PEOrESSOE  plOckee  on  the  magnetic  induction  of  CETSTALS.  559 
whence,  by  substitution, 
{a'^—b'^)=z±{a‘^- If)  ^1^  cos  (p, 
30.  Again,  the  formula  (7.)  may  be  expanded  thus : — 
where 
tan  2x= 
sin  2«  cos  ip 
cos  2«  + A:sin^ip’ 
j sin^  cos^  a — „ 
K— 2 — To =sim  a 
— 0^ 
(8.) 
(9.) 
Denoting  the  angle  between  the  two  magnetic  axes,  perpendicular  to  the  circular 
sections  of  the  auxiliary  ellipsoid  (3,),  by  la  (fig.  21),  we  obtain 
^ , }f—(f  . cf—W 
^23^=tan"^y,  ^2z:^=sin"4;,  ^23^=008=* 
(10.) 
whence 
^=sin^  05+tan^(i/. 
(11.) 
Fig.  21.  Fig.  22.  Fig.  23. 
31.  It  will  be  in  some  cases  more  convenient  to  refer  the  horizontal  intersecting 
plane  to  another  principal  section  of  the  auxiliary  ellipsoid.  Hitherto  we  have  sup- 
posed— and  so  we  shall  do  again  in  the  following  articles — the  shortest  axis  \ the  mean 
and  the  longest  ^ to  coincide  with  OX,  OY,  and  OZ.  Now  let  ^ fall,  as  before,  on  OX, 
but  ^ on  OY,  and  ^ on  OZ,  and  accordingly  let  a be  taken  in  the  plane  containing  the 
shortest  and  the  longest  axis  (fig.  22),  from  the  former  towards  the  latter.  In  this  case 
b is  to  be  replaced  by  c,  and  vice  versd ; therefore 
^=sin^a— sin^(y (12.) 
32.  When,  thirdly,  - falls  on  OZ,  T on  OX,  and  - on  OY,  and  accordingly  a is  taken 
doc 
in  the  plane  YOX  (fig.  23),  from  the  mean  axis  ^ towards  the  longest  we  get 
A:=sin^  a— cos^o/.  . 
4 E 
MDCCCLVIII. 
(13.) 
