28* MOLECULAR FREE PATHS 413 



the case when all the particles are in motion than in the other. 

 The value of this free path is simply found by dividing the whole 

 distance travelled in the unit of time (which is measured by the 

 velocity) by the number of collisions experienced in the same 

 time. We thus obtain 



for the value of the free path of a particle in a swarm of particles 

 at rest, but 



G 3 



A 



for its value in a swarm of particles in motion. This calculation 

 shows the correctness of the value given in Chapter VI. 67 for 

 the ratio of the free paths in the two cases. 



29*. Mean Collision -frequency according to 

 Maxwell's Law 



The assumption that all the particles possess equal velocities 

 is not, however, strictly true : we should rather take Maxwell's 

 law, proved in Appendix II., according to which, if there are N 

 molecules of a gas in unit volume and the gas has no progressive 

 motion, the number of them with velocity-components U, V, W is 



Introducing this value of n into the formula ( 27*) 

 irn&r = TmsV {(u - U) 2 + (v - V) 2 + (w - TF) 2 }, 



which gives the frequency of collision of a particle, whose velocity- 

 components are u, v, w, with n others which move about in unit 

 volume with velocity-components U, V, W, and integrating we 

 obtain 



for the number of collisions which a particle with velocity-com- 

 ponents u, v, w makes in one second with all the N molecules 

 contained in unit volume. Since 



r = V {(u - U) 2 + (v- VY + (w - T7) 2 }, 



this value B depends on u, v, w, and consequently on the velocity 

 <o = \/ (u 2 -f- v 2 + w 2 ) with which the particle in question moves. 



