﻿302 
  Dr. 
  Russell 
  and 
  Mr. 
  Wright 
  : 
  The 
  Wright 
  Electrical 
  

  

  IX. 
  Equations 
  of 
  Degrees 
  higher 
  than 
  the 
  Third, 
  

  

  The 
  device 
  can 
  be 
  usefully 
  employed 
  to 
  find 
  the 
  values 
  o£ 
  

   the 
  roots 
  of 
  equations 
  higher 
  than 
  the 
  third 
  with 
  sufficient 
  

   accuracy 
  for 
  practical 
  work. 
  When 
  calculating, 
  for 
  example, 
  

   the 
  requisite 
  resistances 
  for 
  the 
  motor 
  controller 
  of 
  an 
  electric 
  

   car, 
  the 
  following 
  equation 
  * 
  has 
  to 
  be 
  solved 
  

  

  x'h'in-i) 
  + 
  aia! 
  — 
  ao 
  = 
  0, 
  

  

  where 
  n 
  is 
  the 
  number 
  of 
  steps 
  in 
  the 
  rheostat. 
  

  

  To 
  solve 
  this 
  equation 
  three 
  only 
  of 
  the 
  slides 
  shown 
  in 
  

   fig. 
  7 
  are 
  required. 
  AB 
  is 
  connected 
  with 
  Q3, 
  and 
  AiBi 
  and 
  

   A2B2 
  are 
  connected 
  with 
  P3. 
  The 
  finger 
  making 
  contact 
  

   with 
  A2B2 
  is 
  inclined 
  at 
  an 
  angle 
  tan~^?2/(?2 
  — 
  1 
  ) 
  with 
  II', 
  

   and 
  the 
  fingers 
  on 
  AiB^ 
  and 
  AB 
  are 
  inclined 
  at 
  angles 
  of 
  

   45° 
  and 
  0° 
  respectively. 
  The 
  value 
  of 
  .v 
  is 
  then 
  increased 
  

   until 
  a 
  balance 
  is 
  obtained. 
  

  

  When 
  solving 
  equations 
  containing 
  large 
  numbers 
  of 
  terms 
  

   it 
  is 
  sometimes 
  convenient 
  to 
  divide 
  the 
  equation 
  by 
  a 
  suitable 
  

   power 
  of 
  X. 
  In 
  this 
  case 
  the 
  fingers 
  making 
  contact 
  with 
  

   slide 
  resistances 
  representing 
  terms 
  containing 
  negative 
  powers 
  

   of 
  cT; 
  are 
  inclined 
  to 
  the 
  left 
  of 
  II' 
  (fig. 
  8). 
  

  

  Let 
  us 
  consider, 
  for 
  example, 
  a 
  sextic 
  equation. 
  As 
  shown 
  

   above, 
  we 
  alter 
  it 
  so 
  that 
  the 
  required 
  root 
  or 
  roots 
  are 
  

   multiples 
  or 
  submultiples 
  of 
  the 
  equation 
  

  

  .y^ 
  — 
  a-^.v^ 
  + 
  a^d^ 
  + 
  a^x^ 
  — 
  a'^'xP' 
  — 
  aiX 
  + 
  ao 
  = 
  0, 
  

  

  which 
  lie 
  between 
  1 
  and 
  10. 
  

  

  Dividing 
  this 
  equation 
  by 
  x^ 
  we 
  get 
  

  

  x^ 
  — 
  a^x^ 
  + 
  a^x 
  + 
  as 
  — 
  a2X 
  " 
  ^ 
  — 
  a^x'^ 
  + 
  aQX~^ 
  = 
  0. 
  

  

  In 
  general, 
  seven 
  slide 
  resistances 
  will 
  be 
  required, 
  but 
  if 
  

   when 
  X 
  is 
  put 
  equal 
  to 
  unity 
  any 
  term 
  is 
  less 
  than 
  the 
  

   hundredth 
  part 
  of 
  the 
  sum 
  of 
  the 
  terms 
  of 
  the 
  same 
  sign 
  

   preceding 
  it, 
  that 
  term 
  can 
  be 
  neglected. 
  The 
  slide 
  resis- 
  

   tances 
  are 
  first 
  moved 
  until 
  the 
  readings 
  on 
  the 
  index-line 
  

   are 
  ao, 
  «!, 
  ^2) 
  ••• 
  &c., 
  as 
  shown 
  in 
  fig. 
  <S. 
  The 
  contact-fingers 
  

   are 
  next 
  turned 
  round 
  to 
  the 
  requisite 
  angles 
  tan~^3, 
  tan~^2,. 
  . 
  . 
  

   tan 
  "^ 
  — 
  3, 
  respectively. 
  The 
  ends 
  of 
  the 
  contact-fingers 
  are 
  

   then 
  connected 
  with 
  suitable 
  points 
  on 
  the 
  arms 
  of 
  the 
  

   bridge. 
  

  

  At 
  first 
  sight 
  it 
  might 
  be 
  thought 
  thyt 
  two 
  extra 
  pairs 
  of 
  

   contact-points 
  P4, 
  Q4 
  and 
  P5, 
  Q5 
  would 
  be 
  required 
  in 
  the 
  

   bridge 
  arms. 
  If 
  a 
  one 
  per 
  cent, 
  inaccuracy, 
  however, 
  is 
  

  

  * 
  E. 
  Wilson, 
  ' 
  Electrical 
  Traction,' 
  vol. 
  i. 
  p. 
  42. 
  

  

  