PEOFESSOE  PLOCKEE  ON  THE  MAGNETIC  INDUCTION  OF  CEYSTALS.  685 
the  second  is  perpendicular  to  it.  On  the  other  hand,  when  a crystal  is  perpen- 
dicularly suspended  between  the  two  magnetic  poles  along  any  diameter  of  the  magnetic 
auxihary  ellipsoid,  one  of  the  two  axes  of  the  horizontal  elliptical  section  of  this  ellipsoid 
will  point  axially,  the  other  equatorially.  Feesnel  showed  already  that,  by  a simple 
geometrical  construction,  the  two  directions  of  vibration  of  any  plane  luminous  wave 
may  be  deduced  from  the  position  of  the  two  optic  axes.  I showed  in  the  preceding 
paper  and  proved  it  by  observation,  that  exactly  the  same  construction  gives  the  position 
of  a crystal  freely  oscillating  between  the  two  poles  of  a magnet ; the  two  horizontal 
lines  pointing  axially  and  equatorially  are  immediately  obtained  by  projecting  the  two 
magnetic  axes  on  the  horizontal  plane,  and  by  bisecting  the  angle  between  these  pro- 
jections. In  treating  the  inverse  problem,  “ to  find  the  magnetic  axes  of  a crystal  after 
having  determined  by  observation  the  position  which  the  crystal  takes  between  the  two 
poles,  when  suspended  along  any  known  direction,”  I proposed  trigonometrical  formulae, 
which  may  be  generalized  in  the  following  manner.  Suppose  OX,  OY,  OZ  to  be  the 
three  principal  axes  of  induction,  without  knowing  their  relative  values ; let,  within  the 
suspended  crystal,  MON  be  the  horizontal  plane  passing  through  the  centre  O and 
intersecting  the  principal  plane  OZ  along  OM ; let  ON  be,  within  the  horizontal  plane, 
the  line  pointing  either  axially  or  equatorially.  Denote  by  the  angle  between  the 
planes  MON  and  XOZ,  by  a and  "k  the  angles  MOX  and  NOM.  Thus  the  position, 
within  the  ciystal,  of  the  horizontal  plane  is  determined  by  the  angles  (p  and  a,  the 
position  of  the  crystal  by  the  angle  After  having  determined,  by  means  of  the 
following  two  equations. 
tan  — j— oij 
tan  X 
cos  <p’ 
tan  ['/l-\-a)  = 
cos  A 
cos  (f 
the  two  auxiliary  angles  rj  and  rj,  calculate  the  value  of  the  expression  tan  rj  tan  rj.  This 
value  may  be  foirnd  to  be  either  positive  or  negative ; if  negative,  the  absolute  value  may 
be  either  <1  or  >1.  Thus  we  obtain  three  difierent  cases. 
The  position  of  the  two  magnetic  axes  is  fixed  by  knowing  the  principal  plane  con- 
taining both  axes,  and  the  angle  between  each  of  them  and  a given  one  of  the  two  axes 
of  magnetic  induction  lying  in  the  same  plane.  Denoting  this  angle  by  co,  we  have,  in 
the^/’5^  case, 
tan  7j  tan  ^'=tan^  a. 
Both  axes  lie  in  the  plane  XOZ,  a being  the  angle  between  each  of  them  and  the  axis 
OX.  The  last  formula  holds  in  both  cases ; the  axis  OX  may  be  the  axis  of  the  greatest 
or  of  the  least  induction.  From  this  axis  the  angle  co  is  to  be  measured. 
In  the  second  of  the  three  abov-e-mentioned  cases,  we  obtain 
tan  n tan  — sin^  oj. 
The  two  magnetic  axes  lie  in  the  plane  XOY,  perpendicular  to  the  plane  XOZ,  and 
4 H 2 
