Prof.  Jeans's  Dynamical  Theory  of  Gases.  163 
states.  To  assert  that  one  is  in  a  high  degree  more  probable 
than  the  other,  would  be  contrary  to  Maxwell's  law. 
16.  The  real  defect  in  the  H  Theorem,  and  in  every  proof 
yet  given  of  Maxwell's  law,  consists  in  my  opinion  in  this, 
that  it  ignores  the  continuity  of  the  motion  altogether.  The 
H  Theorem  is  based  wholly  on  the  assumption  mentioned 
in  art.  (11)  above.  Now  let  us  suppose  that  at  a  given 
initial  instant  the  state  of  the  system  is  formed  as  follows  : — - 
One  person  assigns  component  velocities  u,  v,  to  to  each 
molecule  according  to  any  law  he  pleases,  subject  only 
to  the  condition  that  Sm(u2  +  v2  +  w2)  =  E.  And  another 
person,  to  whom  these  assigned  velocities  are  wholly  un- 
known, scatters  the  molecules  through  the  space  O.  In 
that  case  Condition  A  is  satisfied  at  the  initial  instant, 
Boltzmann's  assumption  art.  (11)  above  is  satisfied,  and  if  H  is 
not  minimum,  — j~  is  at  that  instant  negative.     But  in  the 
'    dt  ^ 
supposed  distribution,  the  velocities  are  just  as  likely  to  have  all 
the  opposite  values  —u  —v  —w  as  u,  v,  w.     In  either  case  at 
the  initial  instant- ,-    is  negative.     If,  however,  the  system 
w^ere  left  to  itself  for  a  finite  time,  and  then  the  final  velocities 
were  reversed,  the  system  would  retrace  its  course  back  to 
the  initial  state.     And  immediately  before  it  arrives  at  that 
state  — j-  is  evidently  positive.     But  when  it  has  arrived  at 
that  state  —r*  becomes  discontinuously  negative,  because  then 
the  original  velocities  k,  v,  w  would  have  become  —  u  —v  —  w. 
7TT 
And,  as  we  have  seen,  -j-  is  negative  in  that  state  as  well  as 
m 
'      dt  7TT 
the    original    state.       -j,  •>  if    not    zero,    must  be    dis- 
continuous when  the    system   in   its  reversed  course  passes 
through  its  initial  position. 
17.  That  is  the  history  of  the  initial  state  formed  as  I 
have  supposed  it  to  be  formed.  In  dealing  with  the  subse- 
quent course  of  the  system,  to  prove  the  H  Theorem,  or 
Maxwell's  law,  we  assume,  and  if  we  are  to  prove  the  pro- 
position we  must  assume,  that  the  independence  of  the 
molecular  velocities,  or  as  I  call  it  Condition  A,  exists  at  every 
instant  during  the  motion.  But  if  the  motion  be  continuous, 
the  state  of  the  system  at  any  instant  is  a  determinate 
function  of  the  initial  state  and  of  the  time  elapsed,  and  of 
