﻿Measurement 
  by 
  Alternating 
  Currents. 
  67 
  

  

  minute 
  current 
  through 
  the 
  movable 
  coil, 
  thus 
  multiplying 
  

   the 
  sensitiveness 
  possibly 
  1000 
  times 
  over 
  the 
  zero 
  current 
  

   method. 
  

  

  I 
  have 
  also 
  found 
  that 
  many 
  of 
  the 
  methods 
  become 
  very 
  

   simple 
  if 
  we 
  use 
  mutual 
  inductances 
  made 
  of 
  wires 
  twisted 
  

   together 
  and 
  wound 
  into 
  coils. 
  In 
  this 
  way 
  the 
  self-induc- 
  

   tances 
  of 
  the 
  coils 
  are 
  all 
  practically 
  equal 
  and 
  the 
  mutual 
  

   inductances 
  of 
  pairs 
  of 
  coils 
  also 
  equal. 
  Hence 
  we 
  have 
  only 
  

   to 
  measure 
  the 
  minute 
  difference 
  of 
  these 
  two 
  to 
  reduce 
  the 
  

   constants 
  of 
  the 
  coil 
  to 
  one 
  constant, 
  and 
  yet 
  by 
  proper 
  con- 
  

   nexions 
  we 
  can 
  vary 
  the 
  inductances 
  in 
  many 
  ratios. 
  Three 
  

   wires 
  is 
  a 
  good 
  number 
  to 
  use. 
  However, 
  the 
  electrostatic 
  

   induction 
  between 
  the 
  wires 
  must 
  be 
  carefully 
  allowed 
  for 
  or 
  

   corrected 
  if 
  much 
  greater 
  accuracy 
  than 
  yqq 
  is 
  desired. 
  

  

  By 
  these 
  various 
  methods 
  the 
  measurement 
  of 
  capacities 
  

   and 
  inductances 
  has 
  been 
  made 
  as 
  easy 
  as 
  the 
  measurement 
  

   of 
  resistances, 
  while 
  the 
  accuracy 
  has 
  been 
  vastly 
  improved 
  

   and 
  many 
  sources 
  of 
  error 
  suggested. 
  

  

  Relative 
  results 
  are 
  more 
  accurate 
  than 
  absolute 
  as 
  the 
  

   period 
  of 
  an 
  alternating 
  current 
  is 
  difficult 
  to 
  determine, 
  and 
  

   its 
  wave-form 
  may 
  depart 
  from 
  a 
  true 
  sine-curve. 
  

  

  •Let 
  self-inductances, 
  mutual 
  inductances, 
  capacities, 
  and 
  

   resistances 
  be 
  designated 
  by 
  L 
  or 
  Z, 
  M 
  or 
  m, 
  C 
  or 
  c, 
  and 
  R 
  or 
  

   r 
  with 
  the 
  same 
  suffixes 
  when 
  they 
  apply 
  to 
  the 
  same 
  circuit, 
  

   the 
  mutual 
  inductance 
  having 
  two 
  suffixes. 
  Let 
  b 
  be 
  27r 
  

   times 
  the 
  number 
  of 
  complete 
  periods 
  per 
  second, 
  or 
  b 
  = 
  27rn. 
  

  

  The 
  quantities 
  6L, 
  bM 
  or 
  -77 
  are 
  of 
  the 
  dimensions 
  of 
  resist- 
  

  

  L 
  

   ance 
  and 
  thus 
  ^ 
  , 
  b 
  2 
  LG 
  or 
  5 
  2 
  MChave 
  no 
  dimensions. 
  /> 
  2 
  LM, 
  

  

  L 
  M 
  

  

  ~ 
  or 
  7— 
  have 
  dimensions 
  of 
  the 
  square 
  of 
  resistances. 
  

  

  Where 
  we 
  have 
  a 
  mutual 
  inductance 
  M 
  12 
  , 
  we 
  have 
  also 
  the 
  

   two 
  self-inductances 
  of 
  the 
  coils 
  1^ 
  and 
  L 
  2 
  . 
  When 
  these 
  coils 
  

   are 
  joined 
  in 
  the 
  two 
  possible 
  manners, 
  the 
  self-inductance 
  of 
  

   the 
  whole 
  is 
  

  

  L 
  1 
  + 
  L 
  S 
  + 
  2M 
  12 
  or 
  L 
  1 
  + 
  L 
  2 
  -2M 
  12 
  . 
  

  

  In 
  case 
  of 
  a 
  twisted 
  wire 
  coil 
  the 
  last 
  is 
  very 
  small. 
  Likewise 
  

   L|L 
  2 
  — 
  M 
  2 
  12 
  will 
  be 
  very 
  small 
  for 
  a 
  twisted 
  wire 
  coil, 
  as 
  is 
  

   found 
  by 
  multiplying 
  the 
  first 
  two 
  equations 
  together. 
  

  

  If 
  there 
  are 
  more 
  coils 
  we 
  can 
  write 
  similar 
  equations. 
  

   For 
  three 
  coils 
  we 
  have 
  

  

  L 
  x 
  + 
  L 
  2 
  + 
  L 
  3 
  + 
  2M 
  l2 
  + 
  2M 
  13 
  + 
  2M 
  23 
  

  

  1. 
  L 
  1 
  + 
  L 
  2 
  +L 
  3 
  -2M 
  18 
  -2M 
  13 
  + 
  2M 
  23 
  

  

  2. 
  L 
  1 
  + 
  L 
  2 
  + 
  L 
  3 
  -2M 
  12 
  + 
  2M 
  13 
  -2M 
  23 
  

  

  3. 
  L 
  x 
  + 
  L 
  2 
  + 
  L 
  3 
  + 
  2M 
  12 
  -2M 
  13 
  -2M 
  23 
  

  

  F 
  2 
  

  

  