160, 161, 162] ENERGY OF DISTRIBUTIONS. 425 



from which, by 148, 130), -we obtain finally, 



70) F.g 



and this series is convergent for points on the surface of the ellipsoid 

 itself, if & 2 < 2 a 2 . The series converges extremely rapidly if ^ differs 

 little from unity. 



162. Energy of Distributions. Gauss's theorem. If a 



particle of unit mass be at P, (x, y, g) at a distance r from a particle 

 of mass m (J , the work necessary to bring the unit particle from an 

 infinite distance against the repulsion of the particle m q will be 



71) W= 7 m J=yV( X ,y,z)= r r p . 



If, instead of a particle of unit mass, we have one of mass m p 

 the work necessary will be m p times as great, 



m 



72) W pq = r -?m p = ym p V p = ym q V q , 



where _ % 



**-% 



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 q , 



73) w ft = 



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 ^?. 

 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 



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



because every pair would thus appear twice. This expression has 

 been given in 28, 36). 



If the systems are continuously distributed over volumes t, t 1 

 we have 



75) W pq = 



