560       Prof.  A.  W.  Porter  on  the  Inversion-Points  for 
Consider  any  two  points  A  and  B  in  the  same  vertical  on 
inversion-curves,  such  as  figs.  1  or  2  ;  for  these  the  pressure 
is  the  same.  They  will  therefore  correspond  to  two  points 
A'  and  B'  on  a  constant-pressure  line  upon  a  r-T  diagram. 
But  for  both  A  and  B,  T2 
BT 
©-»• 
Therefore  a  curve 
v  =  Tf(p),  obtained  by  integrating  this  equation,  passes 
through  A'  and  B'  on  the  v-T  diagram  ;  the  form  of  this  curve 
is  immaterial  to  the  present  argument.  That  is  to  say,  the  two 
inversion  volumes  at  a  given  pressure  are  proportional  to  the 
corresponding  inversion  temperatures,  since  f(p)  is  the  same 
for  both  ;  or  A!  and  B'  he  on  a  straight  line  passing  through 
the  origin  as  shown  in  fig.  3. 
For  the  same  reason,  if  there  were  n  inversion  temperatures 
for  the  same  pressure,  the  constant-pressure  line  on  a  v-T 
diagram  wTould  be  cut  n  times  by  a  line  through  the  origin. 
Again,  in  general  for  any  fluid  we  may  write 
T^T(|)=FfeT), 
3T 
and  F(_p,  T)=0   for   inversion-points.     But  the   maximum 
pressure  for  which  inversion  occurs  corresponds  to 
or 
9F/BF 
3T/3;. 
=  0; 
