±62  Mr.  S.  H.  Burbury  on  the  H  Theorem  and 
general   very    small,    and  (FT7  —  F/)log  -^L  of  the  second 
'    dH  ■' 
order  of  small  quantities.     But    -—  consists  of  terms  either  of 
this  form  or  of  the  form  (¥f—  F'f)  log  p^  and  is  therefore, 
dm 
dt2 
whatever  its  sign,  very  small,  and  so  is 
It  follows  that  any  given  system  will  remain  for  very  long 
periods  of  time  with  H  nearly  minimum,  though  H  may 
occasionally  for  short  periods  deviate  into  higher  values.  As 
for  each  system  on  average  of  time,  so  at  each  time  on  average 
of  all  systems,  H  is  nearly  minimum,  which  agrees  with  Jeans's 
conclusion. 
Let  us  suppose  there  are  N  states  in  which  H  =  H0,  and 
is  minimum.  Of  these  N  states  there  are  say  n1  states,  each 
of  which  is  the  end  of  a  course  in  which  H  has  fallen  from 
Hj,  its  last  maximum,  to  H0,  and  n2  in  which  it  has  fallen 
from  H2,  its  last  maximum,  to  H0.  And  if  H2<Hl5  n.2>nl. 
Similarly  for  each  of  the  N  states  there  is  a  last  maximum  of 
H,and  N  =  ?z1-f  n2+  .  .  .  +n^.    Then  I  should  say  ^measures 
the  chance   of  H  having  the    value   H];  -—    the   chance    of 
its  having  the  value   H2,    and  so  on  ;     and  it    may  be  that 
n 
^  is  infinitesimal  for  all  values  of  H  not  infinitely  near  to 
.N 
H0.     That  gives  Jeans's  result. 
14.  If,  however,  there  are  in  fact  as  many  courses  in  which 
H  passes  from  H0  to  Hl5  as  in  which  it  passes  from  Ha  to  H0, 
it  is  not  true  that  the  variations  of  H  constitute  an  irreversible 
process.  It  is  not  necessary  to  assume  that  the  fall  from  a 
maximum  H1  to  H0  is  always  followed  by  a  rise  to  precisely 
the  same  value  H}  as  the  next  maximum,  so  that  the  system 
oscillates.  It  is  sufficient  to  say,  as  we  have  said,  that,  on 
average  of  all  systems  and  all  time,  motions  in  one  direction 
happen  as  frequently  as  in  the  other  direction. 
15.  Boltzmamr's  own  explanation  of  our  paradox  was 
that  the  reversed  motion,  in  which  H  increases,  although 
possible,  is  in  a  very  high  degree  less  probable  than  the 
direct  motion  in  which  H  diminishes,  and  on  this  he  founded 
his  doctrine  of  the  connexion  between  reversibility  and  proba- 
bility. I  think  this  explanation  cannot  be  reconciled  with 
the  theorem  ;  l'or  if  Maxwell's  law  holds  when  H  is  minimum, 
which  the  theorem  is  supposed  to  prove,  the  state  in  which 
the  velocities  are  typically  +  «,  -f  r,  and  -f  w,  and  the  state 
in  which  they  are  —  w,  —v,  and    ~w  are  equally  probable 
