﻿of 
  Solving 
  Algebraical 
  Equations. 
  805 
  

  

  If 
  Ci, 
  C2, 
  ... 
  C« 
  be 
  the 
  values 
  in 
  amperes 
  of 
  the 
  currents 
  in 
  

   the 
  wires 
  and 
  H 
  the 
  horizontal 
  intensity 
  o£ 
  the 
  earth's 
  

   magnetic 
  field, 
  the 
  components 
  X 
  and 
  Y 
  of 
  the 
  resultant 
  

   magnetic 
  force 
  at 
  P(ci'i, 
  yi) 
  will 
  be 
  given 
  by 
  

  

  and 
  XT 
  . 
  ^1 
  ^i~^i 
  , 
  ^2 
  Xi-'h2 
  , 
  

  

  i 
  = 
  xl+ 
  -^ 
  . 
  -hT~' 
  f- 
  ..., 
  

  

  ori 
  ri 
  or 
  2 
  t^ 
  

  

  Avhere 
  rj 
  = 
  {x^ 
  -- 
  J„^)2 
  +^^2^ 
  

  

  Hence 
  multiplying 
  X 
  by 
  ^ 
  and 
  subtracting 
  we 
  get 
  

  

  Ci/5 
  C2/5 
  

  

  Y 
  + 
  X. 
  = 
  H+ 
  ^^0.,-6,-y. 
  + 
  ^, 
  (^i-&2"2/iO 
  + 
  ... 
  

  

  = 
  H 
  + 
  

  

  At 
  a 
  neutral 
  point 
  the 
  resultant 
  magnetic 
  force 
  is 
  zero, 
  and 
  

   therefore 
  both 
  X 
  and 
  Y 
  are 
  zero. 
  Hence, 
  if 
  x^ 
  and 
  y^ 
  are 
  

   the 
  coordinates 
  of 
  a 
  neutral 
  point, 
  iCi 
  + 
  z/it 
  is 
  a 
  root 
  of 
  the 
  

   equation 
  

  

  x-bi 
  x 
  — 
  b^ 
  

  

  Comparing 
  this 
  with 
  equation 
  (3) 
  we 
  see 
  that 
  if 
  we 
  adjust 
  

   the 
  values 
  of 
  the 
  currents 
  so 
  that 
  

  

  Ci 
  = 
  5H 
  . 
  Ai/aH, 
  C2= 
  5H 
  . 
  Ag/an, 
  . 
  . 
  . 
  

  

  a=5H.AJa„, 
  

  

  then 
  Xx-^y\i 
  is 
  a 
  root 
  of 
  the 
  equation 
  f 
  {x) 
  =0, 
  and 
  therefore 
  

   Xi 
  — 
  y^i 
  is 
  also 
  a 
  root. 
  

  

  It 
  follows 
  from 
  (4) 
  that 
  2C 
  = 
  0, 
  and 
  therefore 
  only 
  « 
  — 
  1 
  

   ammeters 
  and 
  only 
  n 
  — 
  \ 
  rheostats 
  are 
  required. 
  

  

  As 
  an 
  introduction 
  to 
  the 
  method 
  let 
  us 
  consider 
  the 
  theory 
  

   of 
  quadratic 
  equations. 
  

  

  3. 
  Quadratic 
  Equtions, 
  

   Let 
  us 
  suppose 
  that 
  the 
  equation 
  is 
  

  

  lujthis 
  case 
  it 
  is 
  convenient 
  to 
  write 
  

  

  x^ 
  + 
  bx-^-c 
  _^ 
  clh 
  c/b 
  

   x{x-\-b) 
  X 
  "^ 
  x-\-b' 
  

  

  