408 MATHEMATICAL APPENDICES 26* 



If we denote by a the probability that this molecule will 

 traverse a path equal to 1 unhindered, a is a proper fraction 

 which, from the assumption made, is so far of constant magnitude 

 that for every position of the starting-point it has one and the 

 same value. If the gas as a whole has no motion of translation, 

 the value of a is also the same for every direction in which the 

 molecule considered can move. 



It therefore follows that the probability of traversing a path 

 equal to 2, that is, the path 1 twice over, is a.a or a, 2 . So, too, 

 the probability of its traversing without collision a path three 

 times as long is o?\ and we thus see that in general the pro- 

 bability of an unhindered passage through a length x is given by 



the function 



a*, 



which we may more conveniently write 



where e is the base of natural logarithms and 



i=- JL 



loga' 



so that, as a is a proper fraction and thus log a negative, I is 

 positive. 



This formula agrees in form and meaning with the expression 

 established in the elementary theory ( 66), viz. 



in which q denotes the ratio of the path traversed to the mean 

 free path. We can also now easily see that the constant I means 

 nothing else than the mean probable value of the molecular free 

 path which the molecule considered can attain. 



For out of n molecules which move in the same way as the 

 given molecule, that is, with the same velocity and in the same 



direction, the number 



ne -xii 



traverse the length x without collision, but only 



ne 



(x + dx)ll 



pass over the length x + dx ; hence in the length dx 



= ne 



