576 PROCEEDINGS OF THE AMERICAN ACADEMY. 



but that if we draw two surface elements, dSi and dSg, parallel to each 

 other and perpendicular to the same line of the vector gxh at a dis- 

 tance dx apart, we see that, if dSa is in the direction of gxh from dSj, 

 the energy radiated from a mass that is accelerated for an infinitesimal 



time crosses dSg at a time — units of time later than that at which it 



c 



crosses dSi, so that the transfer is actually in the direction of gxh. 

 But such radiated energy has lost all connection with its source, and 

 there is no positive energy radiated, hence it is negative energy that is 

 transferred in the direction of gxh across these surfaces. 



It is also interesting to notice that in the case of a mass moving with 

 uniform velocity the vector gxh has the same direction as the motion 

 at all points in the plane through this mass perpendicular to this direc- 

 tion. This also indicates that the mass carries negative energy with it 

 as it moves. But if the mass is not charged with electricity, the elec- 

 tric and magnetic forces due to the electrons and corresponding positive 

 charges within it are zero at points outside the mass. Therefore, the 

 electromagnetic energy is wholly within it, and the external gravita- 

 tional energy cannot be considered as a mere diminution of the electro- 

 magnetic energy on any theory that assumes the existence of positive 

 energy only. To calculate the energy in terms of the electric and 



magnetic forces we must distinguish between the forces due to positive 



+ + 

 charges which we may call E and H, and those due to negative charges, 



which we may call E and H. In this case the total energy of the dis- 

 tribution is 



00 



= hffl^^^ ~ ^^ ^^' ^ ^'^ "^ ^^' + H^) -I- 2(E • E + H . H)} dr, 



CO 



where we have exactly the same distribution of total energy in each 

 element ; and we have made a change only in assuming that the whole 

 positive charge and whole negative charge have each a much larger 

 amount of positive energy than the total energy of the system, but 

 that negative mutual energy of the two distributions more than neu- 

 tralizes their positive energy in all volume elements outside the mass, 

 and fails to do so only within the mass. 



Stresses in the Gravitational Field. — We have by no means yet ex- 

 hausted the store of interesting facts connected with this negative 



