Prof.  Jeans' 's  Dynamical  2  heory  oj  Gases.  457 
This  means  that  the  molecules  are  scattered  at  random 
through  the  space  XI.  The  most  probable  distribution  is 
found  by  making  Q  maximum,  and  is  when 
2  as  log  as  is  minimum,  that  is  when  a,  =  a2  =  .  .  =  an. 
s=l 
Subsequently  (art.  55)  he  discusses  the  Partition  of  the 
generalized  space  as  regards  the  velocities.  This  he  says  is 
to  be  effected  in  the  same  way  as  in  the  case  of  the  position 
coordinates.  That  is,  we  are  to  suppose  another  three- 
dimension  space,  or  diagram  of  velocities,  and  u,  v,  ic  are  the 
coordinates  in  that  diagram  of  points  scattered  at  random 
through  it,  but  they  are  to  be  subject  to  one  new  condition, 
namely,  that  %m(u2  +  v2  -f  id1)  =  constant  (equation  08). 
This  new  space,  Jet  us  call  it  IV,  may  be  divided  into  n  equal 
cells,  ft)/  a)/  .  .  &>„',  each  of  volume  ft)',  and  the  number  of 
ways  in  which  N  molecules  may  be  distributed  through  12', 
so  that  there  shall  be/i  in  &>/,  f2  in  a>2',  &c.,with  S/=  »,  is 
N  ! 
0'  = 
A I  /.!..-. /J' 
It  is  shown  that  6'  is  maximum  when 
}\\f\ogfdu  dv  div  is  minimum. 
So  the  solution  is  given  by  making 
\\\  f  ^°S  fdu  dv  dw  minimum, 
given 
(1)  (*(T /  du  dv  dw  =  1  (97).  expressing  only  the  fact  that 
the  number  of  molecules  is  constant  ; 
(2)  jjf  m  (u2  +  ^2  +  iv2)  du  dv  dw  =  constant  (98). 
6.  The  usual  method  leads  to 
No  mention  has  been  made  of  any  possible  relation 
between  velocities  and  space  coordinates,  such  as  that 
referred  to  by  Jeans's  art.  15.  If  any  such  existed  it 
would  manifestly  vitiate  the  reasoning.  It  is  therefore 
assumed  by  necessary  implication  (a)  that  no  such  relation 
exists,  and  therefore  that  the  chance  of  a  molecule  having 
velocities  within  assigned  limits  is  independent  of  the  positions 
of  all  the  other  molecules  for  the  time  being.  Further,  it  is 
expressly  stated   that  the  only  relation  existing   among   the 
Phil.  Maq.  S.  6.  Vol.  11.  No.  61.  April  1906.       2  H 
