Dielectric  Strength  of  Air.  249 
substituting  for  vn+\  and  vn  their  values  and  simplifying,  that 
and  ,  an 
U*  =  "9  1  _Ln-2  > (5) 
V„,     = 
Ull 
1  _  «4n— 2 
=  ^t4=- (6) 
Let  us  now  suppose  that  charges  Ql5  .  .  .  Qn,  are  placed  at 
the  points  A1?  .  .  .  An  (fig.  2),  and  that  charges  Q/,  .  .  .  Q„', 
are  placed  at  the  points  B1?  .  .  .  Bn.  We  shall  find  the  values 
of  these  charges  so  that  the  potential  of  the  spherical  surface 
X  is  Vi  and  that  of  the  spherical  surface  Y  is  zero. 
Consider  the  potential  at  a  point  P'  at  a  distance  a  from  B. 
The  potential  at  this  point  will  obviously  be 
*VP'A„  +  FBj' 
If  therefore  we  choose  the  ratio  of  Q«  to  Q„'  so  that 
QJQJ  =  —  P'Aa/P'B?l  =  a  constant  for  every  point  on  Y,  the 
potential  at  P'  will  be  zero,  and  therefore  also  the  potential 
of  the  spherical  surface  Y  will  be  zero.  Since  An  and  Bre  are 
conjugate  points  with  respect  to  the  sphere  Y  we  have  * 
Cy_  _FA,       _BB,       _ul 
QB  7      P'B,  ~         a     ~       a  '    '     '     *     t<) 
Again,  the  potential  at  a  point  P  distant  a  from  A  is 
given  by 
Ql  ^  (    Qra  +  l       ,       Q,n    \ 
a  +  *lPAn+1  +  PBj- 
This  will  be  Y1  if  we  make  Qx  =  Vxa  and 
(8) 
Qn'  PBn  a  a     '    * 
Hence  if  we  determine  the  charges  Qn  and  Qn'  by  means 
of  (7)  and  (8)  the  potential  of  the  spherical  surface  X  will 
be  Vj  and  that  of  the  spherical  surface  Y  will  be  zero. 
From  (7)  and  (8),  we  have 
0,11  +  1         Vn+\Vn 
Qn  CL' 
>2w-2 
*  Russell,  '  Alternating  Currents,'  vol.  i.  p.  101. 
