290-292] Persistence of Molecular Velocities 241 



of which the value, when integrated from K = 1 to K, = oo , is unity, as it 

 ought to be. 



From expressions (555) and (556) we find that when two molecules 

 collide, having velocities in the ratio K : 1, the "persistences" of the two 

 velocities are 



15/c 4 + l , 5* 2 + 3 



10 2 (3* 2 + 1) 5(3 2 + 1) 



respectively. Hence the mean " persistence " for the two molecules of the 

 collision, the mean of the two expressions just written down, is 



20/c 2 (3/c 2 + 1) ' 



Multiplying this by expression (561) and integrating from K = 1 to K = oc , 

 we find, for the mean persistence of all velocities after collision 



p (25/e 4 

 Ji 4 V2 t 



i 4 V2 tc (1 + * 2 ) 7 / 2 

 of which the value is found to be 



, 



or '406. Thus the average value of the persistence is very nearly equal 

 to f , the value when two molecules collide with equal velocities. 



292. For other laws of force between molecules the persistence will 

 clearly be different from what it is for elastic spheres. Everything will, 

 however, depend on our definition of a collision. If we suppose that a very 

 small interaction is sufficient to constitute a collision, then the mean free 

 path will be very short while the persistence will be nearly equal to unity. 

 If, on the other hand, we require large forces to come into play before calling 

 a meeting of two molecules a collision, then the free path will be long, but 

 the persistence very small, or perhaps negative. The variation in the value 

 of the persistence of velocities just balances the arbitrariness of the standard 

 we set up in defining a collision. 



J. 16 



