﻿Device 
  for 
  evaluating 
  Formulce 
  and 
  solving 
  Equations, 
  299 
  

  

  where 
  Ri 
  and 
  R2 
  are 
  the 
  resistances 
  of 
  the 
  circuits* 
  RPi 
  and 
  

   RP2. 
  Hence 
  it 
  readily 
  follows 
  that 
  

  

  E 
  - 
  Vi 
  ~ 
  10 
  • 
  Ri 
  "^ 
  100 
  VRi 
  ^ 
  R2/ 
  ■*" 
  roOO 
  ' 
  R1R2' 
  

  

  If 
  R 
  be 
  the 
  resistance 
  of 
  a 
  slide, 
  the 
  minimum 
  possible 
  

   value 
  of 
  both 
  Rj 
  and 
  Ro 
  is 
  R/10. 
  Hence 
  the 
  maximum 
  

   possible 
  value 
  of 
  (Yi~V3)7(E-V) 
  is 
  

  

  and 
  this 
  is 
  less 
  than 
  1/53 
  when 
  2r/R 
  is 
  not 
  greater 
  than 
  

   1/287. 
  

  

  Similarly, 
  if 
  p^ 
  R/, 
  and 
  R2' 
  be 
  the 
  resistances 
  of 
  the 
  slides, 
  

   the 
  readings 
  on 
  which 
  are 
  x^ 
  q^j 
  and 
  q^ 
  respectively, 
  we 
  see 
  

   that 
  

  

  E-Vi 
  10 
  Vp^Ri7^100V/3 
  ' 
  R/^RsV 
  

  

  -10 
  K'p'^^^r^ 
  

  

  81 
  

  

  1000 
  

  

  

  •D 
  

  

  In 
  the 
  very 
  unlikely 
  case 
  when 
  p 
  = 
  Ri' 
  = 
  —^ 
  = 
  R2', 
  

  

  (V^ 
  — 
  V3^)/(E 
  — 
  Vj) 
  has 
  its 
  maximum 
  possible 
  value, 
  and 
  in 
  

   this 
  case 
  it 
  will 
  be 
  less 
  than 
  1/53 
  if 
  r/R 
  be 
  not 
  greater 
  

   than 
  1 
  1100. 
  Hence 
  the 
  desired 
  accuracy 
  can 
  be 
  attained 
  

   by 
  making 
  the 
  resistance 
  R 
  of 
  each 
  slide 
  550 
  times 
  greater 
  

   than 
  the 
  resistance 
  of 
  the 
  bridge 
  wire. 
  

  

  It 
  is 
  to 
  be 
  noticed 
  that 
  the 
  effect 
  of 
  connecting 
  a 
  slide 
  

   resistance 
  with 
  P2 
  instead 
  of 
  Pj 
  is 
  to 
  divide 
  its 
  value 
  by 
  10. 
  

   Hence, 
  if 
  we 
  were 
  adding 
  8*8 
  to 
  0*75, 
  we 
  w^ould 
  put 
  the 
  

   contact-finger 
  of 
  one 
  slide 
  on 
  8*8 
  and 
  connect 
  it 
  with 
  Pj, 
  and 
  

   the 
  contact-finger 
  of 
  another 
  slide 
  on 
  7*5 
  and 
  connect 
  it 
  

   with 
  P2. 
  Suppose 
  also 
  that 
  the 
  value 
  of 
  x 
  was 
  123. 
  In 
  this 
  

   case 
  we 
  could 
  only 
  obtain 
  a 
  balance 
  when 
  it 
  was 
  connected 
  

   with 
  Q3. 
  

  

  VIII. 
  Cubic 
  Equation, 
  

   Let 
  us 
  suppose 
  that 
  the 
  cubic 
  equation 
  is 
  

  

  .'c^ 
  + 
  02^;^ 
  — 
  a,.i' 
  + 
  tto 
  = 
  0, 
  

  

  where 
  ^2? 
  «i? 
  and 
  Aq 
  are 
  positive 
  numbers. 
  

  

  The 
  arrangement 
  of 
  the 
  apparatus 
  for 
  solving 
  this 
  equation 
  

  

  