﻿of 
  Solving 
  Algebraical 
  Equations. 
  807 
  

  

  respectively. 
  As 
  we 
  increase 
  the 
  value 
  of 
  c, 
  and 
  therefore 
  

   of 
  the 
  current 
  in 
  the 
  wires, 
  the 
  neutral 
  points 
  Ni 
  and 
  Ng 
  

   approach 
  one 
  another 
  and 
  coincide 
  at 
  P. 
  In 
  this 
  case 
  the 
  

   roots 
  are 
  each 
  equal 
  to 
  — 
  ?>/2. 
  For 
  greater 
  values 
  of 
  c 
  the 
  

   neutral 
  points 
  cannot 
  possibly 
  lie 
  on 
  BO 
  as 
  the 
  resultant 
  

   magnetic 
  force 
  due 
  to 
  the 
  currents 
  at 
  all 
  points 
  of 
  this 
  line 
  

   is 
  greater 
  than 
  H. 
  From 
  symmetry 
  the 
  neutral 
  points 
  lie 
  

   on 
  the 
  line 
  N1PN2 
  bisecting 
  OB 
  at 
  right 
  angles. 
  For 
  a 
  

   particular 
  value 
  of 
  the 
  current 
  they 
  are 
  at 
  the 
  points 
  Nj 
  and 
  

   N2 
  on 
  this 
  line, 
  [f 
  PNi 
  = 
  ?/i 
  then 
  PN2= 
  — 
  ?/i 
  and 
  the 
  cor- 
  

   responding 
  roots 
  of 
  the 
  equation 
  are 
  — 
  Z>/2 
  + 
  ?/i\/^^l 
  and 
  

   —hj'l 
  — 
  i/iV—l. 
  The 
  real 
  part 
  of 
  the 
  imaginary 
  roots 
  is 
  

   therefore 
  independent 
  of 
  c, 
  but 
  the 
  coefficient 
  of 
  \/— 
  -1 
  

   increases 
  as 
  c 
  increases. 
  

  

  When 
  c 
  is 
  negative 
  the 
  curved 
  arrow-heads 
  in 
  fig. 
  2 
  must 
  

   be 
  drawn 
  in 
  the 
  opposite 
  directions. 
  Ni 
  will 
  now 
  be 
  at 
  a 
  

   distance 
  <i'i 
  from 
  along 
  OX, 
  and 
  N2 
  will 
  be 
  at 
  a 
  distance 
  

   — 
  h 
  — 
  Xi 
  from 
  along 
  OX^ 
  The 
  roots 
  in 
  this 
  case, 
  there- 
  

   fore, 
  are 
  always 
  real 
  and 
  of 
  opposite 
  sign, 
  and 
  continually 
  

   increase 
  numerically 
  as 
  c 
  increases. 
  

  

  When 
  h 
  is 
  negative 
  the 
  point 
  B 
  in 
  fig. 
  2 
  will 
  be 
  to 
  the 
  

   right 
  of 
  OY, 
  and 
  the 
  discussion 
  of 
  the 
  roots 
  in 
  this 
  case 
  is 
  

   equally 
  simple. 
  

  

  -1. 
  The 
  Equation 
  to 
  Curves 
  passing 
  through 
  the 
  

   Neutral 
  Points. 
  

  

  From 
  the 
  preceding 
  discussion 
  it 
  will 
  be 
  seen 
  that 
  the 
  

   locus 
  of 
  the 
  real 
  roots 
  of 
  an 
  equation 
  determined 
  in 
  this 
  way 
  

   is 
  the 
  axis 
  of 
  x. 
  Fig. 
  2 
  shows 
  that 
  for 
  a 
  quadratic 
  equation 
  

   when 
  h 
  is 
  positive 
  the 
  locus 
  of 
  the 
  points 
  giving 
  the 
  imagi- 
  

   nary 
  roots 
  is 
  the 
  straight 
  line 
  Ni'N2'. 
  Similarly 
  when 
  h 
  is 
  

   negative 
  it 
  is 
  the 
  parallel 
  line 
  given 
  by 
  equation 
  x 
  = 
  hl2. 
  It 
  

   is 
  important 
  to 
  know 
  the 
  equations 
  to 
  curves 
  on 
  which 
  the 
  

   neutral 
  points 
  must 
  lie 
  in 
  the 
  general 
  case. 
  

  

  Let 
  us 
  suppose 
  that 
  x+yi 
  is 
  a 
  root 
  of 
  the 
  equation 
  

   /'(^) 
  = 
  0. 
  In 
  this 
  case 
  we 
  have 
  

  

  and, 
  therefore, 
  by 
  Taylor's 
  theorem, 
  

  

  /«-j|V"w+fjf'W-... 
  

  

  + 
  t2/|/'(^)-'J/"'(a;)+...}=0. 
  

  

  