418 MATHEMATICAL APPENDICES 31* 



31*. Number of Molecular Collisions in a 

 Current of Gas 



To be able to apply these formulae to the theory of internal 

 friction we have yet to determine the influence which a forward 

 motion of a gas exerts on the collision-frequency of its molecules. 

 If this motion at all points of the gas is characterised by the 

 same speed and the same direction, the frequency of collisions 

 can neither increase nor dimmish. But a perceptible influence 

 may result if layers move near each other with different velocities, 

 as is shown by the experiments made to determine the viscosity. 

 A state of things then arises by the mixing of layers, which we 

 can represent with tolerable accuracy by supposing two masses 

 of gas of the same kind and at the same temperature to be flowing 

 in the same enclosure with unequal speeds. 



Consider, therefore, two groups of gaseous molecules in the 

 same vessel, which they fill with unequal densities ; they further 

 differ in the unequal speed of their flow, but are otherwise com- 

 pletely alike : Maxwell's law of distribution of speeds therefore 

 holds in both groups in exactly equal fashion, provided that we 

 apply it only to that part of the molecular motion which shows 

 itself as heat, and therefore provided that from the motions of the 

 individual molecules we subtract the progressive motion of the 

 group as a whole. In the formulae referring to the separate 

 groups we have consequently to introduce the same value, not 

 only for the molecular weight m, but also for the constant k, and 

 this holds, too, for the radius of the sphere of action s. Suppose, 

 further, that the flow has the same direction for both groups, and 

 take this direction to be that of one of the axes of coordinates. 

 Then the number of collisions per unit time of a particle of the 

 first kind, of which there are N l per unit volume, is 



and that of a particle of the second kind, of which there are 

 per unit volume, is 



r 2 = 7T5 2 ( x /2AT 2 a + A r l7 ), 



wherein y is given by 



/co ,-oo pco /.co /-co 



y==(&ra/7r) 3 duA dvA dwA du^\ d 



J oo J co J _ co J co J co 



r = >/ {(u, - . 2 ) 2 + (v, - v^ + K - 



