61-63] The Assumption of Molecular Chaos analysed 55 



and from equations (109) and (110) we now find 



dxdydz 1 1 ... dx b dx c . . . 

 Vp_ ,- JJ dudvdw 



\\\ dx a dx b dx c ... \ du I dv I dw 



J J J J 00 J 00 J 00 



This vanishes through the last factor, for the obvious reason that when 

 the molecules are equally likely to have all velocities, the probability is 

 infinitely against a single molecule belonging to any specified class. 



63. Let us now suppose that the velocities of the individual molecules are 

 given, and let us calculate the probability in this case that condition p is 

 satisfied. Let us suppose that the velocities of molecule A are known to lie 

 within the limits 



u a and u a -f 8u a , v a and v a + 8va, w a and w a + &w a (H2), 



and that we have similar knowledge for the other molecules. The value of 

 v, the whole space representing systems for which the molecules have the 

 given velocities, is given by equation (110) if the integration is from u a to 

 u a + 8u a for u a instead of from oo to +00, and similarly for the other 

 velocities. 



Thus we have as the new value of v, 



ere 



v=8u a 8v a ... I II ...dx a dx b dx c (113). 



As before, the systems for which condition p is satisfied by molecule A, 

 are represented by those parts of the space v for which x a lies between x and 

 x + dx, and similar conditions are satisfied by y a , z a , u a , v a , w a . We shall 

 suppose, as we legitimately may, that the Bu a , 8v a , 8w a of the limits (112) are 

 infinitesimal in comparison with dudvdw. Then provided that the range for 

 u a given by (112) lies within the range u and u + du, and that similar 

 conditions are satisfied by v a , w a , the contribution to v p corresponding to 

 molecule A is given by the right-hand of (113) provided the integration with 

 respect to x a extends only from x to x + dx, and similarly for y a , z a . Thus if 

 the velocities of molecule A lie within the specified ranges, there is a contri- 

 bution from molecule A to v p of amount 



dxdydz8u a 8v a ... 1 1 1 1 ... dx b dx e ... dy b dy c 



If the velocities given by the limits (112) do not lie within this range 

 dudvdw the contribution is nil. The number of molecules of which the 

 velocities satisfy the condition of lying within this range in other words, the 

 number of molecules capable of taking the rdle of molecule A in expression 

 (114) may be taken to be 



Nf(u, v, w) dudvdw (115)- 



