﻿Resistance 
  in 
  Telephone 
  and 
  other 
  Circuits. 
  431 
  

  

  Iron 
  Wires 
  with 
  High 
  Frequency. 
  

   When 
  IX 
  is 
  large, 
  we 
  may 
  write 
  from 
  (45), 
  

  

  ^p 
  = 
  -3 
  i^2/3-a7 
  log 
  7ic-2^2_ 
  log 
  _j^ 
  . 
  (47) 
  

  

  and 
  by 
  (35), 
  

  

  so 
  that 
  

  

  Neglecting 
  xj^j? 
  as 
  before, 
  and 
  writing 
  the 
  first 
  approxi- 
  

   mations 
  to 
  {oi.p<y) 
  in 
  the 
  second 
  term, 
  and 
  the 
  second 
  in 
  

   ihe 
  first, 
  we 
  finally 
  obtain 
  

  

  AF 
  + 
  BE 
  

   CD 
  

  

  the 
  term 
  in 
  X^//,"^ 
  vanishing 
  identically. 
  

   Thus 
  the 
  effective 
  resistance 
  becomes 
  

  

  This 
  result 
  is 
  much 
  more 
  accurate 
  than 
  the 
  corresponding- 
  

   formula 
  (37) 
  for 
  inductance, 
  owing 
  to 
  the 
  presence 
  of 
  //- 
  in 
  

   the 
  numerator 
  of 
  the 
  first 
  term. 
  Moreover, 
  X 
  here 
  denotes 
  

   the 
  previous 
  X 
  multiplied 
  by 
  //,*. 
  It 
  is 
  applicable 
  to 
  all 
  iron 
  

   circuits 
  of 
  importance 
  in 
  practice, 
  for 
  a 
  frequency 
  greater 
  

   than 
  about 
  a 
  hundred 
  with 
  a 
  radius 
  of 
  1 
  millimetre. 
  

  

  Iron 
  Wires 
  with 
  Low 
  Frequency. 
  

  

  This 
  result 
  has 
  little 
  interest, 
  but 
  may 
  be 
  obtained 
  at 
  once. 
  

   For 
  (a/37) 
  ^^® 
  ^^1 
  comparable 
  with 
  unity 
  in 
  this 
  case, 
  and 
  x 
  

   is 
  not 
  greater 
  than 
  2. 
  We 
  therefore 
  ignore 
  x'^jjj?, 
  and 
  

   obtain 
  

  

  This 
  term 
  also 
  may 
  be 
  ignored 
  unless 
  a 
  very 
  high 
  order 
  of 
  

  

  