Statistical mechanics 



29 



We observe that the function /, is not dependent on u^, v^, iv.^ and may 

 hence be placed before the integral. 



23. Number of stars »thrown in» at a collision. 



Even here it is possible to use, at least in part, the same reasonings as with 

 regard to passages. Let us consider two stars Wj and situated within As/' and 

 Asg" respectively, so that the components of the velocity of ni^ lie between the limits 



^i" i i "i" 



Wj" ± ^ A tv^" 



and the coniponents of the velocity of between the limits m/' ± | Au^" etc. 



Varying zlsj" and zls.," in all possible manners we find, for the number of 

 stars thrown in by collisions into Je^, the expression 



(43) (in) = Aii At f zls," Ab^" d'Y' db" h" (""//'./g", 



where the integration is to be performed over all such values of the variables as 

 after the collision give rise to velocities within zls^ . 



As, according to (40), the relations between the velocities before and after a 

 collision do not depend on <\ and h — in this case not on 'ji" and h" — it is there- 

 fore possible to perform the integrations regarding these variables, and we get 



(43*) (\u) = AÜAtTid^ j As^" AB.;\f\" f^" (»", 



where 



If the velocity component, parallel to the X-axis, before the collision are 

 and u^", so have these components after the collision, according to (40), the com- 

 mon value 



The integration over the variables u^" , w,", etc. is to be performed in such a 

 manner that the expression above gets the value u^^ . Hence we have the relation 



(44) m,u,"^m,u," ^ 



-\- 



where is to be considered unaltered at the integration. We now exchange m/' 

 and Mg" for two new variables; as one of these we may most suitably choose m, 

 defined through (44); as the other we choose a variable equal to the relative 

 velocity before the collision : 



(45) ^2 = ■Wg" — M^" • 



