THE DYNAMICAL THEORY OF GASES. 45 



In like manner we should find for the number of encounters between 

 molecules whose original velocities corresponded to elements equal to dV described 

 about A' and R, and whose subsequent velocities correspond to elements equal 

 to dV described about A and B, 



f 1 (aU(b')(dVYF'de .............................. (21), 



where F' is the same function of RA' and A'GA that F is of BA and AGA\ 

 F is therefore equal to F', 



When the number of pairs of molecules which change their velocities from 

 OA, OB to OA', OR is equal to the number which change from OA', OB' 

 to OA, OB, then the final distribution of velocity will be obtained, which 

 will not be altered by subsequent exchanges. This will be the case when 



Now the only relation between a, b and a, V is 



Mtf + Mp^MjP + MJP, ......................... (23), 



rt J 



whence we obtain /(<*) = ^e ""',/,(&) = (7 3 e"^ .............. . ......... (24), 



where Mjt^Mft ................................... (25). 



(*+**+{ 

 By integrating \l\Cf~ ** dgdr) d, and equating the result to N v we 



obtain the value of C v If, therefore, the distribution of velocities among N, 

 molecules is such that the number of molecules whose component velocities are 

 between and +c/, rj and rj + d-rj, and and + d is 



TV 



(2G), 

 o 



then this distribution of velocities will not be altered by the exchange of 

 velocities among the molecules by their mutual action. 



This is therefore a possible form of the final distribution of velocities. It 

 is also the only form ; for if there were any other, the exchange between 

 velocities represented by OA and OA' would not be equal. Suppose that the 

 number of molecules having velocity OA' increases at the expense of OA. 

 Then since the total number of molecules corresponding to OA' remains constant, 

 OA' must communicate as many to OA", and so on till they return to OA. 



Hence if OA, OA', OA", &c. be a series of velocities, there will be a 

 tendency of each molecule to assume the velocities OA, OA', OA", &c. in order, 

 returning to OA. Now it is impossible to assign a reason why the successive 



