﻿1875.] 
  

  

  Equipotential 
  Curves 
  and 
  Surfaces, 
  fyc. 
  

  

  283 
  

  

  ducting 
  liquid 
  and 
  placing 
  two 
  points, 
  the 
  ends 
  of 
  two 
  covered 
  w 
  7 
  ires, 
  for 
  

   battery-electrodes, 
  at 
  a 
  given 
  depth 
  in 
  the 
  liquid 
  and 
  away 
  from 
  the 
  sides 
  

   and 
  ends 
  of 
  the 
  vessel, 
  taking 
  similar 
  covered 
  wires, 
  immersed 
  to 
  the 
  

   same 
  depth, 
  for 
  galvanometer-electrodes. 
  

  

  For 
  two 
  electrodes, 
  the 
  equipotential 
  surfaces 
  will 
  be 
  surfaces 
  of 
  revo- 
  

   lution 
  around 
  the 
  straight 
  line 
  joining 
  them, 
  and 
  so 
  will 
  cut 
  any 
  plane, 
  

   drawn 
  through 
  this 
  straight 
  line 
  or 
  axis, 
  everywhere 
  at 
  right 
  angles. 
  

  

  Hence 
  we 
  may 
  suppose 
  sections 
  of 
  the 
  liquid 
  made 
  along 
  such 
  planes 
  

   without 
  altering 
  the 
  forms 
  of 
  the 
  equipotential 
  surfaces. 
  This 
  shows 
  

   that 
  we 
  may 
  place 
  our 
  battery-electrodes 
  at 
  the 
  side 
  of 
  a 
  rectangular 
  box 
  

   containing 
  the 
  liquid, 
  and 
  with 
  the 
  points 
  only 
  just 
  immersed 
  below 
  the 
  

   surface 
  of 
  the 
  liquid 
  ; 
  and 
  the 
  equipotential 
  surfaces 
  will 
  be 
  the 
  same 
  as 
  if 
  

   the 
  liquid 
  were 
  of 
  unlimited 
  extent 
  in 
  every 
  direction 
  about 
  the 
  elec- 
  

   trodes. 
  

  

  We 
  shall 
  obtain 
  the 
  section 
  of 
  the 
  equipotential 
  surface 
  by 
  taking 
  for 
  

   galvanometer-electrodes 
  two 
  points 
  in 
  the 
  surface 
  of 
  the 
  liquid, 
  keeping 
  

   one 
  fixed 
  and 
  tracing 
  out 
  points 
  of 
  equal 
  potential 
  with 
  the 
  other. 
  

  

  The 
  potential 
  at 
  any 
  point 
  in 
  space, 
  due 
  to 
  two 
  equal 
  and 
  opposite 
  

   electrodes, 
  is 
  

  

  A 
  (r 
  rj' 
  

  

  where 
  r 
  and 
  r 
  x 
  are 
  the 
  distances 
  of 
  the 
  point 
  from 
  the 
  electrodes 
  ; 
  so 
  that 
  

   for 
  an 
  equipotential 
  surface 
  

  

  - 
  — 
  i 
  = 
  constant. 
  

   r 
  r, 
  

  

  These 
  surfaces 
  are 
  cut 
  at 
  right 
  angles 
  by 
  the 
  curves 
  cos0 
  — 
  cos^ 
  = 
  c, 
  

   which 
  are 
  also 
  the 
  magnetic 
  lines 
  of 
  force, 
  6 
  and 
  <j> 
  being 
  the 
  angles 
  

   which 
  the 
  distances 
  from 
  the 
  electrodes 
  make 
  with 
  the 
  axis. 
  That 
  the 
  

   lines 
  of 
  force 
  in 
  a 
  vessel 
  of 
  finite 
  size 
  should 
  agree 
  with 
  the 
  lines 
  of 
  force 
  

   in 
  space, 
  the 
  form 
  of 
  the 
  boundary 
  of 
  the 
  vessel 
  in 
  a 
  plane 
  through 
  the 
  

   axis 
  should 
  everywhere 
  be 
  a 
  line 
  of 
  force 
  ; 
  but 
  the 
  ends 
  of 
  a 
  rectangular 
  

   vessel 
  coincide 
  very 
  closely 
  with 
  certain 
  lines 
  of 
  force, 
  either 
  when 
  the 
  

   electrodes 
  are 
  at 
  the 
  ends, 
  or 
  when 
  there 
  are 
  two 
  electrodes 
  within 
  the 
  

   vessel, 
  and 
  two 
  supposed 
  electrodes 
  at 
  their 
  electrical 
  images 
  at 
  an 
  equal 
  

   distance 
  outside 
  the 
  ends 
  of 
  the 
  vessel. 
  

   The 
  equipotential 
  surfaces 
  are 
  given 
  in 
  this 
  case 
  by 
  the 
  equation 
  

  

  1 
  + 
  -, 
  — 
  ^ 
  — 
  \= 
  constant, 
  

   r 
  r 
  9\ 
  r 
  x 
  ' 
  

  

  and 
  the 
  lines 
  of 
  force 
  by 
  the 
  equation 
  

  

  COS0 
  + 
  COS 
  — 
  COS0 
  — 
  cos 
  X 
  = 
  C 
  

  

  The 
  curve 
  for 
  which 
  c=2 
  coincides 
  very 
  closely 
  with 
  the 
  ends 
  of 
  the 
  

   box. 
  

  

  The 
  equipotential 
  surfaces 
  were 
  traced 
  out 
  in 
  sulphate 
  of 
  copper 
  and 
  in 
  

   sulphate 
  of 
  zinc 
  by 
  the 
  following 
  method 
  : 
  — 
  

  

  z 
  2 
  

  

  