250 Free Path Phenomena [CH. xn 



It follows that if we trace the motion a sufficient distance back, each 

 molecule must be supposed to have come, not from a distance measured 

 along the axis of z, but from a distance 



(576), 

 of which the value is 



1-8' 

 If we take # = f, this becomes 



It would not, however, be legitimate to assume that each molecule on 

 arriving at the plane z = z has, on the average, a value of yu. appropriate to 

 the plane z = z %. For the molecule has not travelled a distance f 

 undisturbed, and at each collision a certain amount of its excess of momentum 

 will have been shared with the colliding molecule. Of the various simple 

 assumptions possible, the most obvious one to make is that at each collision, 

 the excess of momentum above that appropriate to the point at which the 

 collision takes place is halved, half going to the colliding molecule and half 

 remaining with the original molecule. Making this assumption, it is clear 

 that the excess of momentum to be expected is not that due to having 

 travelled undisturbed a distance equal to that given by expression (576), but 

 a distance 



r+K0r+ *(*+*(*?+))) .................. (578), 



of which the value is 



Again taking 6 = f , this becomes 



K .................................... (579). 



It follows that the persistence of velocities, when the molecules are elastic 

 spheres, can roughly be allowed for by supposing the free path in the viscosity 

 formula to be the mean free path multiplied by a factor 1*25. Combining 

 this with expression (539), we find that the free path in the viscosity formula 

 must be taken to be 



V27TI/0- 2 



(580). 



Hence for a gas, in which the molecules are elastic spheres, the viscosity 

 coefficient (equation (573)) is given by 



..... (581). 



Vtorvo* 



