﻿Device 
  for 
  evaluating 
  Fornniltu 
  and 
  solving 
  Equations. 
  30? 
  

  

  In 
  this 
  case 
  (fig. 
  10) 
  the 
  scale 
  S 
  is 
  an 
  ordinary 
  scale, 
  the 
  

   length 
  KL 
  being 
  equal 
  to 
  lix. 
  The 
  various 
  contact-fingers 
  

   are 
  adjusted 
  so 
  that 
  the 
  angles 
  they 
  make 
  with 
  the 
  index- 
  

   line 
  ir 
  are 
  tan-i 
  (^ilogji), 
  tan-i 
  (63 
  log 
  Q, 
  tan-i 
  (^3 
  log 
  ^ 
  

   and 
  45^ 
  respectively. 
  We 
  connect 
  the 
  contact-fingers 
  as 
  in 
  

   fio-. 
  10 
  and 
  vary 
  x. 
  The 
  values 
  of 
  x 
  for 
  which 
  there 
  is 
  no 
  

   deflexion 
  on 
  the 
  galvanometer 
  are 
  the 
  roots 
  of 
  the 
  given 
  

   equation 
  lying 
  between 
  1 
  and 
  10. 
  By 
  altering 
  the 
  given 
  

   equation 
  to 
  one 
  whose 
  roots 
  are 
  ten 
  times 
  smaller 
  we 
  find 
  

   the 
  roots 
  lying 
  between 
  10 
  and 
  100, 
  and 
  proceeding 
  in 
  this 
  

   wav 
  we 
  can 
  get 
  approximate 
  values 
  of 
  all 
  the 
  roots. 
  

  

  By 
  similar 
  settings 
  of 
  the 
  contact-fingers, 
  and 
  by 
  using 
  

   both 
  logarithmic 
  and 
  ordinary 
  scales, 
  approximate 
  values 
  of 
  

   the 
  roots 
  of 
  very 
  complicated 
  equations 
  can 
  sometimes 
  be 
  

   easily 
  found. 
  

  

  XIII. 
  Tracing 
  Curves 
  Electrically. 
  

   Suppose 
  we 
  desire 
  to 
  find 
  the 
  graph 
  of 
  the 
  curve 
  

  

  y 
  = 
  apXP-{-aqX^—arX'^+ 
  ... 
  =/{x), 
  

  

  where 
  p, 
  q, 
  r, 
  ... 
  may 
  be 
  positive, 
  fractional, 
  or 
  negative 
  

   indices. 
  We 
  set 
  the 
  contact-fingers 
  at 
  angles 
  tan~^j'j>, 
  

   tan"^ 
  g, 
  ... 
  with 
  the 
  index-line, 
  and 
  move 
  the 
  slide 
  resistances 
  

   until 
  the 
  readings 
  on 
  the 
  scales 
  are 
  a^, 
  aq, 
  ... 
  . 
  The 
  contact- 
  

   fingers 
  on 
  the 
  slide 
  resistances 
  representing 
  the 
  positive 
  terms 
  

   are 
  then 
  connected 
  with 
  P, 
  and 
  the 
  other 
  contact-fingers 
  with 
  

   Q. 
  In 
  addition 
  we 
  have 
  a 
  slide 
  resistance 
  Y 
  with 
  a 
  vertical 
  

   finger, 
  which 
  is 
  also 
  connected 
  with 
  Q. 
  The 
  fingers 
  are 
  

   moved 
  down 
  through 
  a 
  given 
  distance 
  a? 
  on 
  the 
  logarithmic 
  

   scale, 
  and 
  the 
  value 
  of 
  the 
  reading 
  y 
  on 
  the 
  slide 
  resistance 
  Y 
  

   when 
  there 
  is 
  no 
  deflexion 
  of 
  the 
  galvanometer 
  is 
  then 
  found. 
  

   In 
  this 
  way 
  simultaneous 
  values 
  of 
  x 
  and 
  y 
  can 
  be 
  rapidly 
  

   obtained, 
  and 
  hence 
  we 
  can 
  readily 
  plot 
  the 
  curve. 
  

  

  The 
  curve 
  could 
  also 
  be 
  traced 
  automatically 
  by 
  making 
  

   the 
  spot 
  of 
  light 
  from 
  a 
  mirror 
  galvanometer, 
  connected 
  

   between 
  P 
  and 
  Q, 
  fall 
  on 
  a 
  strip 
  of 
  sensitive 
  paper 
  which 
  is 
  

   constrained 
  to 
  move 
  so 
  that 
  its 
  velocity 
  is 
  always 
  proportional 
  

   to 
  the 
  rate 
  at 
  which 
  x 
  is 
  increasing. 
  The 
  points 
  where 
  the 
  

   trace 
  cuts 
  the 
  line 
  of 
  zero 
  deflexion 
  would 
  give 
  the 
  roots 
  of 
  

   the 
  equation 
  /(a')=0, 
  and 
  the 
  turning 
  points 
  of 
  the 
  trace 
  

   would 
  give 
  the 
  roots 
  of 
  /' 
  (x)=0. 
  In 
  the 
  same 
  way 
  the 
  

   intecrral 
  curve 
  

  

  Jo 
  

   could 
  be 
  drawn 
  automatically. 
  

  

  