564  PEOFESSOE  PLOCKEE  ON  THE  MAGNETIC  ENDTJCTION  OP  CETSTAXS. 
sections  of  the  auxiliary  ellipsoid,  the  following  resulting  moments  of  rotation  are  ob- 
tained : — 
<p{a^—b^)  sin  2fjb,  sin  2[Jj,  sin  2i>. 
42.  Now  let  us  suppose  the  given  elhpsoid  to  rotate  round  its  vertical  diameter,  the 
horizontal  plane  being  determined  by  any  two  angles  a and  <p.  Let  the  eUiptical  section 
of  the  auxihary  ellipsoid  within  the  horizontal  plane  (4.)  be  represented  by 
<z'V+5'^3/^=1, 
its  shorter  semi-axis  ^ lying  in  the  axis  of  abscissae.  Let  / be  the  length  of  the  semi- 
diameter OM  of  this  elliptical  section  passing,  if  prolonged,  through  the  infinitely 
distant  pole,  and  af  and  y'  the  coordinates  of  its  extremity,  M.  Then 
tan 
whence  the  moment  of  rotation  round  the  vertical  axis 
2<p  tan 
the  angle  between  the  radius  vector  r’  and  the  shorter  axis  ^ being 
43.  The  oscillations  of  the  influenced  ellipsoid,  when  infinitely  small,  may  easily  be 
analytically  determined.  The  ellipsoid,  supposed  to  be  paramagnetically  induced,  is  in 
stable  equilibrium  when  the  shorter  semi-axis  of  the  elliptical  section  of  the  auxi- 
liary ellipsoid  within  the  horizontal  plane  points  towards  the  infinitely  distant  pole. 
When  rotated  through  an  infinitely  small  angle,  this  angle  being  the  angle  between 
r'  and  the  axis  , the  corresponding  moment  of  rotation  becomes 
2(p{a!'^-V^)^ (25.) 
This  expression  will  remain  unchanged  when  the  paramagnetic  induction  becomes  a 
diamagnetic  one ; cp  becoming,  in  this  case,  negative  and  the  longer  axis  , instead  of 
the  shorter  , directed  towards  the  infinitely  distant  pole. 
We  obtain  therefore,  in  both  cases, 
df—  MK2 
denoting  the  mass  of  the  influenced  ellipsoid  by  M,  and  its  moment  of  inertia  with 
regard  to  the  vertical  axis  by  MK^.  Consequently  the  elhpsoid,  under  the  influence  of 
the  infinitely  distant  pole,  oscillates  like  a common  pendulum.  Denoting  the  time  of 
one  oscillation  by  0,  we  get,  in  the  ordinary  way,  by  integration, 
MKV  \ 
® 
44.  When  we  suppose  the  three  semi-axes  A,  B,  C of  the  influenced  ellipsoid  to  be 
successively  vertical,  the  corresponding  values  of  become 
i(B^+C^)=Kf,  i(A^+C^)=K^  i(A^+B^)=K;, 
a/y' 
:2(p(a'^—b'^)  sin  S-  cos 
1 
(24.) 
