﻿875.] 
  

  

  Equipotential 
  Curves 
  and 
  Surfaces, 
  fyc. 
  

  

  281 
  

  

  The 
  forms 
  of 
  the 
  equipotential 
  curves 
  may 
  be 
  traced 
  out 
  experimen- 
  

   tally 
  by 
  attaching 
  two 
  battery-electrodes 
  to 
  a 
  disk 
  of 
  tinfoil, 
  and 
  having 
  

   two 
  similar 
  electrodes 
  attached 
  to 
  a 
  delicate 
  galvanometer 
  ; 
  one 
  of 
  these 
  

   electrodes 
  being 
  fixed 
  at 
  a 
  point 
  through 
  which 
  the 
  equipotential 
  curve 
  is 
  

   to 
  be 
  drawn, 
  the 
  other 
  may 
  be 
  moved 
  from 
  point 
  to 
  point 
  to 
  trace 
  out 
  the 
  

   successive 
  points, 
  so 
  that 
  no 
  current 
  may 
  pass 
  through 
  the 
  galvanometer. 
  

   A 
  comparison 
  of 
  the 
  experimental 
  results 
  with 
  the 
  theory 
  shows 
  a 
  com- 
  

   plete 
  agreement. 
  

  

  In 
  a 
  large 
  square 
  sheet 
  310 
  millims. 
  in 
  diameter, 
  with 
  the 
  electrodes 
  

   126 
  millims. 
  apart, 
  the 
  curves 
  in 
  the 
  centre 
  and 
  near 
  the 
  electrodes, 
  

   which 
  are 
  drawn 
  by 
  pricking 
  fine 
  holes 
  through 
  the 
  tinfoil 
  on 
  a 
  sheet 
  of 
  

   paper 
  below, 
  are 
  very 
  accurately 
  circular, 
  and 
  mostly 
  coincide 
  with 
  

   circles, 
  until 
  the 
  points 
  are 
  so 
  far 
  from 
  the 
  centre 
  that 
  the 
  form 
  of 
  the 
  

   equipotential 
  curves 
  is 
  affected 
  by 
  the 
  edge 
  of 
  the 
  disk. 
  In 
  a 
  circular 
  disk 
  

   with 
  the 
  electrodes 
  on 
  the 
  edge 
  subtending 
  60° 
  at 
  the 
  centre, 
  the 
  experi- 
  

   mental 
  curves 
  are 
  shown 
  to 
  be 
  accurately 
  arcs 
  of 
  circles, 
  with 
  their 
  centres 
  

   on 
  the 
  line 
  joining 
  the 
  electrodes. 
  

  

  In 
  an 
  unlimited 
  sheet, 
  when 
  there 
  are 
  several 
  electrodes 
  by 
  which 
  

   currents 
  enter 
  and 
  leave 
  the 
  sheet, 
  the 
  potential 
  at 
  any 
  point 
  is 
  

  

  Alo 
  g 
  (!LI^A 
  

   \Vi 
  r 
  i 
  • 
  • 
  •/ 
  

  

  where 
  r, 
  r, 
  r" 
  . 
  . 
  . 
  are 
  the 
  distances 
  to 
  the 
  electrodes 
  of 
  one 
  kind, 
  and 
  

   r 
  v 
  r 
  x 
  ', 
  r" 
  are 
  the 
  distances 
  to 
  the 
  electrodes 
  of 
  the 
  other 
  kind. 
  Taking 
  

   the 
  case 
  of 
  one 
  positive 
  electrode 
  at 
  the 
  centre 
  and 
  four 
  negative 
  elec- 
  

   trodes 
  round 
  it 
  at 
  the 
  corners 
  of 
  a 
  square, 
  the 
  curves 
  are 
  traced 
  and 
  are 
  

   seen 
  to 
  be 
  the 
  same 
  as 
  the 
  curves 
  at 
  the 
  corner 
  of 
  a 
  square 
  sheet 
  with 
  

   a 
  positive 
  electrode 
  at 
  the 
  corner 
  and 
  two 
  negative 
  electrodes 
  on 
  the 
  

   edges 
  ; 
  the 
  curves 
  are 
  also 
  the 
  same 
  for 
  a 
  square 
  sheet 
  with 
  a 
  positive 
  

   electrode 
  at 
  the 
  corner, 
  and 
  one 
  negative 
  electrode 
  along 
  the 
  diagonal. 
  

   The 
  equation 
  for 
  these 
  equipotential 
  curves 
  is 
  

  

  and 
  is 
  derived, 
  in 
  the 
  case 
  of 
  the 
  limited 
  sheets, 
  by 
  considering 
  that, 
  to 
  

   every 
  electrode 
  on 
  the 
  limited 
  sheet, 
  there 
  corresponds 
  an 
  equal 
  and 
  like 
  

   electrode 
  at 
  each 
  of 
  the 
  electrical 
  images 
  of 
  that 
  electrode 
  formed 
  by 
  the 
  

   edges 
  of 
  the 
  sheet. 
  If 
  we 
  trace 
  the 
  curves 
  for 
  this 
  arrangement 
  of 
  elec- 
  

   trodes 
  in 
  the 
  unlimited 
  sheet, 
  the 
  edges 
  of 
  the 
  limited 
  sheet 
  will 
  be 
  some 
  

   of 
  the 
  lines 
  of 
  force 
  ; 
  and 
  so 
  we 
  may 
  divide 
  the 
  sheet 
  along 
  these 
  edges, 
  

   without 
  altering 
  the 
  form 
  of 
  the 
  equipotential 
  curves. 
  Where 
  an 
  elec- 
  

   trode 
  and 
  its 
  images 
  coincide 
  in 
  position, 
  the 
  index 
  of 
  r 
  is 
  equal 
  to 
  one 
  

   more 
  than 
  the 
  number 
  of 
  images. 
  

  

  When 
  there 
  are 
  four 
  electrodes, 
  two 
  of 
  each 
  kind 
  on 
  an 
  unlimited 
  

   sheet, 
  an 
  equipotential 
  curve 
  is 
  given 
  by 
  the 
  equation 
  

  

  VOL. 
  XXIII. 
  Z 
  

  

  