562  PEOFESSOE  PLtiCKEE  ON  THE  MAGNETIC  INHUCTION  OF  CETSTALS. 
system  of  four  harmonic  planes.  Such  a system  is  intersected  by  any  plane  in  four 
harmonic  lines.  OS,  OS'  and  OQ,  OQ'  are  therefore  four  harmonic  lines,  whence  the 
angle  2a)  between  OQ  and  OQ'  being  bisected  by  OA, 
tan  i]  tan  rj = tan’^  (18.) 
denoting  the  angles  OS  and  OS'  by  ri  and  rj.  Again,  in  the  triangle  PDS,  obsening  that 
PDS=-|‘r— 9,  DPS=‘r— X,  DS=;;+a,  we  get 
whence 
tan 
tan(;?'+a)= 
tan  X 
•) 
cos  p 
cotx 
cos 
(19.) 
The  position  of  the  horizontally  oscillating  plane  being  determined,  within  the 
influenced  ellipsoid,  by  the  two  angles  a and  9,  and  the  two  directions  within  this  plane 
pointing  axially  and  equatorially  by  the  last  two  equations  furnish  the  values  of  t]  and 
rl^  whence,  by  means  of  (18.),  we  obtain  the  value  of  o),  and  consequently  the  directions 
of  the  two  magnetic  axes. 
38.  We  have  hitherto  considered  as  known  the  direction  of  the  three  axes  of  the 
auxiliary  ellipsoid.  Such  is  the  case  in  the  question  of  a given  ellipsoid,  influenced  by 
an  infinitely  distant  magnetic  pole,  where  these  three  axes  are  coincident  with  the  thi'ee 
axes  of  the  influenced  ellipsoid.  But,  when  treating  on  the  magnetic  induction  of 
crystals,  we  shall  meet  with  questions  where  the  direction  of  the  axes  of  the  auxiliary' 
ellipsoid  is  to  be  determined  by  experiment.  If  a given  ellipsoid  be  suspended  along 
any  diameter,  we  can  find  the  two  axes  of  the  horizontal  section  of  the  auxiliaiy  ellip- 
soid, these  axes  pointing,  one  axially,  the  other  equatorially.  The  new  question  there- 
fore is  a geometrical  one,  “ To  determine  the  three  axes  of  an  ellipsoid,  knowing  the  two 
axes  of  each  of  its  sections,”  and  may  be  resolved  in  the  following  way. 
Let  the  given  influenced  elhpsoid  revolve  round  any  one  of  its  diameters,  supposed  to 
lie  in  the  horizontal  plane,  and  mark  in  each  of  its  positions  the  axial  as  well  as  the 
equatorial  line.  These  two  lines,  two  conjugate  axes  of  the  auxiliaiy  ellipsoid,  will 
describe  during  one  revolution  a conic  surface  of  the  third  order,  containing  the  three 
axes  of  the  auxiliary  ellipsoid ; for  these  axes  ivill  successively  pass  through  the 
horizontal  plane,  and  then  point  either  axially  or  horizontally.  Hence  two  such  conic 
surfaces  will  determine  the  three  axes  of  the  auxiliary  ellipsoid. 
39.  Let  any  two  conjugate  axes  of  the  auxiliary  ellipsoid  be  represented  by 
x=gz,  cc=g’z, 
y—hz,  y=l^z, 
while  this  ellipsoid  is  always  represented  by 
gg'-{-  M'+1=0, 
a^gg’  -f  bVil^ + 0, 
Then 
