280, 281] Calculation of Mean Free Path 231 



again integrate with respect to c, we obtain for the aggregate chance per 

 unit time of a collision between a given molecule of the first type moving 

 with velocity c, and a molecule of the second type, 



r f re 2c'- (c' 1 + 3c 2 ) ~l 



2v.,S 12 " V7rh 3 m. 3 I -ic (c c + 3c' 2 ) e~ hm * c ~ dc + I e km z c- ^ c f 



\_Jc Jo 3c J 



(532). 



281. The former of the two integrals inside the square bracket can be 

 evaluated directly, and is found to be equal to 



e -hw 



The second integral cannot be evaluated in finite terms. If, however, we 

 replace hm. 2 c~ by x 2 , the integral reduces to 



2 (x- + 3/wwoC'-) e~ x " dx, 



and this again, after continued integration by parts with regard to a? y is found 

 to be equal to 



chm- ] 



~ x * dx . 



( fc^hm-t 



J _ i e -hm^ c V/ tm , (4/im.,c 2 + |) + i (2/i?/i :j c 2 + 1 ) 

 ( J o 



c V h 

 The sum of the two integrals in expression (532) is accordingly 



1 f /-cVfeWa 1 



^= \G \fhm 2 e~ hm ^ + | (2Am 2 c 2 + 1 ) e~ x ' 2 dx\ ...... (533). 



h*m./ L J o J 



c V h*m. 

 If we introduce a function* i/r (x) defined by 



(534), 



expression (533) may be expressed in the form 



- > ilr (c V/iW 2 ), 

 2c \/h 5 m 2 5 



and hence if we denote expression (532) by n , its value is found to be 



(535). 



With this definition of @ 12 we see that when a molecule of the first kind 

 is moving with a velocity c, the chance that it collides with a molecule of the 

 second kind in time dt is 12 dt. 



* The value of I e^ 2 dx cannot be expressed in simpler terms, so that ^ (a;) as defined by 



/ 

 equation (534) is already in its simplest form. Tables for the evaluation of ^ (.c) are given in an 



appendix. 



