514 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. 



so that our equation (9) becomes 



do) I*, a 



dudv 



Of this equation the general solution is 

 (11) ^ = 9i(u)+g 9 (v) t 



where g^ and g 2 are perfectly arbitrary functions of their arguments. 

 We consequently have for the solution of (9), 



( 1 2) r<j> r = g 1 (at + r)+ g 2 (at - r). 



When r = we have 



Q = g 1 (at)+g 2 (at), 



and this being true for all values of t the functions g l} g 2 are not 

 independent, but one is the negative of the other, whatever the 

 value of the argument. Putting then 



gi = g, 92=-g> 



we have 



(13) r$ r = g(at + r)-g (at - r). 

 Differentiating by r, 



(14) ^ r + r d ^ = g'(at + 

 and again putting r = 0, we obtain 



_ 



But <j> r is the mean value of </> over the surface of a sphere of radius 

 r with center at P, and the mean value over a sphere of radius 

 zero is the value at P itself. Accordingly 



(16) <J, P = 2 



Now differentiating ( 1 3) by r and t, 



so that 



and for t = 0, 



_ , 



dr a dt ] t =o 



