Kinetic Theory of Gases. 601 



to begin by postulating that there should be no " regularity " 

 in the throw o£ the dice. A limitation of this kind could not, 

 in any case, add anything to the certainty of the ultimate 

 result. It may be noticed that the difficulty of giving a 

 precise definition of a moleJzidar-geordnet state of a gas is 

 exactly paralleled by the difficulty which would be experienced 

 in attempting to define " regularity " in the case of the dice. 



Outline of the New Theory. 



§ 7. Let us now suppose that we are dealing with a great 

 number N of molecules inclosed in a vessel of volume £L. We 

 shall consider the simplest case first, and shall accordingly 

 suppose the molecules to be incompressible elastic spheres of 

 the usual type. Each molecule possesses three degrees of 

 freedom, those of molecule A being represented by ,v a , y a , z a , 

 the coordinates of its centre. The corresponding velocity 

 coordinates will be denoted by u a , v a , w a . The whole gas, 

 regarded as a single dynamical system, will possess 3N degrees 

 of freedom, and its state will be specified by the 6N coordinates 



11 ol % li 'ai ®a, Ha, Za, U b , V & , W b (1) 



Let us imagine a generalized space of 6N dimensions. In 

 this space, the system of which the coordinates are given by 

 (1) can be completely and uniquely represented by a single 

 point, namely the point of which the Cartesian coordinates 

 referred to b\N definite rectangular axes are given by (1). 

 If our gas is to be entirely inside a certain containing vessel, 

 we shall only need a certain portion of this space, say that 

 bounded by 



f A®a, y a , --„)=-- ° ; /(**> y» z *) =°; — 



I ^=± GO ; i?a=+GO (2) 



If the molecules are incompressible spheres of radius R, we 

 shall not need to consider the possibility of a system in which 

 the centres of any two molecules are within a distance smaller 

 than 2R. We may therefore exclude from consideration all 

 those portions of our generalized space which are bounded by 



(*.--k) , +(y.-*)*+(«.-*t) , =*H» l • • (3) 



and other similar equations, one for every possible pair of 

 molecules. The simplest case of all is that in which R = 0; 

 and in this case this last limitation may be disregarded. 

 Taking R = is the same thing as supposing the diameter to 

 vanish in comparison with the mean tree path. 



