PEOrESSOE  plCckee  on  the  mag-netic  induction  of  CETSTALS.  563 
whence 
99 
hh!= 
h^—c^  ^ 2 
^2 ^2 — tan  (t)^ 
€?■  — (?■ 1 
a^—b^  cos^w 
(20.) 
The  two  conjugate  axes,  when  lying  in  the  above-mentioned  conic  surface,  are  contained 
in  a plane  passing  also  through  the  horizontal  diameter  round  which  the  given  ellipsoid 
revolves.  This  diameter  being  represented  by 
x—mz^ 
we  get,  therefore. 
y=nz, 
{g'—9)n-^{h—h’)m-\-{gh'—]ig’)=Q. 
Ehminating  g and  h'  by  means  of  (20.),  and  putting  - and  ^ instead  of  g and  A,  we 
obtain  the  following  equation : 
, . (21.) 
{nx—my)  , {y—nz)  . , ^ x—mz  ^ 
=-  COS^&;+^^^ ^Sim<y-h = 0,  . 
representing  the  conic  surface  of  the  third  order. 
40.  Eeverting  to  the  rotation  of  the  influenced  ellipsoid,  we  immediately  obtain  the 
instantaneous  axis  of  rotation.  The  given  diameter  OM  passing,  when  prolonged, 
through  the  infinitely  distant  pole,  and  this  axis  being  two  conjugate  axes  of  the 
auxiliary  elhpsoid,  each  of  them  is  determined  by  the  other  by  means  of  (20.). 
Denoting  the  angles  between  OM  and  the  three  axes  of  coordinates  OX,  OY,  OZ  by 
V,  and  those  between  the  instantaneous  axis  of  rotation  and  the  three  same  axes  of 
coordinates  by  yJ,  v\  we  have 
cos  fx.  cos  p. 
:COS  V COS  v'=z  ■ 
cos  g cos  g 
(22.) 
41.  The  absolute  moment  of  rotation  found  to  be 
2<p  tan  ^ 
r 
may  easily  be  expanded : | being  the  angle  between  the  diameter  OM  passing  through 
M,  whose  coordinates  may  be  denoted  by  2,  and  the  perpendicular  to  the  plane 
touching  the  auxiliary  ellipsoid  in  this  point,  we  get  by  well-known  formulae, 
tan®  l=\_{a^  — l/)xyf-\-\_{a^  — d^)xz'f-\-\_{lf—c^)yz'f=[a^—c^f[oify‘^  cos®  a>-\-0tfz^-\-'ifz^  sin®  xyj, 
whence 
( ^ \ ^ 
^ \ =(«®— 6‘®)[cos®  yj  cos®  V cos®  <y-f-cos®  (Jj  cos®  ^-j-cos®  V cos®  § sin®  co]  ; 
and  by  eliminating  cos®  v by  means  of 
cos®|yj-l-cos®  j'd-cos®^=l, 
and  by  reducing, 
=;|-(a®— c®)®[sin®  2p!»4-sin^  xy(sin®  2^— sin®  2/!a)] (23.) 
When  the  infinitely  distant  pole  falls  successively  within  each  of  the  three  principal 
