﻿V 
  \BP 
  

  

  that 
  of 
  Three 
  Electrified 
  Spherical 
  Conductors. 
  395 
  

  

  The 
  potential 
  at 
  P 
  is 
  

  

  jj( 
  o: 
  ab 
  a 
  2 
  b 
  

  

  U 
  VAP 
  ~ 
  BP 
  . 
  AF\ 
  + 
  AP 
  . 
  BQ 
  1 
  . 
  AQ 
  2 
  

  

  BP 
  . 
  AP 
  X 
  . 
  AP 
  :3 
  . 
  BP 
  2 
  AP 
  . 
  B'Q! 
  . 
  BQ, 
  . 
  AQ, 
  . 
  AQ 
  4 
  

  

  aW 
  \ 
  

  

  BP 
  . 
  APj 
  . 
  AP 
  3 
  . 
  AP 
  5 
  . 
  BP 
  2 
  . 
  BP 
  6 
  + 
  • 
  • 
  • 
  ) 
  

  

  b 
  ah 
  ah' 
  2 
  

  

  AP 
  . 
  BQx 
  + 
  BP 
  . 
  APj 
  . 
  BP 
  2 
  

  

  a 
  2 
  b 
  2 
  aW 
  

  

  AP. 
  BQi 
  . 
  BQ 
  3 
  . 
  AQ 
  2 
  + 
  BP.APx 
  . 
  AP 
  3 
  . 
  BP 
  2 
  . 
  BP 
  4 
  

  

  AP.BQ 
  1 
  .BQ 
  3 
  .BQ 
  5 
  .AQ 
  2 
  .AQ 
  6 
  + 
  J' 
  

  

  If, 
  now, 
  the 
  image 
  of 
  A 
  in 
  B 
  is 
  I 
  l5 
  the 
  image 
  of 
  I 
  x 
  in 
  A, 
  I 
  2 
  , 
  

   the 
  image 
  of 
  I 
  2 
  in 
  B, 
  I 
  3 
  , 
  and 
  so 
  on, 
  and 
  if 
  the 
  image 
  of 
  B 
  in 
  

   A 
  is 
  J|, 
  the 
  image 
  of 
  J 
  1 
  in 
  B, 
  J 
  2 
  , 
  the 
  image 
  of 
  J 
  2 
  in 
  A, 
  J 
  3 
  , 
  

  

  and 
  so 
  on, 
  it 
  is 
  easy 
  to 
  show 
  from 
  similar 
  triangles 
  that 
  

  

  BP.AP 
  1=c 
  .PI 
  l5 
  

   AP.BQi.AQ^c.PI^AIx, 
  

  

  BP 
  . 
  AP 
  X 
  . 
  BP 
  2 
  . 
  A 
  P 
  3 
  = 
  c 
  . 
  PI 
  3 
  . 
  BI 
  2 
  . 
  AI 
  X 
  , 
  

  

  AP 
  . 
  BQ, 
  . 
  AQ 
  2 
  . 
  BQ 
  3 
  . 
  AQ 
  4 
  = 
  C 
  . 
  PI 
  4 
  . 
  AI 
  3 
  . 
  BI 
  2 
  . 
  AI 
  l5 
  

  

  &c. 
  

  

  Hence 
  the 
  potential 
  is 
  

  

  TJ 
  / 
  a 
  ah 
  a 
  2 
  b 
  

  

  U 
  [ 
  AP 
  ~~ 
  cTPi; 
  + 
  e 
  . 
  PI 
  2 
  . 
  AIj 
  

  

  a 
  2 
  h 
  2 
  a?b 
  2 
  

  

  "" 
  c 
  . 
  PI 
  3 
  . 
  BI 
  2 
  . 
  AIj 
  + 
  c 
  . 
  PI 
  4 
  . 
  AI 
  3 
  . 
  BI 
  2 
  . 
  AI, 
  

  

  c 
  .PI 
  5 
  .BI 
  4 
  .AI 
  3 
  .BI 
  2 
  .AIi 
  '•') 
  

  

  (b 
  ab 
  ab 
  2 
  a 
  2 
  b 
  2 
  \ 
  

  

  + 
  V 
  Up 
  cPJ1C.PJ2.BJi 
  cPJ3.AJ2.BJ! 
  + 
  "v, 
  

  

  which 
  shows 
  that 
  the 
  potential 
  outside 
  the 
  spheres 
  has 
  the 
  

   same 
  value 
  as 
  that 
  due 
  to 
  a 
  series 
  of 
  point 
  charges 
  at 
  

  

  