﻿778 
  Lord 
  Rayleigh 
  on 
  the 
  

  

  of 
  notice. 
  It 
  suffices 
  that 
  

  

  W=g, 
  V% 
  = 
  §, 
  VH 
  = 
  f, 
  . 
  (9). 
  

  

  where 
  H 
  is 
  a 
  single-valued 
  function, 
  subject 
  to 
  V 
  2 
  H 
  = 
  0. 
  

   1^ 
  fo? 
  ^o? 
  ?o 
  ^ 
  e 
  ^ 
  ne 
  rotations, 
  

  

  and 
  thus 
  (9) 
  requires 
  that 
  

  

  V 
  2 
  fo=0, 
  V 
  2 
  %=0, 
  V 
  2 
  ro=0. 
  . 
  . 
  (10) 
  

  

  In 
  two 
  dimensions 
  the 
  dynamical 
  equation 
  reduces 
  to 
  

   D^ 
  jDt 
  = 
  *, 
  so 
  that 
  f 
  is 
  constant 
  along 
  a 
  stream-line. 
  

   Among 
  the 
  cases 
  included 
  are 
  the 
  motion 
  between 
  two 
  

   planes 
  

  

  u 
  = 
  A 
  + 
  Bz/ 
  + 
  (y, 
  v 
  =0, 
  ^tfo=0, 
  . 
  . 
  (11) 
  

  

  and 
  the 
  motion 
  in 
  circles 
  between 
  two 
  coaxal 
  cylinders 
  

   (£o 
  = 
  constant). 
  Also, 
  without 
  regard 
  to 
  the 
  form 
  of 
  the 
  

   boundary, 
  the 
  uniform 
  rotation, 
  as 
  of 
  a 
  solid 
  bod} 
  r 
  , 
  expressed 
  

  

  by 
  

  

  u 
  Q 
  = 
  Cy, 
  v 
  = 
  — 
  O 
  (12) 
  

  

  In 
  all 
  these 
  cases 
  F 
  is 
  an 
  absolute 
  minimum. 
  

  

  Conversely, 
  if 
  the 
  conditions 
  (9) 
  be 
  not 
  satisfied, 
  it 
  will 
  

   ~be 
  possible 
  to 
  find 
  a 
  motion 
  for 
  which 
  F<F 
  . 
  To 
  see 
  this 
  

   choose 
  a 
  place 
  as 
  origin 
  of 
  coordinates 
  where 
  d\/ 
  2 
  u 
  /dy 
  

   is 
  not 
  equal 
  to 
  d\/ 
  2 
  vjdx. 
  Within 
  a 
  small 
  sphere 
  described 
  

   round 
  this 
  point 
  as 
  centre 
  let 
  u' 
  = 
  Cy, 
  v' 
  = 
  — 
  C#, 
  w' 
  — 
  0, 
  

   and 
  let 
  it! 
  = 
  0, 
  v 
  ] 
  = 
  0, 
  w' 
  = 
  outside 
  the 
  sphere, 
  thus 
  satis- 
  

   fying 
  the 
  prescribed 
  boundary 
  conditions. 
  Then 
  in 
  (2) 
  

  

  I 
  (u'\7 
  2 
  u 
  + 
  v' 
  V 
  2 
  t'o 
  + 
  M)'VH) 
  ^ 
  dy 
  dz 
  

  

  = 
  C 
  I 
  (yV\ 
  - 
  a 
  VX) 
  *» 
  % 
  <&?, 
  • 
  • 
  (13) 
  

  

  the 
  integration 
  being 
  over 
  the 
  sphere. 
  Within 
  this 
  small 
  

   region 
  we 
  may 
  take 
  

  

  V*«„ 
  = 
  (V-««)o 
  + 
  ^~ 
  * 
  + 
  -^- 
  y 
  + 
  -£- 
  *, 
  

  

  v-,„ 
  = 
  (v%) 
  + 
  -a^- 
  *■+ 
  -^- 
  y 
  + 
  -^- 
  * 
  J 
  

  

  * 
  Where 
  D/Df= 
  d//fc+w 
  d/dk+ 
  y 
  d/dy+w 
  d/dz. 
  

  

  