716  Mr.  J.  W.  Nicholson  on  the  Symmetrical 
these  high  frequencies  are  less  interesting  physically.  The 
fundamentals  of  series  denned  by  large  values  of  n  are 
themselves  high  modes,  in  comparison  with  the  fundamental 
of  the  system. 
We  now  proceed  to  treat  the  higher  vibrations  of  spheroids 
corresponding  to  any  values  of  n.     The  equation  in  0  is 
^-cot/3^  +  (X-/,Vcos2^=0. 
dp2  dp 
Adopting  the  substitution  * 
tj)=elkc(a.^,    .......     (4l>) 
we  obtain 
(yjr"— cot  fif)  -  ihc(2w  'yfr' '  +  ^ay" -w'yjr  cot  j3) 
+  ^V2(^2  -  cos2/3-a/2)  =  0. 
Neglecting  the  first  bracket,  and  equating  the  brackets  of 
different  orders  in  kc  to  zero, 
--,  —  cos-  p, 
!±:+^_eot/3=o. 
Thus  if  o)=  \    a  /  — 2—  cos2  /3  dp,  the  solution  becomes 
%}      v     c 
A  cos  Jcccd  -f  B  sin  kcco       ,  - — ~-  ,.A^ 
</>  = .  vsm/3,       .     .     (oO) 
(J-co^)* 
except  close  to  the  axis  of  the  spheroids. 
From  this  we  may  obtain  the  limiting  value  of  \  when  n, 
although  possibly  large,  is  small  in  comparison  with  e  or  kc. 
The  denominator  of  <f>  as  written  has  a  period  2ir,  and  there- 
fore <j>  has,  if  co  has.  This  occurs  if  X  =  e2,  if  the  solution 
remains  finite  and  uniform  at  /3  =  0.     Now  writing 
X  =  e2  +  n  .  ?i  +  1 , 
and  putting  cos  /3  =  /a,  we  obtain 
(w2)JJ +(»•»+) +e2-e>2)<£=o,.  •  (51) 
This  is  satisfied  by  a  convergent  series  in  rising  powers 
of  /A,  even  in  //.  if  n  is  odd,  and  odd  if  n  is  even.  .  By  putting 
jub=  +1 — 7],  the  solution  is  finite  at  fi=  +  1,  and  therefore  on 
the  axis  of  the  spheroids.     Thus  a  solution  of  the  required 
*  Webb,  Proc.  Roy.  Soc.  1904,  p.  315. 
