256 PROPAGATION OF ELECTROSTATIC FORCE. 



units of electricity, or the reciprocal of the velocity of light. For 

 this satisfies the general equation 



-V 2 = c 2 d 2 V<ft 2 , (5) 



as well as the so-called "equation of continuity," and also satisfies 

 the special conditions that when t < 



outside of the unit sphere, and that at any time at the surface of 

 this sphere 4 ^ = m ^ Xf) _ ^ 



if we consider the terms containing the factor c as negligible, when 

 not compensated by large values of r. That equation (4) satisfies 

 the general conditions is easily verified, if we set 



u = r- l (t-cr), (6) 



and observe that 



-?%<**<*/<&*, (7) 



and that the three components of $) are given by the equations 



4<7rg = d 2 u/dxd,y (8) 



4-Tr/t = d 2 u/dxdz 



Equation (4) shows that the changes of the electrical displacement 

 are represented by three systems of spherical waves, of forms 

 determined by the rapidity of the discharge of the system (A, B), 

 which expand with the velocity of light with amplitudes diminishing 

 as r~ 3 , r~ 2 , and r~ l , respectively. Outside of these waves, the electrical 

 displacement is unchanged, inside of them it is zero. 



If we write (with Maxwell) dtyt/dt for the force of electrodynamic 

 induction at any point, and suppose its rectangular components 

 calculated from those of d 2< )/dt 2 by the formula used in calculating 

 the potential of a mass from its density, we shall have by Poisson's 

 theorem 



or by (5), V 2 (d%/dt)= -47rc- 2 V 2 ), 



whence d%/dt= -47rc- 2 ). (9) 



From this, with (4), and the general equation 



we see that during the discharge of the system (A, B) the electrostatic 

 force V V vanishes throughout all space, while its place is taken by 

 a precisely equal electrodynamic force dtyi/dt. 



This electrodynamic force remains unchanged at every point until 

 the passage of the waves, after which the electrostatic force, the 

 electrodynamic force, and the displacement, have the permanent value 

 zero. 



