If 
Vibrations  of  Conducting  Surfaces  of  Revolution.        721 
I£  cr=t-|-g,  Ave  find 
8=§A;   .-. (71) 
The  vectors  therefore  decay  more  rapidly  than  in  the  case 
(2    \ 
1  +  re2  \ . 
For  the  other  type,  the  period  equation  reduces  to 
'(^rJ  =  1  +  rU-i)- 
s=(V3-7)s73 (72) 
Thus  the  new  rate  of  decay  is  —  ( -~  +  v  J ,  and  the  decay 
is  again   more   rapid  than  in  the  case  of  the  sphere.     The 
real  value  of    ha   becomes    — r — 7=,  and   is   decreased. 
-;        5^3 
For  the  oblate  spheroid,  the  sign  of  e2  must  be  altered.     A 
particular  case  of  the  oblate  spheroid  is  the  circular  disk, 
which  is  worthy  of  separate  treatment. 
All  formulae  above  for  the  high  periods  may  be  transformed 
as  shown,  so  as  to  be  suitable  for  the  external  space. 
Vibrations  of  other  Surfaces  of  Revolution. 
The  cone  is  best  treated  by  spherical  polar  coordinates. 
The  transformation  appropriate  to  the  paraboloid  is 
p  +  £*=(a+«0)2 (73) 
In  two  dimensions,  this  will  solve  the  problem  of  the 
parabolic  cylinder.  The  appropriate  functions  for  this  case 
have  been  treated  by  Whittaker  *. 
Hyperboloids  of  revolution  form  the  conjugate  case  to 
spheroids,  but  are  of  little  physical  interest.  The  hyperbolic 
cylinder  may  be  solved  in  the  same  manner. 
*  Proc.  Lond.  Math.  Soc.  190: 
