Vibrations  of  Conducting  Surfaces  of  Revolution.       Ill 
form  exists.  The  asymptotic  expansion  therefore  represents 
(except  on  the  axis)  a  function  finite  at  all  points,  if  it  satisfies 
the  odd  and  even  conditions. 
Thus  B  =  0  if  n  is  odd,  and  A  =  0  if  n  is  even.  The  con- 
tinuation of  the  function  on  the  axis  is  not  given  by  the 
expansion. 
When  kc  is  great  in  comparison  with  n,  X  therefore  be- 
comes (kc)2,  and  the  equation  in  a  for  (j>  then  has  the  simple 
solution  given  by  (31).     Thus 
cf>  =  C  cos  (kc  cosh  a)  +D  sin  (kc  cosh  a)  .     .     (52) 
This  is,  of  course,  another  asymptotic  expansion,  suitable 
for  the  space  between  two  spheroids,  and  representing  the 
proper  function  except  at  the  origin,  where  its  continuation 
must  be  differently  expressed.  D  =  0  if  n  is  odd,  and  C  =  0 
if  n  is  even,  if  we  are  treating  the  interior  of  a  spheroid. 
When   the   eccentricity  is  very  small,  the  /3  factor  becomes 
incorrect,  and   must   be   replaced   by   (1— fi2)  — ~- -  where 
fi  =  cos  j3.  But  the  a  factor  remains  correct,  for  c  cosh  a 
becomes  r  when  the  eccentricity  is  small,  and  the  a  factor 
becomes  the  ordinary  expansion  of 
r*{Jn+±{kr)  +vJ  _n_,(kr)\, 
which  is  the  true  factor  for  a  sphere  {cf  supra). 
If  n  is  retained,  the  more  accurate  expression  for  </> 
becomes 
cos  1 
A^/sinh a.  sin/3  cos  (eco'  —  e2)  g-n  \  (eo> — e2) 
^=  (e2  sinh2*-?i  .  n  +  l)i(n  .  n-fl  +  e2  sin2/3)i  '  ^  ^  ^ 
where  e];  e2  are  arbitrary  constants,  and 
ea)/=Ja(e2  sinh2  a— n  .  n+  l)ida,  .      .     .     (54) 
eo  =^  (n  .  n  +  1  +  e2  sin2/3)id/3,     .      .      .     (55) 
This  is  proved  for  the  a  factor  just  as  in  (50). 
If  n  be  neglected,  the  high  periods  between  two  spheroids 
(#i,  «2)  m  Case  2  are  given  by 
A  cos  (kc  cosh  ax.  2)  -f  (3  sin  (kc  cosh  a1?  2)  =  0, 
whence  k(a1  —  a2)  =  S7r. 
77" 
In  Case  1,  k(a1  —  a2)  =  (2s  +  l)-, (5G) 
where  alf  a2  are  the  major  axes.  These  are  replaced  by  the 
minor  axes  in  the  case  of  oblate  spheroids. 
