258 PHENOMENA DEPENDENT ON MOLECULAR PATHS 9o 



itself during the time in which the molecule wanders past 

 it, and therefore comes more easily into the condition of 

 colliding with it. The probability is increased in the same 

 measure as the relative velocity of the two moving particles 

 with respect to each other is greater than that of the 

 particle moving alone. In the case supposed the magnitude 

 of the relative velocity is easily obtained ; the absolute 

 velocities of the tw T o particles (since they are perpendicular 

 to each other) form the shortest pair of sides of a right- 

 angled triangle, the hypotenuse of which represents the 

 relative velocity of each with respect to the other. If these 

 absolute velocities are represented by l } and O 2 as hitherto, 

 the magnitude of the relative velocity is 



+ a, 2 ). 



This consideration may be directly extended to the more 

 general case such as really occurs. If the molecules do not 

 move only at right angles to the particles coming in among 

 them, but hither and thither in all possible directions, the 

 probability of a collision is increased by this motion in 

 exactly the same measure as by that to-and-fro motion 

 which we have hitherto assumed. We have, therefore, in 

 this case too, nothing further to alter than to substitute the 

 .above expression for the relative velocity instead of the 

 absolute velocity of 'the particles which throng into the 

 medium at rest. 



This consideration, which Maxwell 1 seems to have 

 employed several times, puts us now in a position to find 

 an expression for the free path not only for a homogeneous 

 gas, but also for the case before us of a molecule of one gas 

 moving in a different gas. 



The probable number of collisions experienced by a 

 particle in unit time is equal to the mean number of the 

 other particles which come within the range of its sphere 

 of action during this time as it moves along. The path of 

 the particle in unit time is measured by its velocity, for 

 which, in the case in which all the particles are in motion, 

 we must substitute the expression for the relative velocity 



1 Phil. Mag. [4] xix. p. 28 ; Scientific Papers, i. p. 387. 



