Prof.  Jeans' s  Dynamical  Theory  of  Gases.  45D 
cell  above  the  base  of  the  column.  We  might  say  that  the 
value  of  a  cell  for  the  purposes  of  this  distribution  is  propor- 
tional to  e~2h3s.  On  this  principle  we  find  that  the  density  at 
height  s  is  proportional  to  e~2h-gs  as  Boltzmann  teaches,  and 
is  not  affected  in  any  way  by  Jeans's  law  of  distribution. 
9.  In  arts.  62-65  Jeans  discusses  the  assumption  of 
molecular  chaos.  The  number  of  molecules  whose  space  co- 
ordinates lie  between  x  and  x  +  dx,  &c,  and  their  velocities 
between  u  and  u  +  du,  &c,  whatever  be  the  positions  and 
velocities  of  the  other  molecules,  is  denoted  by  vP.  It  can 
evidently  be  written 
vp=  N/(w  v  w)  du  dv  dw  dxdydz  \\\  dxb  dxc  .  .  dyb  dyc  .  .  , 
in  which  xb  yb  zby  xc,  &c.  denote  the  space  coordinates  of  the 
molecules  not  included  in  vp.  Similarly  the  number  whose 
position  coordinates  lie  between  x'  .  .  x  -\-  dx\  &c,  and  their 
velocities  between  u'  .  .  it'  +  du',  &c,  is  given  by 
Vp—  N/(?/  v'  id')  du!  dvf  dw'  dx'  dy'  dzf  jjf  dxc .  .  dyc .  .  dzc, 
xc  yc,  &c.  now  denoting  the  position  coordinates  of  all  the 
molecules  except  those  which  are  included  in  vp  or  vq. 
If  v  =  ("(T  dxdydz  fff  dudvdw,  over  all  the  molecules, 
u  includes  all  possible  states  of  the  system  and  means  the 
same  thing  as  the  volume  of  Jeans' s  generalized  space.     And 
— ,  —   represent  respectively  the  chances  of  a  molecule  being 
included  in  vp  or  in  vq. 
The  question  is  whether  these  chances  are  independent  of 
each  other.  Now  the  motion  being  continuous,  the  velocities 
and  positions  of  all  the  molecules  at  any  instant  are  deter- 
minate functions  of  their  velocities  and  positions  at  the  initial 
instant  t  seconds  ago.  Consider  two  molecules  m  and  m1  : 
let  u0  and  u0'  denote  their  initial  velocities.  If  m  and  m' 
exert  no  forces  on  each  other  during  the  time  /,  and  no  third 
body  sensibly  influences  both  during  the  time  t,  and  if  u  u' 
were  initially  independent,  then  at  the  given  instant  u  cannot 
be  a  function  of  u'.  If  <f>  (u)  du  be  the  chance  of  u  lying 
between  assigned   limits,    (f>  (V)  du    the   same    for    uf ,  it  is 
certain  that  d^M,=  0.      Similarly  #M_  0.      $  (u)  and 
du'  du 
cp  (u!)  are  then  independent  chances. 
Clearly,  then,  if  there   be  no  intermolecular  forces  at  all. 
the  chances  are  independent.     And  they  may  be  independent 
if  t  be  small  enough  and  the  initial  distance  between  m  and 
2H2 
