PEOEESSOE  PLCCKEE  ON  THE  MAGNETIC  INDUCTION  OF  CEYSTALS.  561 
35.  This  construction  of  the  two  axes  OM  and  OM'  is  easy  to  execute  on  a given 
sphere ; we  may  also  without  difficulty  transform  it  into  analytical  expressions. 
Considering  the  three  rectangular  spherical  triangles  AED,  QRD,  and  Q'E'D,  the 
angle  ADE  being  T—(p,  we  get 
tan  DE  = — tan  a cos  p,  1 
tanDR  =— tan  (a— ^y)  cos<p,  V (14.) 
tan  DR'=  — tan  (a+<y)  cos 
whence 
X+^r=i(DR+DR')=DM', 
S=DE-DM', 
denoting  by  S the  angle  between  OE,  the  projection  of  OA  on  the  horizontal  plane,  and 
OM'  pointing  within  this  plane  axially  or  equatorially. 
36.  The  following  relations  too  will  afterwards  be  employed.  From  the  two  triangles 
QED  and  Q'E'D,  just  considered,  we  deduce  also, 
■r\T^  COs(o(  ttj)  , , ty  • / \ 
cosI)E  = — r-^5  sinDE  =smcsin(a— (w), 
sin  'o  \ y ’ 
-pvTj/ COS  (fli -f- Co)  , -pvx>^ • W • / I \ 
cos  i)E  = ^ sm  DE  = sin  ^ sin  (a + (u), 
denoting  by  4 and  the  angles  PQ  and  PQ',  between  the  two  magnetic  axes  (OQ  and 
OQ')  and  the  vertical  line  OP,  and  by  ^ and  the  angles  DQP  and  DQ'P.  Hence 
Sin  ^ sin, 
sin  2K=  —sin  (DE+DE')=^j^^  sin  (a — a;)  cos  cos  (a— a;)  sin 
But  from  the  triangles  DQP  and  DQ'P  we  get,  remembering  that  PDQ=^5r— 9 and 
PD=i^, 
whence 
sin  ^ sin  4'=sin  sin  \p'=cos  <p, 
sin  ^ sin  /sin  f sin  ^ cos<p 
sin  4/  sin  4^  v sin  \{/  sin  4^'  sin  4/  sin  4^'  ’ 
therefore,  expanding  the  last  equation, 
• ) • ,,  sin  2a  cos  0 /n  e \ 
(1®-) 
Reverting  to  the  equation  (8.),  in  which  here,  according  to  art.  (31),  6^  is  to  be  replaced 
by  (f,  we  may  write  it  now  thus : — 
(a'^—6'^)=(a^—c^)  sin  -ip  sin (l*^-) 
37.  By  means  of  the  angle  a/  we  have  determined  the  position  of  the  influenced  ellip- 
soid : this  position,  reciprocally,  being  determined  by  observation  in  any  particular  case, 
we  can  And  the  angle  o).  We  may  use  for  this  purpose  the  formulse  (9.)  and  (11.) — (13.)- 
But  a simple  geometrical  consideration  will  equally  lead  us  to  the  determination  of  the 
value  of  <y. 
The  two  vertical  planes  SPM  and  S'PM',  containing  the  two  axes  of  the  horizontal 
section,  and  the  two  planes  QPR  and  Q'PR'  bisecting  them,  constitute  what  is  called  a 
4 E 2 
