39* MOLECULAR FREE PATHS 433 



mean values, just as in the case of the free path. We might first 

 consider all particles to start new paths at the same moment, and 

 then ask how long the interval is, on the average, before one of the 

 particles undergoes collision. Since a particle whose speed is w 

 collides on the average B times in unit time, the time that passes 

 between successive collisions of this particle is on the average 



Hence we should obtain the mean value for all the particles by 

 multiplying this expression by Maxwell's expression for the 

 probability of occurrence of the speed w, and by integrating from 



w = to w = oo. 



The calculation is much simpler if we do not aim at finding 

 the mean interval from any one collision to the next, but seek the 

 average interval between two collisions that occur within a finite 

 period, as, for instance, the unit of time. We then bring into 

 reckoning, not the time of a single path of each molecule, but the 

 time of all its paths ; and to find the mean value divide, not by 

 the number of particles, but by the number of paths. The mean 

 value of the time between two successive collisions is then nothing 

 else than the ratio of the whole interval to the mean number of 

 the collisions that have occurred in this interval, viz. 



or the ratio of the mean free path to the mean speed. 



This second mean value T is less than that first named. For 

 in its calculation the interval 1/B is taken, not once, but B times 

 for each particle, so that a smaller interval is taken oftener, and a 

 larger interval more seldom. 



4O*. Calculation of the Pressure 



Since the collision-frequency B is a transcendental function of 

 the speed w, the theory frequently leads to formulae that seem very 

 complicated. But in a series of cases the calculation gives quite 

 simple results. 



As an instance, I proceed to calculate anew the pressure exerted 

 by a gas, and this calculation can of course lead to no other result 

 than that given by the general theory which was investigated in 

 the first of these Mathematical Appendices. 



We seek the pressure at an element of surface df, which we 



F F 



