28* MOLECULAR FREE PATHS 411 



molecules possess, the components into which the velocity of any 

 particle can be resolved are expressed by the formulae 



U = G cos s 

 V = G sin s cos 

 W == G sin s sin $, 



in which s and denote two angles which determine the direction 

 of the motion. By formulae of the same kind we may express the 

 velocity-components u, v, w of the particle whose collisions with 

 others we wish to count ; but these formulae are substantially 

 simplified if we so choose the system of coordinates that one 

 of the three axes coincides with the direction of motion of this 

 molecule. We may therefore put 



u= G, v 0, w = 0, 



and the relative velocity of this particle with respect to the other 

 taken is 



r GV(2 2 cos s) = 2G sin %s. 



On substitution, then, we get 



sin s, 



and this magnitude denotes the number of particles with which in 

 unit time any particle so collides that the directions of motion of 

 the colliding particles make the angle s with each other. 



In order to calculate the total number of collisions which a 

 particle suffers in the unit of time we have to take the sum of the 

 values of the above expression for all values of the angles s. It 

 is therefore necessary to know how great is the number n of the 

 particles for which the angle of encounter with the particle con- 

 sidered has the value s, or, better expressed, a value differing 

 infinitely little from s, so as to lie between s and s + ds as its limits. 

 We find this number by making use of the property of heat-motion, 

 that it goes on in the same way in all directions without distinction, 

 so that equal numbers of particles move in every direction. 



Consider all the particles with their directions of motion to be 

 so displaced the latter parallel to themselves that all move 

 towards the colliding molecule, which for the instant is considered 

 at rest ; then the paths of all the particles which make an angle 

 between s and s + ds with the colliding particle fall in the space 

 included between two infinitely close cones whose vertices lie on 

 the colliding particle and whose axes coincide with the direction 



