the  Plug  Experiment.  681 
By  substituting  in  equation  (2)  the  values  given  by 
equations  (1),  (3),  (4),  (5),  and  (6),  we  get 
de  =  —jxIv  —  vdp—/jbGpdp  ;       ....      (7) 
and  since  the  differentials  which  appear  in  this  equation  all 
refer  to  the  isothermal  change  AB,  we  may  write  it  in 
the  form  * 
This  equation  is,  of  course,  not  new,  but  few  writers  of 
text-books  on  thermodynamics  seem  to  think  it  worth 
deducing. 
The  equations  for  the  other  two  experiments  come  out  much 
more  directly.  In  the  case  of  the  isothermal  plug  experi- 
ment, in  which  the  gas  passes  at  once  from  the  state  A  to  the 
state  B,  let  p  be  the  ratio  of  the  energy  abstracted  from  the 
gas  to  the  fall  of  pressure  dp.     Then  we  have,  obviously, 
de——pdv — vdp  +  pdp, (8) 
or  by  the  same  considerations  as  before, 
(!:).=-*-<-»©). m 
In  the  irreversible  isothermal  free  expansion  from  A  to  B, 
the  external  work  is  zero.  If  we  let  X  be  the  ratio  of  the 
energy  (heat)  which  must  be  added,  per  unit  mass  of  gas,  to 
the  increase  of  specific  volume  di\  in  order  to  make  the  final 
temperature  of  the  gas,  after  its  tumultuous  motions  have 
subsided,  the  same  as  its  initial  temperature,  we  have 
de  =  \dv, (9) 
or 
xbv\ 
(C) 
§4.  In  deducing  equations  (A),  (B),  and  (C),  no  use  is 
made  of  the  second  law  ;  hence  they  do  not  involve  the 
thermodynamic  temperature  0  at  all,  and  cannot,  by  them- 
selves, tell  us  anything  about  it,  But  suppose,  finally,  that 
the  gas  passes  from  the  state  A  to  the  state  B  by  a  reversible 
isothermal  expansion.  For  any  reversible  change  of  state 
of  a  system  having  only  two  degrees  of  freedom"  and  acted 
*  Constancy  of  temperature  is  indicated  by  the  subscript  9 :  if  9  is  the 
thermodynamic  temperature,  T  is  evidently  constant  whenever  9  is 
constant. 
Phil.  Mag.  S.  6.  Vol.  11.  No.  65.  May  1906.  2  Y 
