Closing Years of the Nineteenth Century. 421 



kinetic potential which concerns any one of them say, e may 

 be written 



L e = e (a x x + a y y + a z z - c 2 <), 



where a and c denote potential functions, defined by the 



equations 



/ f c r r f 



dxdy'dz, </> = \\\ p - dx'dy'dz' ; 



p denoting the volume-density of electric charge, and v its 

 velocity, and the integration being taken over all space. 



We shall now reject Clausius' assumption that electrons act 

 instantaneously at a distance, and replace it by the assumption 

 that they act on each other only through the mediation of an 

 aether which fills all space, and satisfies Maxwell's equations. 

 This modification may be effected in Clausius' theory without 

 difficulty ; for, as we have seen,* if the state of Maxwell's 

 aether at any point is defined by the electric vector d and 

 magnetic vector h,f these vectors may be expressed in terms 

 of potentials a and ^ by the equations 



d = c" grad < - a, h = curl a ; 



and the functions a and < may in turn be expressed in terms of 

 the electric charges by the equations 



a - JTJ \((**)'lr\ dx'dy'dz', </> = J/J |(J5) dxdtfdsf, 



where the bars indicate that the values of (pv r )' and (p)' refer 

 to the instant (t - r/c). Comparing these formulae with those 

 given above for Clausius' potentials, we see that the only change 

 which it is necessary to make in Clausius' theory is that of 

 retarding the potentials in the way indicated by L. Lorenz.J 

 The electric and magnetic forces, thus defined in terms of the 



* Cf. pp. 298, 299. 



t We shall use the small letters d and h. in place of E and H, when MC are 

 concerned with Lorentz' fundamental case, in which the system consists solely of 

 free aether and isolated electrons. 



% Cf. p. 298. 



