Dielectric  Strength  of  Air. 
247 
7.  Proof  of  the  Series- For  mulct  for  the  Maximum  Electric 
Intensity  between  Two  Equal  Spheres. 
Let  us  suppose  that   the  radii  of  the  conducting  spheres 
X  and  Y  (fig.  2)   are  each  equal  to  a,   that    the   distance 
Fis.  2. 
AlB=d;  AIL=BM=a;  \M.=x=d-2a. 
between  their  centres  A,  and  B  is  d,  and  that  the  minimum 
distance  LM  between  them  is  x,  so  that  d=x  +  2a.  Let  us 
suppose  also  that  these  spheres  are  at  potentials  Vx  and  V2. 
We  picture  Faraday-tubes  starting  from  their  surfaces.  If 
their  potentials  are  of  opposite  sign  some  of  these  tubes 
connect  the  two  spheres  and  others  connect  them  with  neigh- 
bouring conductors.  We  suppose  that  these  other  conductors 
are  so  far  away  that  they  do  not  appreciably  affect  the  distri- 
bution of  the  tubes  in  the  field  between  the  two  spheres. 
Now,  if  the  spheres  be  removed  we  shall  show  that  this  field 
can  be  exactly  reproduced  by  a  series  of  point  charges  placed 
at  definite  points  on  the  lines  AL  and  BM  (fig.  2).  The 
point  charges  will  have  the  spherical  surfaces  X  and  Y  for 
the  equipotential  surfaces  V1  and  V2  respectively.  We  can 
therefore  write  down  at  once  the  potentials  and  the  electric 
intensities  at  all  external  points. 
We  shall  first  consider  the  series  of  points  Al}  A2  .  .  . 
Bj,  B2  .  .  .  (fig.  2)  which  are  connected  by  the  following- 
relations, 
BAX .  BBX  =  a2  =  AXA2 .  A^ 
BA2 .  BB2  =  a2  =  AiA3 .  AiBa. 
We  see  that  the  points  A2,  B,  ;  ...  A„+i,  B„,  are  con- 
jugate with  respect  to  the  sphere  X  and  the  points 
Bl5   Aj  ;    .  .  .  B„,    A«,    arc   conjugate    with    respect    to    the 
