisive determinations.

Although the æther has been likened above to a metal at absolute zero, it is different inasmuch as it probably always contains as much electrokinetic energy as electrostatic. If the two electrons are rotating round their centre of inertia with linear velocities u, their total electrostatic energy can be immediately written down and their total electrokinetic energy obtained according to Heaviside (Phil. Mag. [5] xxvii.). First, for the electrokinetic energy of each electron due to its own translatory motion we have eu/3V2a. Again, if each electron has an angular velocity of rotation w about an axis, its kinetic energy will be aw21/3.

The potential energy of the electricity of an electron is 2/2a. Thus, then, the self-energy of the two electrons is

For their mutual kinetic energy we have e2u2/V2r; if r is the distance between their centres, and their mutual potential energy is r, so that the total mutual energy is

According to the investigations of Thomson, Heaviside, Searle, and Abraham the formula for the electrokinetic energy hold only when u/V is small. For larger values of u/V the kinetic energy is no longer given by half the product of an inertia and the square of the velocity. It seems to me, however, that there is a promising line of research in assuming that for all values of u/V kinetic energy is given by the expressions used above, and in deducing what modifications are required in the fundamental laws of electromagnetism to bring them into harmony with the principle that electric kinetic energy is always the product of the

square of the velocity and half the constant inertia. Thus in (34), when u=V the total mutual energy is nil. The kinetic and potential parts of the self-energy become equal if

a2w2I=u2e2/2V2a, i. e. if a w2=3u2/4.. (35)

The velocity of light then is such, that if possessed by the electrons of a neutron it would make their mutual energy nil, and, subject to (35), would make their total energy consist of two equal parts, kinetic and potential. I have shown in "The Electric Origin of Molecular Attraction" that the energy of an electrostatic field, according to the neutron theory of the æther, is stored in the ether half as kinetic and half as potential energy. It would seem then as though the velocity of light through the æther is connected with the velocity of its electrons in much the same way as the velocity of sound through a perfect gas is related to the translatory mean velocity of its molecules, a possibility contemplated by the founders of the kinetic theory of gases, with the æther a

gas.

One other point demands immediate attention. According to the electromagnetic theory the velocity of light is (Kμ), and much discussion has centred round the dimensions and the nature of K and . FitzGerald suggested (Phil. Mag. [5] xxvii.) that both K and are the inverse of a velocity. Let us express this by putting

K=c/U, μ =1/cU'................in electrostatic units.
K=1}¢U, μ=c/U' ......in electromagnetic units

} (36)

The ratio of the two units is c, and for the free æther we have c UU' V. But in the ether of matter, by which we mean the ether enclosed by the smallest spheres circumscribing each atom, we cannot write U=U', but if r is the velocity of light through matter, we must have 2=UU'. FitzGerald considered that possibly the velocities 1/K and 1/μ are proportional to the square root of the mean turbulence of the æther. So we shall take U to be the velocity of the electrons in the æther. In the æther of matter the velocity of light is different from that in free æther, so that for it we cannot write c=U=V. But for most substances μ retains nearly the same value as in ather, so that U', even in the æther of matter, generally has a value close to that of U in free æther. We might provisionally regard U' as a velocity derived from the angular velocity of rotation of electrons round their own centres. In free æther it is equal to the translatory velocity U of the electrons, and in the æther of matter retains the same value as in free æther, because

velocities of rotation are not changed by the proximity of matter in the same way as those of translation. These stipulations then give us Maxwell's law K=n2. As is now well known, many substances like water have different values of K, according to the circumstances of measurement, this phenomenon being connected with molecular structure and to be allowed for in the interpretation of Maxwell's law. In "The Dielectric Capacity of Atoms" (Austr. Assoc. Adv. Sc. Jan. 1904) I have shown that values of K for the stuff of ions can be calculated from the ionic velocities, and that for the values found for many atoms the following law holds : (K/v) B=constant, .

where B is the volume of the atom and v its valency.

(37)

(38)

When v=1 we have U/B constant, and IU/IB also constant; and as we assume I to be constant, we have the kinetic energy of translation of the æther in all monad atoms the same per unit volume. For atoms of higher valency it would appear that the effective inertia of the æther of the atom for translatory motion is proportional to v2. This relation would establish a connexion between the æther of an atom and its valency, that is between the special doublets which confer its valency on an atom and the neutrons of the æther. This requires further investigation. It is noteworthy that the translatory kinetic energy of the neutrons in a monad atom is the same per unit volume in all atoms, just as in all gases at the same pressure and temperature the translatory kinetic energy per unit volume is the same. FitzGerald's idea is substantially the same as Fresnel's, who took the ratio of density of æther in matter to density of free ather, or pipe, to be equal to the square of the index of refraction, which by Maxwell's law becomes equal to K. But if the translatory momentum of the neutrons per unit volume is the same in æther everywhere, then Up=Vp, and p/pe=V/U=K, which shows that FitzGerald's principle brings us back to Fresnel's important law. It should be noticed that in the foregoing we have taken account of three distinct ways in which kinetic energy can exist in the ather: first by rotation of an electron round its centre, connected with magnetic permeability of the æther; second by rotation. of neutron, which is the same as translation of electron; and third by motion impressed on the doublets of matter by the atoms, this being the origin of radiation.

Melbourne, February 1904.

LV. Kinetics of a System of Particles illustrating the Line and the Band Spectrum and the Phenomena of Radioactivity. By H. NAGAOKA, Professor of Physics, Imperial University, Tokyo*.

SINCE the discovery of the regularity of spectral lines, the kinetics of a material system giving rise to spectral vibrations has been a favourite subject of discussion among physicists. The method of enquiry has been generally to find a system which will give rise to vibrations conformable to the formulæ given by Balmer, by Kayser and Runge, and by Rydberg. The characteristic difference between the lineand the band-spectrum in the magnetic field has scarcely been touched upon in these theoretical investigations. Instead of seeking to find a system whose modes of vibration are brought into complete harmony with the regularity observed in spectral lines, inasmuch as the empirical formulæ are still a matter of dispute, I propose to discuss a system whose small oscillations accord qualitatively with the regularity observed in the spectra of different elements and by which the influence of the magnetic field on band- and line-spectra is easily explicable. The system here considered is quasistable, and will at the same time serve to illustrate a dynamical analogy of radioactivity, showing that the singular property is markedly inherent in elements with high atomic weights. It must, however, be borne in mind that out of the manifold structure of systems that may exist enjoying the said properties, the one here presented is perhaps the most easily conceivable, although the actual arrangement in a chemical atom may present complexities which are far beyond the reach of mathematical treatment.

The system, which I am going to discuss, consists of a large number of particles of equal mass arranged in a circle at equal angular intervals and repelling each other with forces inversely proportional to the square of distance; at the centre of the circle, place a particle of large mass attracting the other particles according to the same law of force. If these repelling particles be revolving with nearly the same velocity about the attracting centre, the system will generally remain stable, for small disturbances, provided the attracting force be sufficiently great. The system differs from the Saturnian system considered by Maxwell in having repelling particles instead of attracting satellites. The present case will evidently be approximately realized if we replace these satel lites by negative electrons and the attracting centre by a * Read before the Physico-Mathematical Society, Tokyo, Dec. 5th, 1903. Communicated by the Author.