117, 118] ENERGY. 221 



Now since outside of T, AF= and outside of T', AF' = the 

 integrals may be extended to all space. But by Green's theorem, 

 both these integrals are equal to 



i rfr (9F9F' 9F9F' 



- I*- 1 ^- + ^~ "a" 



4-7T JjJ^ (9# 9# 9y 9y 

 since the surface integrals 



vanish at infinity. Gauss's theorem accordingly follows from Green's 

 theorem and Poisson's equation. 



118. Energy in terms of Field. For the energy of any 

 distribution consisting of both volume and surface distributions, 

 the sum (4) becomes the integrals 



Now at a surface distribution Poisson's equation is 



4-7T (9% 



If, as in 85, we draw surfaces close to the surface dis- 

 tributions, and exclude the space between them, we may, as above, 

 extend the integrals to all other space, so that 



the normals being from the surfaces $ toward the space T. But 

 by Green's theorem, as before, this is equal to the integral 



Thus the energy is expressed in terms of the strength of the 

 field 



at all points in space. This integral is of fundamental im- 

 portance. 



It is at once seen that this is always positive. 



We may obtain the same expression as follows. Suppose 

 that the matter at a point x, y, z is displaced to a point x + &e> 



