232  Royal  Society  : — 
in  the  model  Dr.  Andrews's  curves  for  the  temperatures  13°'l,  210,5, 
3l°l,  32°-5,  35°-5,  and  48°*1  Centigrade,  which  afford  the  data  for 
its  construction,  may  with  advantage  be  all  shown  drawn  in  their 
proper  places.  The  model  admits  of  easily  exhibiting  in  due  re- 
lation to  one  another  a  second  set  of  curves,  in  which  each  would 
be  for  a  constant  pressure,  and  in  each  of  which  the  coordinates 
would  represent  temperatures  and  corresponding  volumes.  It  may 
be  used  in  various  ways  for  affording  quantitative  relations  inter- 
polated among  those  more  immediately  given  by  the  experiments. 
We  may  now,  aided  by  the  conception  of  this  model,  return  to  the 
consideration  of  continuity  or  discontinuity  in  the  curves  in  crossing 
the  boiling  stage.  Let  us  suppose  an  indefinite  number  of  curves, 
each  for  one  constant  temperature,  to  be  drawn  on  the  model,  the 
several  temperatures  differing  in  succession  by  very  small  intervals, 
and  the  curves  consequently  being  sections  of  the  curved  surface 
by  numerous  planes  closely  spaced  parallel  to  one  another  and  to 
the  plane  containing  the  pair  of  coordinate  axes  for  pressure  and 
volume.  Now  we  can  see  that,  as  we  pass  from  curve  to  curve  in  ap- 
proaching towards  the  critical  point  from  the  higher  temperatures, 
the  tangent  to  the  curve  at  the  steepest  point  or  point  of  inflection 
is  rotating,  so  that  its  inclination  to  the  plane  of  the  coordinate  axes 
for  pressure  and  temperature,  which  we  may  regard  as  horizontal,  in- 
creases till,  at  the  critical  point,  it  becomes  a  right  angle.  Then  it 
appears  very  natural  to  suppose  that,  in  proceeding  onwards  past 
the  critical  point  to  curves  successively  for  lower  and  lower  tem- 
peratures, the  tangent  at  the  point  of  inflection  would  continue  its 
rotation,  and  the  angle  of  its  inclination,  which  before  was  acute, 
would  now  become  obtuse.  It  seems  much  more  natural  to  make 
such  a  supposition  as  this  than  to  suppose  that  in  passing  the  cri- 
tical point  from  higher  into  lower  temperatures  the  curved  line,  or 
the  curved  surface  to  which  it  belongs,  should  break  itself  asunder, 
and  should  come  to  have  a  part  of  its  conceivable  continuous  course 
absolutely  deficient.  It  thus  seems  natural  to  suppose  that  in 
some  sense  there  is  continuity  in  each  of  the  successive  curves 
by  courses  such  as  those  drawn  in  the  accompanying  diagram  as 
dotted  curves  uniting  continuously  the  curves  for  the  ordinary  gaseous 
state  with  those  for  the  ordinary  liquid  state. 
The  physical  conditions  corresponding  to  the  extension  of  the 
curve  from  a  to  some  point  b  we  have  seen  are  perfectly  attainable 
in  practice.  Some  extension  of  the  gaseous  curve  into  points  of  tem- 
perature and  pressure  below  what  I  have  called  the  boiling-  or  con- 
densing-line  (as,  for  instance,  some  extension  such  as  from  c  to  d  in 
the  figure)  I  think  we  need  not  despair  of  practically  realizing  in 
physical  operations.  As  a  likely  mode  in  which  to  bring  steam 
continuing  gaseous  to  points  of  pressure  and  temperature  at  which 
it  would  collapse  to  liquid  water  if  it  had  any  particle  of  liquid  water 
present  along  with  it,  or  if  other  circumstances  were  present  ca- 
pable of  affording  some  apparently  requisite  conditions  for  enabling 
it  to  make  a  beginning  of  the  change  of  state  *,  I  would  suggest  the 
*  The  principle  that  "the  particles  of  a  substance  when  existing  all  in  one  state 
