458 MR. S. CHAPMAN ON THE KINETIC THEORY OF A GAS 



Since the forces are attractive, - is negative, so that 



ar 



(51) , rt 



is positive ; we shall denote it by ffl for convenience. If we write p v for 



< r + <r /;,2 , v ay/ 2 

 ~~~~ ~ 



o 

 o 



the condition for a collision becomes 



P<Pv a - 



Next, by elementary geometry, the angle x, between the radii vectores r = <T+<T' 



and r = , is given by 



dd 

 tanx-r^, 



so that for p < p Va we have 



fi/tM* J- 1 yv, 



nn' v = < - I -I- I ? - - i! 



Consequently we have 



Q' 13 (V ) = 47rV f sin 3 X -pdp = 4^V f ?v -^ ^> d!p +,4,rV f sin 2 x . 

 Jo ,Jo _pv J p v 







where in the latter integral x is given by equation (50). We denote the latter 

 integral by f' 12 (Vo)> the form of the function depending on the law of attraction 

 between the molecules. 

 Thus 



v ( v fl ) = -v Or+Trff+v.') + 4xVo /' 12 ( V B ). 



v o 

 Similarly we have 



Q"(v.) = ^Y O (*+V(+ y o 2 ) +7r y /^ 13 (v ). 



where 



/"(V )= f sm*2 x .pdp. 



J rvo 



Substituting these values in the integrals P, E, S, we find that 

 F 12 = 7 r(^+ ( r7JrV 5 e"'^ ;Vo ^V +6 2 f V 3 <f ^ Vo '(2V }+ f/' ia (Vo) V 5 e~^' Vo2 rf V,,, 



Uo Jo J Jo 



where P^j^denotes the last integral on the preceding line. 



