460  Mr.  S.  H.  Burburv  on  the  II  Tlieorem  and 
ru'  great  enough.  But  if,  during  the  time  t,  m  and  m'  exert 
sensible  forces  on  each  other,  it  is  certain  that  in  general  the 
two  chances  <j)  (w)  and  <£  (u1)  are  not  independent.  The 
question  depends  on  the  forces,  and  on  nothing  else  whatever. 
It  cannot  be  affected  by  any  change  in  notation.  There  is 
indeed  an  attempted  justification  of  the  assumption  of  inde- 
pendence in  the  latter  part  of  art.  65,  p.  57.  But  it  is  only 
by  reference  to  the  analysis  of  Chapter  III.,  the  Partition  of 
the  generalized  space,  in  which  was  assumed  by  necessary 
implication  that  molecular  chaos  or  Condition  A  exists.  It 
is  impossible  to  prove  a  physical  fact  by  merely  using  a  new 
notation. 
10.  Jeans  then,  pp.  59,  60,  gives  an  "Analysis  of  the 
H  Theorem/'  He  reminds  us  that  to  obtain  strictly  accurate 
results,  the  quantities  he  deals  with  must  not  be  regarded 
as  applying  to  a  single  system,  but  to  the  average  of  all 
systems  satisfying  certain  conditions.  These  conditions  I 
suppose  are  that  each  system  consists  of  the  same  number  N 
of  similar  molecules,  and  has  the  same  mean  kinetic  energy. 
Then  he  assumes  (art.  (38)  that  the  function  /  (art.  1  above) 
consists  of  two  numerically  equal  parts,  one  half  containing 
those  molecules  whose  velocities  are  u,  r,  w,  and  the  other 
half  those  whose  velocities  are  —a,  —  v,  —to.  If  one  of 
these  equal  parts  corresponds  to  a  direct  motion,  the  other 
corresponds  to  the  reversed  motion  (art.  1  above).     Whence 
7TT 
it  follows  that,  taking  both  motions  into  account.  —  is  on 
6  at 
average  as  often  positive  as   negative,  is  in  fact  on  average 
zero. 
11.  This  argument,  it  should  be  noted,  does  not  affect  the 
mathematical  correctness  of  the  H  Theorem.  The  theorem 
is  founded  on  a  certain  assumption,  namely,  that  the  number 
per  unit  of  time  of  collisions  between  pairs  of  molecules,  one 
from  the  class  F,  and  other  from  the  class  /,  the  two  mole- 
cules being  so  situated  that  alter  collision  they  are  in  the 
classes  F'  and/7  respectively,  is  proportional  to  F/,  that  is 
to  the  number  of  such  pairs  which  exist — and  is  not  generally 
proportional  to  F///.  This  assumption,  though  I  deny  its 
legitimacy,  justifies,  if  true,  the  H  Theorem.  But  if  it 
applies  to  the  direct,  it  cannot  apply  to  the  reversed  motion, 
because  the  pairs  of  molecules  which  in  the  reversed  motion 
pass  from  the  classes  Fand/'  to  the  classes  F'  and/',  are  the 
identical  pairs  which  in  the  direct  motion  passed  from  F' 
and  /'  to  F  and  /',  and  their  number  is  therefore  by  the 
assumption  proportional,  not  to  F/,  but  to  Wff.  Therefore 
the  H  Theorem  is   inapplicable  to  the  reversed  motion  by 
