220 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. 



In other words, this is the amount of loss of the potential 

 energy of the system on being allowed to disperse to an infinite 

 distance from a distance apart r. Similarly, for any two systems 

 of particles m p , m<j, 



(3) W pq = 2A ^^ = 2 p m p V p ' = 2 q m q V q , 



Tpq 



Vp being the potential at any point p due to all the particles q 

 and V q being the potential at any point q due to all the particles 

 p. This sum is called the mutual potential energy of the systems 

 p and q. If however we consider all the particles to belong to one 

 system, we must write 



(4) TF=i22?^ = i2mF, 



r pq 



where every particle appears both as p and q, the J being put in 

 because every pair would thus appear twice. This expression has 

 been given in 59, (33). 



If the systems are continuously distributed over volumes r, T' 

 we have 



(5) F 



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 q , that at q due to 

 the whole mass M p * 



The above equality may be also proved as follows. Since 



(6) Pp = - 



and . 



the triple integrals in (5) become respectively, 



(7) - 



and - 



* Gauss. " Allgemeine Lehrsatze in Beziehung auf die im verkehrten Verhalt- 

 nisse der Entfernung wirkenden Anziehungs- und Abstossungs-Krafte." Werke, 

 Bd. v. p. 197. 



