65 MOLECULAR FREE PATHS 159 



A molecule therefore passes by many molecules like itself 

 before it collides with another. 



66. Probability of Particular Free Paths 



Now that we have determined the mean value of the 

 molecular free path, the probability-formulae obtained in 

 64 present a simple meaning which makes it possible for 

 us to numerically answer the question, How much more 

 probable is a shorter path than a longer ? 



The expression M(I 7T9 2 /X 2 ) a; , which we have found 

 for the probable number of those among the M projected 

 particles that traverse a path greater than x\, becomes 



M(l r- X/I/)* 



when the mean free path L is introduced into it, so that we 

 can see how the number of the particles which collide in each 

 layer and that of those which pass through it unobstructed 

 depend only on the ratio of the average distance of molecular 

 separation to the mean free path. 



If we wish also to refer to the mean free path the actual 

 path traversed, which hitherto has been given by the 

 number x, we can put for the path 



y = x\ = qL, 



where the number q gives the number of times by which 

 the path already traversed by the particle under consideration 

 exceeds the mean free path. If we also put 



L = Q\, 



we shall have 



x=Qq, 



and the probable number of particles which do not undergo 

 collision in a path of length qL is given by 



Jf (1- 



We do not indeed know the number Q, i.e. the ratio of 

 the mean free path to the distance of molecular separation, 

 which occurs in this expression, but we do know that its 

 value must be very great, so great indeed that we may look 



