OA,  taken  positive  towards  the  space  BOA  in  which  P
is  situated,  we  have


Electric  ]      Since  the  space  BOA  may  be  regarded  as  the  space
at  any     external  to  a  closed  conductor,  N  is  the  actual  force
pomt'     just  outside  the  conductor  at  E,  and  we  have  for  the
electric  density  there

(85)


(85  Us)
fulfil  the


=   —       -


Similarly   for   the   density  at   any   point   F  of   OB
we  have

«,-_££  (J L

27T    VP^3  PJ

In  Fig.  23  the  distances  CP,  CP',  (
Inversion,   relation

CP.  CP  =  CQ .  CQ'  =  a\     ...     (86)


GEOMETRICAL  INVERSION.  91

The  point  Pf  is  called  the  inverse  of  the  point  P  with      Geo-
respect  to  0,  which  is  called  the  centre  of  inversion;  and
similarly  Q'  is  the  inverse  of  Q  with  reference  to  C.     If


p'


Q'  Q

FIG.  23.
we  have  any  system  of  points  P,  Q,  &c.,  given,  we  can
find  in  this  manner  a  corresponding  system  of  points
P',  Q',  &c.  If  P,  Q,  &c.,  determine  a  surface  or  a  solid
figure,  P',  Q',  &c.,  determine  a  corresponding  surface  or
solid  figure.  The  system  P/,  Q',  &c.,  is  called  the  inverse
of  the  system  P,  Q,  &c.,  with  reference  to  C ;  and  it
follows  that  the  system  P,  Q,  &c.,  is  the  inverse  of  P',  Q',
&c.,  with  reference  to  the  same  point.
From  what  has  gone  before  it  is  evident  that  any  inverse  of
point  P'  is  the  image  of  the  point  P  in  the  sphere  of  ^^Ij  |ns

radius  a  and  centre  C.     This  sphere  is  called  the  sphere  Sphere  of
'  /.  .  .  j  .,          -,.        ,-,  ,.          *  .  .  Inversion,

o/  inversion  and  its  radius  the  radius  of  inversion.
