﻿Resistance 
  in 
  Telephone 
  and 
  other 
  Circuits. 
  421 
  

  

  and 
  after 
  reduction 
  

  

  AE-BF 
  

  

  /, 
  3 
  5 
  3 
  \, 
  a, 
  , 
  

  

  We 
  proceed 
  to 
  examine 
  the 
  limiting 
  frequency 
  for 
  which 
  

   the 
  fonr-figure 
  accuracy 
  of 
  these 
  results 
  is 
  preserved. 
  For 
  

   a 
  copper 
  wire 
  of 
  low 
  resistance 
  as 
  ordinarily 
  used, 
  the 
  

   resistivity 
  may 
  be 
  taken 
  as 
  1696 
  C.G.S. 
  If/be 
  the 
  frequency 
  

  

  of 
  alternation, 
  71 
  = 
  271/, 
  leading 
  to 
  x= 
  ^ap 
  approximately. 
  

  

  Thus 
  <r 
  = 
  8 
  leads 
  to 
  the 
  determining 
  relation 
  

  

  a/^ 
  = 
  40. 
  

  

  I 
  

  

  (22) 
  

  

  If 
  a/^ 
  is 
  less 
  than 
  40, 
  the 
  four-figure 
  accuracy, 
  in 
  so 
  far 
  

   as 
  it 
  depends 
  on 
  x^ 
  is 
  lost. 
  When 
  /' 
  is 
  only 
  400 
  per 
  second, 
  

   the 
  limiting 
  diameter 
  of 
  a 
  wire 
  becomes 
  so 
  large 
  as 
  4 
  centi- 
  

   metres,, 
  so 
  that 
  the 
  formulae 
  are 
  unsuited 
  for 
  practice 
  until 
  

   the 
  frequency 
  is 
  really 
  great. 
  The 
  usual 
  range 
  of 
  diameter 
  

   for 
  wires 
  applied 
  to 
  such 
  purposes 
  as 
  telephonic 
  communi- 
  

   cation 
  is 
  from 
  1 
  to 
  3 
  millimetres 
  in 
  the 
  case 
  of 
  cables, 
  and 
  to 
  

   5*7 
  millimetres 
  in 
  that 
  of 
  overhead 
  wires. 
  The 
  frequency 
  in 
  

   cases 
  of 
  transmission 
  of 
  speech 
  will 
  not 
  be 
  greater 
  than 
  2000. 
  

   Taking 
  /=1600 
  as 
  perhaps 
  the 
  real 
  maximum, 
  the 
  limiting 
  

   radius 
  of 
  a 
  wire 
  becomes 
  1 
  centimetre. 
  The 
  results 
  there- 
  

   fore 
  cannot 
  have 
  a 
  great 
  accuracy 
  in 
  such 
  a 
  case, 
  and 
  the 
  

   formula 
  (43) 
  below 
  must 
  be 
  used. 
  But 
  it 
  will 
  appear 
  that 
  

   the 
  corresponding 
  formula 
  for 
  iron 
  leads 
  can 
  be 
  so 
  employed. 
  

   The 
  present 
  results 
  have 
  a 
  two-figure 
  accuracy 
  even 
  when 
  

   a? 
  = 
  3, 
  leading 
  to 
  a 
  radius 
  of 
  3' 
  75 
  millimetres, 
  and 
  therefore 
  

   will 
  apply 
  to 
  many 
  cases 
  even 
  of 
  telephony 
  when 
  thicker 
  

   wires 
  are 
  used. 
  In 
  such 
  cases, 
  their 
  use 
  is 
  preferable 
  to 
  

   that 
  of 
  (43), 
  in 
  that 
  the 
  calculation 
  is 
  more 
  rapid. 
  

  

  It 
  is 
  necessary 
  not 
  to 
  overlook 
  the 
  other 
  sources 
  of 
  error. 
  

   Firstly, 
  in 
  the 
  proof 
  of 
  (1) 
  ■^, 
  AV 
  was 
  neglected 
  in 
  com- 
  

  

  * 
  Phil. 
  Mag. 
  Feb. 
  1909. 
  

   FUh 
  Mag. 
  S. 
  6. 
  Vol. 
  18. 
  No. 
  105. 
  Sept. 
  1909. 
  2 
  F 
  

  

  