Vibrations  of  Conducting/  Surfaces  of  Revolution.       70D 
Vibrations  of  Confocal  Spheroids. 
The  proper  transformation  in  the  case  of  spheroids  of 
ovary  form  is 
z  +  ip  =  c  cosh  (<z  +  ift),       ....     (19) 
which  gives  i 
o  =  c  sinh  a  sin  ft—  — , 
|^4|  =r(cosh2«-  cos2/3). 
Thus  the  characteristic  equation  (10)  becomes 
5^  +  §^~COthaH  -cot/3||  +  Pr(cosh2«-cos2/3)c/)  =  0. 
To  obtain  a  solution  of  the  proper  form,  we  may  write, 
if  X  is  some  constant, 
|^t  -  coth  cc  |^  +  (Pc2  cosh2  a  -X)  </>  =  0,      .     (20) 
|^-cotyg||  +  (\-Fc2cos2/S)^=:0.     .     .      (21) 
In  the  sphere,  \  =  n  .  (n  +  1).  Its  proper  values  for  the 
spheroid  are  to  be  determined  so  as  to  make  <£  recur  when 
ft  increases  by  27T,  and  are  of  the  form 
\=n(n+l)  +en{kc)2  +  8n(Icc¥+ .... 
where  e„,  Sn,  . . .  are  functions  of  n  only, 
Write  <£  =  sinh  a  .  sin  ft  .  A  .  B,  where  A  is  a  function  of 
a  only,  and  B  of  ft  only. 
We  thus  obtain 
-J-*  +  coth  a -r-  -h  A(k2c2  cosh2  a  —  cosech2a— \)  =  0  .  (22) 
rta2  dec  K  v 
f™  +  cot/3~  +  B(\-Pc2cos2/3-cosec2/3)=0     .     (23) 
These  equations  are  included  in  a  form  considered  by 
Niven*  when  discussing  the  conduction  of  heat  in  ellipsoids 
of  revolution.  His  solution  is  unsuited  to  the  present  problem, 
but  we  may  quote  his  value  of  X.  In  the  present  case,  it  is 
found  that 
.        /     ,in,/i.m      2rc2  +  2rc-3  1/7  ,4  f  n(n+l)(n  +  2)(n  +  3) 
K-^-m  +  W 2(2n-.l)(2,  +  3)  "*W   l(2n+l)(2n  +  3y(2n  +  5) 
_   M)(n-l)(n)(n  +  l)1  , 
(2n-3)(2n-l)3(2*i  +  l)J  +  '  •  •      "      K     J 
*  Phil.  Trans.  1880,  p.  138. 
