124, 125] 

 the shell, 



POLARIZED DISTRIBUTIONS. 



(16) V=<h 



The geometrical integral, 



(17) 



237 



dS. 



cos 



has been found in 39 to be equal to the solid angle o> sub- 

 tended at P by the surface 8, if n^ points 

 toward the side of 8 on which P lies. Con- 

 sequently the potential at any point P due 

 to the shell is equal to the product of the 

 difference of potential of polarization on the 

 two sides of the shell by the solid angle 

 subtended by the shell at P, the potential 

 being positive if P is on the positive side of 

 the shell, that is, the side toward which the 

 polarization is directed. Now we have seen in 39 (5) that the solid 

 angle integral is equal to 4?r for a point inside a closed surface, 

 and to zero for an outside point, that is, it experiences a dis- 

 continuity of 4-7r as P crosses the surface. When the surface is 

 not closed the same thing takes place. For the integral 



FIG. 54. 



a) = 



is a continuous function of P so long as r is not zero, that is, so long 

 as P does not lie on the surface. If P lies on the surface, the in- 

 tegral has an infinite element. We remove this by cutting out a 

 small area around P. If now o>' be that part of the integral due 

 to the remainder of the surface, o>' is finite and continuous even 

 when P passes through the surface. As P approaches the surface 

 the solid angle subtended by the small area cut out, which may be 

 treated as plane, approaches 2vr, so that at the surface on the side 

 1, G) 1 = ft) / +27r. At an infinitely near point on the side 2, how- 

 ever, the cosine in the numerator has changed sign, for the small 

 area, so that the solid angle subtended by the latter is to have the 

 negative sign. Accordingly on the side 2, o> 2 = &>' 2?r, and ac- 

 cordingly, 



ft)j ft> 2 = 4-7T, 



