Mr. J. C. Maxwell on the Dynamical Theory of Gases. 187 

 subsequent exchanges. This will be the case when 



mm=fM)m (22) 



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



M^ + M/^M/^-fM/ 2 , . . . (23) 



whence we obtain 

 where 



/iW = C 1 «-S, /,(*) = C*-S, . . (24) 



M ia 2 =N 2 /3 2 (25) 



By integrating I ll C x e ^ d^drjd^, and equating the 



result to Nj, we obtain the value of C r If, therefore, the dis- 

 tribution of velocities among N x molecules is such that the num- 

 ber of molecules whose component velocities are between £ and 

 | + off, ?7 and 97 + ^77, and £ and f -f- df is 



,*]¥,= -Ll. «-%&,<& . . (26) 

 a d 7H2 



then this distribution of velocities will not be altered by the ex- 

 change 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' re- 

 mains 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 velocities of a molecule 

 should be arranged in this cycle, rather than in the reverse order. 

 If, therefore, the direct exchange between OA and OA' is not 

 equal, the equality cannot be preserved by exchange in a cycle. 

 Hence the direct exchange between OA and OA' is equal, and 

 the distribution we have determined is the only one possible. 



This final distribution of velocity is attained only when the 

 molecules have had a great number of encounters ; but the great 

 rapidity with which the encounters succeed each other is such 

 that in all motions and changes of the gaseous system except 

 the most violent, the form of the distribution of velocity is only 

 slightly changed. 



