Prof.  Jeans'*  Dynamical  Theory  of  Gases.  461 
failure  of  the  assumption  on  which  it  is  based.  And  the 
mathematical  correctness  of  Boltzmann's  theorem,  granted 
his  assumption,  is  preserved.  The  doubt  is  whether  the 
assumption  itself  is  valid. 
12.  It  is  proved  only  in  the  normal  state  that  u,  v,  w  are  as 
likely  to  occur  as  —  u,  —v,  —w.  We  have  then,  it  may  be 
said,  no  right  to  assume  it  in  an  analysis  of  the  H  Theorem, 
because  that  theorem  deals  only  with  systems  in  abnormal 
states.  I  understand,  however,  Jeans's  meaning  to  be  this. 
According  to  the  H  Theorem  the  system,  given  in  an 
abnormal  state,  passes  into  the  normal  state.  Bat  when  it  is 
normal,  u,  v,  id  are  as  probable  as  —  u,  — v,  — w,  and  therefore, 
the  motion,  being  continuous.,  the  passage  from  the  abnormal  to 
the  normal  is  just  as  frequent,  and  no  more  so,  than  the  reverse. 
In  this  form  the  argument  is  good  against  Boltzmann  as  an 
argumentum  ad  hominem.  But  it  makes  the  continuity  of 
the  motion  a  vital  point.  For  if  there  be  any  the  least 
discontinuity  in  the  original  motion,  caused,  say,  by  some 
external  disturbance,  it  ceases  to  be  true  that  the  system, 
starting  with  velocities  — u,  — v,  — w,  will  retrace  the  course 
of  a  system  which  ended  with   it,  v,  u\     And  therefore  we 
7TT 
cannot  prove  that  on  average  of  both  motions    y-  is  zero. 
The  discontinuity  would  be  fatal  to  Jeans's  argument,  but  not 
to  the  H  Theorem. 
13.  I   agree  with  Jeans  that  as  a  fact  on  average  of  all 
systems  and  of  all  time,  —  =0,  provided  that  the  motion  be 
at 
continuous   and    protected    from    external    disturbance.      I 
agree  with   him  also  that,  as  a  fact,  on  the  average  of  all 
systems  and  all  time,  the  states  it,  v,   w,  and   —  it,  —  r,  —  w 
occur  with  equal  frequency.     I  agree  further  with  him  that 
on  average  of  all  systems  and  all  time  H  is  nearly  minimum. 
I  expressed  that  myself  in  the  form  that  when  H  differs  little 
c         -l       •   •  dll        ,    d'2K  n      i  -p 
from  its  minimum,  — —  and    —rw  are  very  small,  whereas  if 
dt  af 
H  differs  much  from  its  minimum     -    raav   be  very  great. 
dt  J   & 
For  by  consideration  of  the  direct  and  reverse  motions,  we  see 
that  the  number  per  unit  of  time  of  collisions   by  which  two 
molecules  pass  from  the  classes  F  and  f  to  the  classes  F'  and 
/v,  is  proportional,  not  indeed  necessarily  to  F/ as  Boltzmann 
assumes,  but  either  to  F/  or  else  to  Ff,  and  as  often  to  one  as 
the  other.     But  when  [[   is  nearly  minimum  FY'  — F/'  is   in 
