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  XLIX. 
  Inductance 
  and 
  Resistance 
  in 
  Telephone 
  and 
  other 
  

   Circuits. 
  By 
  J. 
  W. 
  Nicholson, 
  M.A,, 
  D.Sc. 
  ; 
  Trinity 
  

   College, 
  Cambridge*, 
  

  

  I. 
  Effective 
  Inductance. 
  

  

  A 
  GENERAL 
  formula 
  for 
  the 
  effective 
  inductance 
  of 
  a 
  

   circuit 
  consisting 
  of 
  two 
  long 
  parallel 
  wires 
  has 
  been 
  

   given 
  by 
  the 
  author 
  f, 
  and 
  is 
  suitable 
  for 
  cases 
  in 
  which 
  the 
  

   current 
  distribution 
  in 
  either 
  wire 
  is 
  greatly 
  affected 
  by 
  the 
  

   frequency 
  of 
  alternation. 
  In 
  its 
  general 
  form, 
  although 
  its 
  

   limitations 
  are 
  clearly 
  defined, 
  the 
  result 
  is 
  not 
  well 
  adapted, 
  

   in 
  the 
  absence 
  of 
  tables, 
  to 
  rapid 
  calculation. 
  The 
  main 
  object 
  

   of 
  the 
  present 
  paper 
  is 
  to 
  examine 
  certain 
  important 
  cases 
  in 
  

   detail, 
  and 
  to 
  obtain 
  formulae 
  capable 
  of 
  immediate 
  use. 
  A 
  

   calculation 
  of 
  the 
  effective 
  resistance 
  is 
  also 
  made 
  in 
  each 
  

   case. 
  A 
  problem 
  to 
  which 
  attention 
  has 
  been 
  mainly 
  

   directed, 
  which 
  includes 
  several 
  practical 
  cases 
  of 
  great 
  

   interest, 
  is 
  that 
  of 
  the 
  simple 
  telephone 
  circuit, 
  in 
  which 
  the 
  

   leads 
  are 
  not 
  twisted 
  round 
  each 
  other 
  in 
  order 
  to 
  annul 
  the 
  

   inductive 
  effects 
  of 
  the 
  earth 
  and 
  of 
  neio'hbourino; 
  circuits. 
  

  

  In 
  the 
  proof 
  of 
  the 
  general 
  formula, 
  the 
  influence 
  of 
  

   electrostatic 
  capacity 
  was 
  ignored. 
  This 
  imposes 
  a 
  great 
  

   limitation 
  upon 
  the 
  types 
  of 
  circuit 
  for 
  which 
  the 
  expression 
  

   may 
  be 
  used. 
  An 
  estimation 
  of 
  the 
  maximum 
  capacity 
  

   causing 
  no 
  alteration 
  to 
  a 
  given 
  order 
  of 
  accuracy 
  is 
  given 
  

   in 
  this 
  paper. 
  

  

  Throughout 
  the 
  investigation, 
  only 
  iron 
  and 
  copper 
  wires, 
  

   as 
  the 
  two 
  extreme 
  cases, 
  are 
  considered. 
  The 
  large 
  perme- 
  

   ability 
  of 
  iron 
  completely 
  changes 
  the 
  character 
  of 
  the 
  effect 
  

   of 
  frequency 
  on 
  its 
  self-induction, 
  as 
  compared 
  with 
  other 
  

   metals. 
  To 
  all 
  metals 
  except 
  iron 
  greatly 
  used 
  in 
  practice, 
  

   the 
  formulae 
  developed 
  for 
  copper 
  wires 
  may 
  be 
  applied 
  with 
  

   a 
  nearly 
  identical 
  order 
  of 
  accuracy. 
  

  

  Let 
  a 
  be 
  the 
  radius 
  of 
  either 
  wire, 
  c 
  their 
  distance 
  apart, 
  

   and 
  (//,, 
  a) 
  their 
  permeability 
  and 
  resistivity. 
  They 
  are 
  equal 
  

   in 
  all 
  respects. 
  

  

  The 
  value 
  of 
  the 
  inductance 
  per 
  unit 
  length 
  is 
  then 
  

  

  T 
  ., 
  c 
  4yLt 
  ber 
  a;ber'.2?+ 
  beiojbei^^ 
  ,, 
  ,^. 
  

  

  ^ 
  = 
  ^'°ga 
  + 
  T- 
  (ber-.-/+(bei'.0^ 
  +^' 
  ' 
  ^^^ 
  

   where, 
  if 
  n/27r 
  be 
  the 
  frequency, 
  

  

  JC='2a(7rfjLn/a)% 
  (2) 
  

  

  * 
  Communicated 
  by 
  the 
  Physical 
  Society 
  : 
  read 
  June 
  11, 
  1909. 
  

   t 
  Phil. 
  Mag. 
  Feb. 
  1909 
  ; 
  Proc. 
  Phys. 
  Soc. 
  vol. 
  xxi. 
  

  

  