247] ELECTROMAGNETIC WAVES. 513 



If the surface S is a sphere of radius r with its center at the point 

 P. we have 



where by </> r we denote the values of < at points on the surface of 

 the sphere of radius r, with center P. 



Introducing polar coordinates into the left-hand side of equation 

 (4) also we may write it 



Now differentiating this and the transformed right-hand 

 member (5) by the upper limit r changes our equation (3) into 



The surface integral 



which appears on both sides is 4?r times the mean value of the 

 function < on the surface of the sphere of radius r. Calling this 

 mean value r we have the equation 



which, on performing the differentiations and dividing by r, may be 

 written 



_., 



If we now introduce two new independent variables 



u = at -f r, v = at r, 

 we have, putting r<f> r ty, 



dt ~~ dudt 3vdt~ du dv ' 



_ _ _ __ 



dr ~~ du dr dv dr ~~ du dv ' 



dt z 



= _ , 



r W dudv 

 w. E. 33 



