119] ENERGY. 225 



Since the unvaried distribution is a Newtonian one, by Poisson's 

 equation the factors multiplying SV in the first two integrands 

 are zero. Consequently the variation of W is equal to minus the last 

 integral, which, as the integrand is a sum of squares, is necessarily 

 positive. Accordingly V W < and we may state the theorem : 



If the potential due to any given distribution of matter acting 

 according to the Newtonian Law is known, the energy calculated 

 by the formula (3) is a maximum for the actual distribution of 

 potential as compared with arbitrary distributions differing by 

 an infinitesimal amount from the actual. 



We may state this theorem in physical language,, avoiding 

 specification of the form in which W is to be expressed, as follows. 

 We may consider V + S V as the potential due to a Newtonian 

 distribution whose densities differ at each point of space by an 

 infinitesimal amount from the densities of the given distribution, 

 the differences being otherwise perfectly arbitrary. We will call 

 the supposed distribution 2, the original distribution being 1. 

 Then the terms 



8F) dS+fjjp (F+ 87) dr, 



are the mutual energy TF 12 of the distributions 1 and 2, by 

 117, (5). 



The integral 



is the energy W z of the distribution 2, by 118, (10). 



Accordingly equation (4) is 



Tf 1 + S F F=TT 12 -Tr 2 , 

 so that 



$ V W = W 12 -(W 1 +W,)<0, W 12 < W,+ W,. 

 w. E. 15 



