21b'  Mr.  A.  Russell  0/2  ^e 
Let  V/2  and  —  V/2  be  the  potentials  on  any  two  equal 
equipotential  surfaces  surrounding  P  and  N  respectively. 
We  shall  find  a  formula  to  determine  the  maximum  value  of 
the  electric  intensity  between  these  two  surfaces.  From 
symmetry  the  maximum  value  of  the  electric  intensity  Rm  in 
the  space  between  the  two  will  be  at  the  points  A  and  B, 
where  the  line  joining  P  and  N  cuts  the  surfaces.  If 
PN  =  cZ  and  PA=«,  we  have 
V_9_     g 
2       a      d  —  a 
and  therefore  q        a(d  —  a) 
Y  =  2(d-2a)' 
We  also  have  ^  r  1  1       } 
and  hence  Rw?  =  (Y/as)f, (&) 
where  #,  which  equals  d  —  2a,  is  the  minimum  distance 
between  the  two  surfaces  and  /is  given  by 
'~K1+'M+i(ihl3) (c) 
Now  V/o?  is  the  average  value  of  the  electric  intensity  along 
the  line  joining  the  nearest  points  of  the  surfaces,  and  is  the 
number  which  electricians  ordinarily  give  as  a  measure  of 
the  dielectric  stress  on  the  insulating  medium.  We  see  that 
/  is  the  factor  required  to  convert  this  number  into  the  maxi- 
mum electric  intensity. 
When  as/a  is  large  the  surfaces  are  very  approximately 
spheres  of  radius  a,  and  (c)  can  therefore  be  used  to  calculate 
the  value  of  /  for  two  spheres  when  their  distance  apart  is 
large  compared  with  the  radius  of  either. 
When  as/ a  is  small  we  can  show  that 
/=l  +  */3p, {d) 
approximately,  where  p  is  the  radius  of  curvature  of  the 
equipotential  surfaces  at  the  points  where  the  intensity  is  a 
maximum.  We  should  expect  therefore  that,  if  we  had  two 
spheres  the  radius  of  each  of  which  was  p,  (d)  would  give 
the  value  of  /  approximately  when  as/p  was  small.  We  shall 
show  later  on  that  (d)  gives  the  value  of/  in  this  case,  to  an 
accuracy  of  one  in  a  thousand  when  x/p  is  0*1  or  less.  Even 
when  as ]p  is  unity  the  error  is  only  about  2  per  cent. 
The  two  terms  given  on  the  right-hand  side  of  (d)  are  the 
first  two  terms  in  the  important  Kirchhoff- Schuster  formula 
quoted  below. 
