MOTION OF ELECTEIFIED SYSTEMS OF FINITE EXTENT, ETC. 189 



Thus a uniformly accelerated rotation is possible, and the reaction of the medium is 

 equivalent to an increase of the effective inertia of the sphere = fm', m' being the 

 electric inertia for linear motion at slow speeds. 



The moment of inertia of the mass ^m' uniformly distributed is |W x f 2 or ^m'fi 2 , 

 and is the same as that of a thin shell of mass ^m' and radius a. Thus for both 

 linear and rotary uniform acceleration the dynamical effect of the aether is represented 

 by the addition of a uniform thin shell of mass fyn' on the surface of the sphere, along 

 with a particle of mass fyn' at the centre of the sphere. 



As in the case of linear motion, this state is not attained without the production of 

 initial vibratory disturbance which may be supposed to decay rapidly. The deter- 

 mination of these turns on the occurrence of free modes of motion, the arbitrary 

 constants being determined by the initial conditions. 



The integral of (4), when L 0, is 



and thus the free vibrations are determined by 



Assuming the forms 



we get 



A( 1-X) = B-i ( 1 - K 1 'X) - ( L + K 1 a X) e-* K> *}, 



A ( 1 - X) -3 + X s = HKX a ( 1 - e^ K ' *). 

 L m j 



Thus X is determined as a root of 



(1 -K 12 X) I ( I -X) :} '"' + X 2 1 - ( 1 -X) KX 2 



-SK>/aA _ i 



-- - _ _ 



-(l-X)KX 2 



tanh K'/'X = - L , 



m 

 The first of these forms shows that there is a real root for X which will not differ 



