570 PROCEEDINGS OF THE AMERICAN ACADEMY. 



undisturbed by external influences, about an axis of symmetry that is 

 at rest in the gravity ether, we may determine whether it can satisfy 

 the condition 



'^ =0. 



d{Ct) 



It might now be supposed that we could not do this by determining 

 whether 



because the radiation from an accelerated electron already in motion is 

 not the same as if it had the same acceleration when at rest. But it is 

 obvious that, if we consider any direction making given angles with the 

 directions of the velocity and acceleration, the ratio of the rate of 

 radiation in that direction in the case with velocity to the corresponding 

 rate in the case with no velocity is a function of only the magnitude of 

 the velocity and the angle between the velocity and acceleration, and 

 not a function of the charge or acceleration of the electron. Hence we 

 see that, if we consider a case where the two variables that determine 

 this ratio are the same for every point of the infinite sphere, we may 

 then, and only then, say that there is no radiation of negative energy 

 from a pair of uniformly accelerated gravitating particles if, and only if, 



midii = — m^^^. 



Returning to our rotating body, which may be any solid, liquid, gas, 

 or collection of very minute particles like Saturn's rings, provided only 

 that all points in any ring of infinitesimal cross-section around the axis 

 shall have the same density and the same constant angular velocity 

 around it, we see that the condition that no energy shall be radiated to 

 infinity from the body is that there shall be no radiation to infinity from 

 any of these rings. 



First let us assume that the axis is stationary in the gravity ether 

 and perpendicular to the direction of motion of the electric ether 

 through it, and let us see when this condition will be fulfilled. For 

 simplicity let us consider first only the case of a collection of particles 

 such as Saturn's rings or a rotating nebula with no internal forces but 

 those of gravity. In such a case it is obvious from symmetry that if 

 the inertia of a particle were proportional to its mass and independent 

 of its velocity, the system could rotate indefinitely, preserving always 

 its symmetry about the stationary axis, and, since the accelerations at 

 opposite points would be equal, opposite, and constant, not radiating 



