﻿Electromagnetic 
  Method 
  of 
  Solving 
  Algebraical 
  Equations. 
  803 
  

  

  the 
  same 
  as 
  in 
  the 
  method 
  described 
  below. 
  The 
  equi- 
  

   potential 
  lines 
  are 
  then 
  traced 
  out 
  either 
  by 
  an 
  electro- 
  

   chemical 
  method 
  or 
  by 
  the 
  Kirchhoff 
  and 
  Carey 
  Foster 
  

   method. 
  If 
  the 
  line 
  of 
  sources 
  and 
  sinks 
  be 
  taken 
  as 
  the 
  

   axis 
  o£ 
  cV, 
  and 
  the 
  origin 
  be 
  chosen 
  midway 
  between 
  the 
  

   outer 
  wires, 
  the 
  coordinates 
  of 
  the 
  nodal 
  points, 
  that 
  is, 
  the 
  

   points 
  where 
  an 
  equipotential 
  line 
  intersects 
  itself, 
  enable 
  

   us 
  to 
  write 
  down 
  all 
  the 
  roots, 
  real 
  or 
  imaginary, 
  of 
  the 
  

   given 
  equation 
  at 
  once. 
  If 
  {xi^ 
  i/i) 
  be 
  the 
  coordinates 
  of 
  a 
  

   nodal 
  point, 
  we 
  can 
  see 
  from 
  the 
  symmetry 
  of 
  the 
  arrange- 
  

   ment 
  that 
  (.^1, 
  — 
  ?/i) 
  will 
  be 
  the 
  coordinates 
  of 
  another 
  nodal 
  

   point. 
  It 
  follows 
  from 
  the 
  method 
  of 
  adjusting 
  the 
  currents 
  

   that 
  ^^'i 
  + 
  ?/i'^ 
  — 
  1 
  gives 
  a 
  pair 
  of 
  conjugate 
  roots 
  of 
  the 
  given 
  

   equation. 
  The 
  real 
  roots, 
  therefore, 
  are 
  all 
  given 
  by 
  the 
  

   abscissas 
  of 
  the 
  nodal 
  points 
  lying 
  on 
  the 
  axis 
  of 
  X. 
  The 
  

   method 
  is 
  not 
  rigorous, 
  as 
  Mr. 
  Lucas 
  has 
  not 
  considered 
  the 
  

   magnitude 
  of 
  the 
  error 
  introduced 
  owing 
  to 
  the 
  finite 
  size 
  of 
  

   the 
  conducting 
  sheet. 
  It 
  would 
  be 
  very 
  laborious 
  to 
  apply 
  

   in 
  practice. 
  In 
  solving 
  a 
  biquadratic, 
  for 
  instance, 
  the 
  

   currents 
  in 
  five 
  wires 
  would 
  have 
  to 
  be 
  adjusted 
  to 
  given 
  

   values 
  before 
  the 
  equipotential 
  lines 
  could 
  be 
  mapped 
  out. 
  

  

  We 
  shall 
  now 
  describe 
  an 
  electromagnetic 
  method, 
  in 
  which 
  

   the 
  horizontal 
  field 
  due 
  to 
  the 
  earth^s 
  magnetism 
  is 
  used 
  in 
  

   an 
  analogous 
  manner 
  to 
  the 
  conducting 
  sheet 
  in 
  Mr. 
  Lucas's 
  

   method. 
  A 
  drawing-board 
  with 
  a 
  slit 
  cut 
  in 
  it, 
  a 
  few 
  pieces 
  

   of 
  bell-wire, 
  any 
  form 
  of 
  " 
  charm 
  ^'' 
  compass, 
  ordinary 
  

   ammeters 
  and 
  rheostats 
  or 
  lamp-resistance 
  boards 
  such 
  as 
  

   are 
  found 
  in 
  every 
  physical 
  laboratory 
  can 
  be 
  utilized 
  at 
  

   once 
  for 
  the 
  experiment 
  (see 
  § 
  7). 
  

  

  2. 
  The 
  Electromagnetic 
  Method. 
  

  

  Let 
  us 
  suppose 
  that 
  we 
  have 
  to 
  find 
  the 
  roots 
  for 
  the 
  

   equation 
  

  

  f{x)^^aax" 
  + 
  an-ix''-^+... 
  +ao 
  = 
  0. 
  .. 
  • 
  (1) 
  

  

  We 
  first, 
  by 
  the 
  ordinary 
  methods 
  given 
  in 
  books 
  on 
  

   algebra, 
  resolve 
  the 
  expression 
  

  

  IM 
  

  

  lcc—hy){X'-b^) 
  ... 
  {x 
  — 
  hn) 
  

  

  into 
  partial 
  fractions. 
  The 
  numbers 
  6,, 
  1)2^ 
  ... 
  hn 
  are 
  any 
  

   convenient 
  numbers 
  so 
  chosen, 
  however, 
  that 
  

  

  hi 
  + 
  h2+ 
  ... 
  '\-hn='-an-\lan 
  (2) 
  

  

  For 
  example, 
  if 
  the 
  second 
  term 
  of 
  f 
  [a") 
  be 
  missing 
  we 
  

   must 
  choose 
  6i, 
  62, 
  ... 
  so 
  that 
  2/^ 
  = 
  0. 
  

  

  