Kinetic Theory of Gases. 617 



Recapitulation and Discussion. 



§ 33. The whole of what has been proved amounts to the 

 following : Firstly, a gas not in the normal state tends to 

 approach that state ; and, secondly, in examining the physical 

 behaviour of a gas, departures from the normal state are 

 insignificant, and we may legitimately proceed as if the gas 

 were in the normal state throughout. 



These results have only been obtained for the simplest type 

 of gas, but it is obvious that they can be extended so as to 

 apply to any kind of gas. The normal state is in each case 

 found by assigning the minimum value to the function 

 analogous to the H-function already discussed — a function 

 which may be conveniently referred to as Boltzmann's 

 minimum-function. 



The whole of the physical behaviour and statistical pro- 

 perties of a gas can be deduced from a knowledge of the 

 H-function, just as the behaviour of a dynamical system can 

 be deduced from a knowledge of the energy-function. 



Some examples of the use of this function, illustrating some 

 of the peculiarities which may arise, are given in the remain- 

 ing sections : — 



Examples of the Minimum-theorem. 

 I. — Field of External Force. 



§ 34. Let us suppose the molecules to move in a field of 

 external force of potential %, a function of x, y, z. The 

 analysis is exactly similar to that of § 29, except that equation 

 (37) must be replaced by 



{im(u 2 + v 2 + w*) + xifd* d V dz du dv dw = E > ' ( 39 ) 

 and from this we immediately obtain the law 



/=Ae- 7 ^K+^+w 2 )-2% > . , . (40) 



II. — Mixture of Gases. 



§ 35, Let the N molecules be of different kinds, aN of one 

 kind, /3N of another, and so on. The various coordinates can 

 be represented in a generalized space as before. The chance 

 that the whole gas shall be in a specified state is the product 

 of the separate chances that the gas of each kind shall be in 

 the corresponding state, whence we find, as the correct form 

 for H, 



H = SajjVm , / a log,/; thvdyd: du dv dw . . (41^ 



(a) * 



