! r eetor Potentials due to Motion of Electric Charges. 231 



The co-ordiuates, velocities, and accelerations of Q' are 



r 

 all functions of t— , and, therefore, we have a number of 

 c 



expressions satisfying the equation. Thus, if the co- 

 ordinates of Q' are a?', y\ z\ its velocities u Xi u y , u Zi and 

 its accelerations f x , f yi f z , 



&c. . . . 



(i-ti ^-r> K 1 -?)' 



all satisfy the equation of wave-propagation at any point P 

 at a distance r from the position Q/ of the moving electron, 

 u r being the velocity of Q' resolved along Q'P, and c the 

 velocity of light. 



Suppose, now, we have a distribution of moving electricity, 

 and we wish to find solutions of the equations of wave- 

 propagation that will hold at any point P. 



If A= " * , then V 2 A — 2 . ~ will vanish 



at any point P where there is no electricity. If there is 

 electricity at P moving with velocity u, we draw a small 

 sphere round P as centre. As this sphere diminishes it 



is clear that A and -=—jp contributed by the charges inside 



the sphere tend to zero, but V 2 A tends to the value 



2lTpC , c + u 

 u ° c — u 



Hence A = I 1 1 ■ ' — ^— is the solution of the equation 



r J $t z U ° C — U 



T 



But, since u is a function of t , it follows that 



-.«t 



pudx' dy' dz' 



"•^H 1 -?)] 



