247] ELECTROMAGNETIC WAVES. 515 



Now inserting the value of <j> r , 



Suppose that for a certain initial instant, for which we shall 

 take t = 0, the values of the function < and of its time derivative 



~ are given as functions of a point in space, 



(JU 



(20) 



Inserting in the equation (19) it becomes 



but when r at the value of the left-hand side is by (16) equal 

 to <f) P . 



Accordingly we have finally, 



This solution was given by Poisson*. It shows that the value of </> 

 at all times may be calculated for every point P if we know the 

 mean value of d(f>/dt at a time earlier by the interval at for all points 

 on the surface of a sphere of radius at about P. as well as the rate 

 of variation of the mean value of (f> as the radius of the sphere is 



altered. Suppose that initially < and ^ are both zero except for 



ot 



a certain region whose nearest point lies at a distance r : from P 

 and whose farthest at a distance r 2 . Then as long as t<r l /a 

 the mean value of < on the sphere of radius at is zero, and after 

 t > rja as well. Accordingly there is no disturbance except 

 between the times rja and rja t or the quantity (f> is propagated 

 in all directions with the velocity a. It may be easily shown that 

 < P is finite if F and / are finite everywhere. 



We might have obtained the same result in a more simple 

 manner by transforming A< to polar coordinates in equation (3), 

 making < independent of the angular coordinates, when the equa- 

 tion becomes 



* Nouveaux Memoires de VAcademie des Sciences, t. in. 



332 



