636 Prof. 0. W. Richardson on Distribution of the 

 from (2). This equation can now be integrated, giving 

 Ui-U 2 



M>! — W 2 



N 



3BT{2ip + logg^} 



=m(-H ( ,,-,,^Xr^ 



I 16 



, 9 1 f'i My 



2^2 3 Jo ^-1 B l-^ 2 ) 



) 



The values of Y a and V 2 corresponding to x x and x 2 respec- 

 tively can be found by eliminating U\ and U 2 from the 

 corresponding pair of equations (2) and (3). Thus, by 

 substitution in equations (13), w 1 —w 2 will be expressed as a 

 function of V l5 V 2 , and T. This relation involves in addition 

 only the universal constants h, Is, and R and the molecular 

 weight M of the gas. In general, the necessary algebraic 

 processes are of an involved character and the solutions can 

 only be obtained graphically. 



There are, however, three comparatively simple cases : — 



(1) When Xi and x 2 are both small, corresponding 

 either to small densities or to high temperatures or to 

 both these conditions. 



In this case equations (2), (3), and (13) reduce to 

 equation (1) in agreement with the classical dynamics. 



(2) When X\ and x 2 are both large, corresponding 

 either to high densities or to low temperatures or to a 

 combination of these conditions. 



In this case equations (2), (3), and (13) lead to 



w 1 — w 2 = 



1 K T? _2 _ 



.±^ 5 (Y*—Y 

 "16 c {y/2 Vl 



" f ) + ^Rc 3 T 4 (V 2 2 - 



-V: 2 ), 



(14) 



ere 



c 



3 terms ] 



8 MR /4tt \f 

 "5 m 2 V9N/ " 



neglected in (14 



3NR 



= 8 a * * * ' 



) involve e~ Xl , e~ 



. . . (15) 

 x 2, and their 



positive integral powers. 



(3) When x L is very large and x 2 is very small. This 

 case can arise when equilibrium occurs between two 

 phases of a gas in which the concentrations are enor- 

 mously different ; for example, a gas in contact with a 

 liquid in which it is exceedingly soluble, or the atmo- 

 spheres of electrons within and without a conductor. 



