Vibrations  of  Conducting  Surfaces  of  Revolution.       713 
appropriate  functions  for  internal  vibrations  are 
</>i  =11+^  sech2  a.  J  cos  (he  cosh  a) 
/■,      2    7         1  )   sin  {he  cosh  a)         ■_  . 
—  <  1  —  -  Arc-  +  v  sech2  a   >  — =± n '- ,     (36) 
I         5  o  J        he  cosh  a       '     v     J 
i         /o      2  70  9      9       19    \  cos  (he  cosh  a) 
(/>2  =  (  o  —  =  &V  +  -sech-  a     — ^ c ^ 
\         <  /  /       A'c  cosh  a 
J    ("          21-5Pc2        9sech^)     .     ..         .     ,       ,. 
+  1l""'-722       U2" 7    ,■;  ■>     >  sin  (fee  cosh  a),      (37) 
f,        75-22h2c2         10  seen4*) 
'03  =    1  1  —  t^Tt iT~^2 79  9 —    f  cos  (Ac  COsh  a) 
L        lo(A'ccosha)-  h2c2       )  y 
fr       ^  //  ^2 75  — 24Pc2   __  10  sech3  u \  sin  {he  cosh  a)     ,,_. 
~~  \  b  ~~  15  ^    j"      (5lc  cosh  a)2  F?~  J  ~  &c  cosh  a      •  (      ^ 
In  finding  periods  corresponding  to  any  value  of  n,  the 
equation  in  /3  need  not  be  solved,  as  it  is  known  from  the 
case  of  the  sphere  that  the  forces  will  be  finite  when  n  is  not 
zero.  We  shall  only  examine  the  periods  corresponding 
to  n  =  1. 
If  a  is  the  major  semi-axis,  and  e  the  eccentricity,  c  =  ae} 
e  =  sech  a,  where  a  relates  to  the  boundary.  The  periods 
belonging  to  case  2  correspond  to  the  surface  condition  <£i  =  0, 
when  n  =  0. 
Thus  if  a  =  ka,  the  period  equation  becomes 
tan  a  2    2  2  .     , 
_=l+soV.       ....     (39) 
the  corresponding  equation  for  the  sphere  being  tan  cr  =  cr. 
If  8  be  the  positive  correction  for  eccentricity  of  any  root  6 
of  tan  a  =  o\ 
*       2    9Z19  /d  /tan  #\ 
/ 
=  §f^       (40) 
writing  tan  0  =  0,  sec2  <9  =  1  +  02. 
The  effect  of  a  small  eccentricity  is  therefore  to  increase 
2 
ha  in  the  ratio  1  +  -=  c2,  if  a  sphere  become  a  spheroid  with 
the    same    major    axis.       The   wave-lengths  are  less  in   the 
2 
spheroid  in  the  ratio  1  —  f  ^2,  and  the  correction  is  therefore 
Phil.  Mag.  S.  6.  Vol.  11.  No.  05.  May  1906.  3  A 
