﻿780 
  , 
  Lord 
  Rayleigh 
  on 
  the 
  

  

  limit 
  — 
  a 
  conclusion 
  which 
  could 
  not 
  be 
  admitted. 
  But 
  it 
  

   may 
  be 
  worth 
  while 
  to 
  examine 
  this 
  case 
  more 
  closely. 
  

  

  We 
  suppose 
  as 
  before 
  that 
  u 
  , 
  v 
  , 
  w 
  are 
  the 
  velocities 
  in 
  

   the 
  steady 
  motion 
  M 
  and 
  u, 
  v, 
  w 
  those 
  o£ 
  the 
  motion 
  M, 
  

   both 
  motions 
  satisfying 
  the 
  dynamical 
  equations, 
  and 
  giving 
  

   the 
  prescribed 
  boundary 
  velocities; 
  and 
  we 
  consider 
  the 
  

   expression 
  for 
  the 
  kinetic 
  energy 
  T 
  / 
  of 
  the 
  motion 
  (1) 
  which 
  

   is 
  the 
  difference 
  of 
  these 
  two, 
  and 
  so 
  makes 
  the 
  velocities 
  

   vanish 
  at 
  the 
  boundary. 
  The 
  motion 
  M' 
  with 
  velocities 
  

   u', 
  v', 
  w' 
  does 
  not 
  in 
  general 
  satisfy 
  the 
  dynamical 
  equations. 
  

   We 
  have 
  

  

  loir 
  Cf 
  ,du' 
  , 
  ,dv' 
  ,dw'\. 
  . 
  . 
  /1/n 
  

  

  p 
  -it 
  =}{ 
  u 
  w 
  +v 
  dt 
  +w 
  nrj 
  dx 
  d 
  * 
  dz 
  - 
  ^ 
  

  

  In 
  equations 
  (7) 
  which 
  are 
  satisfied 
  by 
  the 
  motion 
  M 
  we 
  

   substitute 
  u 
  — 
  u 
  -{-u', 
  &c. 
  ; 
  and 
  since 
  the 
  solution 
  M 
  is 
  

   steady 
  we 
  have 
  

  

  du 
  q\ 
  dw 
  _ 
  nK 
  

  

  dt 
  ~ 
  dt 
  = 
  ^df- 
  {) 
  (1Dj 
  

  

  We 
  further 
  suppose 
  that 
  V 
  2 
  w 
  , 
  V 
  2 
  ^o> 
  V 
  2 
  Wo> 
  are 
  derivatives 
  

   of 
  a 
  function 
  H, 
  as 
  in 
  (9). 
  This 
  includes 
  the 
  case 
  of 
  uniform 
  

   rotation 
  expressed 
  by 
  

  

  u 
  =y, 
  v 
  ——x, 
  tt'o 
  = 
  0, 
  . 
  . 
  . 
  (16) 
  

  

  as 
  well 
  as 
  those 
  where 
  there 
  is 
  a 
  velocity-potential. 
  Thus 
  

   (7) 
  becomes 
  

  

  with 
  two 
  analogous 
  equations, 
  where 
  

  

  w 
  = 
  V+pIp 
  — 
  vH, 
  v 
  = 
  fjujp. 
  . 
  . 
  . 
  (18) 
  

  

  These 
  values 
  of 
  du'/dt, 
  &c, 
  are 
  to 
  be 
  substituted 
  in 
  (14). 
  

  

  In 
  virtue 
  of 
  the 
  equation 
  of 
  continuity 
  to 
  which 
  u' 
  , 
  v' 
  , 
  w' 
  

   are 
  subject, 
  the 
  terms 
  in 
  37 
  contribute 
  nothing 
  to 
  dT' 
  /dt, 
  as 
  

   appears 
  at 
  once 
  on 
  integration 
  by 
  parts. 
  The 
  remaining 
  

   terms 
  in 
  dT 
  f 
  /dt 
  are 
  of 
  the 
  first, 
  second, 
  and 
  third 
  degree 
  in 
  

   u' 
  , 
  v 
  ! 
  , 
  w' 
  . 
  Those 
  of 
  the 
  first 
  degree 
  contribute 
  nothing, 
  since 
  

   a 
  , 
  v 
  , 
  w 
  satisfy 
  equations 
  such 
  as 
  

  

  dit 
  du 
  dit 
  dvr 
  , 
  

  

  