240 Free Path Phenomena [CH. xi 



We have now proved the statement made in 287, to the effect that there 

 is a tendency for the original velocity to persist after collision. It will be 



convenient to refer to the ratio - as the "persistence" of the velocity a. 



The persistence, then, is measured, as we have found, by a quantity which 

 varies from 33^ to 50 per cent, of the original velocity, according to the 

 velocity of the colliding molecule. 



291. By averaging over all possible velocities we can obtain the mean 

 value of this persistence averaged over all collisions. 



From expression (529), we find as the chance per unit time that a 

 molecule moving with velocity a shall collide with a second molecule moving 

 with velocity b, in such a direction that their relative velocity is V, 



2va n - frttm* e~ hmb * -dbV*-dV... . . .(557), 



a 



and hence for the total number of collisions of this type in the gas, per unit 



time, 



8v*(T*h 3 m s e- hm < a * +b *>abdadbV 2 dV .................. (558). 



Let us suppose that of the two velocities a and b, a is the greater. Then 

 the limits for V are a + b and a b. Integrating expression (558) with 

 respect to V we find as the total number of collisions per unit time in which 

 the velocities of the two colliding molecules are a, b, 



iv*a*h 3 m 3 e - hm W + W atf (b 2 + 3a 2 ) da db. 

 If we put a = xb, so that K is greater than unity, then since 



_ J 



d (*, 6) 



this becomes 



ig-itaWm 3 e -* m <* + *> tcb 6 (3/c 2 + 1) die db ............ (559). 



Integrating with respect to b from 6 = to b = oo , we find, for the total 

 number of collisions per unit time in which K, the ratio of the greater to the 

 less velocity, lies between K and K + die, 



/i j7/a 



hm (1 + /c 2 ) 7 ' 2 



Since the total number of collisions, by formula (46), is 



. 



~ v a > 

 hm 



the fraction of the whole number of collisions, for which K lies between K and 

 K + die, is 



a +l) rf/c ...(561), 



