﻿438 
  Mr. 
  R. 
  Hargreaves 
  on 
  a 
  

  

  of 
  the 
  normal 
  drawn 
  outwards 
  to 
  the 
  liquid, 
  and 
  dS 
  is 
  an 
  element 
  

  

  o£ 
  surface: 
  thus 
  near 
  the 
  surface 
  the 
  integral 
  is 
  j 
  rfS 
  1 
  p(f)dv. 
  

  

  dv 
  

   Since 
  -r-=-v 
  v 
  the 
  normal 
  component 
  of 
  velocity 
  of 
  the 
  solid, 
  

   at 
  

  

  the 
  second 
  part 
  of 
  the 
  rate 
  of 
  change 
  is 
  — 
  §pv 
  v 
  <j>d$, 
  <j> 
  

  

  having 
  a 
  surface 
  value. 
  If 
  </> 
  is 
  many-valued 
  we 
  have 
  to 
  

  

  consider 
  also 
  the 
  surfaces 
  of 
  barriers 
  moving 
  with 
  the 
  solids. 
  

  

  For 
  a 
  barrier 
  cr 
  we 
  get 
  an 
  integral 
  — 
  §pv 
  v 
  <f)d<r 
  on 
  one 
  face 
  

  

  of 
  the 
  barrier, 
  and 
  on 
  completing 
  the 
  circuit 
  so 
  as 
  to 
  come 
  

  

  to 
  the 
  other 
  face 
  we 
  have 
  (p 
  + 
  k 
  for 
  </>, 
  k 
  being 
  a 
  constant 
  of 
  

  

  circulation, 
  and 
  — 
  v 
  v 
  for 
  v 
  vi 
  the 
  sum 
  for 
  the 
  two 
  faces 
  giving 
  

  

  §pv 
  v 
  icda. 
  Thus 
  with 
  circulation 
  admitted 
  the 
  whole 
  value 
  

  

  of 
  the 
  rate 
  of 
  change 
  is 
  

  

  j- 
  \p<j> 
  dr 
  = 
  I 
  p 
  ^r 
  dr 
  — 
  pr 
  v 
  <j> 
  eZS 
  + 
  pv 
  v 
  K 
  da, 
  . 
  

  

  (3). 
  

  

  where 
  of 
  course 
  a 
  summation 
  for 
  several 
  surfaces 
  and 
  barriers 
  

   may 
  take 
  the 
  place 
  of 
  the 
  one 
  written. 
  Again 
  we 
  have 
  

  

  = 
  "~ 
  I 
  p?v<l> 
  d$ 
  + 
  I 
  p 
  ~r^ 
  fc 
  da, 
  

  

  since 
  ^ 
  = 
  v 
  v 
  at 
  a 
  surface 
  S. 
  This 
  in 
  conjunction 
  with 
  (3) 
  ? 
  

  

  yields 
  

  

  d_ 
  

   dt 
  

  

  \p<j>dT= 
  \Pj^ 
  dT 
  + 
  \p(v 
  v 
  -^)/cd*. 
  

   Adding 
  this 
  to 
  (2) 
  written 
  in 
  the 
  form 
  

  

  (4) 
  

  

  we 
  obtain 
  

  

  fad? 
  + 
  jAp4>dr 
  = 
  J|2« 
  2 
  rfT-Jp(M-t 
  V 
  )^<r. 
  (5). 
  

  

  The 
  potential 
  <f> 
  may 
  be 
  written 
  as 
  ^ 
  + 
  &), 
  where 
  yjr 
  is 
  a 
  

   linear 
  function 
  of 
  the 
  velocities 
  of 
  the 
  solids, 
  and 
  a> 
  of 
  the 
  

   constants 
  of 
  circulation, 
  with 
  coefficients 
  depending 
  in 
  each 
  

   case 
  on 
  the 
  coordinates 
  of 
  the 
  solid 
  as 
  well 
  as 
  on 
  (xyz). 
  

   In 
  virtue 
  of 
  Kelvin's 
  extension 
  of 
  Green's 
  theorem, 
  there 
  is 
  

  

  