﻿Resistance 
  in 
  lelephone 
  and 
  other 
  Circuits. 
  429 
  

  

  Heaviside 
  * 
  has 
  also 
  given 
  the 
  value 
  o£ 
  y8/a_, 
  and 
  

   Lord 
  Rayleigh 
  t 
  that 
  of 
  7/a, 
  by 
  different 
  modes 
  o£ 
  proof. 
  

   Thus 
  finally, 
  when 
  

  

  -- 
  = 
  a7r(2M//<7)*>l, 
  (41) 
  

  

  L 
  = 
  41og£^ 
  + 
  1- 
  _^.,4+ 
  -|^^,8__^^^U._ 
  

  

  1 
  ,4 
  , 
  13 
  ,, 
  647 
  .i,_4a^ 
  

  

  !4' 
  "^4320" 
  122.360.56' 
  c^ 
  

  

  i:z*a\ 
  , 
  r, 
  11 
  , 
  , 
  473 
  . 
  304107 
  j^l 
  

  

  

  + 
  SaYi+ 
  1.4+1- 
  l^A 
  .... 
  (42) 
  

  

  Small 
  Frequency 
  J 
  Copper 
  Wires, 
  

   In 
  the 
  notation 
  of 
  (18), 
  with 
  p 
  = 
  loge 
  /la, 
  

  

  PD 
  

  

  ^.2^=l_2/S.r^ 
  + 
  a.t'y. 
  

  

  The 
  frequencies 
  for 
  which 
  this 
  formula 
  will 
  ordinarily 
  be 
  

   required 
  lie 
  between 
  200 
  and 
  1600, 
  and 
  the 
  radii 
  between 
  1 
  

   and 
  3 
  millimetres. 
  These 
  values 
  cause 
  p 
  to 
  vary 
  between 
  

   about 
  13 
  and 
  16 
  only, 
  and 
  z 
  between 
  \ 
  and 
  unity, 
  so 
  that 
  

   P/CD 
  cannot 
  be 
  expanded 
  in 
  ascending 
  or 
  descending 
  powers 
  

   of 
  /3 
  or 
  2 
  in 
  general. 
  The 
  result 
  may 
  be 
  most 
  simply 
  written 
  

   in 
  the 
  form 
  

  

  L=41og^ 
  +1 
  ^ 
  + 
  %(l-2Pp,-l)l(i-4.Pp,+i^ph^) 
  

  

  4. 
  -flog- 
  . 
  (2«/J22_2ap<y.-/3^+2^7)/(l-4^p^ 
  + 
  4a/3V), 
  (43) 
  

  

  where 
  {aByz) 
  are 
  given 
  by 
  (39), 
  and 
  p 
  = 
  loge/ia. 
  

  

  The 
  two 
  final 
  terms 
  in 
  the 
  last 
  numerator 
  are 
  only 
  about 
  

   ^Q 
  of 
  the 
  other 
  two, 
  and 
  may 
  be 
  ignored. 
  

  

  * 
  Elec. 
  Papers, 
  ii. 
  d. 
  64. 
  

  

  t 
  Phil. 
  Mag. 
  xxii. 
  p. 
  381 
  (1886) 
  ; 
  Scientific 
  Papers, 
  vol. 
  ii, 
  

  

  