426 VIII. NEWTONIAN POTENTIAL FUNCTION. 



The theorem expressed by the equality of the two integrals is 

 known as Gauss's theorem on mutual energy, where V p ' represents 

 the potential at p due to the whole mass M q , V qj that at q due to 

 the whole mass M p }) 



The above equality may be also proved as follows. Since 



76) Q f --{ 



and 



the triple integrals in 75) become respectively, 



77) ~- 4 



and 



, /// 



Vq^Vq'dlq. 



Now since outside of r, 4V = and outside of T', z/F f = the 

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

 both these integrals are equal to 



dV dV dV dV 3V cV 

 dx dx dy dy dz dz 



since the surface integrals 



on 



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

 theorem and Poisson's equation. 



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

 distribution consisting of both volume and surface distributions, the 

 sum 74) becomes the integrals 



78) W = 



Now at a surface distribution Poisson's equation is 



1 f dV cV\ 



" 1 



1) Gauss, u Allgemeine Lehrsatze in Beziehung auf die im verkehrten Ver- 

 haltnisse der Entferming wirkenden Anziehungs- und Abstossungskrafte." Werke, 



T J TT _ + r\m 



Bd. Y, p. 197. 



