﻿1008 
  Lord 
  Rayleigh 
  on 
  the 
  Stability 
  of 
  the 
  

  

  instability. 
  A 
  complete 
  theoretical 
  proof 
  of 
  instability 
  

   requires 
  not 
  only 
  a 
  method 
  capable 
  of 
  dealing 
  with 
  finite 
  

   disturbances 
  but 
  also 
  a 
  definition, 
  not 
  easily 
  given, 
  of 
  what 
  is 
  

   meant 
  by 
  the 
  term. 
  Iu 
  the 
  case 
  of 
  stability 
  we 
  are 
  rather 
  

   better 
  situated, 
  since 
  by 
  absolute 
  stability 
  we 
  may 
  understand 
  

   complete 
  recovery 
  from 
  disturbances 
  of 
  any 
  kind 
  however 
  

   large, 
  such 
  as 
  Reynolds 
  showed 
  to 
  occur 
  in 
  the 
  present 
  case 
  

   when 
  viscosity 
  is 
  paramount 
  *. 
  In 
  the 
  absence 
  of 
  dissipation, 
  

   stability 
  in 
  this 
  sense 
  is 
  not 
  to 
  be 
  expected. 
  

  

  Another 
  manner 
  of 
  regarding 
  the 
  present 
  problem 
  of 
  the 
  

   shearing 
  motion 
  of 
  an 
  inviscid 
  fluid 
  is 
  instructive. 
  In 
  the 
  

   original 
  motion 
  the 
  vorticity 
  is 
  constant 
  throughout 
  the 
  

   whole 
  space 
  between 
  the 
  walls. 
  The 
  disturbance 
  is 
  repre- 
  

   sented 
  by 
  a 
  superposed 
  vorticity, 
  which 
  may 
  be 
  either 
  posi- 
  

   tive 
  or 
  negative, 
  and 
  this 
  vorticity 
  everywhere 
  moves 
  with 
  

   the 
  fluid. 
  At 
  any 
  subsequent 
  time 
  the 
  same 
  vorticities 
  exist 
  

   as 
  initially 
  ; 
  the 
  only 
  question 
  is 
  as 
  to 
  their 
  distribution. 
  And 
  

   when 
  this 
  distribution 
  is 
  known, 
  the 
  whole 
  motion 
  is 
  deter- 
  

   mined. 
  Now 
  it 
  would 
  seem 
  that 
  the 
  added 
  vorticities 
  will 
  

   produce 
  most 
  effect 
  if 
  the 
  positive 
  parts 
  are 
  brought 
  together, 
  

   and 
  also 
  the 
  negative 
  parts, 
  as 
  much 
  as 
  is 
  consistent 
  with 
  

   the 
  prescribed 
  periodicity 
  along 
  x, 
  and 
  that 
  even 
  if 
  this 
  can 
  

   be 
  done 
  the 
  effect 
  cannot 
  be 
  out 
  of 
  proportion 
  to 
  the 
  magni- 
  

   tude 
  of 
  the 
  additional 
  vorticities. 
  If 
  this 
  view 
  be 
  accepted, 
  

   the 
  temporary 
  large 
  increase 
  in 
  Prof. 
  Orr's 
  example 
  would 
  

   be 
  attributed 
  to 
  a 
  specially 
  unfavourable 
  distribution 
  initially 
  

   in 
  which 
  (m 
  large) 
  the 
  positive 
  and 
  -negative 
  parts 
  of 
  the 
  

   added 
  vorticities 
  are 
  closely 
  intermingled. 
  We 
  may 
  even 
  

   go 
  further 
  and 
  regard 
  the 
  subsequent 
  tendency 
  to 
  evanescence, 
  

   rather 
  than 
  the 
  temporary 
  increase, 
  as 
  the 
  normal 
  pheno- 
  

   menon. 
  The 
  difficulty 
  in 
  reconciling 
  the 
  observed 
  behaviour 
  

   of 
  actual 
  fluids 
  with 
  the 
  theory 
  of 
  an 
  inviscid 
  fluid 
  still 
  

   seems 
  to 
  me 
  to 
  be 
  considerable, 
  unless 
  indeed 
  we 
  can 
  admit 
  

   a 
  distinction 
  between 
  a 
  fluid 
  of 
  infinitely 
  small 
  viscosity 
  and 
  

   one 
  of 
  none 
  at 
  all. 
  

  

  At 
  one 
  time 
  I 
  thought 
  that 
  the 
  instability 
  suggested 
  by 
  

   observation 
  might 
  attach 
  to 
  the 
  stages 
  through 
  which 
  a 
  

   viscous 
  liquid 
  must 
  pass 
  in 
  order 
  to 
  acquire 
  a 
  uniform 
  

   shearing 
  motion 
  rather 
  than 
  to 
  the 
  final 
  state 
  itself. 
  Thus 
  

   in 
  order 
  to 
  find 
  an 
  explanation 
  of 
  " 
  skin 
  friction 
  " 
  we 
  may 
  

   suppose 
  the 
  fluid 
  to 
  be 
  initially 
  at 
  rest 
  between 
  two 
  infinite 
  

   fixed 
  walls, 
  one 
  of 
  which 
  is 
  then 
  suddenly 
  made 
  to 
  move 
  in 
  

   its 
  own 
  plane 
  with 
  a 
  uniform 
  velocity. 
  In 
  the 
  earlier 
  stages 
  

   the 
  other 
  wall 
  has 
  no 
  effect 
  and 
  the 
  problem 
  is 
  one 
  considered 
  

   by 
  Fourier 
  in 
  connexion 
  with 
  the 
  conduction 
  of 
  heat. 
  The 
  

   * 
  See 
  also 
  Orr, 
  Proc. 
  Koy. 
  Irish 
  Acad. 
  1907, 
  p. 
  124. 
  

  

  