﻿448 
  Mr. 
  J. 
  R. 
  Wilton 
  on 
  

  

  wherefore 
  ~ 
  contributes 
  to 
  ( 
  .-— 
  ) 
  ( 
  -.-r, 
  ) 
  a 
  second 
  order 
  

   term 
  

  

  T^ 
  a 
  . 
  [(Y 
  / 
  + 
  lV) 
  + 
  (Z 
  1 
  ^Z/)][(Y 
  1 
  '' 
  + 
  Y 
  2 
  '') 
  + 
  (V-Z 
  2 
  '')] 
  J 
  

   o£ 
  which 
  only 
  the 
  parts 
  ^±i 
  a 
  2 
  (Y 
  1 
  ' 
  + 
  Y 
  2 
  / 
  )(Y 
  1 
  ' 
  , 
  + 
  Y 
  2 
  "), 
  and 
  

  

  r 
  ^a 
  2 
  (Z 
  1 
  '-Z 
  2 
  / 
  )(Z 
  1 
  ''-Z 
  2 
  ''), 
  are 
  accounted 
  for 
  without 
  

   4 
  

  

  invoking 
  the 
  aid 
  of 
  the 
  function 
  </>. 
  This 
  function 
  must 
  

  

  therefore 
  be 
  such 
  that 
  

  

  ^ 
  i 
  t-^|^ 
  = 
  Z 
  f- 
  1 
  [(Y 
  1 
  '+Y 
  2 
  ')(Z 
  1 
  ''-Z 
  2 
  '')+(Y 
  1 
  '' 
  + 
  T 
  2 
  '')(Z/-Z 
  2 
  ')],(ll. 
  

  

  with 
  the 
  conditions 
  that 
  6 
  ^— 
  , 
  and 
  ~\- 
  are 
  all 
  zero 
  when 
  t 
  = 
  0. 
  

   T 
  " 
  0% 
  d* 
  

  

  ~Put 
  w 
  + 
  at 
  = 
  0, 
  x-~at 
  = 
  T, 
  so 
  that 
  Yi 
  and 
  Zj 
  are 
  functions 
  

  

  of 
  0, 
  Y 
  2 
  and 
  Z 
  2 
  are 
  functions 
  of 
  t. 
  Equation 
  (11) 
  then 
  

  

  becomes 
  

  

  ^= 
  2 
  Sccyi'+t.')(v-z.")+cti"+tv:)((z 
  1 
  '-z,')]. 
  

  

  with 
  the 
  conditions 
  

  

  0=|| 
  = 
  |*=O, 
  when0=r. 
  

   00 
  Or 
  

  

  The 
  solution 
  is 
  

   ^,= 
  'y±i[^)-t(T)-eT 
  2 
  'Z 
  2 
  '+TT 
  1 
  'Z 
  1 
  ' 
  + 
  Y 
  s 
  Z 
  1 
  -Y 
  1 
  Z 
  2 
  '+T 
  2 
  'Z 
  1 
  -Y 
  I 
  'Z 
  2 
  ], 
  

   where 
  the 
  function 
  ifr 
  is 
  to 
  be 
  determined 
  by 
  the 
  conditions 
  

   || 
  = 
  M 
  = 
  0. 
  when 
  6 
  = 
  r, 
  

  

  00 
  OT 
  

   which 
  reduce 
  to 
  one. 
  We 
  find 
  

  

  ^r{0) 
  = 
  2$Y 
  1 
  'Z 
  1 
  'd0-0Y 
  1 
  'Z 
  1 
  '-Y 
  1 
  Z 
  1 
  ' 
  + 
  Y 
  J 
  'Z; 
  

  

  and, 
  finally, 
  we 
  obtain 
  as 
  the 
  second 
  order 
  approximation 
  to 
  

   the 
  solution 
  of 
  equation 
  (1) 
  

  

  y 
  = 
  x+ 
  iCY, 
  + 
  Y 
  2 
  ) 
  + 
  KZ, 
  - 
  Z,) 
  - 
  ^ 
  (Yl 
  ' 
  _ 
  Y 
  2 
  ') 
  [ 
  Y, 
  - 
  Y 
  2 
  + 
  at 
  fT,' 
  + 
  Y, 
  ') 
  ] 
  

  

  - 
  2at(Y 
  1 
  'Z 
  l 
  ' 
  -4- 
  Y 
  2 
  'Z 
  2 
  ') 
  + 
  (Y 
  1 
  ' 
  + 
  Y 
  s 
  ') 
  (Z, 
  - 
  Z 
  2 
  ) 
  - 
  ( 
  Y, 
  - 
  Y 
  2 
  ) 
  (Z/ 
  + 
  Z/) 
  j 
  

  

  .... 
  (12) 
  

  

  