63-66] The Assumption of Molecular Chaos analysed 57 



Writing v for N/l we see that the right-hand of equation (118) becomes 

 identical with our former expression (2) (p. 14). In other words, in the case 

 in which the density is constant throughout the gas, the probability that 

 condition p shall be satisfied for a system selected at random, is equal to the 

 probability calculated in 14. 



From equations (118) and (119) we have the important result 



"pq_"p / (120). 



v v v" 



Thus the fulfilment of conditions a and /3 are now independent events. 

 In other words, we have proved that, relatively to our present basis of pro- 

 bability, the assumption of molecular chaos enunciated in 15 is justifiable in 

 the case in which the radii of the molecules are vanishingly small. 



To justify the way in which this assumption was used in Chapter II. we 

 must go somewhat further. In expression (4) we found a value which we 

 supposed to be equal to the number of collisions of a certain class a. The 

 actual value of this expression was, however, equal to 

 (the number of possible collisions of class a) 



x (the probability of each collision happening). 



This quantity therefore expresses the probable number of collisions, the 

 number which actually occur averaged over a large number of cases, or the 

 "expectation" of collisions, but does not necessarily express the actual number 

 in any particular case. Looked at from another point of view, however, the 

 quantity expressed 



= (the sum of a number of small elements of volume) 



x (the density of molecules of class B)... (121). 



Now there was nothing in analysis of Chapter III. to compel us to take 

 the " cells " to be continuous in space. We may accordingly regard the first 

 factor in expression (121) as one of these cells. We now see in accordance 

 with the results of Chapter III. that expression (121) not only gives the 

 " expectation " of molecules of class B in these elements of volume, but that 

 it is infinitely probable that it gives the actual number, to within an infini- 

 tesimal fraction of the whole. It follows that the number of collisions of a 

 given type found in Chapter III. not only gives the most probable number of 

 collisions, but that it is infinitely probable that it gives the true number, to 

 within an infinitesimal fraction of itself. 



66. When the molecules are not vanishingly small, let us denote the 

 integral 



J J 1 1 . . . dx b dx c . . . dy b dy c . . . 



taken over all values of the variables which are not excluded by 33, by 

 I (b, c ...). The value of the integral can only depend on x a , y a , z a , and 



