354 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. 



The theorem on mutual potential energy of two distributions, 

 117 (5), also holds, 



(2) (jV'crdS + 



By making use of these theorems, we find by the process of 

 131, that the potential in conductors is constant. All the 

 theorems on systems of conductors, 136 140, also remain un- 

 changed, except that the coefficients of capacity and potential 

 receive different values from those there given. 



180. New Law of Force. Action of Medium. We 



have found in 118 that the energy of any distribution acting 

 according to the Newtonian Law was exactly accounted for by 

 supposing each element of volume in all space to contain a 

 quantity of energy equal, per unit volume, to l/8?r times the 

 square of the total force of the field. Since the phenomena now 

 to be considered resemble the phenomena of Newtonian distri- 

 bution to such an extent that it was long before any difference 

 was discovered, this proposition must be nearly true. We have 

 found however that the energy depends on the medium as well 

 as on the distribution. We shall therefore, in order to explain 

 the phenomena, make an assumption deviating as little as possible 

 from the above proposition in regard to the energy, and con- 

 taining it as a particular case, but allowing us to take account of 

 the medium. 



The assumption will be justified if its consequences accord 

 with experiment. We shall assume merely that each element 

 of volume contributes to the energy an amount per unit volume 

 proportional to the square of the force of the field, the factor of 

 proportionality being a property of the medium, which may vary 

 from point to point. The whole energy is accordingly 



where //< is a positive point-function, which is given as soon as 

 the substances occupying each region of space are known. For 

 a homogeneous medium //, is constant. We shall now insert this 

 form of the energy in the expression W=2W d W f , and apply 



