38* MOLECULAR FREE PATHS 431 



are traversed between successive collisions by the particles in 

 question is 



The sum, therefore, of all the paths which all the N particles 

 traverse in a straight line, i.e. between successive collisions, is 

 given by the integral 



From this total length of the paths of all the particles we obtain 

 their average length, which we shall express by M(l), by dividing 

 by the number of particles N, viz. 



The mean value given by this formula is thus expressible as 

 the arithmetic mean of all the values of the free path I at any 

 moment for the whole number N of the molecules contained in 

 unit volume. We may thus take all the N particles as starting 

 at a given moment, each with its speed w, and then determine the 

 mean value of the lengths of the paths attained at this single 

 start. 



We must distinguish this mean value from that which we 

 obtain by considering the paths traversed by the particles in the 

 course of a prolonged time. To find the mean in this other case 

 we have to consider not only a single path traversed by any 

 particle, but the whole of the B paths which it passes over back- 

 wards and forwards in the unit time. The sum of all the paths 

 traversed in unit time is therefore given by the integral 



which is at once integrable, and leads to the value 



= NCI. 



But, according to 32*, the number of these paths is 



