﻿436 
  Method 
  of 
  comparing 
  the 
  Self- 
  Inductance 
  of 
  a 
  Coil. 
  

  

  Rearranging 
  equations 
  (8), 
  (9), 
  and 
  (10) 
  we 
  have 
  

  

  (B 
  + 
  P 
  + 
  Q)^-(P 
  + 
  Q)2/-Pi 
  =E, 
  . 
  . 
  . 
  (8') 
  

  

  -(P+Q)i;+(P 
  + 
  Q 
  + 
  R 
  + 
  S)2/ 
  + 
  (P4R)^=rw-Ly, 
  . 
  (90 
  

  

  -Pi;+(P 
  + 
  Il)^ 
  + 
  (R 
  + 
  G)i 
  =.-Xz.. 
  . 
  (10') 
  

  

  In 
  the 
  steady 
  state 
  when 
  w, 
  y, 
  and 
  z 
  are 
  each 
  =0, 
  the 
  

   current 
  z 
  will 
  be 
  =0 
  i£ 
  the 
  minor 
  of 
  E 
  in 
  the 
  determinant 
  

   for 
  z 
  is 
  zero, 
  that 
  is 
  if 
  P/Q=R/S. 
  

  

  Integrating 
  the 
  above 
  equations 
  with 
  respect 
  to 
  the 
  time 
  

   between 
  the 
  limit 
  0, 
  at 
  which 
  the 
  currents 
  and 
  quantities 
  are 
  

   zero, 
  and 
  the 
  time 
  ^i, 
  at 
  which 
  the 
  currents 
  are 
  steady 
  and 
  the 
  

   quantities 
  of 
  electricity 
  which 
  have 
  passed 
  have 
  attained 
  the 
  

   values 
  A'l, 
  yi, 
  Zi, 
  we 
  have 
  : 
  — 
  

  

  (B 
  + 
  P 
  + 
  QK-(P 
  + 
  Q)yi-P^^i 
  =pE^^, 
  . 
  (12) 
  

  

  Jo 
  

   -(P 
  + 
  Q)^i 
  + 
  (P 
  + 
  Q 
  + 
  R 
  + 
  S)3/i 
  + 
  (P 
  + 
  R)^i=(Kr2 
  -L)2/i, 
  (13) 
  

  

  ~P^i 
  + 
  (P 
  + 
  R)yi 
  + 
  (R 
  + 
  G)^i 
  =0. 
  . 
  . 
  . 
  (14) 
  

  

  From 
  the 
  second 
  of 
  which 
  the 
  relation 
  Ui 
  = 
  Kryi 
  given 
  by 
  

   (11) 
  has 
  been 
  used 
  to 
  eliminate 
  u^. 
  

  

  From 
  these 
  equations 
  it 
  is 
  seen 
  by 
  inspection 
  that 
  Zi 
  = 
  if 
  

  

  Kr^ 
  — 
  L 
  = 
  0, 
  and 
  the 
  minor 
  of 
  I 
  Eii 
  in 
  the 
  determinant 
  for 
  

  

  r 
  

  

  Zi 
  is 
  zero, 
  that 
  is 
  if 
  P/Q 
  = 
  R/S, 
  i. 
  e. 
  the 
  condition 
  for 
  a 
  steady 
  

   balance 
  previously 
  obtained. 
  

  

  Since 
  the 
  only 
  conditions 
  assumed 
  to 
  hold 
  in 
  the 
  above 
  

   proof 
  are 
  that 
  the 
  current 
  in 
  each 
  mesh 
  of 
  the 
  network 
  is 
  

   initially 
  zero 
  and 
  finally 
  steady^ 
  the 
  question 
  whether 
  

   oscillations 
  take 
  place 
  in 
  the 
  interval 
  or 
  not, 
  does 
  not 
  

   influence 
  the 
  result. 
  

  

  Whether 
  ti 
  can 
  be 
  so 
  chosen 
  that 
  the 
  currents 
  throughout 
  

   the 
  network 
  have 
  become 
  sufficiently 
  steady, 
  without 
  the 
  

   condition 
  that 
  there 
  has 
  been 
  no 
  motion 
  of 
  the 
  galvanometer- 
  

   needle 
  or 
  coil 
  during 
  that 
  time 
  being 
  violated, 
  is 
  quite 
  another 
  

   question. 
  With 
  a 
  modern 
  ballistic 
  galvanometer 
  of 
  the 
  type 
  

   recently 
  constructed 
  by 
  Prof. 
  B. 
  0. 
  Peirce 
  *, 
  of 
  Harvard^ 
  

   having 
  a 
  period 
  of 
  10 
  minutes, 
  there 
  will 
  be 
  very 
  few 
  cases 
  

   in 
  which 
  there 
  is 
  any 
  doubt 
  that 
  both 
  conditions 
  are 
  satisfied. 
  

  

  It 
  is 
  well 
  to 
  remember 
  that 
  even 
  then, 
  the 
  needle 
  or 
  coil 
  

   should 
  start 
  from 
  a 
  symmetrical 
  position 
  if 
  the 
  absence 
  of 
  

   reflexion 
  is 
  to 
  be 
  taken 
  as 
  a 
  proof 
  that 
  the 
  time 
  integral 
  of 
  the 
  

   current 
  throughout 
  the 
  instrument 
  is 
  zero 
  f. 
  

  

  * 
  B. 
  0. 
  Peirce, 
  Proc. 
  Amer. 
  Acad. 
  xliv. 
  p. 
  283 
  (1909). 
  

   t 
  A. 
  Russell, 
  Phil. 
  Mag. 
  xii. 
  p. 
  202 
  (1906). 
  

  

  