720  Mr.  J.  W.  Nicholson  on  the  Symmetrical 
contain  an  exponential  factor  of  negative  imaginary  argument. 
By  this  consideration,  all  previous  formulae  for  periods  may 
be  at  once  transformed  into  those  proper  to  the  external 
space. 
In  the  case  o£  the  sphere,  when  n  =  l,  the  proper  form  of 
cj>  is,  so  far  as  concerns  r, 
*=Ae-'ir(1+i) w 
In  the  second  type  of  vibration,  6  =  0  when  r  =  a. 
1 
Thus 
ika 
and 
_  Yt 
ikYt                    a 
e       =  e         : 
so  that  there  is  really  no  vibration  at  ail,  but  a  simple  decay 
at  a  very  rapid  rate,  on  account  of   the  large  value  of  V. 
In  the  first  type,  >~-  =  0,  when  r=a,  leading  to 
l  +  (Jc-k2a9  =  0, 
or  fta=±(*W3) (69) 
The  time  factor  becomes 
e     'Ia  .  e       2a, 
/3 
and  therefore  the  real  value  of  ka  is  ^~— ,  but  there  is,  in 
— 
addition,  a  rapid  decay  of  the  vectors  at  half  the  rate  of  the 
previous  type.     In  the  case  of   the  prolate  spheroid,  when 
n=l,  by  changing  the  sign  of  i  in  (33), 
2 
I  —16  cosh  a  J   -J   j ^ ^ &_ I    /rjf\\ 
^~  \        tecosha      5(tecosha)2      5(ie  cosha)uJ  ^     * 
Taking  the   second   type    of  vibration,    corresponding   to 
$  =  0    at   c  cosh  a  =  a,  where    a    is  the  major   semiaxis,  and 
putting  cosh  a  =  -  ,     ka  =  cr, 
1-fffV 
-2^2 
i+ __?__  + ££  +  ^=0, 
ia  5ct"        DiaJ 
or  neglecting  <?4, 
1  +  icr-  |oV  =  0. 
5 
