Kinetic Theory of Gases. 615 



Assume the number of molecules having coordinates lying 

 between x and x-\-dx, . . . u and u-\- du . . . &c, to be 



N/(#, y, z, w, i', w?) efa? ^3/ <is rZ?i 6?v div, . . (34) 



and introduce a new function H defined by 



H= \\\N\flogfdx dy dz du dv die . . . (35) 



Then it will easily be seen that the propositions of § 24 

 apply to the region of our generalized space for which the 

 total energy is E, provided we replace K by H, and instead 

 of K = 0, write H = H , where H is the minimum value of 

 H defined by equation (35), when / is subject to the con- 

 ditions 



dx dy dz du dv dio=l, (36) 



\m(u 2 + v 2 + iv 2 ) f d% dy dz dudv dw = ~E. . (37) 

 The resulting law is found to be 



f(u, v, w, x, y,z) =Ae- hm (™ 2 +v 2 +™ 2 ), . . . (38) 



giving uniformity of density for all values of x, y, z and 

 Maxwell's law of distribution of velocities in a single 

 equation. 



The Normal State. 



§ 30. When the 6N coordinates of the molecules of a gas are 

 such that H has its minimum value, or a value which differs 

 from this minimum by an infinitesimally small amount, the 

 gas will be said to be in its ''normal state/' For gases in 

 states other than the normal, the difference between the 

 value of H for the gas and the minimum possible value cor- 

 responding to the same energy, may be taken to supply a 

 measure of the divergence of the gas from the normal state. 



We now find that proposition (i.) of § 24 may be replaced 

 by 



(iv.) All except an infinitesimally small fraction of the 

 whole of the generalized space represents gases which are in 

 their normal states. 



§ 31. Let us now examine what becomes of the two 

 remaining propositions. 



Suppose that we start the gas from a configuration about 

 which nothing is known except that the total energy is E 

 (speaking physically, we have a gas of which the temperature 

 and pressure are known), then the representative point may 

 be supposed to be selected at random from that part of our 

 generalized space for which the energy is E. (Suppose we 



2 T 2 



