Fundamental Law of Electrical Action. 351 



Neglecting the terms due to the Mass-motion, the 

 Lagrangian Function T-U, for the two charged particles, is 

 easily seen to be equivalent to 



fjb / uv! cos 6\ 1 

 3\ H / kR' 



The similarity of this form with the Clausius form is 

 apparent. There is of course discrepancy in the ( ^) term. 



These formulae are all limited to the case where the 

 velocities of the moving charges are small compared with 

 the velocity of light. 



From what has been said before, it will be seen that the 

 problem is still an open one. The investigations hitherto 

 given are largely empirical, and not based on sufficient 

 theoretical basis. In view of the recent extraordinary 

 development of electronic physics, it cannot be said that the 

 importance of the problem has been in any way diminished. 

 On the contrary, a knowledge of the laws of electronic 

 attraction and a clear formulation of the dynamics of the 

 electron are necessary before we can satisfactorily handle 

 any problem on electronic physics, — such as the atomic 

 model, or radiation from atoms and electrons. 



In the present investigation I have throughout used the 

 New Electrodynamics [i. e., as modified by Lorentz, Einstein, 

 and Minkowski according to the Principle of Relativity). 

 I have particularly used the method of four-dimensional 

 analysis which was first initiated by Minkowski *. A large 

 amount of work in this line has been done by Born f and 

 Sommerfeld i, though not always with the same specific 

 purpose which I have in this paper. Sommerfeld in 

 particular, in connexion with his development of four- 

 dimensional analysis,- has investigated the law of attraction 

 between two moving electrons; but the result obtained is so 

 cumbrous as to make further progress almost impossible. 

 This is due to the fact that for the scalar and vector potential 

 of the field produced by a moving electron, they arrived at 

 an expression which is only a partial statement of the com- 

 plete result (see remarks at the end of § 8). When this 

 complete result is introduced, the electric and magnetic 

 forces as well as the ponderomotive force acting on an 



* H. Minkowski, Mathematische Annalen, vol. lxviii. p. 472 et seq. 

 + Born, Ann. d. Physik, vol. xxviii. p. 571. 



+ Sommerfeld, Ann. d. Physik, vol, xxxiii. pp. 649 et seq. ; vol. xxxii. 

 pp. 749 et seq. 



