Vibrations  of  Conducting  Surfaces  of  Revolution.       707 
The  two  latter  conditions  may  be  combined  to 
*>'=/•(»)  -M/3) (18) 
The  general  solution  of  (11)  has  been  given  by  Prof. 
Michell  *.  For  our  present  purpose,  it  is  necessary  to  select 
those  solutions  which  satisfy  the  further  conditions  (12). 
Excluding  the  cases  in  which  the  conjugate  transformation 
involves  elliptic  functions,  it  appears  that  the  surfaces  for 
which  our  problem  may  be  solved  are  the  sphere,  spheroid, 
cone,  paraboloid  and  hyperboloid  of  revolution.  Since  a 
cylinder  is  a  particular  form  of  surface  of  revolution,  in 
which  the  bounding  curve  describes  a  circle  of  infinite 
radius,  it  is  evident  that  the  cases  of  the  circular,  elliptic, 
parabolic,  and  hyperbolic  cylinders,  as  well  as  that  of  two 
infinite  planes  meeting  at  any  angle,  may  also  be  solved. 
It  is  found  that  the  anchor-ring,  a  surface  of  considerable 
physical  interest,  does  not  admit  of  such  a  solution. 
We  proceed  to  discuss  the  vibrations  between  two  surfaces 
of  revolution  of  spherical  and  spheroidal  shape.  When  they 
are  perfectly  conducting,  the  surface  conditions  are  that 
the  resultant  magnetic  induction  and  electric  force  at  the 
surface  are  tangential  and  normal  respectively.  There  are 
thus  two  distinct  classes  of  vibrations,  corresponding  to  <£  =  0, 
and  «? — =0,  if  a  =  const,  is  one  of  the  surfaces. 
Vibrations  of  Concentric  Spheres. 
The  periods  of  this  simple  system  have  been  obtained  by 
Macdonald  f>  as  the  roots  of  a  complicated  transcendental 
equation.  If  (r  0  to)  are  spherical  polar  coordinates  referred 
to  the  centre  of  the  spheres, 
z  =  r  cos  0,         p  =  r  sin  0, 
whence  (10)  becomes 
"2P  +  P-Cot^||+^  =  0;    .     .     (13) 
whose  solution,  finite  for  all  inclinations  0,  is,  if  (jl  =  cos  0, 
<j>= J/sin'fl^p  {  AJ„+,(/,r)  +  BJ_„_#r)  }  .     (14) 
*  Messenger  of  Math.  vol.  xix. 
f  Loc.  cit. 
