As   another  example  consider  an  electrified  circular  disc  of  Force  at

uniform  electric  density  o-  and  radius  r  acting  on  a  unit  of  positive  point  on

electricity  placed  on  the  axis  of  the  disc  at  a  distance  h  from  its  axis  of

plane.    The  potential  of  a  concentric  ring  of  radius  x  and  breadth  Electrified
dx  is  Inirxdxl  *J/i2  +  x*.     For  the  whole  disc  therefore


=  27TO-  (  V^2  +  /-2  -  K)     .....    (22)
For  the  repulsion  on  the  particle  we  have  the  value


=  27ro-(l  -  /S/VA2  +  r2)     .    .     .     (23)

The  following  proposition  (due  to  Gauss)  is  interesting
and  important :  The  average  potential  over  a  spherical
surface  due  to  electricity  entirely  without  the  sphere  is
equal  to  the  potential  at  the  centre.
Suppose  the  electric  distribution  to  be  a  quantity  q  situated  at
a  point  0.  The  potential  at  a  point  P  on  the  spherical  surface

distant  r  from  0  is  ?.    The  average  potential  over  the  surface  is,

if  R  be  its  radius, I  /  -  ds.     But  /  I  -  ds  is  the  potential

at  0  due  to  a  uniform  distribution  of  surface  density  q  over  the
spherical  surface  and  is  therefore  (p.  14)  — - — -,  if  D  be  the
distance  of  0  from  the  centre  of  the  sphere.  The  average
potential  is  therefore  -,  that  is,  the  potential  at  the  centre  of  the

sphere  due  to  q  at  0.  The  same  result  can  be  obtained  for  every
part  of  the  distribution,  and  the  proposition  is  established.
From  this  theorem  follows  obviously  the  result  already
proved  that  there  is  no  place  of  maximum  or  of  minimum
potential  in  space  void  of  electricity.
16  ELECTROSTATIC  THEORY.
