410 Mr. S. H. Burbury o?i Boltzmanns Law of 



according to the foregoing theory approximately 



The chance that each shall contain n is 



Now if n be great enough, the first o£ these expressions 

 vanishes compared with the second so long as q is appreciable. 

 But any quantity of gas which we can measure contains a 

 practically infinite number of molecules. We should expect 

 then, according to the theory, to find the density the same 

 for all measurable quantities of gas taken from the same 

 level of potential of external forces. Nevertheless inequalities 

 of density if they exist, whether w r e measure them or not, 

 have the physical consequences above mentioned, namely, the 

 variation in the values of the mean potential or mean virial. 

 We cannot measure the velocity which the Kinetic Theory of 

 Gases attributes to a molecule; but we assert notwithstanding 

 the physical consequences of it. 



13. In the limiting case, when the density attains its 

 utmost possible limit, it may be that any inequalities of 

 density, even affecting spaces too small for observation, would 

 involve an immense increase of potential, and therefore, 

 according to our law, cannot occur. In such a limiting case 

 perfectly uniform density may exist notwithstanding, or even 

 in consequence of, the law e~ 2h x, I think, however, that for 

 the reason to be given later, the law itself ceases to be valid 

 before any such extreme density is attained. It is evident 

 also that as h increases, i. e. temperature diminishes, the 

 distribution tends to become more uniform. The opposite 

 effect follows from increase of temperature. 



14. As the case of binary encounters approaches the limit- 

 ing case of ideal elastic spheres, let us consider that case 

 separately. There being N elastic spheres of equal mass 

 moving in the space S, with mean kinetic energy for each 

 T = 3/4A, what is the chance that at or near a point P within 

 S the density shall at a given instant be p, where p> or 

 <N/S? \ 



At a given instant let all the spheres, except one which we 

 will call m, have positions oc x y^ % .' . . ^^_i j/n-i ^n— i- Let 

 m be at x y z. Then let all the spheres, so placed, have 

 velocities distributed at haphazard acccording to the law 

 6 -fc2»i(« 2 +«> 2 +«> 2 ) w ith mean kinetic energy T. And in this dis- 

 tribution let u v w be the velocities of m. So far as regards 



die 

 finite forces, — — =0. But u will change by the collisions of 



