FOUNDATIONS OF THE KINETIC THEORY OF GASES. 267 



free path of a P : whose speed is v. We obtain this by remarking that, in the 

 present problem, h x is regarded as constant, so that there is no term in v{. 



Hence, if G x be the mass of the first gas on the negative side of the section, 

 divided by the area of the section, we have 



^h=-F 1 (n 1 « 1 -r,; i € 1 l3) . . . . (3.) 



If G 2 be the corresponding mass of the second gas, we have (noting that, 

 by (1), < + < = 0) 



^-P^V^+V.^/3) (4.) 



From the definitions of the quantities G l3 G 2 , we have also 



dG,_ v fW,_ p , >j 



~dx~ ~ 1% ' dx* " ^ ' I 



> ■ • ■ • (5.) 

 dG,_ rf 2 G 2 _ p , | 



dx~ 1 ^' dx*-~^ n '- J 



47. We have now to form the equations of motion for the layers of the two 

 gases contained in the section of the tube between x and x + Sx. The increase 

 of momentum of the P x layer is due to the difference of pressures, behind and 

 before, caused by P x s ; minus the resistance due to that portion of the impacts 

 of some of the PjS against P 2 s in the section itself, which depends upon the 

 relative speeds of the two systems, each as a whole. This is a small quantity 

 of the order the whole pressure on the surfaces of the particles multiplied by 

 the ratio of the speed of translation to that of mean square. The remaining 

 portion (relatively very great) of the impacts in the section is employed, as we 

 have seen, in maintaining or restoring the "special state" in each gas, as well 

 as the Maxwell condition of partition of energy between the two gases. If R 

 be the resistance in question, the equations of motion are 



a up t' ' ' (6 ' } 



where d represents total differentiation. 



48. To calculate the value of R, note that, in consequence of the assumed 

 smallness of a 1} a 2 , relatively to the speeds of mean square of the particles, the 

 number of collisions of a P x with a P 2 , and the circumstances of each, may be 

 treated as practically the same as if a t and a 2 were each zero : — except in so far 



VOL. XXXIII. PART II. 2 Q 



