246 Free Path Phenomena [OH. xn 



299. The calculation of the transport of momentum across the plane 

 z = z in the reverse direction is exactly similar, and we obtain for this 

 quantity the expression 



gj \dt (566). 



On the whole, then, the net excess transfer from above to below, which is 

 the difference of the two expressions (565) and (566), is found to be 



dt (567). 



We can conveniently suppose that 



\c = cl (568), 



where I is a new quantity, which is of course the mean free path of a 

 molecule, this mean being taken in a certain way. The way in which the 

 mean has to be taken is not the same as the way in which it was taken in 

 284, so that we do not obtain an accurate result, in the case of elastic 

 spheres, by replacing I by the X of equation (547). At the same time the mean 

 values calculated in different ways will not greatly differ from one another, 

 and as our present calculation is at best one of approximation, we shall be 

 content for the moment to suppose I to be identical with the mean free path, 

 however calculated. The extent of the error involved in this procedure will 

 be examined later. / 



300. It might be thought that I ought to be replaced by |X instead of 

 by X. For if OQ is the whole free path, tnere is no reason why PO should 

 be less than PQ, so that the probable value of PQ might be thought to 

 be i\. 



The fallacy in this reasoning becomes obvious when we consider that in 

 selecting free paths at random by choosing points on these free paths, the 

 longer free paths have a greater chance of being chosen than the shorter 

 ones, the chance of any path being chosen being in fact exactly proportional 

 to the length of the path. The average path chosen in this way, accordingly, 

 will be much longer than that calculated in 284. To see that \ is the 

 right value to assign to I, we notice that after a molecule has left 0, its 

 chances of collision are exactly the same as if it had just undergone collision 

 at P. Hence PO = X, and therefore, by a similar argument, PQ = X. 



A simple example taken from Boltzmann's Vorlesungen* will perhaps 

 elucidate the point further. In a series of throws with a six-faced die the 

 average interval between two throws of unity is of course five throws. But 

 starting from any instant the average number of throws since a unit throw 

 last occurred will be five, and similarly, working back from any instant, the 

 average number of throws since a unit throw occurred is also five. 



O 



* i. p. 72. 



