Vibrations  of  Conducting  Surfaces  of  Revolution.       719 
and  the  value  of  <f>  may  be  expressed  in  terms  of  elliptic 
functions,  as  was  previously  done  for  the  elliptic  cylinder 
(loc.  cit>).  It  may  be  noted  that  o^  and  w2  are  both  real  for 
all  roots  e.  This  formula  holds  for  all  values  of  /3  if  n  is  not 
very  small  compared  with  kc,  and  does  not  fail  on  the  axis. 
It  is  convenient  to  note  here  that  in  the  case  of  the  elliptic 
cylinder  corresponding  to  the  above,  n2  should  be  —  ?i2,  and 
the  quantity  6  there  employed  must  be  retained  in  full  as  a 
series  in  powers  of  (kc)2.  The  series  in  two  dimensions, 
given  by  Mathieu  in  connexion  with  the  vibrations  of  an 
elliptic  membrane,  is, 
0_„2+     (MY      ,       (*6)'.5»»+7      ,  r,y. 
tf-'l+2(^i)  +  32(n'-l)«(«*-4)+"-'     •     C63) 
corresponding  to 
d2M 
X?  +  (0-2k2b2co§2v)M.=0.      .     .     .     (64) 
The  period  equations  for  the  vibrations  between  two 
spheroids,  corresponding  to  large  values  of  n.  may  be  written 
down  at  once  from  (60). 
Selecting  the  function  corresponding  to  the  internal  Bessel 
function  of  the  sphere,  by  writing  c  =  A/  =  0,  it  appears  that 
IT 
€i  in  (60)  is  -r  for  the  single  spheroid.  Thus  for  the  higher 
series,  the  period  equations  for  a  single  spheroid  are 
(e2  cosh2  a-\'-n2f=  (4s +  1)^,    .     .     (65) 
and 
i 
In  the  case  of  the  elliptic  cylinder,  there  is  only  one  such 
equation,  namely 
d      cosice^  —  — ) 
W)* 
the  value  of  \'  being  given  by  (63)  above 
da.  -0,      .     .     .     .     (67) 
Vibrations  in  the  External  Space. 
The  vibrations  in  the  space  external  to  a  single  conductor 
have,  in  general,  a  large  modulus  of  decay,  and  cannot  be 
maintained  for  a  great  space  of  time.  The  form  of  (f>  suit- 
able for   the  space,   employing   imaginary  quantities,   must 
