456  Mr.  S.  H.  Burbuiy  on  tJie  II  Theorem  and 
There  seems  to  be  no  reason  why  the  density  should  vary  at 
all  on  a  scale  infinitely  larger  than  that  of  the  intermolecnlar 
distances,  unless  in  a  field  o£  external  force  such  as  gravity. 
Jeans's  definition  virtually  assumes  v  to  be  constant,  except 
.as  varied  by  external  forces. 
3.  He  then  gives  the  usual  proof  of  the  H  Theorem  on  the 
hypothesis  of  "molecular  chaos "  (art.  15).  This  is  the 
hypothesis  which  I  call  condition  A.  namely,  that  the  chance 
of  a  molecule  having  velocities  within  assigned  limits  is  in- 
dependent of  the  positions  of  all  the  other  molecules  for  the 
rime  being,  and  also  independent  of  the  velocities  of  all  the 
other  molecules  for  the  time  being,  subject  only  to  the  con- 
stancy of  ^%m(u2  -\-r2  -\-w2).  the  total  kinetic  energy.  The 
legitimacy  of  this  assumption  is,  Jeans  says,  not  self-evident, 
because  it  is  conceivable,  for  instance,  that  molecules  having 
nearly  equal  velocities  should  tend  to  flock  together.  He 
reserves  the  consideration  of  this  question  to  later  chapters. 
I.  In  Chapter  II.  he  introduces  a  new  notation.  Given  a 
system  of  N  molecules,  he  calls  the  3N  position  coordinates 
,1'x  yx .  .  zjg,  and  the  3N  component  velocities  i^  i\  .  .  w-^,  the 
"  coordinates  of  a  point  in  a  generalized  space  of  o'N  dimen- 
sions. "  Every  state  of  the  system  is  represented  by  a  point 
in  that  space,  and  the  series  of  states  which  follow  each  other 
in  natural  motion  are  represented  by  a  line  in  that  space. 
This  is  a  change  in  notation,  advantageous  it  may  be,  but 
still  only  in  notation.  And  as  no  new  hypothesis  is  expressly 
introduced,  the  physical  relations  between  the  molecules  must 
be  understood  to  remain  unaffected.  Therefore  every  pro- 
position concerning  these  relations,  true  in  the  usual  notation, 
is  true  in  the  new  notation,  and  vice  versa.  Also  every  pro- 
position which  can  be  proved  by  the  use  of  the  new  notation, 
can  be  proved — I  do  not  say  proved  equally  well,  but  can 
be  proved — by  the  use  of  the  old  notation.  If  the  new  method 
leads,  or  appears  to  lead,  to  any  physical  results  not  attainable 
by  the  ordinary  method,  that  can  only  be  because  along  with 
the  new  method  we  have  introduced  some  new  hypotheses 
unawares. 
5.  Jeans  then  discusses^  arts.  39-51,  the  "  Partition  of  the 
generalized  space  among  the  position  coordinates.''  We  are 
to  suppose  the  three-dimension  space  Q  in  which  the  mole- 
cules are  moving  to  be  divided  into  n  equal  cells  a),  <w2  •  •  &>/*? 
each  of  volume  <o,  so  that  fl  =  nco.  The  number  of  ways  in 
which  N"  molecules  can  be  assigned  to  the  n  different  cells,  so 
that  there  shall  be  ay  in  &>l5  a2  in  o>2,  &c.  with  a1  +  a2  + .  .  an  =  N 
»3  N! 
*=-    ,   „  ,    ' — ,  (equation  60,  p.  39). 
a\  .    tio  i  .  .  .  .  an  ; 
