54 The Law of Distribution [CH. iv 



Hence the condition that the probabilities of p and q being satisfied may be 

 regarded as "independent" is that expressions (103) and (104) shall be equal, 

 or, written symmetrically, that 



v m = l ^p^ (105). 



Analysis of the Assumption of Molecular Chaos. 



62. The assumption of molecular chaos was tantamount to an assumption 

 that two probabilities might be regarded as independent. Equation (105) 

 accordingly enables us to test whether this assumption is legitimate or not 

 relatively to our present basis of probability namely, the generalised space 

 filled with homogeneous fluid. 



To do this, let us define condition p as the condition that a molecule 

 of class A shall be found in the element dxdydz of the gas in other words, 

 that one of the N molecules shall have coordinates lying between the limits, 



x and x + doc, y and y + dy, z and z + dz 



u and u + du, v and v + dv, w and w + dw\ 



For certain systems this condition is satisfied by molecule A, and these 

 systems are represented in the generalised space by that region for which x a 

 lies between x and x + dx, and for which similar conditions are satisfied by 

 2/a> ^a> u a , v a , w a . This region supplies to v p a contribution of amount 



III ... dx a dx b dx c . . . du a du b du c (107), 



where the integration extends over all values of the variables which are not 

 excluded by 33, except in the case of x a , y a , z a , u a , v a) w a , for which the 

 limits are those given by (106). The integral may be written in the form 



/Y f + f +0 

 dxdydzdudvdw 1 1 ... dx b dx c ... I du b I du e (108). 



J J J oo J --oo 



For other systems, condition p is satisfied by molecule B, and these systems 

 again supply a contribution of amount equal to the above. Each of the N 

 molecules contributes in this way to v p an amount equal to that given by 

 expression (108), so that the total value of v p is 



rr r+<x> [+<* 

 v p = N dxdydzdudvdw 1 1 ... dx b dx c ... I du b I du c (109). 



J J J 00 J 00 



The value of v, the volume of the whole space, is given by (107), if the 

 integrals are taken through all values of all the variables except those values 

 excluded by 33. This integral may of course be written in the form 



v= 



... dx a dx b dx e ... I du a \ du b ......... (HO), 



