﻿1002 
  Lord 
  Rayleigh 
  on 
  the 
  Stability 
  of 
  the 
  

  

  of 
  the 
  motion, 
  we 
  superpose 
  upon 
  it 
  a 
  two-dimensional 
  

   disturbance 
  u, 
  v, 
  where 
  u 
  and 
  v 
  are 
  regarded 
  as 
  small. 
  If 
  

   the 
  fluid 
  is 
  incompressible, 
  

  

  du 
  dv 
  ~ 
  /iN 
  

  

  - 
  r 
  + 
  7-=0; 
  ...... 
  (1) 
  

  

  ax 
  ay 
  

  

  and 
  if 
  the 
  squares 
  and 
  products 
  of 
  small 
  quantities 
  are 
  

   neglected, 
  the 
  hydrodynamical 
  equations 
  give 
  * 
  

  

  (d 
  TJ 
  d\(du 
  dv\ 
  d 
  2 
  XJ 
  . 
  ,_,. 
  

  

  From 
  (1) 
  and 
  (2), 
  if 
  we 
  assume 
  that 
  as 
  functions 
  of 
  

   t 
  and 
  ei', 
  u, 
  v 
  are 
  proportional 
  to 
  e 
  l 
  K 
  nt 
  + 
  Jcx 
  )^ 
  where 
  k 
  is 
  real 
  

   and 
  n 
  may 
  be 
  real 
  or 
  complex, 
  

  

  In 
  the 
  paper 
  quoted 
  it 
  was 
  shown 
  that 
  under 
  certain 
  

   conditions 
  n 
  could 
  not 
  he 
  complex 
  ; 
  and 
  it 
  may 
  be 
  con- 
  

   venient 
  to 
  repeat 
  the 
  argument. 
  Let 
  

  

  njk 
  — 
  p 
  + 
  iq, 
  v 
  = 
  a. 
  + 
  iff, 
  

  

  where 
  p 
  % 
  q, 
  a, 
  ft 
  are 
  real. 
  Substituting 
  in 
  (3) 
  and 
  equating 
  

   separately 
  to 
  zero 
  the 
  real 
  and 
  imaginary 
  parts, 
  we 
  get 
  

  

  <Pa_ 
  2 
  d 
  2 
  V 
  (p 
  + 
  U)x±q/3 
  

   df~ 
  > 
  + 
  dy* 
  0H-U)» 
  + 
  g"' 
  

  

  d 
  ^_ 
  j« 
  R 
  d 
  2 
  U^q«±(p+JJ){3 
  

   dy*-' 
  CP 
  ^ 
  dif 
  (^ 
  + 
  U) 
  2 
  +^ 
  

  

  whence 
  if 
  we 
  multiply 
  the 
  first 
  by 
  /3 
  and 
  the 
  second 
  by 
  a 
  

   and 
  subtract, 
  

  

  d_( 
  R 
  du_ 
  J/3\_d 
  2 
  U 
  q(«* 
  + 
  /3 
  2 
  ) 
  

   dyXdy 
  a 
  dy)~ 
  dif 
  (p+\J)* 
  + 
  q 
  2 
  ' 
  ' 
  ' 
  W 
  

  

  At 
  the 
  limits, 
  corresponding 
  to 
  finite 
  or 
  infinite 
  values 
  

   of 
  y, 
  we 
  suppose 
  that 
  v, 
  and 
  therefore 
  both 
  a. 
  and 
  /?, 
  vanish. 
  

   Hence 
  when 
  (4) 
  is 
  integrated 
  with 
  respect 
  to 
  y 
  between 
  these 
  

   limits, 
  the 
  left-hand 
  member 
  vanishes 
  and 
  we 
  infer 
  that 
  

   q 
  also 
  must 
  vanish 
  unless 
  d^JJ/dy 
  2 
  changes 
  sign. 
  Thus 
  

   in 
  the 
  motion 
  between 
  walls 
  if 
  the 
  velocity 
  curve, 
  in 
  which 
  

   U 
  is 
  ordinate 
  and 
  y 
  abscissa, 
  be 
  of 
  one 
  curvature 
  throughout, 
  

   n 
  must 
  be 
  wholly 
  real 
  ; 
  otherwise, 
  so 
  far 
  as 
  this 
  argument 
  

  

  * 
  Proceeding's 
  of 
  London 
  Mathematical 
  Society, 
  vol. 
  xi. 
  p. 
  57 
  (1880) 
  ; 
  

   Scientific 
  Papers, 
  i. 
  p. 
  485. 
  Also 
  Lamb's 
  ' 
  Hydrodynamics/ 
  § 
  345. 
  

  

  