﻿Laminar 
  Motion 
  of 
  an 
  Inviscid 
  Fluid. 
  1007 
  

  

  dXJ/dy 
  are 
  continuous. 
  And 
  it 
  is 
  further 
  evident 
  that 
  any 
  

   proposed 
  velocity-curve 
  inay 
  be 
  replaced 
  approximately 
  by 
  

   straight 
  lines 
  as 
  in 
  my 
  former 
  papers. 
  

  

  The 
  fact 
  that 
  n 
  in 
  equation 
  (15) 
  appears 
  only 
  as 
  n 
  2 
  is 
  a 
  

   simple 
  consequence 
  of 
  the 
  anti-symmetrical 
  character 
  of 
  U. 
  

   For 
  if 
  in 
  (lo) 
  we 
  measure 
  y 
  from 
  the 
  centre 
  and 
  integrate 
  

   between 
  the 
  limits 
  +-JJ, 
  we 
  obtain 
  in 
  that 
  case 
  

  

  1 
  

  

  * 
  rc 
  2 
  /P-f-U 
  2 
  , 
  n 
  nr 
  , 
  

  

  {iv 
  Ik"— 
  U 
  J 
  )- 
  

  

  in 
  which 
  only 
  ?r 
  occurs. 
  But 
  it 
  does 
  not 
  appear 
  than 
  n 
  2 
  is 
  

   necessarily 
  real, 
  as 
  happens 
  in 
  (15) 
  . 
  

  

  Apart 
  from 
  such 
  examples 
  as 
  were 
  treated 
  in 
  my 
  former 
  

   papers 
  in 
  which 
  d 
  2 
  XJ/dy 
  2 
  vanishes 
  except 
  at 
  certain 
  definite 
  

   places, 
  there 
  are 
  very 
  few 
  cases 
  in 
  which 
  (3) 
  can 
  be 
  solved 
  

   analytically. 
  If 
  we 
  suppose 
  that 
  v== 
  sin 
  (777///), 
  vanishing 
  

   when 
  y 
  = 
  and 
  when 
  y 
  = 
  l, 
  and 
  seek 
  what 
  is 
  then 
  admissible 
  

   for 
  U, 
  we 
  get 
  

  

  U 
  + 
  ^ 
  = 
  Acos{F 
  + 
  7r 
  2 
  //-}^ 
  + 
  Bsin 
  {P 
  + 
  tt 
  2 
  // 
  2 
  }^ 
  (17) 
  

  

  in 
  which 
  A 
  and 
  B 
  are 
  arbitrary 
  and 
  n 
  may 
  as 
  well 
  be 
  sup- 
  

   posed 
  to 
  be 
  zero. 
  But 
  since 
  U 
  varies 
  with 
  k, 
  the 
  solution 
  is 
  

   of 
  no 
  great 
  interest. 
  

  

  In 
  estimating 
  the 
  significance 
  of 
  our 
  results 
  respecting 
  

   stability, 
  we 
  must 
  of 
  course 
  remember 
  that 
  the 
  disturbance 
  

   has 
  been 
  assumed 
  to 
  be 
  and 
  to 
  remain 
  infinitely 
  small. 
  

   Where 
  stability 
  is 
  indicated, 
  the 
  magnitude 
  of 
  the 
  admissible 
  

   disturbance 
  may 
  be 
  very 
  restricted. 
  It 
  was 
  on 
  these 
  lines 
  

   that 
  Kelvin 
  proposed 
  to 
  explain 
  the 
  apparent 
  contradiction 
  

   between 
  theoretical 
  results 
  for 
  an 
  inviscid 
  fluid 
  and 
  obser- 
  

   vation 
  of 
  what 
  happens 
  in 
  the 
  motion 
  of 
  real 
  fluids 
  which 
  

   are 
  all 
  more 
  or 
  less 
  viscous. 
  Prof. 
  MT. 
  Orr 
  has 
  carried 
  this 
  

   explanation 
  further*. 
  Taking 
  the 
  case 
  of 
  a 
  simple 
  shearing- 
  

   motion 
  between 
  two 
  walls, 
  he 
  investigates 
  a 
  composite 
  dis- 
  

   turbance, 
  periodic 
  with 
  respect 
  to 
  x 
  but 
  not 
  with 
  respect 
  to 
  

   t, 
  given 
  initially 
  as 
  

  

  v 
  = 
  B 
  cos 
  Ix 
  cos 
  my, 
  (18) 
  

  

  and 
  he 
  finds, 
  equation 
  (38), 
  that 
  when 
  m 
  is 
  large 
  the 
  dis- 
  

   turbance 
  may 
  increase 
  very 
  much, 
  though 
  ultimately 
  it 
  

   comes 
  to 
  zero. 
  Stability 
  in 
  the 
  mathematical 
  sense 
  (B 
  in- 
  

   finitely 
  small) 
  may 
  thus 
  be 
  not 
  inconsistent 
  with 
  a 
  practical 
  

  

  * 
  Proc. 
  Roy. 
  Irish 
  Academy, 
  vol. 
  xxvii, 
  Section 
  A, 
  No. 
  % 
  1907. 
  

   Other 
  related 
  questions 
  are 
  also 
  treated. 
  

  

  