﻿1006* 
  Lord 
  Rayleigh 
  on 
  the 
  Stability 
  of 
  the 
  

  

  must 
  do 
  when 
  q 
  = 
  Q. 
  In 
  every 
  case 
  then, 
  when 
  k 
  is 
  small, 
  

   the 
  real 
  part 
  of 
  n 
  must 
  equal 
  some 
  value 
  of 
  — 
  HJ 
  *. 
  

  

  It 
  may 
  be 
  of 
  interest 
  to 
  show 
  the 
  application 
  of 
  (13) 
  to 
  a 
  

   case 
  formerly 
  treated 
  t 
  in 
  which 
  the 
  velocity-curve 
  is 
  made 
  

   up 
  of 
  straight 
  portions 
  and 
  is 
  anti-symmetrical 
  with 
  respect 
  

   to 
  the 
  point 
  lying 
  midway 
  between 
  the 
  two 
  walls, 
  now 
  

   taken 
  as 
  origin 
  of 
  y. 
  Thus 
  on 
  the 
  positive 
  side 
  

  

  Yy 
  

   from 
  y 
  = 
  to 
  y 
  = 
  ±b/ 
  U*= 
  -•; 
  ; 
  

  

  from 
  y^\V 
  to 
  y 
  = 
  ib' 
  + 
  b, 
  U=§ 
  + 
  /iV(y-J&') 
  ; 
  

  

  while 
  on 
  the 
  negative 
  side 
  U 
  takes 
  symmetrically 
  the 
  opposite 
  

   values. 
  Then 
  if 
  we 
  write 
  n/kY*=7i', 
  (13) 
  becomes 
  

  

  0: 
  

  

  fib' 
  dy 
  ri> 
  

  

  Jo 
  c^+^ 
  + 
  L 
  

  

  dy 
  

  

  h2 
  

  

  {2y/6+/<y-i4')+.»'} 
  

  

  -f-same 
  with 
  n' 
  reversed. 
  

   Effecting 
  the 
  integrations, 
  we 
  find 
  after 
  reduction 
  

  

  , 
  2 
  _ 
  ?* 
  2 
  2b 
  + 
  i 
  + 
  2iJLb(b 
  + 
  b 
  , 
  )+pW 
  n 
  „ 
  

  

  in 
  agreement 
  with 
  equation 
  (23) 
  of 
  the 
  paper 
  referred 
  to 
  

   when 
  k 
  is 
  there 
  made 
  smalL 
  Here 
  n, 
  if 
  imaginary 
  at 
  all, 
  is 
  

   a 
  pure 
  imaginary, 
  and 
  it 
  is 
  imaginary 
  only 
  when 
  /jl 
  lies 
  

   between 
  — 
  1/b 
  and 
  —1/b 
  —2/6'. 
  The 
  regular 
  motion 
  is 
  then 
  

   exponentially 
  unstable. 
  

  

  In 
  the 
  only 
  unstable 
  cases 
  hitherto 
  investigated 
  the 
  velocity- 
  

   curve 
  is 
  made 
  up 
  of 
  straight 
  portions 
  meeting 
  at 
  finite 
  

   angles, 
  and 
  it 
  may 
  perhaps 
  be 
  thought 
  that 
  the 
  instability 
  

   has 
  its 
  origin 
  in 
  this 
  discontinuity. 
  The 
  method 
  now 
  under 
  

   discussion 
  disposes 
  of 
  any 
  doubt. 
  For 
  obviously 
  in 
  (13) 
  it 
  

   can 
  make 
  no 
  important 
  difference 
  whether 
  dJJ/dy 
  is 
  dis- 
  

   continuous 
  or 
  not. 
  If 
  a 
  motion 
  is 
  definitely 
  unstable 
  in 
  the 
  

   former 
  case, 
  it 
  cannot 
  become 
  stable 
  merely 
  by 
  easing 
  off 
  

   the 
  finite 
  angles 
  in 
  the 
  velocity-curve. 
  There 
  exist, 
  there- 
  

   fore, 
  exponentially 
  unstable 
  motions 
  in 
  which 
  both 
  U 
  and 
  

  

  *■ 
  By 
  the 
  method 
  of 
  a 
  former 
  paper 
  " 
  On 
  the 
  question 
  of 
  the 
  Stability 
  

   of 
  the 
  Flow 
  of 
  Fluids" 
  (Phil. 
  Mag. 
  vol. 
  xxxiv. 
  p. 
  59, 
  1892 
  ; 
  Scientific 
  

   Papers, 
  iii, 
  p. 
  579) 
  the 
  conclusion 
  that 
  p-\-TJ 
  must 
  change 
  sign 
  may 
  be 
  

   extended 
  to 
  the 
  problem 
  of 
  the 
  simple 
  shearing 
  motion 
  between 
  two 
  

   parallel 
  walls 
  of 
  a 
  viscous 
  fluid, 
  and 
  this 
  whatever 
  may 
  be 
  the 
  value 
  

   of 
  k. 
  

  

  t 
  Proc. 
  Lend. 
  Math. 
  Foe. 
  vol. 
  xix. 
  p. 
  67 
  (1887 
  ) 
  ; 
  Scientific 
  Papers, 
  

   iii.p, 
  20, 
  figs. 
  (3), 
  (4), 
  (5), 
  

  

  