MOTION OF ELECTRIFIED SYSTEMS OF FINITE EXTENT, ETC. 175 



On integration for the whole sphere we get the comparatively simple value 



-! v" 

 3 a * ' 



Thus the equation of motion of the sphere is 



a 



Since the initial conditions are = 0, f = 0, when t = 0, and x(~ (( ) = x'(~ a ) = ^> 

 we get the integral 



Hence, eliminating , we get the equations 



Thus, as a particular solution, we get 



,v 2maCt . ., mm 

 J ^-- - TT- sJ- 



In addition, we must have vibratory terms in order to secure the satisfaction of the 

 initial conditions. For this purpose we assume the forms 



^ (C'i-r) = 



lk(CV+>-) = 



These satisfy the condition 



Substituting, we obtain the equations 



. + ^)A = G 

 m/ 



' x\ * 

 4- X A = 



m } 

 The values of X are thus the roots of the equation 



(K-l)(l+-)-( K - 1 )f 1 + K . V2 +K 1/2 V+fK-l-K 1 / 2 



\ Wl I \ / tn I \ 



(K- l) 1 + + (K- 1)K^- 1 + K 1 ' 2 K- 1 + K )K 1 / 2 X 2 +K(K 1 ' a + 1)X 3 

 m) '\ m) \ m] 



