Kinetic Theory of Gases. 613 



if we replace p by — j \\p dx dy dz, this becomes 



K= iMM p py d * dz - ■ ■ ■ ^ 



We have now obtained a value of K which is dependent 

 only on the positions of the molecules, being entirely inde- 

 pendent of the way in which the gas was divided up into cells. 

 The propositions stated in § 24 are therefore independent o£ 

 the arrangement of cells. 



§ 26. Equation (21) shows that the minimum value of K 

 is given when p = po everywhere ; i. e., when the density is 

 constant throughout the gas. Proposition (i.) of § 24 may 

 therefore be taken to state that if a point be selected at 

 random in our generalized space, it is infinitely probable that 

 the corresponding system will be one for which p is constant 

 everywhere. 



This is not difficult to understand. Selecting a point at 

 random in our generalized space is equivalent to placing N 

 molecules at random at points inside the containing vessel. 

 The fundamental principles of the theory of probability lead 

 us to expect that it will be infinitely probable that the dis- 

 tribution of gas will be uniform. 



§ 27. We have discussed the distribution of x, y, z co- 

 ordinates : the distribution of u, r, w coordinates can be 

 treated in the same way. In this case, however, what is 

 required (for reasons which will appear later) is not a 

 knowledge of the partition of the whole space, but only of 

 that part of it for which the kinetic energy has an assigned 

 value E. This is the part given by the equation 



\ m 2 (t,„» + e/+».*)=B. • • • (26) 



a, b,... 



Selecting a point at random from this part of our space, is 

 not equivalent to distributing velocities at random to the 

 N molecules, but' is equivalent to distributing 3N velocities 

 (u a , v a , w a , u b , v b , &c.) about a mean value 2E/3»*N. We 

 shall therefore not expect the resulting distribution to be 

 uniform, but shall expect the velocities to be grouped about 

 this mean value according to the law of trial and error. 



We could treat the whole question on the lines on which 

 the former question was treated. We should find that 

 instead of the old function K defined by equation (25) , we 

 should have a function H defined by 



H.z=ffiflogfdudvdw .... (27) 

 where / is a function of u, r } w such that the number of 

 Phil Mug. S. 6. Vol. 5. No, 30. June 1903. l> T 



