﻿14 
  Lord 
  Kelvin 
  : 
  The 
  Problem 
  of 
  

  

  ■j- 
  at 
  the 
  end 
  of 
  the 
  interval 
  with 
  any 
  accuracy 
  desired 
  ; 
  the 
  

  

  accuracy 
  depending 
  on 
  the 
  smallness 
  of 
  the 
  interval 
  chosen. 
  

  

  t 
  K 
  dt 
  

  

  By 
  taking 
  the 
  curves 
  for 
  — 
  ^ 
  and 
  ^ 
  as 
  straight 
  lines 
  

  

  within 
  each 
  successive 
  interval 
  treated, 
  and 
  by 
  making 
  a 
  

   roughly 
  estimated 
  allowance 
  at 
  each 
  step 
  for 
  the 
  error 
  thus 
  

   introduced, 
  the 
  process 
  can 
  be 
  carried 
  out 
  very 
  quickly 
  and 
  

   quite 
  satisfactorily, 
  as 
  a 
  process 
  of 
  step-by-step 
  calculation, 
  

   without 
  the 
  assistance 
  of 
  carefully 
  drawn 
  curves. 
  

  

  § 
  9. 
  The 
  accuracy 
  of 
  the 
  above 
  process 
  can 
  easily 
  be 
  

   proved 
  analytically 
  for 
  the 
  case 
  of 
  any 
  very 
  short 
  interval 
  ; 
  

   but 
  when 
  such 
  a 
  process 
  is 
  applied 
  to 
  a 
  succession 
  of 
  intervals, 
  

   there 
  is 
  certain 
  to 
  be 
  a 
  cumulative 
  error, 
  which 
  may, 
  or 
  may 
  

   not, 
  increase 
  without 
  limit 
  as 
  the 
  work 
  proceeds. 
  So 
  far 
  as 
  

   it 
  is 
  possible 
  to 
  judge, 
  however, 
  the 
  process 
  of 
  § 
  8 
  seems 
  to 
  

   be 
  practically 
  applicable 
  to 
  obtain 
  numerical 
  solutions 
  of 
  

   differential 
  equations 
  of 
  the 
  form 
  

  

  dH 
  

  

  a?=/(M)i 
  oo, 
  

  

  provided 
  /has 
  the 
  opposite 
  sign 
  from 
  t. 
  It 
  has 
  been 
  applied 
  

   with 
  very 
  satisfactory 
  results 
  to 
  the 
  two 
  equations 
  — 
  

  

  d 
  2 
  t 
  

  

  »—' 
  < 
  10 
  >' 
  

  

  and 
  

  

  dH 
  t 
  r 
  ° 
  

  

  ^ 
  = 
  ~^ 
  ( 
  U 
  )' 
  

  

  for 
  each 
  of 
  which 
  the 
  direct 
  verification 
  of 
  the 
  solutions 
  

   obtained 
  is 
  possible 
  ; 
  the 
  solution 
  of 
  the 
  latter 
  equation 
  

   obtained 
  was 
  © 
  5 
  (^)? 
  which 
  is 
  given 
  in 
  § 
  6 
  above. 
  

  

  In 
  these 
  cases, 
  and 
  in 
  the 
  case 
  of 
  any 
  Homer 
  Lane 
  

   Function, 
  if 
  t, 
  throughout 
  a 
  number 
  of 
  intervals, 
  becomes 
  

  

  d?t 
  

   greater 
  than 
  its 
  true 
  value, 
  the 
  absolute 
  value 
  of 
  -=-= 
  also 
  

  

  dt 
  dx 
  

  

  becomes 
  greater 
  than 
  its 
  true 
  value, 
  and 
  -j- 
  is 
  therefore 
  

  

  tending 
  to 
  become 
  less 
  than 
  it 
  would 
  otherwise 
  be. 
  Hence 
  

   the 
  successive 
  additions 
  to 
  t 
  in 
  each 
  interval 
  in 
  which 
  it 
  is 
  

   greater 
  than 
  its 
  true 
  value 
  are, 
  or 
  tend 
  to 
  become, 
  less 
  than 
  

   they 
  would 
  be 
  with 
  a 
  correct 
  t. 
  Thus, 
  when 
  t 
  has 
  become 
  

   too 
  great 
  in 
  any 
  interval, 
  throughout 
  the 
  succeeding 
  intervals 
  

   it 
  tends 
  to 
  return 
  to 
  its 
  true 
  value 
  rather 
  than 
  to 
  go 
  without 
  

  

  