712  Mr.  J.  W.  Xicholson  on  the  Symmetrical 
e2 
true  fundamental  series  of  periods.    When  n  =  1,  \  =  2  +  —    to 
the  second  order,  by  (25),  ° 
e2 
<?3  =  —    to  the  same  order, 
and  c4  and  all  higher  coefficients  commence  with  e4. 
Thus  a  solution  for  n=l,  with  small  eccentricity,  is 
c  K  —  %p2  e-2  e2  -\ 
±     _     ifocosha    '    1— — | I       (33) 
t1  (.         Stecosha       5(tecosha)2       5(tecosha)3J      v 
Another  solution  fa'  is  obtained  by  changing  the  sign  of  i. 
When  7i  =  2,  a      3  „    ,       /ac^ 
\=6+^e2,  by   (26), 
2  ,     0  Q      5  2  9  9  9  2 
Ci  =  ye2-3,       r,  =  3  — -^e,       c3  =  ^e2,       c4=— yc1, 
and  the  higher  coefficients  commence  with  e4. 
Thus 
,      .     f\        21-2e2  21-5e2 
r"  ^  He  cosh  a       /(tecosna 
(te  cosh  a)' 
9  e2  9 
i}    •  (34=) 
7  '  (te  cosh  a)3       7  *  (te  cosh  a) 
and  a  solution  </>/  is  obtained  by  changing  the  sign  of  l. 
Similarly, 
*.*■*    (,         90-4e2  225-22e2  75-24e2 
3  \         lotecosha      15(tecosh«)2       o{ie  cosh  a)- 
10e2 
)sh  a)4        (<e  cosh  a)3  J       V 
(tecosha)4   '   (<e  cosh  a)5  J'  v"  y 
Each  succeeding  function  necessitates  the  addition  of  a 
new  term.  For  vibrations  between  two  confocal  spheroids, 
a  linear  combination  of  any  pair  (cf>,  <£')  must  be  taken. 
Inside  a  single  spheroid,  we  require  a  function  finite  at  the 
centre.  By  comparison  with  the  Bessel  functions,  it  appears 
that  if  n  =  l,  3,  5,...  the  real  part  of  <£  must  be  taken,  and  if 
7t  =  2,  4,    6,...,  the  imaginary  parr.       Thus  the   first  three 
