428 MATHEMATICAL APPENDICES 37 



37*. Free Path of a Particle with a Given Speed 



The values of the molecular free paths calculated in the pre- 

 ceding article are of the nature of average values, since they are 

 deduced from the mean value of the speed li and from that of 

 the collision-frequency r. We may therefore, for instance, look 

 on the value 



found for a simple gas, as a mean value of the paths which are 

 traversed by the whole lot of particles moving with different 

 speeds. But it is in no way to be considered as the probable or 

 mean length of free path which any one single particle, moving 

 with a particular speed, passes over without a collision. 



To find the probability that any particle moving with speed w 

 traverses a path of length x (or rather of a length between x and 

 x + dx) between successive collisions, we go back to the formulae 

 of 26*, which give 



e-* ll dx/l 



for this probability, I being the mean length of the paths traversed 

 by the molecules which move with the speed w. Since a particle 

 with speed w collides on an average B times in unit time with 

 other particles, where B has the value given in 32*, the path 

 travelled in unit time is 



- El = a). 



Thus the mean free path of the particle with speed w is 



I = w/B, 



and the probability of a length x being traversed with speed w 

 without disturbance, and for a collision to occur at its extremity, is 



(B/ w )e- B *^; 



also the probability of the particle's traversing a path which 

 exceeds the limit x is 



So as to show more clearly how the probability and mean free 

 path I depend on the speed w, in accordance with the above 

 formulae, I have calculated a few values of B/T and the corre- 

 sponding values of l/L = (r/B) (<*>/&) from the formula 



B /r = 1 



