156 PHENOMENA DEPENDENT ON MOLECULAR PATHS 63 



or by the ratio of the central section of the sphere of action 

 to the face of the elemental cube. 



On the other hand, the probability that the particle does 

 not undergo collision in its path X, but passes through a 

 layer of thickness X without colliding with the other particles 

 within it, is 



We may therefore, to use an ordinary expression, bet 

 7rs 2 /(X 2 7r$ 2 ) to 1 that the particle will undergo collision 

 before it passes over a distance X, and (X 2 ?rs 2 ) / vrs 2 to 1 

 that it will not collide in this distance. 



In these formulae there lies a simple meaning. If we 

 suppose that the molecule struck were pushed from the 

 interior of the elemental cube containing it into the same 

 face through which the moving particle entered, it would 

 cut out of this face, whose area is X 2 , a portion equal to Trs 2 , 

 through which the entering molecule would not be free to 

 pass ; only the remainder X 2 Trs 2 would allow free entrance 

 for the molecule. The two probabilities therefore have, as 

 indeed they must have, the same ratio as the not-free part to 

 the free part of the face of the elemental cube. 



64. Probability of a Longer Path being Traversed 



From this value of the probability for the traversing of 

 a path of length X, or for the passage without collision 

 through a gaseous layer of a thickness X equal to that 

 separating two neighbouring molecules, may be easily 

 calculated the probability for the passage through a thicker 

 layer or for a path of finite magnitude. 



For this we suppose M moving particles, instead of one, 

 to be simultaneously projected into the medium, consisting 

 of particles at rest, which we suppose to be divided into 

 a number of layers of thickness X. Of these M particles 

 MW/X 2 will probably undergo collision in the first layer, 

 while only the remainder M(l 7rs 2 /X 2 ) will pass through 

 it unhindered. Of these the number that collide in the 

 second layer of the same thickness X is M(l 



