64 MOLECULAR FREE PATHS 157 



while M(l 7rs- 2 /X 2 ) 2 particles pass through this layer. 

 So in the third layer M(l - 7r* 2 /X 2 )W/X 2 collide, and 

 M(\. 7rs 2 /X 2 ) 3 pass through. Proceeding in this manner 

 we see that in the xth layer there are probably 



that collide, and thus have traversed a path of length x\ 

 while the remainder M(\. Trs^/X 2 )* do not probably suffer 

 collision, and therefore traverse still longer paths. 1 



From this we at once obtain the probability that a 

 single molecule will pass over a given path and then collide, 

 by dividing the probability in the case of M particles by the 

 number M, since the probability for one particle must be 

 M times less than for M particles. The probability, there- 

 fore, that a moving particle traverses a path x\ and collides 

 on its completion is 



65. Calculation of the Mean Free Path under 

 Simplified Assumptions 



From the foregoing formulae we can calculate by elemen- 

 tary methods and without great difficulty the probable mean 

 value of the lengths of the paths traversed by all the 

 particles. To find this it is only requisite to calculate the 

 sum of all the different paths traversed by the M molecules 

 and to divide it by their number, that is, by M. 



Of the M particles there remain M7rs 2 /\ 2 in the first 

 layer, and these therefore traverse only the path X ; thus the 

 sum of the paths traversed by these molecules is MW>/X. 

 Similarly, there remain in the second layer, after completing 

 the path 2X, the number M(l - *rs?l\*)irs?l\*, the sum of 

 whose paths is therefore 2Jf(l - 7r* 2 /X 2 )7r* 2 /X. In this way 

 we find in general that the sum of the paths of the particles 

 which collide in the xth layer is xM(l rr^/X 2 )*- W/X, and 

 the total sum of the paths traversed by the whole of the M 

 particles is therefore 



2.xM(l - 7r* 2 /X 2 )*- W/X, 



1 This formula is developed in 26 Mn a mathematically simpler form. 



