558  PEOFESSOE  PLtj'CKEE  OX  THE  jMAGXETIC  ESTDrCTIOX  OF  CETSTALS. 
OX  (fig.  20),  the  mean  axis  of  both  ellipsoids  along  OY,  the  least  axis  of  the  first 
and  the  greatest  of  the  second  along  OZ.  We  may  determine  the  horizontal  plane 
X'OY',  passing  through  the  common  centre  of  both  ellipsoids,  by  two  angles,  a and 
(p ; a being  the  angle  between  the  axis  OX  Pig.  20. 
and  the  line  OX'  in  which  the  plane  XOY 
is  intersected  by  the  plane  X'OY',  and  <p 
the  angle  between  the  two  planes.  This 
determination  admits  no  ambiguity  when 
we  conceive  the  plane  XOY  to  pass  from 
its  original  position  by  a double  rotation 
into  the  horizontal  plane  X'OY';  rotating 
first  round  OZ  through  the  angle  a,  taken 
from  OX  towards  OY,  till  OX  coincides 
with  OX'  and  OY  with  a line  we  shall  de- 
note  by  OY";  rotating  secondly  round  OX' 
through  the  angle  9,  taken  in  the  plane 
ZOY“,  from  OY®  towards  OZ,  till  OY®  coincides  with  OY'  and  OZ  with  OZ'.  The  thi'ee 
new  axes,  OX',  OY',  OZ',  are  perpendicular  to  each  other,  as  the  primitive  ones  are. 
The  equation  of  the  ellipse  in  the  horizontal  plane  X'OY',  referred  to  the  axes  of  coordi- 
nates OX'  and  OY',  is  immediately  obtained,  when  in  the  equation  (3.),  by  means  of  the 
following  relations, 
x—x'  cos  a— y sin  a cos  p, 
y=x'  sin  a+y  cos  a cos  p, 
z =y  sin  p, 
X,  y and  z are  replaced  by  ^ and  y\  It  becomes 
g'ir'^-|-2(ra^'y-l-ry^=l, (4.) 
by  putting,  for  brevity. 
(f  cos®  a -j-  W sin®  a = 
— (a® — 5®)  sin  a cos  a cos  ^ —a 
(«®  sin®  a-j- J®  cos®  a)  cos®  <p-|-c®  sin®  9=r. 
(5.) 
Denoting  the  two  semi-axes  OM  and  OM'  of  this  ellipse  by  ^ and  y>  and  the  two  angles, 
MOX'  and  M'OX',  between  them  and  the  axis  OX'  by  X and  we  get  the  well- 
known  equations — 
(6.) 
(7.) 
tan2X=' 
29.  From  these  two  equations  we  may  first  deduce  the  follorving  one, 
''  ^ — sin  2A 
