FOUNDATIONS OF THE KINETIC THEORY OF GASES. 273 



Equation (12) shows how the gradient of density of either gas varies, in the 

 stationary state, with its percentage in the mixture. For the multiplier of 



-r^ is obviously a maximum when 



dx J 



i i 



s^ + ys-f s 2 +s 2 2 /y' 



in which y=n 1 ln 2 , is so. This condition gives 



n 1 /n 2 =y=s 2 fs 1 . 



Hence the gradient is least steep at the section in which the proportion of the 

 two gases is inversely as the ratio of the diameters of their particles ; and it 

 increases either way from this section to the ends of the tube, at each of which 

 it has the same (greatest) amount. This consideration will be of use to the full 

 understanding of the more complex case (below) in which the masses, as well as 

 the diameters, of the particles differ in the two gases. 



55. Let us now suppose the mass per particle to be different in the two 

 gases. The last terms of the right-hand side of (10), viz., 



may be written in the form 



P, dn^ jn-njhzr* f(y)dy + nfa r°° f(y)dy , 



37m dX \ Jkl ^^M 2 F(2/) + (»-%)VF(y^) ^U(^-* 2 %)+* 2 ^^)| 



where the meanings off and F are as in § 34. 



If we confine ourselves to the steady state, we may integrate (10) directly 

 with respect to x, since dQJdt is constant. In thus operating on the part just 

 written, the integration with regard to x (with the limiting conditions as in (11)) 

 can be carried out under the sign of integration with respect to y : — and then 

 the y integration can be effected by quadratures. 



The form of the x integral is the same in each of the terms. For 



I (n-njtini _ I n x dt^ __ n j 1 , A ^ 



Jn Aiii + Bin-nJ J H A(n-n 1 )+Bn 1 A-B ( A-B b 



B 



This expression is necessarily negative, as A and B are always positive. When 

 A and B are nearly equal, so that B = (l + e)A, its value is 



A\2 3^ /' 

 so that, even when A and B are equal, there is no infinite term. 



