424 MATHEMATICAL APPENDICES 34* 



the colliding particle relatively to that encountered that has to be 

 shortened by f s, and, of this shortening, part falls on the striking 

 particle and part on that struck. If the former moves with a 

 velocity whose components are u, v, w, and the latter with a 

 velocity whose components are U, F, W, the path of the striking 

 particle is diminished by 



J U* + V*+W* 



I ( u - C7) 2 + (v - F) 2 + (w - TF) 



and that of the struck particle by 



2 ,/ __ IP+V*+ W* 



' \ ( u _ 7)2 + ( v _ 7)2 + (w _ W ) 



To find the average shortening of the paths of a particle with 

 velocity components u, v, w for all its collisions, we have to 

 multiply the former number by the collision-frequency ( 29*) 



where 



r=*/{(u- U? + (v- F) 2 + (w - TF) 2 }, 



and integrate : the result is 



If now we multiply this expression by the probability of occurrence 

 of the velocity w = *J(u 2 + -u 2 + w 2 ), viz. 



oo 



( 18*), and integrate with respect to w between the limits and 

 we obtain the value f TrsWto, which we must divide by the mean 

 collision-frequency F = N /2?rsWli in order to find the mean 

 value of the correction 



According to this calculation, on taking account of the magnitude 

 of the molecules, we have to put for the value of the mean free 

 path 



If instead of N, the number of molecules per unit volume, we 



