﻿Motion 
  of 
  a 
  Viscous 
  Fluid. 
  781 
  

  

  The 
  terms 
  of 
  the 
  third 
  degree 
  are 
  

  

  J 
  L 
  L 
  dx 
  dy 
  dz 
  j 
  

  

  , 
  t 
  f,dv'_ 
  , 
  dv' 
  ,dv'^[ 
  

   [_ 
  dx 
  dy 
  dz 
  J 
  

  

  , 
  f 
  f 
  dio' 
  , 
  dw' 
  ,dw 
  f 
  ~\ 
  "j 
  _ 
  

  

  +w 
  r 
  *r 
  +t 
  ' 
  ^ 
  + 
  w 
  m 
  J 
  J 
  *■ 
  * 
  ^ 
  

  

  which 
  may 
  be 
  written 
  

  

  _1(Y 
  , 
  ^ 
  2 
  + 
  ^ 
  2 
  + 
  ^ 
  2 
  ) 
  

  

  2jr 
  ^ 
  

  

  ') 
  r^ 
  + 
  t^ 
  + 
  w") 
  

  

  + 
  iv 
  — 
  -j- 
  '- 
  dx 
  dy 
  dz 
  ; 
  

  

  and 
  this 
  vanishes 
  for 
  the 
  same 
  reason 
  as 
  the 
  terms 
  in 
  ot. 
  

  

  We 
  are 
  left 
  with 
  the 
  terms 
  of 
  the 
  second 
  degree 
  in 
  

   u\ 
  v\ 
  iv'. 
  Of 
  these 
  the 
  part 
  involving 
  v 
  is 
  

  

  v\ 
  [i?VV 
  + 
  v'V 
  2 
  v' 
  i^W]^ 
  dy 
  dz. 
  . 
  (20) 
  

  

  So 
  far 
  as 
  this 
  part 
  is 
  concerned, 
  we 
  see 
  from 
  (3) 
  that 
  

  

  dV/dt 
  = 
  -F', 
  (21) 
  

  

  F' 
  being 
  the 
  dissipation-function 
  calculated 
  from 
  u 
  f 
  , 
  v', 
  w'. 
  

  

  Of 
  the 
  remaining 
  18 
  terms 
  of 
  the 
  second 
  degree, 
  9 
  vanish 
  

   as 
  before 
  when 
  integrated, 
  in 
  virtue 
  of 
  the 
  equation 
  of 
  con- 
  

   tinuity 
  satisfied 
  by 
  w 
  , 
  v 
  , 
  iv 
  . 
  Finally 
  we 
  have 
  * 
  

  

  L 
  ax 
  ay 
  dz 
  J 
  

  

  If 
  the 
  motion 
  u 
  , 
  r 
  , 
  w 
  be 
  in 
  two 
  dimensions, 
  so 
  that 
  

  

  * 
  Compare 
  0. 
  Reynolds, 
  Phil. 
  Trans. 
  1895, 
  Part 
  I. 
  p. 
  146. 
  In 
  

   Lorentz's 
  deduction 
  of 
  a 
  similar 
  equation 
  (Abhandlungeii, 
  vol. 
  i. 
  p. 
  46) 
  

   the 
  additional 
  motion 
  is 
  assumed 
  to 
  be 
  small. 
  This 
  memoir, 
  as 
  well 
  as 
  

   that 
  of 
  Orr 
  referred 
  to 
  below, 
  should 
  be 
  consulted 
  by 
  those 
  interested. 
  

   See 
  also 
  Lamb's 
  Hydrodynamics, 
  § 
  346. 
  

  

  