158 PHENOMENA DEPENDENT ON MOLECULAK PATHS 65 



where the sign of summation denotes a summation for all 

 integral values of x from 1 to oo. The mean probable 

 length of the free path L of a molecule is therefore given 

 by the formula 



L = 



This sum may be easily calculated. For 



or, n our case, 



1 = (X 2 / 2 ) 2 . 



Hence it results from this calculation of probability that 

 the length of the path which a moving particle would 

 traverse without collision amid a multitude of particles at 

 rest is on the average 



L = \*/7TS 2 . 



This formula, which Clausius deduced in the memoir 

 referred to in a similar way, but with the use of the integral 

 calculus, assumes an intelligible form if we write it 



L : X = X 2 : TT^ ; 



it then expresses that the mean free path bears the same 

 ratio to the mean distance separating two neighbouring 

 particles as the area of a face of the elemental cube has to 

 the central section of the sphere of action. 



From this proposition Clausius draws a very important 

 conclusion. The above proportion shows indeed that the 

 free path L is greater than the distance of molecular separa- 

 tion X, and that it must be very much greater than the 

 latter in a rarefied gas. For by definition X is the edge of 

 the elemental cube in which a single molecule is contained, 

 and s, the radius of the sphere of action, is a distance 

 within which the force exerted by the molecule is sensible. 

 It would be in contradiction of our theory, no less than of 

 experiment, which has shown an almost perfect absence of 

 cohesion in gases, if we were not to assume the latter length 

 s to be considerably smaller than the former X ; consequently 

 also the proportion shows that L is considerably larger 

 than X. 



