﻿450 
  Mr. 
  J. 
  R. 
  Wilton 
  on 
  

  

  given 
  by 
  equation 
  (15) 
  or 
  (16) 
  when 
  und 
  t 
  are 
  given 
  by 
  

   equation 
  (13) 
  or 
  (14) 
  and 
  

  

  A 
  .= 
  i(^ 
  + 
  T 
  )_iJ(l 
  + 
  Y 
  1 
  , 
  )Z 
  1 
  W 
  + 
  ij(l 
  + 
  lV)Z 
  2 
  VZT 
  

  

  If 
  we 
  retain 
  only 
  the 
  terms 
  of 
  lowest 
  order, 
  

   at 
  = 
  \(0-r) 
  = 
  1/(Y 
  2 
  "-Z 
  2 
  "), 
  ■) 
  (u) 
  

  

  or 
  

  

  at 
  = 
  W-r)=-%l(Y 
  1 
  " 
  + 
  Z 
  1 
  "), 
  V 
  < 
  (lg) 
  

  

  with 
  the 
  same 
  value 
  of 
  ,r 
  as 
  in 
  equation 
  (17). 
  J 
  

  

  8. 
  A 
  glance 
  at 
  equation 
  (1*2) 
  shows 
  that 
  the 
  first 
  order 
  

   approximation 
  

  

  y 
  = 
  x 
  + 
  L[Y(x 
  + 
  at) 
  + 
  Y(z 
  — 
  at) 
  + 
  Z(z 
  + 
  at) 
  — 
  Z(x 
  — 
  at)] 
  

  

  ceases 
  to 
  hold 
  when 
  

  

  ^(Y 
  1 
  ' 
  2 
  -Y 
  2 
  ' 
  2 
  + 
  Z/ 
  2 
  -Z 
  2 
  ' 
  2 
  + 
  2Y 
  1 
  'Z 
  1 
  ' 
  + 
  2Y 
  2 
  'Z 
  2 
  ')-2 
  ( 
  T/Z/rffl 
  

  

  J 
  x-at 
  

  

  ceases 
  to 
  be 
  small 
  compared 
  with 
  

  

  Yi 
  + 
  Y 
  2 
  + 
  Z 
  1 
  — 
  Z 
  2 
  , 
  

  

  where 
  Yj 
  and 
  Z 
  x 
  are 
  functions 
  of 
  x 
  + 
  at, 
  Y 
  2 
  , 
  Z 
  2 
  of 
  x—at. 
  

  

  It 
  is 
  evident 
  that 
  t 
  is 
  in 
  general 
  of 
  the 
  order 
  of 
  magnitude 
  

   of 
  the 
  least 
  value 
  of 
  the 
  reciprocal 
  of 
  Y, 
  w 
  hich 
  is 
  the 
  same 
  

   order 
  as 
  that 
  given 
  by 
  equation 
  (17) 
  or 
  (18). 
  

  

  In 
  the 
  particular 
  case 
  when 
  Y 
  + 
  Z 
  = 
  0, 
  which, 
  in 
  the 
  

   ordinary 
  approximate 
  solution, 
  represents 
  a 
  single 
  pro- 
  

   gressive 
  wave, 
  

  

  y 
  = 
  x 
  + 
  Y(x—at), 
  

  

  the 
  second 
  order 
  approximation 
  is 
  

  

  ■y 
  = 
  x 
  + 
  Y(x-at) 
  + 
  r) 
  ^hatY 
  ,2 
  +( 
  Y'*(0)d0J, 
  

  

  which 
  ceases 
  to 
  hold 
  after 
  a 
  time 
  t 
  whose 
  order 
  of 
  magnitude 
  

   is 
  given 
  by 
  

  

  at 
  = 
  Y^ 
  2 
  /Y 
  2 
  ' 
  2 
  ; 
  

  

  i.e., 
  at 
  is 
  of 
  the 
  order 
  of 
  the 
  reciprocal 
  of 
  the 
  amplitude 
  

   of 
  the 
  wave. 
  

  

  