﻿Resistance 
  in 
  Teleplione 
  and 
  other 
  Circuits. 
  419 
  

  

  from 
  which 
  may 
  be 
  deduced 
  

  

  ber^ 
  X 
  bei^' 
  x 
  — 
  bei' 
  x 
  ber" 
  x 
  = 
  ber 
  x 
  ber 
  ' 
  x 
  + 
  bei 
  x 
  bei' 
  <r, 
  

   ber' 
  X 
  ber" 
  ci'+ 
  bei^ 
  x 
  bei" 
  <2? 
  = 
  ber 
  x 
  bei' 
  ^— 
  bei 
  x 
  ber' 
  .2? 
  

  

  -.t'-i(ber'^)2-a7-i(bei'^')', 
  }' 
  (9) 
  

   (a?ber".r)2+(^bei''a?)2=(ber'a^)2+ 
  (bei'A')H 
  (a^ber^)^ 
  j 
  

   -f 
  (x 
  bei 
  a?) 
  2 
  — 
  2<r(ber 
  x 
  bei' 
  <r 
  — 
  bei 
  ^ 
  ber^ 
  .2?) 
  . 
  J 
  

   IV^riting 
  

  

  P=ber2a7 
  + 
  bei2^, 
  ^ 
  

  

  aP=(ber^<r)2 
  + 
  (bei'^)2, 
  I 
  

  

  yiSP=berA'ber'^+ 
  bei.:cbei'a7, 
  I 
  ' 
  ' 
  " 
  ^^ 
  

   7P 
  = 
  ber 
  a^bei'a?— 
  beiicber'^. 
  J 
  

   Then 
  E^7P 
  = 
  a(l-/.2) 
  +^,2_27^, 
  

  

  F^/P 
  = 
  2/ty8, 
  

   Dx^lV 
  = 
  ct(l-iJLY 
  + 
  x^-2x^{l-fi), 
  

  

  "^^^^^ 
  DXi==E 
  + 
  .F 
  (11) 
  

  

  For 
  the 
  special 
  case 
  //-=!; 
  D 
  = 
  P, 
  so 
  that 
  

  

  Xi 
  = 
  l-2(7-./3)/a-, 
  (12) 
  

  

  u 
  very 
  simple 
  result, 
  which 
  is 
  true 
  in 
  practice 
  for 
  all 
  but 
  

   iron 
  wires. 
  Again, 
  on 
  reduction, 
  

  

  log 
  ajc 
  — 
  fi^Jka^Q 
  = 
  log 
  a/c 
  — 
  //,(/8 
  — 
  uy) 
  lax, 
  

  

  log 
  ha 
  — 
  /jiJo/hajQ 
  = 
  log 
  ha 
  —fjb(^^-~ 
  cy^/ax, 
  

  

  and 
  their 
  quotient 
  becomes 
  

  

  (A 
  + 
  .B)/C, 
  

   i,vhere 
  

  

  x^A=(xx^ 
  log 
  ha 
  . 
  log 
  a/c—fi^x 
  log 
  7ia7c 
  + 
  />t^,l 
  

   ^B=/LtylogAc, 
  >. 
  (13) 
  

  

  x^C 
  = 
  a.2?^ 
  log^ 
  Aa 
  — 
  2fjLfix 
  log 
  Zia 
  + 
  //,^, 
  J 
  

  

  and 
  it 
  has 
  been 
  noticed 
  that 
  ^^ 
  + 
  y^ 
  = 
  tx. 
  identically. 
  

  

  With 
  these 
  values, 
  it 
  is 
  found 
  that, 
  M 
  and 
  N 
  being 
  given 
  

   by 
  (3), 
  

  

  c2CD(M 
  + 
  tX) 
  = 
  4a2(E 
  + 
  tF)(AH-tB). 
  

  

  Thus 
  M 
  = 
  4a2(AE-BF)/c2CD,-i 
  

  

  N 
  = 
  4a2(AF 
  + 
  BE)/c2CDJ 
  * 
  * 
  ' 
  ' 
  ^^^^ 
  

  

  