and  the  Constitution  of  the  Atom.  121 
motion  in  a  sense  divergent,  the  disturbance  increases  expo- 
nentially with  r  ;  and  thus  (12)  cannot  apply  to  a  problem 
where  the  disturbance  is  supposed  to  originate  in  the  neigh- 
bourhood of  the  sphere. 
The  analysis  of  the  electrical  problem  is  necessarily  rather 
elaborate,  and  in  illustration  it  may  be  well  to  consider  the 
analogous  question  for  sound.  For  the  term  of  order  zero, 
the  velocity  potential  ^0  of  a  divergent  wave  takes  the  form* 
^o=^*C«*-,) (13)5 
where  a  denotes  the  velocity  of  sound.  In  the  usual  theory 
the  divergent  vibration  is  supposed  to  be  maintained  by  forces 
operative  at  r  =  Q  with  a  prescribed  frequency.  At  present 
we  regard  (13)  as  applicable  to  the  space  outside  a  certain 
sphere  of  radius  c.  whose  surface  remains  at  rest,  so  that  the 
case  is  that  of  air  vibrating  round  a  solid  ball  of  radius  c. 
The  condition  to  be  satisfied  at  r  —  c  is  d^0/dr  =  0  ;  so  that 
1-H&c=0 (11), 
and  f0=  V(«'->/« (15). 
In  like  manner  for  the  term  of  order  unity  we  have 
ryp-1=Acos6e-ikr(^l  + ~)eikat    .     .     .    (16), 
and  the  surface  condition  gives 
ikc  +  2+~=0 (17), 
ike  v     ' ? 
whence 
ikc~-l±i '.     (18). 
"When  r  is  great,  (16)  becomes  accordingly 
,        A  cos  6  ,   .  ,  -,  ,    w 
Y'l= ^      ±t)(at— r)/c       ^        ^        ^  (19). 
Both  iu  (15)  and  (19)  -f  diminishes  exponentiallv  with  the 
time  but  increases  exponentially  with  the  distance  r.  The 
case  is  not  mended  if  we  start  with  eik(-af+rK  Instead  of  (15) 
we  then  find 
<f0=-V^)/<     ....         (20), 
increasing  exponentially  with  r  as  before  and  now  also  with  /. 
It  does  not   appear  that  any  solution   exists  of  the  kind 
*  'Theory  of  Sound,'  §  325. 
