230 Free Path Phenomena [CH. xi 



probable number of molecules of this second kind in a cylinder of base 7r$ 12 s 

 and height F, where F is the relative velocity of the two molecules. If c and c' 

 are the resultant velocities of the two molecules, and if these velocities 

 are inclined at an angle $, the value of F is given by 



F 2 = c 2 + c' 2 -2cc'cos< (526). 



The number per unit volume of molecules of the second kind moving with 

 a resultant velocity intermediate between c' and c' + dc' is 



7T 



.(527). 



These are moving in all directions equally, so that the proportion for which 

 the direction makes an angle intermediate between </> and </> + d<f) with that 

 of the first molecule is 



Multiplying expression (527) by this factor we obtain for the number per 

 unit volume of molecules of the second kind for which the velocity is such 

 that c, (f> lie within limits dc', d(j>, 



- e- hm ^'' sm6d<b c' 2 dc. 



The number of molecules satisfying these conditions to be found inside a 

 cylinder of volume 7r$ 12 2 F is therefore 



sin ^d^c' z dc' ............... (528). 



From equation (526), we obtain by differentiation, keeping c and c' 



constant, 



Vd V = cc' sin < d<f>, 



so that expression (528) can be replaced by 



" 2 



2*/ 2 12 2 VTJ^W e~ hm * c " 2 -dc'V'dV. .................... (529). 



c 



Keeping c and c' constant, the value of F changes as < changes. Integrated 

 with respect to this variation we have 



in which the limits for F are c + c and c ~ c. 



Separating the two cases of c' being greater and less than c, we find 



) when c>c (530), 



or =c'(c' 2 +3c 2 ) w henc'<c (531). 



If, then, we integrate expression (529) with respect to F, and if, after 

 substitution from the appropriate one of the equations (530) and (531), we 



