708  Mr.  J.  TV.  Nicholson  on  the  Symmetrical 
In  the  mode  in  which  i/r  =  0,  the  only  non-zero  electrical 
components  are  (x,  y,  c).  Since  a  =  0,  the  magnetic  force 
at  the   surfaces  is  tangential,  and  the  surface  condition  is 
merely  y=0,  or  ^  =0.     This  leads  to  Macdonald's  equation 
for  the  periods.  The  mode  denned  by  i/r  leads  to  the  other 
set  of  periods.  The  case  n  =  0  makes  all  the  components 
zero,  and  leads  to  no  vibration. 
The  most  interesting  case  is  that  of  a  single  sphere.  The 
most  general  components  in  a  symmetrical  oscillation  inside 
the  conductor  are,  if  c  is  the  radius, 
taken  for  all  roots  &  of 
£  {<*.«(*•)  H«      ....     (15) 
together  with 
Bw(«  +  1) 
r~2 
Jn+Akr)Vn(ii)  cos  (kVt  + e), 
taken  for  all  roots  £  of 
J„+i(^)  =  0 ■•     (16) 
The  gravest  modes  of  the  first  and  second  systems 
(corresponding  to  n  =  l  in  each  case)  are  given  by  the  least 
roots  of 
(kc-~j)tznkc+ 1  =  0,     ....      (17) 
and  tan  hc=kc, (18) 
respectively. 
