﻿On 
  the 
  Motion 
  of 
  a 
  Viscous 
  Fluid. 
  Ill 
  

  

  if 
  the 
  space 
  occupied 
  by 
  the 
  fluid 
  is 
  simply 
  connected, 
  or 
  in 
  

   any 
  case 
  if 
  V 
  be 
  single- 
  valued. 
  Hence 
  

  

  F 
  = 
  F 
  + 
  F', 
  . 
  (6) 
  

  

  or 
  since 
  F 
  ; 
  is 
  necessarily 
  positive, 
  the 
  motion 
  M 
  makes 
  F 
  an 
  

   absolute 
  minimum. 
  It 
  should 
  be 
  remarked 
  that 
  F' 
  can 
  

   vanish 
  only 
  for 
  a 
  motion 
  such 
  as 
  can 
  be 
  assumed 
  by 
  a 
  solid 
  

   body 
  (Stokes), 
  and 
  that 
  such 
  a 
  motion 
  could 
  not 
  make 
  the 
  

   boundary 
  velocities 
  vanish. 
  The 
  motion 
  M 
  determined 
  by 
  

   (4) 
  is 
  thus 
  unique. 
  

  

  The 
  conclusion 
  expressed 
  in 
  (6) 
  that 
  M 
  makes 
  F 
  an 
  

   absolute 
  minimum 
  is 
  not 
  limited 
  to 
  the 
  supposition 
  of 
  a 
  slow 
  

   motion. 
  All 
  that 
  is 
  required 
  to 
  ensure 
  the 
  fulfilment 
  of 
  (5), 
  

   on 
  which 
  (6) 
  depends, 
  is 
  that 
  V 
  2 
  wo> 
  V 
  2 
  ^ 
  , 
  V 
  2 
  w 
  should 
  be 
  

   the 
  derivatives 
  of 
  some 
  single-valued 
  function. 
  Obviously 
  

   it 
  would 
  suffice 
  that 
  ^7 
  2 
  u 
  , 
  V 
  2 
  »'o> 
  V 
  2 
  ^o 
  vanish, 
  as 
  will 
  happen 
  

   if 
  the 
  motion 
  have 
  a 
  velocity-potential. 
  Stokes 
  * 
  remarked 
  

   long 
  ago 
  that 
  when 
  there 
  is 
  a 
  velocity-potential, 
  not 
  only 
  

   are 
  the 
  ordinary 
  equations 
  of 
  fluid 
  motion 
  satisfied, 
  but 
  the 
  

   equations 
  obtained 
  when 
  friction 
  is 
  taken 
  into 
  account 
  are 
  

   satisfied 
  likewise. 
  A 
  motion 
  with 
  a 
  velocity-potential 
  can 
  

   always 
  be 
  found 
  which 
  shall 
  have 
  prescribed 
  normal 
  velocities 
  

   at 
  the 
  boundary, 
  and 
  the 
  tangential 
  velocities 
  are 
  thereby 
  

   determined. 
  If 
  these 
  agree 
  with 
  the 
  prescribed 
  tangential 
  

   velocities 
  of 
  a 
  viscous 
  fluid, 
  all 
  the 
  conditions 
  are 
  satisfied 
  

   by 
  the 
  motion 
  in 
  question. 
  And 
  since 
  this 
  motion 
  makes 
  F 
  

   an 
  absolute 
  minimum, 
  it 
  cannot 
  differ 
  from 
  the 
  motion 
  

   determined 
  by 
  (4) 
  with 
  the 
  same 
  boundary 
  conditions. 
  We 
  

   may 
  arrive 
  at 
  the 
  same 
  conclusion 
  by 
  considering 
  the 
  general 
  

   equation 
  of 
  motion 
  — 
  

  

  *(J+"s+'|+-£)-^-^- 
  (7) 
  

  

  If 
  there 
  be 
  a 
  velocity-potential 
  </>, 
  so 
  that 
  u 
  = 
  d<j>/da:, 
  &c, 
  

  

  4>^^£--=2-i{(S) 
  2+ 
  (|) 
  2+ 
  © 
  2 
  }^ 
  8) 
  

  

  and 
  then 
  (7) 
  and 
  its 
  analogues 
  reduce 
  practically 
  to 
  the 
  

   form 
  (4) 
  if 
  the 
  motion 
  be 
  steady. 
  

  

  Other 
  cases 
  where 
  F 
  is 
  an 
  absolute 
  minimum 
  are 
  worthy 
  

  

  * 
  Camb. 
  Trans, 
  vol. 
  ix. 
  (1850); 
  'Math, 
  and 
  Phvs. 
  Papers,' 
  vol. 
  iii. 
  

   p. 
  73 
  

  

  