566 PROCEEDINGS OF THE AMERICAN ACADEMY. 



not, though it does reduce the component of force in the direction of 

 motion to an infinitesimal of the third order. For if we consider a 

 system where Ai and A2 are the positions of the two masses m^ and 7^2, 

 revolving in circular orbits at the time t, B^ and B2 their positions at 



the time it y~ ) and ( t ^r^ )' ^^^ ^1 ^^^ ^2 the positions 



they would have occupied if they had kept since the above-mentioned 

 times the velocities they then had, we see that Ci and Bi must always 

 lie on the same side of the line A^Ai produced. If the velocity C 

 becomes infinite, the distances from C^ and C2 to the line Aj^A^ are 

 obviously infinitesimals of the third order, but they can never be ex- 

 actly zero ; and hence there will always be a forward component of 

 force that must ultimately produce the same disastrous results that 

 we have seen in Theory II. Hence we see that Theories II and III are 

 almost certainly false. 



Theory IV. — Let us now suppose that Theory IV is correct, and 

 examine its consequences. By Green's Theorem it may easily be 

 proved that the gravitational energy liberated in scattering any distri- 

 bution of gravitating matter to infinity is 



-8S///^'* 



if the matter is all at rest. But if the matter is in motion, we must 

 add to this, to get the gravitational energy liberated in bringing it all 

 to rest at infinity, 



Wk J J J^ 



sr ■ ■ ■^^^^- 



And if the ratio of the velocity of matter at every point to that of 

 light is the vector point function p, the total kinetic energy in the uni- 

 verse is 



/// 



pc^ (E-' - 1) dr 



where R = Vl — ^^- We may now assume as in the corresponding 

 electromagnetic case, that the integrands in the above expressions 

 actually represent the energy that would be removed in the scattering 

 process from the volume elements for which they are evaluated, and 



