﻿784 
  Lord 
  Rayleigh 
  on 
  the 
  

  

  multiplier^ 
  (29) 
  vanishes. 
  This 
  corresponds 
  to 
  

  

  ^ 
  = 
  R. 
  o 
  + 
  R 
  1 
  cos(0 
  + 
  e 
  1 
  )+ 
  . 
  . 
  . 
  + 
  R,„cosn(0- 
  + 
  e») 
  + 
  ... 
  , 
  (30) 
  

  

  where 
  R 
  , 
  R 
  x 
  , 
  &c. 
  are 
  functions 
  o£ 
  r, 
  while 
  e 
  u 
  e 
  2 
  , 
  &c. 
  are 
  

   constants. 
  If 
  yjr 
  can 
  be 
  thus 
  limited, 
  dT'/dt 
  reduces 
  to 
  — 
  F', 
  

   and 
  the 
  original 
  motion 
  is 
  stable. 
  

   In 
  general 
  

  

  f- 
  F 
  -^ 
  S 
  ,$-C*)*. 
  (M, 
  

  

  C», 
  S„ 
  must 
  be 
  such 
  as 
  to 
  give 
  at 
  the 
  boundaries 
  

  

  C 
  n 
  = 
  0, 
  dC 
  n 
  /dr 
  = 
  0, 
  S» 
  = 
  0, 
  d$ 
  n 
  /dr 
  = 
  0;. 
  (32) 
  

  

  otherwise 
  they 
  are 
  arbitrary 
  functions 
  of 
  r. 
  It 
  may 
  be 
  

   noticed 
  that 
  the 
  sign 
  of 
  any 
  term 
  in 
  (29) 
  may 
  be 
  altered 
  at 
  

   pleasure 
  by 
  interchange 
  of 
  G 
  n 
  and 
  S„. 
  

  

  When 
  fju 
  is 
  great, 
  so 
  that 
  the 
  influence 
  of 
  F 
  preponderates, 
  

   the 
  motion 
  is 
  stable. 
  On 
  the 
  other 
  hand 
  when 
  //, 
  is 
  small, 
  

   the 
  motion 
  is 
  probably 
  unstable, 
  unless 
  special 
  restrictions 
  

   can 
  be 
  imposed. 
  

  

  A 
  similar 
  treatment 
  applies 
  to 
  the 
  problem 
  of 
  the 
  uniform 
  

   shearing 
  motion 
  o£ 
  a 
  fluid 
  between 
  two 
  parallel 
  plane 
  walls, 
  

   defined 
  by 
  

  

  u 
  = 
  A 
  + 
  By, 
  vo 
  = 
  0, 
  w 
  = 
  0. 
  . 
  . 
  (33) 
  

  

  From 
  (23) 
  

  

  ^ 
  = 
  -F'- 
  P 
  Bffu'v'dvdy. 
  . 
  . 
  • 
  (34) 
  

  

  If 
  in 
  the 
  superposed 
  motion 
  v' 
  = 
  0, 
  the 
  double 
  integral 
  

   vanishes 
  and 
  the 
  original 
  motion 
  is 
  stable. 
  More 
  generally, 
  

   if 
  the 
  stream-function 
  of 
  the 
  superposed 
  motion 
  be 
  

  

  ifr 
  = 
  C 
  cos 
  Jcx 
  + 
  $ 
  sin 
  ka;, 
  .... 
  (35) 
  

  

  where 
  C, 
  S 
  are 
  functions 
  of 
  y, 
  we 
  find 
  

  

  = 
  _ 
  F+ 
  ^J(s|-c|)i 
  y 
  . 
  . 
  . 
  (36) 
  

  

  Here 
  again 
  if 
  the 
  motion 
  can 
  be 
  such 
  that 
  C 
  and 
  S 
  differ 
  

   only 
  by 
  a 
  constant 
  multiplier, 
  the 
  integral 
  would 
  vanish. 
  

  

  