556  PEOFESSOE  PLtiCKEE  ON  THE  MAGNETIC  ENOrCTION  OF  CEXSTALS. 
If  the  induction  of  the  infinitely  distant  pole,  regarded  till  now  to  be  paramagnetic, 
become  a diamagnetic  one,  nothing  is  changed  but  the  sign  of  the  two  forces,  the  first 
ellipsoid,  with  its  centre  in  E,  being  now  filled  with  northern,  the  second  one,  with  its 
centre  in  E',  with  southern  magnetic  fluid. 
25.  Denoting  the  value  of  each  force  by  (p,  the  resulting  moment  of  rotation  is 
immediately  found  to  be 
2®sin^  2<ptan0 
-t ^5  or  — (1.) 
prr^  V y 
The  axis  of  this  moment,  which  we  shall  denote  by  OE,  round  which  the  influenced  ellip- 
soid tends  to  move,  is  perpendicular  to  the  plane  MOP.  The  two  diameters  OM  and 
OR,  possessing  the  property  of  being  axes  of  the  ellipse  fonned  by  the  intersection  of 
the  ellipsoid  with  the  plane  passing  through  them,  are  two  conjugate  axes  of  the  smTace ; 
the  relation  between  the  two  is  a reciprocal  one.  To  any  diameter,  regarded  as  one  of 
two  such  axes,  corresponds  only  one  conjugate  axis.  Therefore  the  axis  round  which  the 
body  tends  to  revolve  is  continually  changed,  if  the  given  ellipsoid  under  the  influence  of 
the  infinitely  distant  pole  freely  move  round  its  centre.  To  any  one  of  the  three  axes  of 
the  auxiliary  ellipsoid  exceptionally  corresponds  an  infinite  number  of  second  conjugate 
axes,  lying  all  in  the  principal  plane  perpendicular  to  it.  Hence  the  influenced  ellipsoid 
>vill  oscillate  continually  round  such  an  axis  if  the  infinitely  distant  pole  he  in  the  con- 
jugate principal  plane. 
26.  When  the  influenced  ellipsoid  is  only  free  to  rotate  round  a vertical  line  passing 
through  its  centre — we  shall  always  suppose  the  infinitely  distant  pole  to  he  in  the 
auxiliary  ellipsoid  along  y,  and  construct  at  its  extremity  the  tangent  plane.  Let  p be  the  length  of  the 
perpendicular  from  the  centre  on  this  plane,  and  y'  its  direction.  Let  the  influenced  ellipsoid  E move  along 
y'  through  the  infinitely  small  distance  — .tr,  and  denote  it  in  the  new  position  by  E^.  By  the  two  eUipsoids 
E and  E',  two  infinitely  thin  sheets  are  determined,  whose  acute  edges  lie  in  the  curve  of  intersection  of 
E and  E^.  One  of  these  two  sheets,  placed  towards  M,  is  exterior  to  E and  interior  to  E^ ; the  other, 
placed  oppositely  to  M,  is  interior  to  E and  exterior  to  E^.  Conceive  both  sheets  fiEed  with  electricity  of 
the  same  density,  - — , but  of  a different  kind,  the  electricity  of  the  second  sheet  being  the  same  as  the 
electricity  of  M. 
“ Such  is  on  the  surface  of  the  influenced  ellipsoid  E the  distribution  of  electricity  induced  by  the 
infinitely  distant  mass  M.” — PoGaENDOEi'1'’s  Annalen,  xciv.  p.  192. 
It  is  well  known  that  the  mathematical  theory  of  magnetic  induction  differs  from  the  theory  of  electrical 
induction  only  by  a constant,  which  Poissox  denotes  by  Tc ; this  constant  being  equal  to  unity  in  the  last 
case.  In  order  to  apply  Professor  Beer’s  construction  to  magnetic  induction,  we  have  only  to  replace 
the  above  defined  auxiliary  ellipsoid  by  another,  whose  semi-axes  -,  i,  - are  connected  with  the  former 
semi-axes  — , i by  the  following  relations 
aP  ¥ co  * ® 
-k=27r(l-/i)  + ^•-E 
