PEOrESSOE  PLiJCEIEE  ON  THE  MAGNETIC  INDUCTION  OE  CETSTALS.  555 
analogous  to  the  action  of  an  infinitely  distant  pole  on  an  ellipsoid  of  finite  dimensions. 
In  both  cases  the  lines  of  the  inductive  force  are  parallel.  Thus  the  solution  of  the 
following  question  coincides  with  the  solution  of  the  question  regarding  magnecrystallic 
action : — “ To  determine  the  couple  of  forces  acting  upon  an  ellipsoid,  when  infiuenced 
by  an  infinitely  distant  pole.” 
23.  Poisson  gave  a complete  analytical  solution  of  this  question,  expressing  by  means 
of  eUiptic  functions,  which  may  be  calculated  in  every  particular  case,  the  intensity 
and  the  direction  of  the  resulting  forces,  and  hence  the  resulting  moment,  relating  to 
any  given  axis  of  rotation.  But  the  comphcated  analytical  expressions  of  his  formulse  will 
scarcely  allow  of  deducing  from  them  the  law  they  express.  Professor  Beee  recently  suc- 
ceeded in  presenting  the  results  of  Poisson’s  theory  in  a most  simple  and  elegant  way. 
making  use  of  an  auxihary  ellipsoid,  whose  three  axes. 
- ) , are  expressed  by 
eUiptic  integrals. 
24.  Let  this  ellipsoid  be  intersected  in  the  two  points  M and  M'  by  the  straight  line 
passing  through  its  centre  O and  the  infinitely  distant  pole.  Construct  the  two  planes 
touching  the  elhpsoid  in  M and  M',  and  perpendiculars  from  the  centre  to  these  planes, 
intersecting  them  in  the  two  points  P and  P'.  Let  the  distances  OM  and  OM'  be 
denoted  by  r,  the  perpendiculars  OP  and  OP'  by  y),  the  angle  between  OP  and  OM  by 
Finally,  determine  two  points  E and  E',  lying  in  OP  and  OP',  on  opposite  sides  of  the 
centre  O,  whose  distance  from  the  centre  equals  — • Now  conceive  two  ellipsoids,  both 
equal  to  the  given  influenced  one,  and  having  then*  axes  similarly  directed,  the  first  one, 
with  its  centre  in  E,  filled  with  southern,  the  second,  with  its  centre  in  E',  filled  with 
northern  magnetic  fluid.  Then  the  resulting  action  exerted  by  the  infinitely  distant 
pole,  supposed  to  be  a northern  one,  on  the  given  paramagnetically  induced  ellipsoid 
equals  a couple  of  forces,  represented  by  the  attraction  of  the  first  and  the  repulsion  of 
the  second  ellipsoid,  both  filled  with  magnetic  fluid*. 
* The  solution  given  above  of  Poissox’s  problem  immediately  results  from  Professor  Eeee’s  com- 
munication, which,  as  it  is  short,  it  is  but  justice  to  translate  here,  merely  changing,  to  avoid  error,  the 
notation  in  some  cases. 
“ Let  A,  B,  C be  the  semi-axes  of  an  eUipsoid,  E,  electrically  influenced  by  an  electric  mass,  M,  infinitely 
distant  along  y,  whose  action  on  the  unit  of  volume,  filled  with  the  unit  of  electricity,  is  M^^.  Let  /xmt'  be 
the  attraction  or  repulsion  between  two  infinitely  small  volumes,  u,  filled  with  electricity  of  the  density  1, 
at  a distance  equal  to  unity. 
.Ill 
“ Construct  an  auxiliary  eUipsoid,  whose  semi-axes  -j,  — are  directed  along  the  semi-axes  A,  B,  C of 
the  influenced  conductor.  Take 
sin^  cos^ 
^ fo  Qv  sin 
Jo  Jo 
A' 
A^ 
/sin^.&  cos^^Y 
where 
. COS"  V sin^  V 
B^ 
In  like  manner  determine  i and  by  replacing  A by  B and  A by  C.  Let  r be  the  radius  vector  of  the 
