﻿782 
  Lord 
  Rayleigh 
  on 
  the 
  

  

  w 
  =0, 
  while 
  u 
  Q 
  and 
  v 
  are 
  independent 
  of 
  z, 
  (22) 
  reduces 
  to 
  

  

  • 
  • 
  (23) 
  

  

  Under 
  this 
  head 
  comes 
  the 
  case 
  of 
  uniform 
  rotation 
  ex- 
  

   pressed 
  in 
  (16), 
  for 
  which 
  

  

  dx 
  ^ 
  dy 
  ^ 
  dy 
  dx 
  ~~ 
  

  

  Here 
  then 
  dT'ldt= 
  — 
  F' 
  simply, 
  that 
  is 
  T' 
  continually 
  dimi- 
  

   nishes 
  until 
  it 
  becomes 
  insensible. 
  Any 
  motion 
  superposed 
  

   upon 
  that 
  of 
  uniform 
  rotation 
  gradually 
  dies 
  out. 
  

  

  When 
  the 
  motion 
  a 
  , 
  v 
  , 
  w 
  has 
  a 
  velocity-potential 
  <£, 
  (22) 
  

   may 
  be 
  written 
  

  

  ^ 
  = 
  " 
  F 
  P 
  )V 
  U 
  d^ 
  +V 
  df 
  +W 
  d** 
  

  

  + 
  2u'v' 
  -fi- 
  + 
  2v 
  V 
  ^- 
  + 
  2w'u 
  f 
  -^H 
  dx 
  dy 
  dz. 
  (24) 
  

   dxdy 
  dy 
  dz 
  dz 
  dxj 
  u 
  y 
  

  

  So 
  far 
  as 
  I 
  am 
  aware, 
  no 
  case 
  of 
  complete 
  stability 
  for 
  all 
  

   values 
  of 
  jju 
  is 
  known, 
  other 
  than 
  the 
  motion 
  possible 
  to 
  a 
  

   solid 
  body 
  above 
  considered. 
  It 
  may 
  be 
  doubted 
  whether 
  

   such 
  cases 
  exist. 
  Under 
  the 
  head 
  of 
  (24) 
  a 
  simple 
  example 
  

   occurs 
  when 
  <p 
  = 
  tan 
  _1 
  (yjx), 
  the 
  irrotational 
  motion 
  taking- 
  

   place 
  in 
  concentric 
  circles. 
  Here 
  if 
  r 
  2 
  — 
  x 
  2 
  + 
  y 
  2 
  , 
  

  

  . 
  . 
  . 
  (25) 
  

  

  If 
  the 
  superposed 
  motion 
  also 
  be 
  two-dimensional, 
  it 
  may 
  

   be 
  expressed 
  by 
  means 
  of 
  a 
  stream-function 
  i/r. 
  We 
  have 
  

   in 
  terms 
  of 
  polar 
  coordinates 
  

  

  , 
  dty 
  dylr 
  . 
  „ 
  1 
  d-\lr 
  - 
  

   u 
  = 
  ~y~ 
  — 
  — 
  J 
  - 
  sm^+ 
  - 
  -^ 
  cos 
  #, 
  

   dy 
  dr 
  r 
  du 
  

  

  , 
  d^lr 
  dylr 
  n 
  1 
  dylr 
  . 
  a 
  

   —v' 
  = 
  -f~ 
  = 
  V- 
  cos 
  a— 
  — 
  J~ 
  sin 
  0, 
  

   doc 
  dr 
  r 
  dd 
  

  

  