The  H  Theorem  and  the  Dynamical  Theory  of  Gases.     455 
stress  is  dissipated  within  a  finite  time,  contrary  to  what  we 
should  expect  from  theoretical  considerations.  At  any  rate, 
whether  or  not  it  is  admitted  that  the  breaks  in  the  curves 
indicate  that  the  elastic  limit  has  been  reached,  it  is  certain 
that  at  these  points  a  discontinuity  occurs,  the  law  on  the  one 
side  beiug  entirely  different  from  that  on  the  other  side. 
In  conclusion  I  wish  to  thank  Professor  Trouton  and 
Professor  Porter  for  the  kind  interest  they  have  shown  in  the 
work,  and  for  the  suggestions  which  have  enabled  me  to 
surmount  many  of  the  difficulties  which  have  arisen  during 
the  investigation. 
XL.    The  H  Theorem  and  Professor  J.  H.  Jeans's  Dynamical 
Theory  of  Gases.      By  S.  H.  Burbury,  F.R.S.* 
B 
OLTZMANN'S  H  Theorem  professes  to  prove  that 
whenever  the  state  of  a  gas  is  other  than  the  normal 
state,  —j—  is  negative,  and  H  is  a  minimum  in  the  normal  state. 
-l-oo 
H  is  the   function  m  (/log/'—/' )  du  dv  dw;fdu  dv  dw  being 
—  oo 
the  number  per  unit  of  volume  of  molecules  whose  com- 
ponent velocities  lie  between  u  and  u  +  du,  v  and  v  +  dvr 
w  and  u  -+-  dw.  In  the  normal  state  as  usually  understood, 
y=Ae-Am(M2+u2+w2).  As  a  consequence  of  the  theorem,  Hy 
once  minimum,  remains  minimum  for  ever,  making  the 
process  irreversible.     Then  the  objection  was  made  that  the 
Theorem,  proving  — —  to  be  necessarily  negative,  if  not  zero,. 
proves  too  much,  because  we  have  only  to  reverse  all  the 
velocities  simultaneously,  and  the  system  will  retrace  its 
course  with  H  increasing.  The  question  is  how  to  explain 
the  paradox. 
2.  Professor  J.  H.  Jeans,  a  very  strong  mathematician, 
has  dealt  with  this  question  in  his  recent  work  on  t he- 
Dynamical  Theory  of  Gases.  He  gives  (art.  11)  a  definition  of 
the  density,  v,  of  a  gas  at  a  point  P  :  namely,  v  is  the  number 
of  molecules  in  a  volume  at  P,  which  volume  is  very  great 
compared  with  the  mean  molecular  distance,  but  very  small 
compared  with  the  scale  of  variation  of  density  of  the  gas. 
The  definition  is  free  from  ambiguity.  But  it  ignores  varia- 
tions of  density  on  the  scale  of  the  intermolecular  distances. 
*  Communicated  l)v  the  Author. 
