48 The Law of Distribution [CH. in 



There is, however, a feature about the u, v. w coordinates which did not 

 occur in the case of x, y, z coordinates. Throughout the motion of a gas the 

 energy remains constant, or, as in 38, the trajectories are confined to single 

 members of the family of loci E = constant. From this it results that we 

 are not interested in the partition of the velocity coordinates throughout 

 the whole of the generalised space, but only throughout the region which 

 corresponds to a single definite value of E. Hence in finding the minimum 

 value of H, we suppose /subject not only to 



fdudvdw= 1 (97), 



the equation corresponding to the former equation (67), indicating that the 

 number of molecules in the gas is equal to N, but also subject to 



E (98), 



expressing that the kinetic energy of the system is equal to E. 



The equation of variation of H, subject to equations (97) and (98), is seen 

 to be 



H=fn{l + log/+ X + fani(u a + v- + w 2 )} 8/dudvdw (99), 



in which X, fi are undetermined multipliers. The minimum value of H is 

 accordingly given by 



l + log/+\ + ^ra(w 2 + z; 2 + O = (100), 



leading, if we change the constants, to the solution 



/= Ae- hm{u2+v *+ w * } (101). 



56. It could now be shewn, exactly as in the former case, that throughout 

 all but an infinitesimal fraction of the whole of that part of the generalised 

 space in which the energy of the corresponding system is E, the law of 

 distribution of velocities is that given by equation (101). It does not seem 

 necessary to reproduce the details of this proof; the mathematician will be 

 able to construct them for himself, while the physicist will probably not wish 

 to be detained over them. 



The law of distribution of velocities expressed by equation (101) is a 

 special case of the general law found for the " steady state " in the last chapter. 

 We are limited to this special case because we have, at the outset, supposed 

 our containing vessel to be fixed in space. If, on the contrary, the vessel is 

 moving in space with a velocity of components u , v , w the analysis of this 

 chapter can be made to apply by supposing all coordinates referred to moving 

 axes, moving with a velocity of components u , v , w . In this case equation 

 (91) expresses the law of distribution of velocities relative to these moving 

 axes. The law of distribution of absolute velocities in space is therefore 



f = ^ e -An[(-Mn) 2 +(-o) S -Kw-Wo) 2 j (102), 



which is the general law for the steady state given by equation (25). 



