436 ME. S. CHAPMAN ON T1IF, KINETIC THEORY OF A GAS 



of the relative velocity V (say) through an angle 2x in the plane of the <>rl>its (where 

 ^TT x is the angle between an asymptote mid the line of apses), the molecules 

 travelling away from one another along the second pair of asymptotes with the 

 relative velocity unchanged. The angle x is a function of p and V a which depends 

 upon the nature of the molecular forces. 



The velocity components (>/,.,, t> 12 , w 12 ) of a molecule m after collision with a 

 molecule m' can therefore be written down,* from merely geometrical considerations, 

 in terms of the original components (u, r, iv), (u 1 , v', it/), m, m', x> and e, the latter 

 being the angle between the plane of the orbits (which contains V and p) and a 

 plane containing V and parallel with the axis of x. Thus 



m' 



(2) r , = u+ - [2 (u'-u) sin 2 x-</{V 2 -(tt'-) 2 } sin 2 X cos (e-w,)], 



"* f 



m' 



(3) = + [ 2 (v' - v) sin 2 x-v/{VJ-( V ' - v)*} sin 2 X cos (,-,,)], 



7/1 ~r *lv 



(4) Wig = w+ - [2 (w'w) sin 2 x \/{^ (w 1 w) 2 } sin 2x cos (e w 3 )~\. 



~ 



The angle w l is introduced only for symmetry, as it is zero ; iv 2 and w 3 are given by 



(5) (u'-u) (v'-v) + S{[V a -(u'-uY] [V 2 -(?/-r) 2 ]} cos ^v 2 = 0, 

 and a similar equation in which iv'w replaces v' v. Of course, 



(6) V 2 = (u'-u) 



This notation having been explained, we return to the consideration of AQ ; we 

 divide this into two parts A n Q and A^Q, the former representing the part due to 

 collisions of the molecules m among themselves, the latter that due to collisions with 

 the molecules m'. Thus 



(7) AQ = A U Q + A 12 Q. 

 Then it is not difficult to provet that 



(8) A 12 Q= Jjj^ jjj + ] W '(* 18 Q)/(M, v, w)f'(u',v',w') du dv dw du'dv'dw^p dp d e , 



where <S 12 Q = Q t2 -Q in which Q 12 is the same function of (w, 2 , r }2 , w ia ) as is Q of 

 (u, v, w) ; thus c$ 12 Q denotes the change in the value of Q for a molecule m, produced 

 by a collision with a molecule m'. 



* See JEANS' "Dynamical Theory of Gases," pp. 284-288; BOLTZMANN'S 'Vorlesungen iiber Gas- 

 theorie, 1 vol. i., 21. 

 J See the original papers by MAXWELL, or the treatises by BOLTZMANN and JEANS, already quoted. 



