﻿Device 
  for 
  evaluating 
  Formula' 
  and 
  solving 
  Equations. 
  301 
  

  

  and 
  thus 
  x^ 
  -\- 
  a.jX^ 
  ^ 
  o^iX 
  -\- 
  a^ 
  = 
  ^ 
  . 
  Hence 
  x 
  is 
  one 
  of 
  the 
  roots 
  

   o£ 
  this 
  equation. 
  

  

  When 
  a 
  contact-finger, 
  Nj 
  for 
  instance, 
  reaches 
  the 
  end 
  

   of 
  its 
  slide 
  NjBi 
  will 
  be 
  R/10. 
  If 
  we 
  now 
  move 
  the 
  slide 
  

   AiBi 
  so 
  so 
  that 
  Aj 
  is 
  on 
  the 
  index-line, 
  unclamp 
  the 
  conkict- 
  

   finger 
  and 
  move 
  it 
  parallel 
  to 
  itself 
  until 
  Ni 
  is 
  over 
  Ai 
  and 
  

   then 
  reclamp 
  it, 
  the 
  resistance 
  of 
  NiBi 
  will 
  be 
  R. 
  If, 
  then, 
  

   we 
  chsconnect 
  the 
  finger 
  Ni 
  from 
  Q3 
  and 
  connect 
  it 
  with 
  Qg 
  

   the 
  deflexion 
  on 
  the 
  galvanometer 
  will 
  not 
  be 
  altered, 
  and 
  so 
  

   we 
  can 
  continue 
  to 
  increase 
  x. 
  It 
  will 
  be 
  seen 
  that 
  as 
  x 
  

   increases 
  from 
  1 
  to 
  10 
  the 
  contact-Hnger 
  !N"3 
  traverses 
  A3B3 
  

   three 
  times. 
  The 
  first 
  time 
  N3 
  gets 
  to 
  the 
  end 
  of 
  its 
  scale 
  it 
  

   is 
  connected 
  wirh 
  Pg 
  and 
  moved 
  back 
  to 
  A3. 
  The 
  second 
  

   time 
  it 
  reaches 
  the 
  end 
  it 
  is 
  connected 
  with 
  Pi 
  and 
  again 
  

   moved 
  back. 
  

  

  The 
  de^dce 
  enables 
  us 
  to 
  find 
  any 
  real 
  root 
  of 
  the 
  equation, 
  

   greater 
  than 
  10'' 
  and 
  less 
  than 
  10"+^ 
  All 
  we 
  have 
  to 
  do 
  is 
  

   to 
  find 
  the 
  root 
  of 
  the 
  equation 
  

  

  x^ 
  + 
  (^2/10").!'^ 
  - 
  (ai/10-^")^^' 
  + 
  «o/10-"'^ 
  = 
  0, 
  

  

  lying 
  between 
  1 
  and 
  10, 
  and 
  multiply 
  the 
  result 
  by 
  10''. 
  

   By 
  solving 
  a 
  similar 
  subsidiary 
  equation 
  we 
  can 
  find 
  the 
  

   approximate 
  values 
  of 
  the 
  roots 
  of 
  the 
  equation 
  which 
  are 
  

   less 
  than 
  unity. 
  

  

  AVe 
  may 
  use 
  ^Newton's 
  rule 
  to 
  find 
  more 
  accurate 
  values 
  of 
  

   the 
  roots. 
  If 
  a, 
  for 
  instance, 
  be 
  an 
  approximate 
  value 
  of 
  a 
  

   real 
  root 
  olf{x)=0,a—f{a)/f'{a) 
  gives 
  usually 
  a 
  very 
  much 
  

   closer 
  approximation 
  x^. 
  The 
  failing 
  case 
  is 
  when 
  we 
  have 
  

   two 
  roots 
  very 
  nearly 
  equal 
  to 
  one 
  another. 
  The 
  device 
  

   always 
  indicates 
  when 
  this 
  occurs. 
  If 
  two 
  roots 
  are 
  each 
  

   equal 
  to 
  a, 
  or 
  if 
  they 
  are 
  approximately 
  equal 
  to 
  a, 
  the 
  gjalva- 
  

   nometer 
  deflexion 
  instead 
  of 
  passing 
  to 
  the 
  other 
  side 
  of 
  zero 
  

   when 
  X 
  becomes 
  greater 
  than 
  a 
  returns 
  to 
  the 
  same 
  side. 
  

  

  In 
  practice 
  ii 
  f(a) 
  is 
  large 
  when 
  /(a) 
  is 
  zero 
  the 
  device 
  is 
  

   very 
  sensitive, 
  but 
  when 
  f'(a) 
  is 
  small 
  and 
  in 
  general, 
  there- 
  

   fore, 
  when 
  the 
  roots 
  are 
  equal 
  the 
  device 
  is 
  unsensitive. 
  

  

  To 
  find 
  approximate 
  values 
  of 
  the 
  negative 
  roots 
  of 
  the 
  

   original 
  equation 
  we 
  find 
  by 
  the 
  device 
  the 
  roots 
  of 
  

  

  c^^ 
  ~ 
  a^x'^ 
  — 
  a^x 
  — 
  Aq 
  = 
  . 
  

  

  The 
  imaginary 
  roots* 
  are 
  most 
  readily 
  found 
  by 
  first 
  

   finding 
  as 
  accurate 
  a 
  value 
  Xi 
  as 
  possible 
  of 
  the 
  real 
  root. 
  

   We 
  then 
  divide 
  the 
  cubic 
  expression 
  by 
  x—Xi 
  and 
  equate 
  

   the 
  dividend 
  to 
  zero. 
  The 
  roots 
  of 
  this 
  quadratic 
  equation 
  

   give 
  the 
  approximate 
  values 
  required. 
  

  

  * 
  Cf. 
  C. 
  P. 
  Steinmetz, 
  'Transient 
  Electric 
  riienomena/ 
  p. 
  13G. 
  

  

  