﻿of 
  Soldng 
  Algebraical 
  Equations, 
  

  

  809 
  

  

  small, 
  two 
  neutral 
  points 
  No 
  and 
  N3 
  obviously 
  lie 
  between 
  

   and 
  B3 
  and 
  a 
  neutral 
  point 
  N"i 
  lies 
  along 
  B^X'. 
  

  

  H 
  

  

  N^^s 
  

  

  @- 
  

  

  Fig. 
  3 
  

   Y 
  

  

  ^N. 
  

  

  X' 
  N/ 
  

  

  / 
  

  

  / 
  

  

  / 
  

  

  X 
  

  

  As 
  the 
  current 
  increases 
  the 
  neutral 
  points 
  N2 
  and 
  N3 
  approach 
  

   one 
  another 
  and 
  the 
  neutral 
  point 
  N"/ 
  moves 
  along 
  BjX^ 
  For 
  

   a 
  particular 
  value 
  o£ 
  c, 
  No 
  and 
  N3 
  coincide 
  at 
  P, 
  and 
  we 
  have 
  

   two 
  equal 
  roots. 
  For 
  greater 
  values 
  of 
  c 
  the 
  neutral 
  points 
  

   N2 
  and 
  N3 
  move 
  uniformly 
  along 
  the 
  curve 
  Ng'PNs', 
  one 
  

   being 
  as 
  much 
  above 
  the 
  line 
  XX' 
  as 
  the 
  other 
  is 
  below 
  it. 
  

   But 
  the 
  neutral 
  point 
  N, 
  always 
  moves 
  along 
  the 
  axis 
  of 
  X. 
  

  

  The 
  equation 
  (12), 
  therefore, 
  has 
  always 
  three 
  roots. 
  

   When 
  c 
  is 
  small 
  they 
  are 
  all 
  real, 
  two 
  of 
  them 
  being 
  positive 
  

   and 
  one 
  negative. 
  For 
  a 
  certain 
  value 
  of 
  c 
  the 
  two 
  positive 
  

   roots 
  become 
  equal, 
  and 
  for 
  greater 
  values 
  of 
  c 
  we 
  have 
  two 
  

   conjugate 
  imaginary 
  roots 
  and 
  one 
  real 
  negative 
  root. 
  Both 
  

   the 
  real 
  and 
  imaginary 
  parts 
  of 
  the 
  conjugate 
  roots 
  continually 
  

   increase 
  as 
  c 
  increases. 
  

  

  The 
  equation 
  to 
  the 
  curve 
  Ng'PNg' 
  in 
  fig. 
  3 
  can 
  be 
  found 
  

   at 
  once 
  from 
  (11) 
  ; 
  it 
  is 
  

  

  ^x'^-f 
  = 
  h' 
  (14) 
  

  

  The 
  negative 
  branch 
  of 
  this 
  hyperbola 
  indicated 
  by 
  the 
  

   dotted 
  curve 
  in 
  fig. 
  3 
  gives 
  the 
  locus 
  of 
  the 
  neutral 
  points 
  

   when 
  c 
  is 
  negative. 
  In 
  this 
  case 
  there 
  is 
  obviously 
  always 
  

   one 
  real 
  positive 
  root. 
  Increasing 
  the 
  value 
  of 
  b 
  in 
  (12) 
  is 
  

   equivalent 
  to 
  putting 
  the 
  wires 
  B^ 
  and 
  B3 
  further 
  apart. 
  

   This 
  obviously 
  increases 
  the 
  limits 
  of 
  the 
  values 
  of 
  the 
  

   Fhil. 
  Mag. 
  S. 
  6. 
  Vol. 
  18. 
  No. 
  107. 
  ^^ov. 
  1909. 
  3 
  H 
  

  

  