of Stationary States of Motion. 245 



It may be objected that the electron mechanics itself pre- 

 supposes the classical expressions for the energy densities 

 and the Poynting flux corresponding to them, so that its 

 agreement with Lienard's expression is to be expected 

 a priori and cannot be used as an argument for the correct- 

 ness of their common foundations. 



5. It is indeed true that Abraham * in his classical deduction 

 of electron mechanics uses the Lagrangian method with the 

 classical expressions for the energy densities, the Poynting flux 

 and the corresponding expression for the density of electro- 

 magnetic momentum, but none of them are really essential for 

 the deduction and they are used merely for shortening thework. 

 The true foundations of the theory are the equations of the 

 electromagnetic field, the expression of Lorentz and Larmor 

 for the mechanical force on a moving charge, and the defini- 

 tions of force and m ss afforded by Newton's first and second 

 law of motion. Using these as a basis I have shown else- 

 where f that the equations of motion of the electron can be 

 obtained by direct integration over the space occupied by it 

 in the form of series, which proceed according to ascending- 

 powers of a length determining the linear dimensions of the 

 electron, with coefficients formed of multiple integrals ex- 

 tending over the electron and depending on its velocity, 

 acceleration, and the other quantities determining its motion. 

 The first approximation gives the electromagnetic momentum 

 and mass ; the second gives in addition the reaction due to 

 radiation in the form first found by Abraham J by an indirect 

 method. 



For our purpose it is important to observe that the equa- 

 tion of energv obtained from these equations of motion 

 enables us to define the kinetic energy of the electron and 

 the radiation from it without any assumptions as to the 

 proper expressions for the energy densities of the sur- 

 rounding field and the energy flux. In fact, the reaction 

 due to radiation consumes work irreversibly as well as 

 reversibly ; the rate at which it consumes work irreversibly 

 is given exactly by Lienard's expression for the radiation. 

 Hence this expression is proved independently to be con- 

 sistent with the ordinary electron mechanics, and the same 

 thing follows for the Poynting flux. 



6. Thus w r e are driven to consider the second alternative 



* Abraham, Ann. d. Pht/s. ser. 4, vol. x. p. 105 (1903). "French 

 translation in Abraham et Langevin, ' Ions, Electrons, Corpuscules,' 

 p. 1 (1905). 



t Schott, * Electromagnetic Radiation/ Ch. XI. and App. C, D, 

 and F. 



+ Abraham, Elektromagnetische Theorie der Strahlung, p. 123 (1905). 



