27, 28] Mass Motion and Molecular Motion 27 



Now RP = RP', and the volume of a has been seen to be equal to the 

 volume of /3. Hence the two densities (35) and (36) are equal. Since the 

 two cylinders have also been shewn to be equal it follows that the number 

 of molecules of class A which collide with the element d<r in the interval dt 

 is equal to the number of molecules of class C which do the same thing. 

 Each of the former molecules is changed by collision from a molecule of class 

 A to one of class C, and each of the latter from a molecule of class C to one 

 of class A. Hence the number of molecules of class A remains unaltered 

 by collisions with the element do: The same is of course true of every other 

 class of molecule, and of every other element of the surface of the containing 

 vessel, whence we see that the whole law of distribution is unaltered by the 

 presence of the walls, or, in other words, that the law of distribution (33) 

 represents a steady state. 



It now appears, however, that there are only the two constants A and h 

 at our disposal in the case of a gas enclosed in a vessel which is either at 

 rest or moving with a known velocity u , v , w , and these two constants are 

 of course connected by the relation (30). By varying these constants we are 

 enabled to assign to our gas different values of the total energy, or, speaking 

 physically, different temperatures. Similarly by varying v we are enabled to 

 assign different densities to the gas. 



Mass Motion and Molecular Motion. 



28. We have seen that the most general "steady state" possible consists 

 of a motion compounded of a mass-motion and a molecular- motion. The 

 mass-motion has velocity components u , v , w , the molecular-motion has 

 velocity components u u , v v , w w , which we have denoted (p. 23) 

 by u, V, w. The number of molecules having molecular velocities lying 

 between u and U+du, V and v + dv, w and w -f dw is the number of 

 molecules having actual resultant velocities lying between u and u + c?u etc., 

 etc., and this by equation (25) 



= Ae~ hm[(u ~ U(i}2+{v ~ v < >) '* +(w ~ w ' >) * ] dudvdw 



Hence we may suppose the molecular velocities distributed according to the 

 law 



Ae- hm v*+*+*>dud\td)N ........................... (37). 



If we adopt the scheme of transformation (28) we may replace the velocity 

 of which the components are u, v, w by a velocity of magnitude c, in a 

 direction which makes an angle 6 with the axis of z, and such that a plane 

 through this direction and the axis of z makes an angle </> with the'axis of x. 

 The law of distribution (37) now becomes 



(38). 



