29, 30] Collisions and Free Path 31 



Hence from expression (43) the number of collisions for which the new 

 variables lie within a range dudvdwrdadftdy is 



(44), 



in which we have 



F 2 = (w - w) 3 + (v 



c 2 + c' 2 = M 2 + u'' 2 +- v* + v"* + w 2 + tu'' 2 = 2 (u 2 + v 2 + w 2 ) + \ (a 3 + /3 2 + 7 2 ), 

 or, if we write u 2 + v 2 + w 2 = c 2 , 



c 2 + c' 2 = 2c 2 + \ F 2 . 



Let us again transform variables according to the schemes 

 u = c sin 8 cos <, a = V sin -v/r cos ^, 

 v = c sin 6 sin </>, IB = Fsin i/r sin ^, 

 w = c cos ^, 7 = V cos A/T. 



In order that u, v, w may have all possible values, 6 must range from 

 to TT, <f) from to 2?r, and c from to oo . If, however, we give a similar 

 range to the new variables in the second scheme of transformation, we shall 

 be counting each collision twice over. For a collision in which a, /3, 7 have 

 given values can, by merely changing the roles of the two molecules, be 

 regarded as a new collision in which the signs of a, {3, 7 are altered. This 

 source of error can be eliminated by limiting the integration with respect to 

 i/r from to 7T/2, instead of from to TT. Hence we obtain, for the number 

 for which c lies between c and c +dc, while V lies between Fand V+dV, 



Integrating with respect to c from to oo , the number of collisions for 

 which Flies between Fand F+rfFis 



or z>V 2 A/ e -^hmv 2 Y 3 dV (45), 



a result which will be required later. 



If we finally integrate this from F= to F= oo , we obtain for the total 

 number of collisions 





"is ......................... ........ (46) - 



There are v molecules per unit volume, and each collision terminates two 

 free paths. Hence the v molecules describe 



free paths per unit time. 



