a  Fluid  passing  through  a  Porous  Plug,  555 
consider  the  difference  of  pressure  on  the  two  sides  of  the 
plug  as  being  infinitesimal  ;  considerable  simplification 
results  from  doing  this.  Of  course,  it  must  I  e  recognized 
that  this  condition  does  not  hold  good  for  the  circumstances 
of  the  Olszewski  experiment,  the  discussion  of  which  is 
undertaken  in  section  (d). 
The  equation  characteristic  of  the  cooling-effect  is 
\dtjp  lc> '< 
(U+pv) 
where  v=  specific  volume,  p  =  pressure,  T  =  temperature, 
Qp  the  specific  heat  at  constant  pressure,  and  U  the  intrinsic 
energy  of  the  fluid. 
If  no  change  of  temperature  takes  place  for  an  infinitesimal 
change  of  pressure 
p 
a  relation  which  is  independent  of  the  calorimetric  properties 
of  the  fluid.  For  any  given  equation  of  state  this  equation 
completely  determines  the  inversion-point  corresponding  to 
each  temperature.  We  proceed  to  connect  it  with  van  der 
Waals's  equation  and  with  an  equation  of  Dieterici. 
(a)    Van  der  Waals's  Equation. 
We  will  write  the  equation  in  its  reduced  form,  i.  e., 
pressures,  volumes,  and  temperatures  will  be  expressed  as 
fractions  a,  j3,  and  <y  of  their  critical  values.  The  results 
obtained  will  then  be  the  same  for  every  fluid  obeying  this 
equation  owing  to  the  applicability  of  the  law  of  corresponding 
states. 
The  reduced  equation  is 
(a+|2)(3/3-l)  =  87, 
and  the  equation  characteristic  of  the  inversion-points  becomes 
or 
y\Ty 
^1  +  |(;^-i)=o; 
whence  the  inversion  temperature  is  given  by  the  equation 
3(3/3-  1)2_ 
2  0  2 
V=        iff 
