18 



The Law of Distribution of Velocities 



[OIL ii 



u' + du', etc., and such that the third condition of p. 15 is satisfied by the 

 line of centres at impact is, by comparison with expression (4), seen to be 



v~f(u, v, w)f(ii', v', w')Va 2 cos ddudvdwdudv'dw'dcadt (7). 



19. These collisions will all belong to class ft, provided that the limits 

 determined by du, dv, etc., are such that the values of u, v given by 

 equations (5) and (6) lie within the appropriate limits u and u + du, v 

 and v + dv, etc. To obtain the whole number of collisions of class ft we must 

 integrate expression (7) over all values of u, v, etc., such that the values of 

 u, v, etc., lie within these limits. 



To do this we need only consider the ratio of the two products of differ- 

 entials dudvdwdu'dv'dw' and dudvdwdu'dv'dw. We use Jacobi's theorem 

 that 



dudvdwdu'dv'dw' = ^dudvdwdu'dv'dw' (8), 



where A denotes the determinant 



I 



Using the values given by equations (5) and (6) we find without trouble 

 that A = 1. This may in fact be seen without actual calculation. For since 

 equations (5) and (6) are linear as regards the velocities, the value of the 

 above determinant cannot depend on the velocities. Also since the relation 

 between the velocities before and after collision is, on account of the reversi- 

 bility of the motion, a reciprocal relation, it is clear from equation (8) that 

 the only possible values for A are + 1, of which the negative value may for 

 obvious reasons be rejected. 



Hence equation (8) becomes 



dudvdwdu'dv'dw' = dudvdwdudv'dw' (9), 



and expression (7) may be written in the form 



v 2 f(u, v, w)f(u', v', w') Fcr 2 cos Odudvdwdu'dv'dw'dwdt (10). 



If this number of collisions is exactly to include all of class ft, the values 

 of du, dv, dw, du, dv, dw' must be those which occur in the specification of a 

 collision of class a (p. 15) and therefore those which occur in expression (4). 



20. Suppose that expression (4) is summed over all possible classes of 

 collisions which can occur to a molecule of class A. Or, what is the same 

 thing, suppose that expression (4) is integrated over all possible values of 



