Vibrations  of  Conducting  Surfaces  of  Revolution.      715 
major  axis  of  the  previous  spheroid  is  to  be  replaced  by  the 
minor  axis  of  the  new  one.  The  sign  of  e2  is  to  be  changed 
wherever  it  occurs.  Thus  the  corrections  of  the  fundamental 
series  of  periods  for  eccentricity  are  equal  and  opposite  to 
those  of  the  other  type  of  spheroids.  The  detailed  theory  for 
the  oblate  spheroids  will  not  be  given,  as  the  previous  formulas 
may  be  at  once  transformed  properly. 
Hie  Higher  Periods  of  Spheres  and  Spheroids. 
In  cases  1  and  2  of  the  sphere,  when  n  is  small,  and  the 
higher  periods  are  being  treated,  the  period  equations  become, 
adopting  the  ordinary  asymptotic  expansion  of  Bessel 
functions, 
°£}-(*.-(»  +  l)£.)=0 (45) 
Thus  if  m  be  a  large  integer,  a  possible  wave-length  for  a 
sphere  of  radius  a  is    —  approximately. 
When  n  is  appreciable,  the  ordinary  expansion  of  Bessel 
functions  ceases  to  apply.  But  it  was  shown  by  the  author  * 
that  the  proper  expansion  in  this  case  is 
cos^^V-«.re+l-»  +  Jcos-lV    •  '   -i 
SW*r)=-v/-— i j^ tttt-  •   (46> 
2V       '  V     7T  (AV-W.W  +  1) 
and  the  two  period  equations  become 
. ,  111  .72+1  /i       .    o\   T7"      /      t 
v/P^-n.n  +  l-^  +  ijcos"1^       /(.2c2      =(45  +  3)  j    (s  large) 
and 
(      /n.n+1       it 
tanj^/Pa2— n.  72  +  1  —  (w  +  i)  cos"1  \/       j^ j 
n  .  n  + 1 
2(ka)i{k2a2-n.n  +  l)i-' 
(48) 
These  two  formulse  will  supply  the  correction  necessary  to 
the  periods  deduced  from  the  ordinary  expansion,  when  n 
n 
becomes  appreciable,  if  expanded  in  powers  of  y- . 
ka 
However  great  n  may  be,  these  equations  will  give  the 
higher  modes  of  the  series  defined  by  n,  and  will  do  so  more 
accurately  as  n  increases,  if  the  higher  roots  are  taken.     But 
*"Phil.  Mag.  Feb.  1906,  p.  195. 
3  A2 
(47) 
