FOUNDATIONS OF THE KINETIC THEORY OF GASES. 271 



the final state ( }/ 9\> ?, 9\) , in which the gases are completely mixed. 

 When the vessels are equal this takes the simple form 



( , / 2SDpA 



52. In the case just treated there is no transmission of energy, so that the 

 fundamental hypotheses are fully admissible. In general, however, it is not so. 

 The result of § 41, properly modified to apply to the present question, shows 

 that the energy which, on the whole, passes positively across the section x is, 

 per unit area per second, 



4 V l h + h 2 J 6 A ll ^ 3 2 ^ i} 



This, of course, in general differs from section to section, and thus a disturbance 

 of temperature takes place. In such a case we can no longer assume that 

 h x and h 2 are absolute constants ; and thus terms in <&$ would come in ; just as 

 a term in C 5 appeared in the expression for energy conducted (§ 42). Thus, in 

 order that our investigation may be admissible, the process must be conducted 

 at constant temperature. This, in general, presupposes conditions external to 

 the apparatus. 



53. Though it appears hopeless to attempt a general solution of equation 

 (10), we can obtain from it (at least approximately) the conditions for a steady 

 state of motion such as must, we presume, finally set in between two infinite 

 vessels filled with different gases at the same temperature and pressure. For 

 the left-hand member is then an (unknown) constant, a second constant is 

 introduced by integrating once with respect to x ; and these, which determine 

 the complete solution, are to be found at once by the terminal conditions 



1 dG, f n for x = , ) /1 -, N 



And, by a slight but obvious modification of the latter part of § 51 above, we 

 can easily extend the process to the case in which the vessels are of finite 

 size : — always, however, on the assumption that their contents may be regarded 

 as promptly assuming a state of uniform mixture. The consideration of § 52, 

 however, shows that the whole of the contents must be kept at constant temper- 

 ature, in order that this result may be strictly applicable. 



54. Recurring to the special case of § 51, let us now suppose that, while 

 the masses of the particles remain equal, their diameters are different in the 

 two gases. Thus, suppose s x >s 2 . Then it is clear that 



s-f—s'*, and s 2 — s 2 , 



