﻿Resistance 
  in 
  Telephone 
  and 
  other 
  Circuits. 
  425 
  

  

  Copper 
  Wires 
  loith 
  High 
  Frequence/. 
  

  

  With 
  the 
  above 
  restrictions 
  as 
  regards 
  capacity 
  and 
  

   frequency 
  (the 
  latter 
  being 
  of 
  little 
  practical 
  import), 
  writing 
  

   in 
  (17), 
  

  

  ,5 
  _ 
  l/i_l_ 
  J_Vl-i 
  -L\ 
  

  

  Then 
  with 
  the 
  values 
  of 
  (21), 
  

   L 
  = 
  41og.^4(l-2 
  + 
  l,)H-^(AE-BF), 
  

  

  where 
  \ 
  = 
  2a(2'n'fjLn/a)i 
  = 
  47ra 
  (8/o-)*, 
  

  

  with 
  a 
  four-figure 
  accuracy 
  so 
  far 
  as 
  X 
  is 
  concerned, 
  if 
  

   'X>S^2. 
  But 
  a 
  further 
  simplification 
  may 
  be 
  introduced. 
  

   Since 
  ha 
  is 
  a 
  very 
  small 
  magnitude 
  in 
  actual 
  cases, 
  its 
  

   logarithm 
  is 
  large 
  and 
  negative. 
  Thus 
  if 
  /o 
  = 
  loge 
  ha, 
  the 
  

   functions 
  may 
  be 
  expanded 
  in 
  a 
  descending 
  series 
  of 
  powers 
  

   of 
  p, 
  and 
  

  

  CD 
  , 
  1/, 
  2\ 
  1/1 
  2 
  2\ 
  1/3 
  1\ 
  

   P^=l--x(^+^) 
  + 
  xA2 
  + 
  p 
  + 
  p^; 
  + 
  X3(i 
  + 
  2p; 
  

  

  or 
  

  

  Po2 
  , 
  1/, 
  2\ 
  1/1 
  2 
  2\ 
  1/3 
  1\ 
  ,oo^ 
  

  

  whence, 
  after 
  considerable 
  reduction 
  

   ^(AE-BF) 
  = 
  ,logf-^(p44,log2+ 
  31og-;) 
  

  

  ■^ 
  1^' 
  { 
  «''- 
  V+ 
  1 
  V- 
  (8-8p+3p^) 
  logf 
  } 
  . 
  

  

  The 
  vanishing 
  of 
  the 
  coefficient 
  of 
  X"^ 
  is 
  curious. 
  

  

  Finally, 
  for 
  a 
  pair 
  of 
  copper 
  wires, 
  the 
  main 
  error 
  being 
  of 
  

   relative 
  magnitude 
  a^c^ 
  when 
  they 
  are 
  close 
  together, 
  if 
  

   X>8>/2, 
  

  

  X 
  = 
  47ra(//^)i, 
  (33) 
  

  

  