CHAPTER III. 



THE LAW OF DISTRIBUTION (CONTINUED). 



. II. THE METHOD OF GENERAL DYNAMICS. 



The Conception of a Generalised Space. 



32. IN the last chapter it was twice found convenient to represent the 

 three velocity coordinates u, v, w of a molecule, by a point in space of which 

 the coordinates referred to three rectangular axes were u, v, w. The principle 

 involved is a useful one, capable of almost indefinite extension, and will be 

 largely used both in the present chapter and elsewhere in the book. 



The space of nature possesses three dimensions, but just as it is open for 

 us to represent any two coordinates in an imaginary space of only two dimen- 

 sions, so in the same way we may represent any four coordinates in an 

 imaginary space of four dimensions. Similarly if a dynamical system is 

 specified by n coordinates, we can represent these coordinates in a space of n 

 dimensions, and the various points in this space will correspond to the various 

 configurations of the dynamical system. 



In the present chapter, we attempt to find the law of distribution of 

 velocities by a method which consists essentially in regarding the whole gas 

 as a single dynamical system, and in representing its coordinates in a single 

 imaginary space of the appropriate number of dimensions. 



Let us suppose that the gas consists of a great number N of exactly 

 similar molecules, enclosed in a vessel of volume O. At the outset we shall 

 suppose these molecules to be elastic spheres of the kind already described. 

 Each molecule will possess six coordinates, the three positional coordinates of 

 its centre referred to three fixed rectangular axes in the containing vessel, 

 and the three components of the velocity of its centre resolved parallel to 

 these three axes. We shall denote the separate molecules by the letters 

 A, B, C, etc., and the six coordinates of molecule A will be denoted by 

 x a, ya> z a , Ua, v a , w a . The whole gas may accordingly be regarded as a single 

 dynamical system possessing QN coordinates, namely, 



#a, Va, z a , u a , v a , w a , x b , y b , z b , u b , v b , w b , # c ...etc (51). 



j. 3 



