34 The Law of Distribution [CH. in 



We can suppose this dynamical system represented in a generalised space 

 of 6N dimensions. The configuration of the system in which the coordinates 

 are those given by (51) will be represented by a single point in this space, 

 namely the point of which the coordinates referred to QN rectangular axes 

 are those given by (51). 



33. In this way every possible configuration is represented by a point, 

 but it does not follow that every point represents a possible configuration. 

 For instance if as before the diameter of each molecule is <r, then a configu- 

 ration in which 



is physically impossible because it represents a configuration in which the 

 centres of molecules A and B are separated by a distance less than a- a con- 

 figuration, therefore, in which parts of these two molecules occupy the same 

 space. We must therefore suppose the region in which the inequality (52) 

 is satisfied to be excluded from our generalised space. If we shut off all such 

 regions, found by substituting for a, b in (52) the suffixes corresponding to all 

 possible pairs of molecules, we see that every point in the space which is left 

 will represent a system which is physically possible in so far that no two 

 molecules overlap. 



There is still the boundary to be considered. For a configuration to be 

 physically possible it is necessary that the centre of each molecule shall be at 

 a normal distance from the boundary which is greater than the radius of the 

 molecule. Thus all parts of the space must be excluded which do not satisfy 

 equations of the form 



^ .(53), 



where </> (x, y, z) is the shortest normal distance from x, y, z to the surface of 

 the enclosing vessel. 



34. If we exclude all the regions just indicated, it is clear that any 

 point in the space which remains will represent a configuration of the system 

 of molecules which is physically possible. In the course of the motion of the 

 gas, this configuration will give place to other configurations, and by tracing 

 out this series of configurations in the generalised space, we should obtain a 

 " path " indicating the motion of the gas. By starting from a great number 

 of points, and tracing the motion backwards as well as forwards, the whole 

 space can be mapped out into paths in this way. Since the motion of the 

 gas is completely determined when all the coordinates (51) are known, it 

 follows that through any point there is one and only one path; two paths can 

 never intersect. Also, of course, the paths are fixed in the generalised space; 

 the motion of a gas, starting from given values of velocity and position 

 coordinates, is always the same. These paths are identical with the 

 " trajectories " of abstract dynamics. 



