566  PEOFESSOE  PLIJCKEE  ON  THE  MAGNETIC  INDUCTION  OF  CETSTALS. 
may  also  be  written  thus, 
sin  ^=prl  sin  21, 
To  being  the  radius  vector  OMo  of  the  first  auxiliary  ellipsoid  l}*ing  along  ^(= OP),  and 
jPo(  = OPo)  the  perpendicular  to  the  plane  touching  it  in  the  extremity  of  this  radius 
vector  and  coinciding  with  r(  = OM). 
47.  Let  § (=OE)  be  equal  to  and  directed  along  OP  and  OMq;  let  y^,  be  the 
coordinates  of  Mg,  and  a^,  jSg,  the  angles  between  gi(  = OE)  and  the  three  axes  of 
coordinates;  then 
— _cosX  cos^/3o  cos^7o 
\g)  a!^  ' ' 
This  relation  shows  that  the  point  E falls  on  the  surface  of  a new  ellipsoid,  which 
may  be  called  the  ellipsoid  of  induction.  Its  three  semi-axes  are  a*,  5®,  c®.  Therefore  the 
ellipsoid  of  induction  and  the  concentric  sphere,  whose  radius  is  equal  to  unity,  are  two 
polar  surfaces,  with  regard  to  the  first  auxiliary  ellipsoid,  the  polar  plane  of  the  point 
E touching  the  sphere  in  a point  K,  in  which  the  sphere  is  intersected  by  OMq. 
By  means  of  the  ellipsoid  of  induction,  which  may  be  represented  by 
+ + — 
(32.) 
we  can  completely  resolve  the  proposed  question,  replacing  in  all  the  former  formulae 
a®.  If,  c®  by  a,  /3,  y.  Thus,  for  instance. 
, « — /3  ^ « + y — 2^ 
COS®(a;= 5 cos  2<y= 1 
a — y a — <y 
whence  the  two  magnetic  axes  are  known. 
48.  With  reference  to  the  relations  between  the  different  ellipsoids  and  the  sphere 
above  mentioned,  we  easily  obtain  various  constructions  of  Poisson’s  problem,  among 
which  I select  the  following  one  (fig.  25). 
