72 PROFESSOR TAIT ON THE 



particles, the virtual thickness of the layer (so far as these particles are con- 

 cerned) is 



v Sx/v, 



where v = Jv 2 +v j 2 —2w 1 cos[3 



is the relative speed, a quantity to be treated as essentially positive. 



Thus the fraction of the impinging particles which traverses this set without 

 collision is 



1 — i^tts^SxVq sin /3 d/3/2v . 



To find the fraction of the impinging particles which pass without collision 

 through the layer, we must multiply together all such expressions (each, of 

 course, infinitely nearly equal to unity) between the limits and -k of {3. The 

 logarithm of the product is 



?2. 1 7TS 2 &£ 



]xT" 



-/ sjv 2 +v 1 2 — 2vv x cos/3. sin /3cl/3> 



2v 

 Making v the variable instead of /3, this becomes 





2v\J u ° dv °- 



If v be greater than v u the limits of integration are v — v lf and v + i\, and the 

 expression becomes 



1 + 3F 2 )' 

 but, if v be less than v u the limits are Vi — v and v x + v, and the value is 



-**<£+?)• 



These give, as they should, the common value 



— 4» 1 7ts 2 &/3 

 when v=Vi. 



9. Finally, suppose the particles in the layer to be in the " special" state. 



If there be n in unit volume, we have for the number whose speed is between 



the limits Vi and v 1 + dv 1 



n^invfdv^-'' ' 



-6 



7T 



Hence the logarithm of the fraction of the whole number of impinging particles, 

 whose speed is v and which traverse the layer without collision, is 



