MOTION OF ELECTRIFIED SYSTEMS OF FINITE EXTENT, ETC. 161 



The limiting value of this expression when k = is found to be f 2 /aC 2 , thus 

 agreeing with the result in Section 3. 



When condition (2) is used the method of procedure is very similar to that already 

 used. 



The components of electrodynamic force are 



Y> .. /i i.2\o$n V' - (-\ l- 2 \ ^" 7' - /I i.'J,ty 

 A o (lk 75 i i o - (lK )= , /j o ( L A ) -TT . 



The condition that the tangential component of (X' , Y' , Z' ) should vanish at 

 r a is equivalent to the condition that i// should be constant at r = a The 

 solution of this problem is given by MACDONALD (' Electric Waves,' p. 172) in the 

 form 



e 



= log COth 77, 



where 



cosh" 77 sinh a 



7? sinn" r) 



It appears that the surface density of electricity is uniform and equal to e/lira*. 



In proceeding to the disturbed state it was found convenient to modify slightly the 

 expressions formerly used. 



With the same restrictions as before the total Meld in the disturbed state is 

 given by 



a = 0, 



n I 8l//,| 1 C'm , O~(l) , / 3 tin ; OV 



P '>' -5 + r, 5rf~ ~* V^ + V 5a * n > 



dc U o^ oi ox 02 ox oz nz 



,8i//,, i oty j vty ,,. r 2 i/>,, ,a x 



~~ A- 5 -- /, ^7^5 -- r * ^ ^ -- ' / ^ -- A- ^ , 

 o (J ot o v.r vi ex d c 



Y = -^L' fL. 



+ 



' 



z .*- 



Hence 



. _ _ 



a^ y a^a? ^^a^ ca^a? c 



VOL. CCX. A. Y 



