﻿446 
  Mr. 
  J. 
  R. 
  Wilton 
  on 
  

  

  function 
  of 
  t 
  when 
  x 
  — 
  0. 
  Such 
  solutions 
  may 
  be 
  obtained 
  

   fairly 
  straightforwardly, 
  but 
  their 
  importance 
  is 
  not 
  such 
  

   as 
  to 
  warrant 
  our 
  delaying 
  to 
  investigate 
  them 
  in 
  this 
  

   connexion. 
  

  

  Putting 
  Q 
  = 
  aZ\ 
  T 
  = 
  0, 
  and 
  therefore 
  P 
  = 
  l 
  + 
  Y', 
  we 
  find 
  

   after 
  a 
  slight 
  reduction 
  that 
  the 
  solution 
  of 
  equation 
  (10), 
  

   subject 
  to 
  the 
  conditions 
  

  

  y 
  = 
  x-\-Y(.x), 
  q 
  = 
  aJZi 
  (a), 
  when 
  £ 
  = 
  0, 
  

  

  is 
  given 
  by 
  

  

  y= 
  — 
  — 
  — 
  5 
  ; 
  — 
  ' 
  

  

  m 
  = 
  i(^ 
  + 
  T 
  )-i(Z 
  1 
  -Z 
  2 
  )-i 
  f 
  (Y 
  1 
  %'<W-Y 
  I 
  'Z/«fr) 
  

  

  -i(^-T 
  + 
  Y 
  1 
  -Y 
  2 
  )( 
  rTY7 
  - 
  rTY 
  - 
  7 
  -Z 
  1 
  '-Z 
  2 
  '), 
  

  

  at 
  = 
  ^_ 
  T+ 
  T 
  1 
  -Y 
  2 
  )/( 
  1 
  -^-,+ 
  T 
  i 
  T 
  --Z 
  1 
  ' 
  + 
  Z/). 
  

  

  This 
  solution 
  cannot 
  always 
  represent 
  the 
  motion, 
  for 
  it 
  

   becomes 
  discontinuous 
  after 
  a 
  certain 
  time. 
  We 
  shall 
  return 
  

   to 
  this 
  point 
  in 
  paragraph 
  7. 
  

  

  6. 
  To 
  compare 
  this 
  with 
  the 
  ordinary 
  approximate 
  solution, 
  

   namely, 
  

  

  y 
  = 
  x 
  + 
  ±Y(.x 
  + 
  at)+±Y(x 
  — 
  at)+\Z(x 
  + 
  at)-^Z(x 
  — 
  at), 
  

  

  we 
  must 
  expand 
  y 
  in 
  terms 
  of 
  functions 
  of 
  x-\-at 
  and 
  of 
  

   x 
  — 
  at. 
  To 
  do 
  this 
  we 
  make 
  use 
  of 
  the 
  fact 
  that 
  y 
  is 
  nearly 
  

   equal 
  to 
  x 
  — 
  i. 
  e., 
  Y 
  is 
  small. 
  

  

  The 
  reduction 
  is 
  long, 
  but 
  it 
  is 
  a 
  good 
  deal 
  simplified 
  by 
  

   taking 
  it 
  in 
  two 
  stages. 
  We 
  consider 
  first 
  the 
  case 
  where 
  

   Z 
  = 
  0, 
  and 
  retain 
  only 
  terms 
  of 
  the 
  second 
  order 
  in 
  the 
  

   values 
  of 
  x, 
  y, 
  and 
  t. 
  We 
  find 
  

  

  + 
  KY 
  l 
  + 
  Y 
  2 
  )+KY 
  l 
  -Y 
  2 
  )(Y/-Y 
  2 
  '), 
  

  

  x 
  + 
  at= 
  tf 
  + 
  KYi-Y^ 
  + 
  iCY-YjjY; 
  

  

  + 
  i(8-T)fT 
  1 
  '-i(Y 
  1 
  '-YO(3t,' 
  + 
  T,')], 
  

  

  x 
  -at 
  = 
  r- 
  A(Y 
  1 
  -Y 
  3 
  )-l(Y 
  1 
  -Y 
  2 
  )Y 
  !! 
  ' 
  

   -^-T)[Y 
  2 
  '+l(Y 
  I 
  '-Y 
  2 
  ')(Y 
  1 
  +3Y/)]. 
  

  

  