Page:LangevinStLouis.djvu/19

 as in the permanent case, an expression which remains stationary. that is to say, the variation of which is zero when supposed slightly modified, can start from its real state. We are thus led to replace the energies We, Wm) which play this role in the permanent case, by an integral taken with respect to the time, and which represents not the sum of the energies, since this quantity, equal to the total energy, ought to remain constant if only electromagnetic action come in, but their difference:

$\int_{t_{0}}^{t_{1}}(W_{e}-W_{m})dt$,|undefined

an integral which remains stationary for all virtual modifications of the system, such modifications being subject to the condition of disappearing at the limits t0 and t1 of the integral, exactly as in the analogous principle of Hamilton in mechanics. The principle of zero variation just announced, and which we will consider as the result of an induction based entirely on electromagnetic principles, allows us in fact to find three of Hertz' s equations, if we admit the three others as an imposed interconnection of the system, and furnishes in the most simple manner the solution which we have obtained for the first problem by means of these equations. Moreover, the motion of the electrons supposed given only at the times t0 t1 comes into the integral, and the condition that this must be stationary allows us to find the law of the motion during the interval, by starting from a principle whose signification is purely electromagnetic. We obtain thus exactly the results of Max Abraham; the equations of motion contain terms which depend first on the motion of the electron, and are proportional, in the hypothesis of quasi-stationary motion, to its acceleration, having coefficients that are functions of the velocity which we will call the longitudinal and transverse masses of the electron; also some terms depending on the charge, and on the external fields, which we will call the forces, and we find that they coincide with those given by Lorentz. The external motion of the electron is thus determined by the actual electromagnetic state of the system.

(27) The Process in the Electron. In order to simplify the analysis and to avoid considering the motion of rotation of the electron, I will consider it as a cavity in the ether; the volume integrals which express the energies We, Wm of the electric and magnetic fields extend only over the space external to the surface which bounds the cavity. We can suppose as a special condition outside of the electric charge that the form of this surface is fixed, spherical for example, due to an unknown action of nature, and we find the equations of Abraham for the longitudinal and transverse masses of a spherical electron.

But we can suppose a more simple condition, implying only a fixed volume of the cavity on account of the incompressibility of the external