Fundamental Law of Electrical Action. 363 



With the above assumption, we have R' = 0, 



.-. | R | = | (W(w) | = u ^ x - a ^+ u 2(y-b)+ u^z—c)-r 



\/(l — u 2 ) 



V(- 



we can write 



ew 



[a,, a 2 , a,, a J = (R ,~ = _ _ [ Ul , u 2 , n* q. 



Using the ordinary lime-coordinate, we have 



»- .. *' . , F,G,H«'- ( 5"3i"*. . ( i 2) 



This result has been obtained in various ways byHerglotz*, 

 Sommerfeld f, and other workers. Sommerfeld effects the 

 integration o£ equation (10 a), with the aid of Cauchy's law 

 of residues, and confirms the result (previously obtained by 

 Herglotz), 



ew 

 a =(RV; W 



But a comparison o£ the methods of arriving at the two 



formulae will show that the expression (11') is but a partial 



statement of the result, it being assumed from the very 



beginning that the time-coordinates are separated by the 



interval r/e, where r — three-dimensional distance between 



ew 

 the points. The result ^ = -5- is perfectly general, and in 



full agreement with the requirements and the spirit of the 

 principle of relativity. This reduces to the expression (11'), 

 when for the purpose of forming an idea of the result in 

 three-dimensions, we make the particular assumption just 

 mentioned about the time-coordinates. Hence it is apparent 

 that w T hen we apply the result to the determination of the 

 magnetic and electric forces, and the ponderomotive force, 

 we must use the expression (11), and not (ll/)- 



9. The Ponderomotive Force on an Electron due to the field 

 produced by the motion of another electron. 



In § 6 we investigated the action of a point-charge on 

 another charge ; in the present section we shall investigate 

 the action of an electron (Bj [coordinates of centre (a, 6, c, A,), 



* Herglotz, Gott. Nach. Heft 6 (1904). 



t Sommerfeld, Ann. d. Physik, vol. xxxiii. p. 666. 



