Vibrations  of  Conducting  Surfaces  of  Revolution.       711 
where  the  series  will  not,  in  general,  terminate.  Denoting 
the  series  in  brackets  by  S,  and  substituting  in  the  equation, 
we  obtain,  if  z  =  tet, 
02+e2)(S  + 2  $)  +  (e2-^)s  -  o. 
From  this  equation,  the  relations  among  successive 
coefficients  become 
2(m+l)cm+i  =  (e3— \  +  m  .  m  +  l)cm  —  2(m  —  l)e2cm-i 
-f  (m-l)(m-2)e2^_2    .     .     (30) 
20i  =  (e2-\), 
±c2=  (2  +  e2-\K 
6c3  =  (6-f  e2~\)c2-262c1. 
In  the  case  of  n  =  0,  e2  =  A,  to  the  seventh  order.  Thus 
cx  =  0,  and  therefore  c2  and  successive  coefficients  are  all  zero. 
In  fact,  the  solution  in  this  case  is,  to  this  order, 
(j>  =  A  cos  (he  cosh  a)  +  B  sin  (kc  cosh  a)     .     .      (31) 
so  far  as  a  is  concerned. 
The  conjugate  equation  in  ft  is 
whose  solution  is  accurately 
<£  =  C  cos(£c  cos  /3)  +  D  sin  (fo?  cos  ft).     .     .     (32) 
Now  this  is  a  function,  which,  even  for  great  values  of  kc, 
is  perfectly  periodic  in  ft,  and  is  therefore  a  possible  solution 
for  n  =  0  for  any  value  of  kc.  But  there  is  only  one  such 
solution.  Thus  we  infer  without  further  calculation  that 
for  n  =  0,  all  the  terms  of  Niven's  series  will  vanish  except 
(kc)2,  which  is  the  accurate  value  of  X0.  Although  this 
result  is  interesting  from  the  point  of  view  of  several  pro- 
blems, it  is  not  to  be  expected,  f  roui  the  analogy  of  the  sphere, 
that  it  corresponds  to  any  vibration  in  this  case,  and  in  fact, 
on  writing  down  the  values  of  the  electric  and  magnetic 
forces  by  previous  equations,  it  is  found  impossible  to  make 
them  finite  on  the  axis  (/3  =  0)  unless  they  are  everywhere 
zero.     We  therefore  pass  to  the  case  n  =  l,  which  leads  to  the 
