710         1  "Mr.  J.  W.  Nicholson  on  the  Symmetrical 
Thus  when  n  =  0,  X  —  (£c)2  up  to  and  including  the 
seventh  order.     This  is  found  to  correspond  to  no  vibration. 
For  the  first  three  types  o£  vibration,  given  by  w=l,  2,  3, 
the  values  of  \  are 
Xl  =  2+16(f:ey--^(kcy+JIi(Mf- (25) 
x^+^kcy-^a-cy+J-^ikcy- (26) 
X3  =  12  +  J  (kef  +  ^^  (koY- (27) 
After  the  first  term,  these  series  converge  very  rapidly  for 
values  of  kc  which  are  not  very  great,  and  more  rapidly  as  n 
increases.  Now  for  the  fundamental,  and  for  the  vibrations 
near  to  it  in  a  sphere  of  radius  a,  ka  is  not  much  greater 
than  1  or  2,  and  unless  the  spheroids  differ  greatly  from  the 
spherical  shape,  these  series  may  be  conveniently  used  in 
calculating  the  gravest  vibrations  of  the  doubly  infinite 
system.  For  c  =  ae,  where  a  is  the  semi-axis  major  and 
e  the  eccentricity. 
Let  kc  be  denoted  by  e.  We  may  pass  from  the  spheroidal 
coordinates  to  spherical  polars  (r,  #,  co)  by  making  a  great  and 
n 
c  small,  so  that  r  —  -  ea,     ft  =  6. 
It  is  known  that  the  equation  corresponding  to  (20)  when 
ol  is  infinite  and  c  zero,  is  satisfied  by  the  form 
cf>  =  e'k%(ikr), 
where  /„  denotes  a  terminating  polynomial  in  descending- 
powers  of  ikr,  which  is  useful  for  calculation  for  only  moderate 
values  of  hr. 
"Write  cosh  «  =  t  in  (20),  and  it  becomes 
^_1>3 .+  (^2-^=°-      •     •     •     (28) 
If  in  this  transformed  equation  we  write,  following  the 
method  suggested  by  the  case  of  the  sphere, 
9     e  \(tety  +  (tety+1  +  • ' 
we  find  that   r  =  0. 
Thus  ,  ,  /.,       c,  c2  \  ,n^ 
.^r'T+^  +  ^-t--   •)'  •    '    ■    (29) 
