PEOFESSOE  PLiiCKEE  ON  THE  MAGNETIC  INDUCTION  OF  CEYSTALS.  557 
horizontal  plane,  which  therefore  contains  OM — we  may  project  OP  on  the  horizontal 
plane,  and  denote  the  angle  between  this  projection  OP'  and  OM  by  Then  the 
moment  of  rotation  round  the  vertical  axis  is  found  to  be 
2<p  tan 
the  radius  vector  r being  always  dmected  towards  the  pole.  The  horizontal  plane  is 
intersected  by  the  auxiliary  ellipsoid  in  an  ellipse  passing  through  M.  According  to 
simple  geometrical  relations,  the  angle  may  be  defined  also  to  be,  in  the  horizontal 
plane,  the  angle  between  OM  and  the  perpendicular  from  the  centre  O on  the  straight 
line  touching  the  ellipse  in  M. 
27.  In  the  case  of  equilibrium,  where  the  moment  of  rotation  disappears, 
tan  |'=0. 
This  condition  is  satisfied  if  one  of  the  two  axes  of  the  ellipse  lying  in  the  horizontal  dia- 
metral plane  points  towards  the  infinitely  distant  pole ; if  the  longer  axis  does,  ^ is  a 
maximum,  the  equilibrium  therefore  an  unstable  one ; if  the  shorter  axis,  ^ becomes  a 
minimum,  the  equilibrium  a stable  one.  Hitherto  the  induction  was  supposed  to  be 
paramagnetic ; if  it  become  a diamagnetic  one,  the  unstable  equilibrium  becomes  stable, 
and  vice  versa.  If  the  section  in  the  horizontal  plane  be  a circle,  the  angle  always 
equals  zero,  the  ellipsoid,  therefore,  however  turned  round  its  vertical  diameter,  will  not 
move.  Hence,  the  magnetic  pole  being  always  situated  in  the  horizontal  plane, 
An  ellipsoid  with  three  uneqiial  axes,  oscillating  round  any  of  its  diameters,  supposed  to 
he  vertical,  when  influenced  either  paramagnetically  or  diamagnetically  by  an  infinitely 
distant  pole,  will  he  so  directed  that  the  auxiliary  ellipsoid  sets  the  shorter  axis  of  its  hori- 
zontal section  either  axially  or  eguatorially.  The  two  diameters  of  the  auxiliary  ellipsoid, 
perpendicular  to  its  circular  sections,  are  the  two  magnetic  axes  of  the  infiuenced  ellipsoid. 
28.  In  order  to  verify  these  results  emanating  fi’om  Poisson’s  theory  in  the  case  of 
an  ellipsoid  with  three  unequal  axes,  infiuenced  by  an  infinitely  distant  pole,  it  will  be 
necessary  to  develop  them  in  the  analytical  way.  The  formulae  we  shall  deduce  will 
find  also  their  immediate  application  in  the  case  of  magnetically  induced  crystals. 
Let  us  suppose  the  infiuenced  ellipsoid  to  rotate  round  any  of  its  diameters,  this 
diameter  being  vertical,  and  the  infinitely  distant  pole  lying  in  the  horizontal  plane. 
Its  position  of  equilibrium  and  the  law  of  its  oscillations  round  the  vertical  diameter 
will  be  determined  by  the  ellipse  in  which  the  horizontal  plane  intersects  the  auxiliary 
elhpsoid.  This  elhpsoid  is  represented  in  the  ordinary  way  by  the  equation 
«v+z»y+c‘V=i, (3.) 
where 
cd  > h^  > (f. 
The  greatest  axis  of  the  influenced  and  the  least  of  the  auxiliary  ellipsoid  lie  along 
