﻿738 
  Prof. 
  C. 
  H. 
  Lees 
  on 
  the 
  Resistance 
  of 
  a 
  

  

  The 
  vector 
  function 
  w 
  = 
  u 
  + 
  iv 
  for 
  this 
  case 
  is 
  given 
  hy 
  the 
  

   equation 
  

  

  w 
  = 
  ^t 
  h 
  ^' 
  w 
  

  

  where 
  t 
  is 
  the 
  thickness 
  of 
  the 
  conductors 
  and 
  h 
  is 
  the 
  con- 
  

   ductivity 
  of 
  the 
  material 
  of 
  which 
  they 
  are 
  composed. 
  

  

  If 
  £p, 
  0, 
  are 
  the 
  coordinates 
  of 
  a 
  point 
  P 
  near 
  in 
  the 
  

   ? 
  plane, 
  and 
  x 
  P 
  is 
  its 
  X 
  coordinate 
  in 
  the 
  z 
  plane 
  (figs. 
  1 
  

   and 
  2), 
  then 
  by 
  equation 
  (3) 
  

  

  x= 
  — 
  argcoth 
  T 
  \ 
  / 
  ^ 
  p 
  — 
  — 
  — 
  argtang\ 
  /— 
  ^ 
  . 
  

  

  which 
  since 
  £ 
  P 
  is 
  small 
  reduces 
  to 
  

  

  b 
  

   c 
  

  

  2b 
  / 
  c 
  2 
  — 
  l> 
  2 
  2c 
  

   x 
  v 
  = 
  — 
  argcoth^y 
  1 
  + 
  —JL 
  f 
  p 
  argtanh 
  

  

  b 
  f, 
  4c 
  2 
  vr^ 
  "1 
  c, 
  c 
  + 
  b 
  

  

  in 
  virtue 
  of 
  equation 
  (4). 
  

   Hence 
  

  

  7T&£ 
  7T 
  , 
  c 
  2 
  — 
  b 
  2 
  c, 
  c 
  + 
  b 
  

  

  -p.u, 
  j^-log-!- 
  1— 
  -jlog^y 
  

  

  Similarly 
  if 
  f 
  Q 
  , 
  0, 
  are 
  the 
  coordinates 
  of 
  a 
  point 
  Q 
  near 
  

   + 
  go 
  in 
  the 
  £ 
  plane, 
  and 
  x 
  q 
  is 
  its 
  X 
  coordinate 
  in 
  the 
  z 
  plane, 
  

   then 
  by 
  equation 
  (3) 
  

  

  x 
  •= 
  — 
  argcoth 
  T 
  \ 
  / 
  ^ 
  Q 
  — 
  _ 
  argtanh 
  \ 
  / 
  _fejtii_. 
  

  

  which, 
  since 
  f 
  Q 
  is 
  large, 
  reduces 
  to 
  

  

  * 
  Q 
  = 
  ^rgcoth| 
  - 
  ^argtanh 
  y/l-^ 
  

  

  & 
  , 
  C+b 
  C 
  f 
  , 
  4& 
  2 
  . 
  7T^ 
  ) 
  

  

  = 
  ^gj^ 
  r 
  ; 
  I 
  log 
  c^ 
  + 
  tt 
  • 
  M 
  4 
  

  

  in 
  virtue 
  of 
  equation 
  (4). 
  

  

  Hence 
  

  

  irkt 
  7r 
  , 
  46 
  2 
  , 
  b. 
  c 
  + 
  b 
  

  

  _ 
  WQ= 
  _-^_lo 
  g 
  _ 
  2 
  H 
  -log^g. 
  

  

  Thus 
  

  

  Hence 
  the 
  resistance 
  between 
  two 
  transverse 
  sections 
  

  

  