﻿Pressure-integral 
  as 
  Kinetic 
  Potential. 
  437 
  

  

  exists, 
  in 
  which 
  each 
  term 
  is 
  of 
  the 
  order 
  energy 
  per 
  unit 
  

   volume. 
  It 
  is 
  reasonable 
  to 
  expect 
  that 
  a 
  volume-integral 
  of 
  

   this 
  equation 
  through 
  the 
  whole 
  liquid 
  will 
  have 
  significance 
  ; 
  

   and 
  the 
  object 
  proposed 
  here 
  is 
  to 
  determine 
  this 
  significance 
  

   and 
  to 
  interpret 
  the 
  pressure-integral. 
  -p 
  -s 
  

  

  Now 
  in 
  the 
  hydro-dynamical 
  equations 
  p^— 
  = 
  —- 
  ^-, 
  ... 
  

  

  the 
  right-hand 
  members 
  have 
  the 
  form 
  taken 
  by 
  Lagrange's 
  

   expressions 
  for 
  force, 
  when 
  it 
  is 
  recognized 
  that 
  p 
  is 
  a 
  

   function 
  of 
  (xyz). 
  The 
  volume-integral 
  J 
  pdr 
  will, 
  however, 
  

   depend 
  on 
  the 
  coordinates 
  defining 
  the 
  positions 
  of 
  the 
  solids, 
  

   on 
  their 
  time-rates, 
  and 
  on 
  the 
  constants 
  of 
  circulation. 
  It 
  

   is 
  permissible, 
  therefore, 
  to 
  conjecture 
  that 
  the 
  pressure- 
  

   integral 
  is 
  a 
  kinetic 
  potential 
  giving 
  the 
  whole 
  dynamical 
  

   effect 
  of 
  the 
  liquid 
  motion. 
  Accordingly, 
  the 
  volume-integral 
  

   of 
  (1) 
  is 
  an 
  equation 
  defining 
  the 
  kinetic 
  potential 
  in 
  terms 
  

   of 
  energy 
  and 
  quantities 
  of 
  like 
  order. 
  

  

  In 
  § 
  1 
  we 
  examine 
  this 
  integral 
  for 
  a 
  problem 
  * 
  propounded 
  

   and 
  solved 
  by 
  Lord 
  Kelvin, 
  that 
  of 
  the 
  motion 
  of 
  solids 
  in 
  

   infinite 
  incompressible 
  liquid, 
  when 
  the 
  solids 
  have 
  apertures 
  

   which 
  permit 
  circulation. 
  The 
  above 
  interpretation 
  presents 
  

   the 
  solution 
  in 
  direct 
  connexion 
  with 
  the 
  fundamental 
  

   equation 
  (1). 
  Section 
  2 
  contains 
  a 
  proposition 
  in 
  general 
  

   dynamics 
  relating 
  to 
  the 
  nugatory 
  character 
  of 
  a 
  term 
  of 
  

  

  the 
  form 
  — 
  appearing 
  in 
  the 
  expression 
  for 
  kinetic 
  potential. 
  

  

  In 
  § 
  3 
  the 
  method 
  of 
  § 
  1 
  is 
  applied 
  to 
  the 
  case 
  of 
  gases, 
  

   where 
  intrinsic 
  energy 
  has 
  to 
  be 
  taken 
  into 
  account. 
  

  

  § 
  1. 
  In 
  the 
  first 
  place 
  we 
  may 
  ignore 
  the 
  arbitrary 
  

   function 
  J 
  (t) 
  in 
  (1), 
  because 
  the 
  integral 
  of 
  this 
  term 
  taken 
  

   through 
  the 
  whole 
  liquid 
  does 
  not 
  depend 
  on 
  the 
  coordinates 
  

   or 
  velocities 
  of 
  the 
  moving 
  solids 
  ; 
  we 
  have, 
  therefore, 
  

  

  f(p 
  + 
  ^| 
  + 
  |2u^r=0 
  (2) 
  

  

  Now 
  consider 
  -=- 
  \p^dr 
  the 
  whole 
  rate 
  of 
  change 
  of 
  the 
  

  

  integral. 
  The 
  change 
  embraces 
  two 
  parts, 
  one 
  in 
  which 
  

  

  the 
  integrand 
  is 
  differentiated, 
  this 
  part 
  being 
  1 
  p~dr; 
  and 
  

  

  another 
  in 
  which 
  the 
  limits 
  are 
  concerned. 
  Near 
  the 
  surface 
  

   of 
  a 
  solid 
  dr 
  may 
  be 
  written 
  as 
  dv 
  dS, 
  where 
  dv 
  is 
  an 
  element 
  

  

  * 
  Phil. 
  Mag. 
  May 
  1873. 
  

  

  