30(5 Application of the Kinetic Theory to Dense Gases. [Feb. 7, 

 Again we find 



3 n 1 



o 



and nT s = nT-nT r = ^ 



and therefore = - , which increases as K increases, that is 

 T n + K 



ceteris paribus as the diameter of the spheres increases. 



Again the mean pressure per unit of area is^? = f (l -f /c)/>T r , which 

 is independent of c. For a system of material points p = -/T r , that 

 is f(l + /c)/jT r , since in this case K = 0. As the spheres increase in 

 diameter with (l + )T r constant, p remains constant. 



The number of collisions per unit of volume and time varies as 



K i 

 <rv T r , that is, as / -- , and is, therefore, less than it would be if, 



V l-f-K 



with the same diameter, the spheres had velocities independent of 

 each other. 



20. It follows from the fact that p is independent of /c, that local 

 variations of density, that is of K, involve, on the whole, no expendi- 

 ture of work, and will, in fact, come into being. 



21. The effect of collisions between the spheres is now considered 

 directly, to show how we obtain the known results that collisions 

 between members of a group of spheres tend to reduce the group to 

 the " special state " in which T r is constant throughout the group. 

 Let the component velocities of two spheres be x\y\Zi x 2 y z z z before 

 collision and oc 1 \y\z\ x\y\z' z after collision. Then, if the two are 

 members of a group and the chance that the members of the group 

 shall have assigned velocities is ce~ /iQ , in which 



Q = axi 2 



the a coefficients being all alike and the 6's all alike, we find that 

 the chance for the velocities after the collision is ce~^', in which 

 Q' is the same function of x\x' z , &c., that Q is of XiX 2 , &c. This 

 shows that the distribution is not disturbed by collisions if all the 

 a's are alike and all the 6's alike. The group is in the special 

 state. 



22. Bat if the a's differ from each other or the 6's differ from each 

 other, it is shown that collisions tend to reduce them to equality, a 

 with a and b with 6 ; that is to reduce the group to the special state. 



23. Boltzmann's minimum function tends to diminish by collisions, 

 finally becoming constant for any group of contiguous spheres, when 

 T r becomes uniform throughout the group. On the other hand, as 

 the group becomes too large, the spheres composing it develop an 

 opposite tendency to split up into smaller groups, each with some 



