172 Accelerated Motion of Rigidly Charged Dielectric Sphere. 



period equation in X, the real part of the sum being alone 

 retained. Thus 



ct- 



-m^>^i 



As we have seen above the exponential terms representing 

 vibrations soon die away, and we are left with the equation 

 of motion in the form 



2^_ 

 3 a& 





from which we deduce that the electromagnetic mass of the 

 sphere is 



P 2 e* 



s 3 ac 2 



which agrees with the value previously obtained for the 

 quite different perfectly conducting sphere. 



A few general remarks on the solution here obtained seem 

 necessary. 



It will be noticed that the general form of the /function 

 involves an infinite number of arbitrary constants B r> 

 corresponding to the infinite number of roots of the period 

 equation. Moreover, we have only two conditions to deter- 

 mine these constants. In addition to those already written 

 down we have the two conditions for the continuity at the 

 outgoing wave boundary. These give 





Z\B-2a\^ 2 =0. 



The problem thus appears to contain a great degree of 

 indeterminateness. It is, however, to be noticed that the 

 conditions of the problem are also really very indefinite. 

 The system really comprises an infinite number of degrees 

 of freedom, corresponding to the bodily motions of the sphere 

 and the infinite number of possible oscillatory motions which 

 may exist in it. The geometrical conditions only involve 

 the motion of the sphere as a whole and leave wholly 

 unspecified the conditions under which the oscillations are 

 taking place. If the method of distribution of the energy 

 and momentum among these possible degrees of freedom 

 could be specified, then the problem would be determinate. 

 A study of the problem discussed by Lamb, where similar 



