﻿Resistance 
  in 
  Telephone 
  and 
  other 
  Circuits. 
  423 
  

  

  whose 
  inductance, 
  capacity, 
  and 
  resistance 
  are 
  neglected. 
  

   Let 
  z 
  denote 
  distance 
  of 
  a 
  point 
  P 
  from 
  the 
  end 
  at 
  which 
  

   the 
  £or3e 
  acts. 
  If 
  R, 
  L, 
  C 
  be 
  the 
  resistance, 
  inductance^ 
  

  

  ^ 
  z 
  

   J 
  ^^ 
  

  

  and 
  capacity 
  per 
  unit 
  length, 
  and 
  (tj, 
  Y) 
  the 
  current 
  and 
  

   potential 
  at 
  a 
  point 
  P 
  ; 
  then 
  

  

  \ 
  

  

  (23) 
  

  

  (25) 
  

  

  Q'b'^[ht=--bH[hz 
  

  

  where 
  "bfbt 
  = 
  m. 
  

  

  The 
  solution 
  of 
  these 
  equations 
  is 
  found 
  to 
  be 
  

  

  i^ 
  = 
  (E/I)e'("^-), 
  . 
  . 
  • 
  . 
  . 
  (24) 
  

   where 
  

  

  I 
  = 
  L*C-*(l 
  + 
  R7^'L2)i(cosh2ZQi-cos2ZQ2)i 
  

  

  Qi, 
  Q2 
  = 
  n(iLC)M(l 
  + 
  RV^^L2)*+l}i 
  

  

  With 
  the 
  value 
  of 
  a 
  we 
  are 
  not 
  concerned. 
  For 
  the 
  effect 
  

   of 
  leakage, 
  reference 
  may 
  be 
  made 
  to 
  Heaviside's 
  paper*. 
  

  

  IfC 
  = 
  0, 
  I 
  = 
  /(R2 
  + 
  LV)^ 
  .... 
  (26) 
  

  

  which 
  is 
  the 
  ordinary 
  impedance 
  assumed 
  to 
  be 
  valid 
  in 
  the 
  

   previous 
  paper. 
  In 
  the 
  electrostatic 
  theory, 
  we 
  write 
  L 
  = 
  

   which 
  is 
  obviously 
  unjustifiable 
  if 
  R 
  is 
  small 
  or 
  n 
  very 
  great, 
  

   so 
  that 
  circuits 
  of 
  the 
  first 
  kind 
  lose 
  their 
  character 
  in 
  these' 
  

   cases. 
  

  

  We 
  proceed 
  to 
  estimate 
  the 
  correction 
  to 
  be 
  made 
  in 
  the 
  

   value 
  of 
  L, 
  to 
  take 
  account 
  of 
  a 
  small 
  capacity, 
  regarding 
  

   only 
  the 
  first 
  order 
  term. 
  Writing 
  

  

  V, 
  X/ 
  = 
  (l 
  + 
  R7n2L2)*+l, 
  

  

  then 
  on 
  reduction, 
  

  

  cosh 
  2ZQi- 
  cos 
  2ZQ2 
  = 
  LC7i2Z2(x,2^x/){l 
  + 
  4LOn2Z2(Xi2-.x/)} 
  

  

  = 
  2C?zZ2(R2 
  + 
  LV)^(l_lZ2^2LC), 
  

   so 
  that 
  

  

  I 
  = 
  /v/2 
  ^ 
  (R^ 
  + 
  LV)i(l-J^VLC). 
  

  

  * 
  Lqc. 
  cit. 
  

  

  2r2 
  

  

  