14 The Law of Distribution of Velocities [CH. n 



these limits, let us agree to say that a molecule of which the components lie 

 between u and u + du, v and v + dv, w and w + dw is a molecule of class A. 



The total number of molecules of class A is of course 

 Nf(u, v, w) dudvdw. 



In accordance with the definition of 12, we can say that the number of 

 molecules belonging to class A which are found within the element of volume 

 dxdydz selected at random is 



vf(u, v, w) dudvdw dxdydz (2). 



Interpreted literally this statement is unintelligible for dudvdw dxdydz is a 

 small quantity of the sixth order ; interpreted in the sense already explained, 

 no exception can be taken either to its intelligibility or truth. 



The assumption of Molecular Chaos. 



15. Let us imagine that instead of the element dxdydz having been 

 selected at random, we had supposed it to be an element in the immediate 

 neighbourhood of a second molecule of which the components of velocity 

 were known to lie between u and u' + du', v and v' -f- dv, w' and w + div', 

 let us say a molecule of class B. We are no longer justified in saying that 

 the probability of finding a molecule belonging to class A inside this element 

 is given by expression (2). If all the molecules of class A were distributed 

 at random, and then those of class B were independently distributed at 

 random, the statement would be true enough. But if the gas is moving in 

 accordance with the dynamical conditions of nature, it is quite conceivable 

 that, for instance, molecules possessing nearly equal velocities tend to flock 

 together. If this were so the probability we are discussing would be greater 

 than that given by expression (2) when the velocities of the two molecules of 

 classes A and B were nearly equal ; in general, it would depend on u', v', w', 

 as well as on u, v, w. 



In the case which is discussed in the present chapter that in which the 

 molecules are hard elastic spheres it is usual to assume that the molecules 



r 



having velocity-components lying within any small specified limit are, at 

 every instant throughout the motion of the gas, distributed at random, in- 

 dependently of the positions or velocities of the other molecules, provided 

 only that two molecules do not occupy the same space. The legitimacy of 

 this assumption is not self-evident. Indeed, nothing but a discussion of the 

 dynamical equations which determine the motion of the molecules can decide 

 whether the assumption is true or not. Such a discussion will be given in 

 the next chapter ; for the present we shall be content to make the assumption, 

 without discussing its validity. 



