244 Free Path Phenomena [CH. xn 



297. As regards any single molecule which meets the plane z = z at any 

 point P, the amount of /* carried across the plane z = z will of course depend, 

 in actual fact, upon the whole past history of the molecule before reaching 

 the point P. We are going to conduct our preliminary calculations upon the 

 supposition that the history of the molecule previous to the last collision 

 before meeting P, say at Q, is immaterial. This would be justifiable if, on 

 the average, all past histories previous to the point Q were equally probable. 

 This, unfortunately, is not so when the molecules are elastic spheres. From 

 the persistence of velocities investigated in the last chapter, it follows that a 

 molecule which is known to have arrived at P from Q has probably started 

 originally from some point further away from P than the point Q. Since, 

 however, the amount of the persistence depends in general on the particular 

 molecular structure assumed, it will be simplest to neglect it altogether at 

 first, and subsequently correct our results for it as far as possible. 



298. Consider a molecule meeting z = z in P, having previously come 

 from a collision at Q. Let the velocity components of the molecule be 

 u, v, w, and let the velocity be regarded as consisting of two parts : 



(i) a velocity u , of components u ot 0, 0, equal to the mass-velocity of the 

 gas at P; 



(ii) a velocity c, of components u u , v, w, the molecular- velocity of the 

 molecule relatively to the gas at P. 



Let QP be the actual path described, and let this make an angle % with 

 the axis of z. Let the length of the path described relatively to the motion 

 of the gas at P be \ r , and let this make an angle with the axis of z. The 

 direction-cosines of the direction of \ r are of course proportional to the com- 

 ponents of the relative velocity c, and therefore to u u , v, w. Since all 

 directions of this molecular velocity may be regarded as equally probable, 

 the probability of 6 lying between 9 and 6 + d6 is proportional to sin 6 dd. 



Also since the projection of the relative path on the axis of z is equal to 

 that of the absolute path, 



PQ cos x = >W cos e (562). 



The z coordinate of Q is therefore 



z - \rcos.0 (563). 



We are working upon the hypothesis that the expectation of p for the 

 molecule in question is that appropriate to the point Q. We shall therefore 

 take it to be the mean value of /* at the point Q. 



Since /Z, the mean value of /*, is a function of z only, we can denote the 

 value of /Z over the plane z = by /Z()> and the value of p, at Q will now be 



j,z \ cos 8. 



