necessary for Equipartition of Energy. 593 



orders may be considered. For instance, to take ternary 

 encounters into account we imagine systems o£ triple mole- 

 cules, these being represented in a space of 6n-dimensions. 

 The density in this space being t we have as conditions 

 additional to those given in § 11 — 



(e) Throughout the 6n-dimensional space t must be con- 

 stant along every stream-line. 



(?) At every point on the boundary o£ the 6rc-dimensional 

 space we must have 



T — p(T. 



§ 13. Encounters of higher orders may be similarly treated. 

 If Pi is used to denote the density in the space of 2kn-dimen- 

 sions, in which £-ple molecules are represented, the complete 

 system of conditions for steady motion is 



(i.) Along every stream-lice in the 2&?i-dimensional space, 



p k = constant (20) 



(ii.) At every point on the boundary of the 2X:n-dimensional 

 space 



Pk = p a ph, ■ (21) 



in which p„, pb refer to the two systems of molecules of orders 

 «, b, of which the encounter results in the particular system 

 of order k which is represented at the point in question (we 

 therefore have always a + b = h) . 



If encounters of all orders are to be taken into account 

 these conditions must be satisfied for all values of k from 

 Jc = l to k = r x> . In the case of k=l, equation (21) must be 

 interpreted so as to become identical with the condition (S) 



oi '§ n : 



It will be noticed that if these conditions are satistied for 

 all values up to k= x> , 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 " molekular-ungeordnet " state. 



Solution of Equations. 



§ 14. 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. 



