RELATIONS TO OTHER SCIENCES 139 



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 W e , W m , 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: 



* 



/ 



/ If 



an integral which remains stationary for all virtual modifications 

 of the system, such modifications being subject to the condition of 

 disappearing at the limits t and ^ 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 t Q ^ 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 ex- 

 ternal fields, which we will call the forces, and we find that they coin- 

 cide 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 W e , W m 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 ex- 



