90 The Law of Distribution [CH. v 



102. Encounters of higher orders may be similarly treated. If p K is 

 used to denote the density in the space of 2Kn + 6K 3 dimensions, in 

 which ^T-ple molecules are represented, the complete system of conditions 

 for steady motion is 



(i) Along every stream line in the 2Kn + 6-fiT 3 dimensional space 



p K = constant .............................. (217). 



(ii) At every point on the boundary of this space 



p K = Pap b ................................. (218), 



in which p a , p b refer to the two systems of molecules of orders a, 6, of which 

 the encounter results in the particular system of order K which is repre- 

 sented at the point in question (we therefore have always a + b = K). 



If encounters of all orders are to be taken into account these conditions 

 must be satisfied for all values of K from K = 1 to K= GO . In the case of 

 K = 1, equation (218) must be interpreted so as to become identical with the 

 condition (B) of 100. 



It will be noticed that if these conditions are satisfied for all values up to 

 K = oo no hypothesis need be made as to the smallness of the radius of 

 molecular action in comparison with the free path. The only assumption 

 now made is that the gas is in a state of molecular chaos. 



Solution of Equations. 



103. As before, let % be a quantity, a function of the coordinates of 

 a molecule or system of molecules, such that throughout the undisturbed 

 motion of the molecule or system, ^ maintains a constant value, and such 

 that when two molecules or systems combine to form a new system, the % of 

 the new system is equal to the sum of the %'s of the component systems. 

 Speaking loosely we may say that % is defined as being capable of exchange 

 between molecules at a collision, but is indestructible. 



Then a solution of equations (217) and (218) will be seen to be 



log Pir =2 % ...(# = l, 2, ...oo).; ................ (219), 



K 



where S^ is the value of ^ for a ./iT-ple molecule, being by definition equal to 



K 

 the sum of the %'s of the K constituent molecules. If % ly ^ 2 ... are all the 



possible values of %, the most general solution is 



2 +...) .................. (220). 



As regards the number and meaning of the %'s the question stands as in 

 76 ; and for the reasons there given we may, in the case of a gas which has 

 no mass-motion, reject all except 



tt = l> 



^ 2 = E, the energy of a molecule. 



