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



explicitly contain the densities. This expression holds good only 

 for a distribution acting according to the Newtonian Law. As 

 these two expressions must be equal for all distributions, we may 

 write 



(3) W 



aFV V aF\ 2 



ff ir^o fff f T r 1 

 = o-VdS + pF- 



JJ JJJn L & 



If in this latter expression for W we make an arbitrary varia- 

 tion in the form of the function V, we obtain for the varied value 

 of W an integral containing the variations of V, a, and p. If 

 we suppose that the new distribution also acts according to the 

 Newtonian law, in virtue of Poisson's equation there will be 

 relations between SF, So-, 8p. 



We shall however remove this restriction, and consider V, cr, p 

 as perfectly independent functions, which can be varied inde- 

 pendently. 



We shall choose Bo- and Bp as zero, in other words we shall 

 suppose V to be varied from the values that it actually has for 

 the original Newtonian distribution, the variation being entirely 

 arbitrary, while the densities are unchanged. Calling the varia- 

 tion under these circumstances 8TF, 



(4) W+& r W=<r(V+*V)d8+ P (V+8V)dr 



a* ) + ( 



T 

 + 



From this we obtain by subtraction of (3), 

 (5) S v W=llaSVdS + jjl pSVdr 



aF 



_ 



dz dz 



asF 



Integrating the third integral by Green's theorem, we have 

 finally 



