﻿442 
  Mr. 
  R, 
  Hargreaves 
  on 
  a 
  

  

  theorem, 
  when 
  the 
  equation 
  

  

  ~d 
  2 
  4> 
  -tf± 
  7F4> 
  _ 
  1 
  V4> 
  

  

  holds 
  in 
  space 
  external 
  to 
  moving 
  surfaces 
  S, 
  viz. 
  

  

  J 
  W 
  + 
  W 
  + 
  ^\-y^ 
  1 
  ) 
  dT 
  ~ 
  -) 
  4 
  '^ 
  dS 
  -} 
  V 
  2 
  * 
  1+ 
  5*7* 
  

  

  -J^-J^g*-^^. 
  • 
  • 
  (13) 
  

  

  When 
  half 
  the 
  left-hand 
  member 
  is 
  used 
  as 
  a 
  kinetic 
  

   potential, 
  its 
  effective 
  section 
  is 
  given 
  by 
  surface 
  integrals. 
  

  

  § 
  3. 
  For 
  a 
  compressible 
  fluid 
  or 
  gas 
  in 
  irrotational 
  motion 
  

   equation 
  (1) 
  is 
  replaced 
  by 
  

  

  and 
  the 
  volume 
  integral 
  is 
  to 
  be 
  taken 
  after 
  multiplication 
  

   by 
  p. 
  When 
  compressibility 
  is 
  admitted 
  we 
  must 
  recognize 
  

   the 
  existence 
  of 
  energy 
  associated 
  with 
  compression, 
  and 
  

   separate 
  the 
  pressure 
  from 
  this 
  element 
  of 
  intrinsic 
  energy. 
  

   Thus 
  using 
  pv=l 
  as 
  in 
  thermodynamics, 
  

  

  p\ 
  —=p 
  l 
  vdp 
  = 
  p(yp 
  — 
  \pdv)=p'—p\pdv='p 
  + 
  pXl 
  say. 
  

  

  The 
  integral 
  then 
  stands 
  

  

  J^ 
  + 
  p 
  U 
  + 
  p|f 
  +£2tt«)iT=0, 
  . 
  . 
  (14) 
  

  

  where 
  J 
  pf(t)dr 
  is 
  neglected 
  for 
  the 
  same 
  reason 
  as 
  before. 
  

   In 
  lieu 
  of 
  (3), 
  recognizing 
  the 
  variability 
  of 
  p, 
  we 
  have 
  

  

  -t 
  \ 
  pcf)dr=\ 
  ^Ap<f>) 
  dT 
  — 
  I 
  pv 
  v 
  <j)d$+\ 
  pv 
  v 
  icdcr. 
  (15) 
  

   Again 
  

  

  = 
  — 
  I 
  pv 
  v 
  <p 
  d$ 
  + 
  j 
  p 
  ^— 
  k 
  da, 
  

   which 
  in 
  conjunction 
  with 
  (15) 
  makes 
  

  

  aJ**"J'£*+JH*-s9«*- 
  • 
  (16 
  > 
  

  

  