﻿444 
  Mr. 
  J. 
  R. 
  Wilton 
  on 
  

  

  These 
  equations 
  give 
  

  

  s 
  = 
  2(>o+<7o)/(£o 
  + 
  >7o) 
  = 
  -2/(\)/5« 
  2 
  {/'(>)} 
  3/5 
  . 
  

  

  The 
  integration 
  of 
  equations 
  (7) 
  cannot, 
  evidently, 
  be 
  

   completed 
  except 
  when 
  /and 
  F 
  are 
  specified 
  functions 
  ; 
  and 
  

   even 
  in 
  the 
  simplest 
  case 
  the 
  labour 
  of 
  determining 
  a 
  solution 
  

   is 
  considerable. 
  

  

  5. 
  It 
  is, 
  however, 
  for 
  our 
  present 
  purpose, 
  unnecessary 
  to 
  

   go 
  through 
  this 
  labour. 
  Equation 
  (2) 
  reduces 
  to 
  

  

  3fc-« 
  ■•■•■•■ 
  C8) 
  

  

  when 
  7 
  = 
  3; 
  and 
  although 
  this 
  is 
  widely 
  different 
  from 
  its 
  

   actual 
  value, 
  yet 
  it 
  will 
  appear 
  that, 
  owing 
  to 
  the 
  fact 
  that 
  

  

  ~ 
  is 
  very 
  nearly 
  unity, 
  the 
  solution 
  so 
  obtained 
  represents 
  

  

  QX 
  

  

  the 
  facts 
  with 
  a 
  fair 
  degree 
  of 
  accuracy, 
  and 
  we 
  shall 
  be 
  led 
  

   to 
  a 
  second 
  order 
  approximation 
  to 
  the 
  solution 
  of 
  equation 
  (1) 
  

   which, 
  it 
  so 
  happens, 
  holds 
  for 
  all 
  values 
  of 
  7. 
  

  

  In 
  this 
  case 
  we 
  may 
  take 
  the 
  conditions 
  to 
  be 
  satisfied 
  in 
  a 
  

   rather 
  more 
  general 
  form 
  ; 
  e. 
  g 
  , 
  we 
  may 
  take 
  

  

  y=\ 
  + 
  YO)=j(PX' 
  + 
  QT')<a, 
  *=T(X), 
  *=X(X),j 
  

   p=-P(X), 
  ?=Q(X), 
  )' 
  

  

  where 
  p 
  and 
  q 
  stand 
  for 
  ^ 
  - 
  and 
  ^*f, 
  respectively. 
  From 
  

  

  these 
  equations 
  we 
  find, 
  using 
  the 
  notation 
  of 
  the 
  second 
  

   paragraph, 
  that 
  

  

  PX 
  + 
  QT-X-Y 
  

   V 
  = 
  -P 
  ' 
  

  

  3e_ 
  T 
  QT-X-Y 
  ch*_ 
  _T 
  QT-X-Y 
  

   B? 
  2P 
  + 
  2a 
  ' 
  B^- 
  2P 
  + 
  2 
  tt 
  

  

  The 
  solution 
  of 
  equation 
  (8) 
  under 
  these 
  conditions 
  is 
  * 
  

   given 
  by 
  eliminating 
  6 
  and 
  t 
  between 
  the 
  equations 
  

  

  * 
  See, 
  for 
  example, 
  Goursat's 
  < 
  Lecons 
  snr 
  l'lntegration 
  des 
  Equations 
  

   aux 
  d&ivees 
  Partielles 
  clu 
  Second 
  Ordre/ 
  torn. 
  i. 
  p. 
  61 
  or 
  121. 
  

  

  