Fundamental Laiv of Electrical Action. 349 



particles in motion, Gauss deduced the following expression 

 for the mutual attraction between two charges : 



*-?;K("-i©')]. 



where e, e' are the charges, r = mutual distance, n = relative 

 velocity. 



But the law was found to be inconsistent with the principle 

 of conservation of energy, and naturally fell through. 



Other physicists in turn took up the problem. The most 

 celebrated formula is that of Weber *, according to whom 

 the mutual potential of two moving charges is given by the 

 expression 



This formula is consistent with the principle of conser- 

 vation of energy, hut was nevertheless found by Helmholtzf 

 to lead to improbable results. 



These laws were all based on the idea of action at a distance. 

 But in 1845, Gauss J again returned to the problem (which 

 he now calls the real keystone of electrodynamics), with 

 the idea that the action, instead of being propagated instan- 

 taneously, may be propagated with a finite velocity in a 

 manner similar to that of light. But he did not succeed, as 

 he himself tells us, in forming any consistent mental picture 

 of the manner in which the action is propagated, and seems 

 to have given up the attempt. 



Three other mathematicians, Riemann, Neumann, and 

 Betti §, followed in the wake of Gauss, and suggested 

 solutions, but these also have been no more successful than 

 their predecessors. According to Riemann ||, the force 

 components between two charges are given by the Lagrangian 

 derivatives of the function 



ee < r (u—uV+Cv-v'y+iw-w'yi 



^ = ^L ? "J' 



where (u, v, ic) are the velocities of the one particle, (u 1 , v' y w') 

 are the velocities of the other. 



* Maxwell, loc. cit. pp. 484 k 485. 

 t Phil. Mag. December 1872. 

 % Maxwell, loc. cit. p. 490. 

 § Maxwell, loc. cit. p. 490. 

 || Clausius, Phil. Mag. 1880. 



