﻿304 
  Dr. 
  Russell 
  and 
  Mr. 
  Wright 
  : 
  The 
  Wright 
  Electrical 
  

  

  on 
  the 
  slide 
  resistance 
  K"'W' 
  to 
  10. 
  When 
  the 
  fingers 
  on 
  

   the 
  resistances 
  A2B2 
  and 
  A3B3 
  get 
  to 
  the 
  ends 
  of 
  their 
  scales 
  

   we 
  connect 
  them 
  with 
  Pj 
  and 
  Qi 
  respectively. 
  When, 
  how- 
  

   ever, 
  the 
  finger 
  on 
  the 
  resistance 
  A3B3 
  gets 
  for 
  the 
  second 
  

   time 
  to 
  the 
  end 
  o£ 
  its 
  scale 
  we 
  disconnect 
  altogether 
  the 
  wires 
  

   connected 
  with 
  P3, 
  and 
  move 
  the 
  wire 
  connected 
  with 
  Pg 
  to 
  

   P3 
  and 
  the 
  wire 
  connected 
  with 
  Qi 
  to 
  Q2. 
  In 
  working 
  the 
  

   device 
  these 
  operations 
  seem 
  quite 
  natural 
  and 
  little 
  thinking 
  

   is 
  required. 
  It 
  is 
  also 
  easy 
  to 
  see 
  that 
  if 
  a 
  one 
  per 
  cent, 
  

   inaccuracy 
  is 
  permissible 
  it 
  is 
  unnecessary 
  to 
  have 
  more 
  than 
  

   three 
  pairs 
  of 
  terminals 
  on 
  the 
  bridge 
  arms. 
  

  

  X. 
  Imaginary 
  Roots. 
  

  

  In 
  solving 
  certain 
  engineering 
  problems 
  in 
  connexion 
  with 
  

   finding 
  the 
  amplitudes, 
  the 
  damping 
  factors, 
  and 
  the 
  periods 
  

   of 
  certain 
  mechanical 
  and 
  electrical 
  oscillations, 
  a 
  necessary 
  

   step 
  is 
  finding 
  the 
  imaginary 
  roots 
  of 
  certain 
  algebraic 
  

   equations. 
  Quadratic 
  equations 
  present 
  no 
  difficulty, 
  and 
  we 
  

   have 
  already 
  shown 
  how 
  approximate 
  values 
  of 
  the 
  imaginary 
  

   roots 
  of 
  cubic 
  equations 
  can 
  be 
  found. 
  

  

  With 
  the 
  biquadratic 
  equations, 
  however, 
  which 
  occur 
  

   when 
  discussing 
  the 
  theory 
  of 
  the 
  parallel 
  running 
  of 
  alter- 
  

   nators*, 
  the 
  oscillations 
  set 
  up 
  in 
  coupled 
  electric 
  circuits 
  in 
  

   wireless 
  telegraphy 
  f, 
  &c., 
  both 
  pairs 
  of 
  roots 
  are 
  sometimes 
  

   imaginary. 
  In 
  this 
  case 
  we 
  proceed 
  as 
  follows: 
  — 
  Let 
  x-{-yL 
  

   be 
  a 
  root 
  of 
  the 
  equation 
  f(z)=0. 
  Then 
  f 
  {a;-\-yi) 
  = 
  0^ 
  and 
  

   hence, 
  expanding 
  by 
  Taylor^s 
  theorem, 
  we 
  have 
  

  

  /(■^•)-^V"«+ 
  - 
  +'2/{/' 
  (.^•)-|^/"' 
  W+ 
  .■•}=0, 
  

   and 
  thus 
  we 
  must 
  have 
  

  

  /G^•)-frW+|^rG^0 
  = 
  O...(a) 
  

  

  and 
  /•/(^^)_|^////(^^,) 
  = 
  0...(6) 
  

  

  From 
  (b) 
  we 
  get 
  y^ 
  in 
  terms 
  of 
  x, 
  and 
  substituting 
  this 
  

   value 
  of 
  y^ 
  in 
  (a) 
  we 
  get 
  an 
  equation 
  of 
  the 
  sixth 
  degree 
  to 
  

   find 
  a.'. 
  The 
  two 
  real 
  roots 
  of 
  this 
  equation 
  can 
  be 
  found 
  by 
  

   the 
  machine 
  arranged 
  in 
  the 
  manner 
  shown 
  in 
  fig. 
  8, 
  and 
  the 
  

   pairs 
  of 
  corresponding 
  values 
  of 
  y 
  are 
  given 
  at 
  once 
  by 
  (b). 
  

   Approximate 
  values 
  of 
  the 
  four 
  imaginary 
  roots 
  can 
  thus 
  be 
  

   rapidly 
  found. 
  If 
  a 
  higher 
  degree 
  of 
  accuracy 
  be 
  required 
  

   we 
  can 
  either 
  use 
  Newton's 
  method 
  of 
  approximation, 
  or 
  better 
  

   apply 
  Horner's 
  method 
  to 
  the 
  auxiliary 
  sextic. 
  

  

  * 
  A. 
  Russell, 
  ' 
  Alternating 
  Currents/ 
  vol. 
  ii. 
  p. 
  184. 
  

   t 
  J. 
  A. 
  Fleming-, 
  ' 
  Electric 
  Wave 
  Telegraphy/ 
  p. 
  209. 
  

  

  