SCATTERING OF PLANE ELECTRIC WAVES BY SPHERES. 201 



Since the co-ordinates of P are (0, mr, nr), the value of B is given by 



R? = y? + (y+mr)>+ (z-nr)\ 

 = r i +2r(mynz)+x > +y > +z a , 



where (x, y, z) is a point on the surface near to 0. Thus, when r is very large in 

 comparison with the dimensions of the surface (as we assumed in the previous 

 investigations, 4-6), we can use the approximate formula 



(7'3) R = r+mynz. 



Thus we can write in (7 '13) 



,, (r+my nz) 

 01 = _ _ r p-^(my-m) 



v i' u e , 



T 



*\ 



if v is the value of v at O. Then the most important term in is seen to be 



ov 



cz 



Accordingly the components of electric force in the reflected wave will be given by 

 the approximation 



47T 



(7-4) 



4?r 



= + ^ f(2nC) 



4?r J 



To evaluate the integrals in (7'4) we must write out the equation to the surface in 

 the approximate form 



2z = -(a. 

 Then* 



L- 



J 





 ticn 



Now ay /3 s is the absolute (or Gaussian) curvature of the surface at the point ; 

 and since the surface is supposed convex, we represent this curvature by l/p 2 . 



* This is most easily found by taking the integral as lim 



2 F 2 



