118,119] -ENERGY. 223 



The whole variation of the energy is 



, asin(i2). 



Applying Gauss's Theorem to the mutual energy of two distri- 

 butions, one of which has density p, producing the potential V, the 

 other the density Bp, producing potential SV, we have 



jjj 



and W = pVdr 



gives in agreement therewith 



(15) 



The integrals may be now restricted to the space occupied by 

 matter. 



119. Maximum theorem for Energy. By making use 

 of the two different expressions for the energy we can deduce 

 an important theorem relating to the energy of a distribution. 

 We may use the form, 118, (8), 



(i) W * = * Vd8 + P VdT > 



which is distinguished by the suffix d to denote that the densities 

 occur explicitly. This form, by the definition of the potential, 

 holds for any law of force, whether the Newtonian or not.* On 

 the other hand we may use the form, 118, (10), 



to which we give the suffix / in order to denote that it is ex- 

 pressed only in terms of the field at all points, and does not 



* By this we mean any conservative law in which the action is proportional to 

 the product of the masses, and to some function of their relative position. 



