572 PROCEEDINGS OF THE AMERICAN ACADEMY. 



obtained by multiplying the radiation from one of them alone by the 

 square of the ratio of their relative velocity to the velocity of light. 

 And we shall prove that while, for small bodies, this radiation is due 

 only to the changes in the accelerations in the time required for 

 propagation of radiations from one to the other, Theory IV involves 

 radiations that may be larger than this in any desired ratio, and that 

 are due to actual differences of mass acceleration. 



To prove this, let us determine, from the equations of Theory IV, the 

 accelerations of two masses, nii and m2, moving through the gravity 

 ether with equal opposite velocities parallel to that of the electric 

 ether, and whose ratios to C are Bi and B2., let the direction of w be that 

 of Bi and let us suppose the two particles to be moving so that niz is in 

 the direction of the axis of?/, at a distance a from mi, at the instant 

 considered. If now we let ri be the distance of any point from mi , in 

 a direction making an angle Oi with that of ^, we have the gravitational 

 force due to mi, neglecting that due to radiation and change of Bx in 

 time Ti/Cy directed towards mi at every point, and of intensity 



l&i I=^(l-B,^)(l-B/^sin^^O"^, 

 t'l 



while the gravimagnetic force due to mi is 



hi = Bixg^i. 



These formulas may be proved by substitution in the equations of 

 Theory IV, assuming Bi constant. 



Therefore, neglecting radiated forces from mi and change of Bi, the 

 force acting on tn^ is 



m^fi = ^2 (gi + Baxhi). 

 Therefore 



Similarly 



m,2i, = - j ^-^ B, (1 - B,^)-' (1 - B2 . BO. 



mi^i = + j ^^^ Bi (1 - B.^r (1 - Bi ■ B2). 



in which case the forces due to radiation are also equal and opposite. 



