Prof.  Thomson  on  the  Abrupt  Change  at  Boiling  or  Condensing .  231 
some  distance  at  least,  into  lower  pressures  in  some  such  way  as  is 
shown  by  the  dotted  continuation  from  a  to  b.  But  now  the  ques- 
tion arises,  Does  this  curve  necessarily  end  at  any  particular 
point  b  ?  We  know  that  the  extent  of  this  curve  in  the  direction 
from  a  towards  or  past  b,  along  which  the  liquid  volume  will  con- 
tinue to  be  represented  before  the  explosive  or  bumping  change  to 
gas  occurs,  is  very  variable  under  different  circumstances,  being  much 
affected  by  the  presence  of  other  fluids,  even  in  small  quantities,  as 
impurities  in  the  fluid  experimented  on,  and  by  the  nature  of  the 
surface  of  the  containing  vessel,  &c. 
The  consideration  of  the  subject  may  be  facilitated,  and  aid  towards 
the  attainment  of  clear  views  of  the  mutual  relations  of  temperature, 
pressure,  and  volume  in  a  given  mass  of  a  fluid  may  be  gained,  by 
actually  making,  or  by  conceiving  to  be  made,  for  carbonic  acid, 
from  the  data  supplied  in  Dr.  Andrews's  experimental  results,  a  solid 
model  consisting  of  a  curved  surface  referred  to  three  axes  of  rect- 
angular coordinates,  and  formed  so  that  the  three  coordinates  of 
each  point  in  the  curved  surface  shall  represent,  for  any  given  mass 
of  carbonic  acid,  a  temperature,  a  pressure,  and  a  volume  which  can 
coexist  in  that  mass.  It  is  to  be  noticed  here  that  in  his  diagram  of 
curves  the  results  for  each  of  the  several  temperatures  experimented 
on  are  combined  in  the  form  of  a  plane  curved  line  referred  to  two 
axes  of  rectangular  coordinates — one  of  each  pair  of  ordinates  repre- 
senting a  pressure,  and  the  other  representing  the  volume  corre- 
sponding to  that  pressure  at  the  temperature  to  which  the  curve 
belongs.  Now  to  form  a  model  such  as  I  am  here  recommending, 
and  have  myself  made,  Dr.  Andrews's  curved  lines  are  to  be  placed 
with  their  planes  parallel  to  one  another,  and  separated  by  intervals 
proportional  to  the  differences  of  the  temperatures  to  which  the  curves 
severally  belong,  and  with  the  origins  of  coordinates  of  the  curves 
situated  in  a  straight  line  perpendicular  to  their  planes,  and  with 
the  axes  of  coordinates  of  all  of  them  parallel  in  pairs  to  one  another, 
and  then  the  curved  surface  is  to  be  formed  so  as  to  pass  through 
those  curved  lines  smoothly  or  evenly*.  The  curved  surface  so  ob- 
tained exhibits  in  a  very  obvious  way  the  remarkable  phenomena  of 
the  voluminal  conditions  at  and  near  the  critical  point  of  temperature 
and  pressure,  in  comparison  with  the  voluminal  conditions  throughout 
other  parts  of  the  range  of  gradually  varying  temperatures  and  pres- 
sures to  which  it  extends,  and  even  throughout  a  far  wider  range 
into  which  it  can  in  imagination  be  conceived  to  be  extended.  It 
helps  to  afford  a  clear  view  of  the  nature  and  meaning  of  the  conti- 
nuity of  the  liquid  and  gaseous  states  of  matter.  It  does  so  by  its 
own  obvious  continuity  throughout  its  expanse  round  the  end  of  the 
range  of  points  of  pressure  and  temperature  where  an  abrupt  change 
of  volume  can  occur  by  boiling  or  condensing.    On  the  curved  surface 
*  For  the  practical  execution  of  this,  it  is  well  to  commence  with  a  rectangular 
block  of  wood,  and  then  carefully  to  pare  it  down,  applying,  from  time  to  time, 
the  various  curves  as  templets  to  it,  and  proceeding  according  to  the  general 
methods  followed  in  a  shipbuilder's  modelling-room  in  cutting  out  small 
models  of  ships  according  to  curves  laid  down  on  paper  as  cross  sections  of 
the  required  model  at  various  places  in  its  length. 
