714  Mr.  J.  T\r.  Nicholson  on  the  Symmetrical 
very  small.    An  eccentricity  of  \  reduces  the  wave-lengths  in 
9 
the  ratio  ~  approximately. 
~\  j 
In  case  1,  n=l,  the  period  equation  is  ^~  =  0. 
This  equation  supplies  the  true  fundamental  of  the  system. 
With  the  previous  notation,  after  some  reduction,  it  becomes, 
Let  6  be  a  root  of  tan  a  =  2 ,  the  period  equation  for 
the  sphere,  and  8  be  the  correction  for  eccentricity.     Then  it 
may  be  at  once  shown  that 
2    9    PS  I  d    ftanfl      Q,      a\ 
--e2       6'~3 
5-^-2) ^> 
2 
For  the  higher  roots,  this  becomes  S  =  —  ~  e2Q.      The  cor- 
rection  is  therefore  equal  and  opposite  to  that  in  case  2,  if 
the  vibrations  are  not  too  near  the  fundamental.  The  funda- 
mental of  the  sphere  corresponds  to  a  value  of  #  approximately 
equal  to  2'744  or  7tt/8. 
The  Oblate  or  Planetary  Spheroid. 
The  proper  substitution  in  this  case  is 
p -\- iz  =  c  cosh  {x  +  i ft),       ....     (43) 
giving  p  =  c  cosh  a  cos  ft, 
dfc'*)  =  c2(cosh2a-cos2/3); 
and  the  equation  for  </>  becomes 
(jar         0/3"  0«  OP 
In  this,  write  (  —  e2,  a — — ,  /3—  ^- )  for  (e2,  a,  /3)  respec- 
tively, and  it  becomes  identical  with  (20).  Thus  to  apply 
the  previous  results  to  an  oblate  spheroid,  it  is  only  necessary 
to  make  these  three  substitutions  in  them.  In  previous 
investigations,  cosh  a  becomes  i  sinh  a  ;  sinh  a,  i  cosh  a  ; 
cos  ft,   —  sin/3;    shift,  cos/3;    and  Ice  cosh  a,  Ice  sinh  a.     The 
