420 MATHEMATICAL APPENDICES 31 



collision-frequency of its molecules as to a gas which possesses 

 no other than its molecular heat-motion. 



32*. Collision-frequency of a Particular Molecule 



The collision-frequency of a molecule which moves with a 

 given speed may be calculated, but not quite so easily as the 

 mean collision-frequency of all the molecules. To calculate this 

 number B we can, in the case of a single gas composed of exactly 

 similar molecules, make use of the formula obtained in 29*, 

 viz. : 



B = 7rsW(ta/7r) f r dU\ 



J _oo J -co 



where 



Since the velocity of the colliding molecule and its axial 

 components are in general of finite magnitude we may put new 

 variables U + u, V + v, W + w f or U, V, W without altering the 

 limits of the integrations ; consequently 



where for shortness is put 

 x = km{(U + > 



Since there is no distinction as regards direction, we may choose 

 our coordinate system as we like, and, therefore, take the 

 direction, in which the particle under consideration moves with 

 the speed 



u = ^/( u t + v 2 + w 2 ), 



as that of one of the axes. If we further substitute polar 

 coordinates ^, s, <j> in place of the Cartesian U, V, TFwe have 



3 rzir fn roo 



B = TT<s^N(km/Tr) J d(f)\ ds sin si 



where 



q = kmfy 2 + o> 2 -f 



On integration with respect to s and <j> this becomes 



-p> 2"^ / 1/, I 



