38 The Law of Distribution [CH. in 



because each molecule of the gas is supposed to be contained in a finite 

 closed vessel. The surfaces E= constant are therefore finite closed surfaces 

 in the generalised space, the surface E = oo alone being, in the limit, infinite 

 and enclosing all the others. 



Hence the motion of the fluid in the generalised space is one of circula- 

 tion in closed surfaces, and, in particular, there is no motion of the fluid across 

 the boundary at infinity. 



Similarly, if there were any other quantities % 1} % 2 ---> functions of the 

 coordinates in the generalised space, which remained constant throughout the 

 motion of the gas, then the motion of the fluid in the generalised space would 

 be confined to the loci 



%! = constant, % 2 = constant, etc. 



The only quantities of which we know, other than the energy, which 

 remain constant over a collision between any two molecules, are the three 

 components of linear momentum, the three moments of angular momentum 

 and the number of molecules in the gas ; of these the components of momen- 

 tum both linear and angular are in general changed by a collision between a 

 molecule and the boundary, and the number of molecules in the gas is not 

 a function of the coordinates in the generalised space. Thus in general the 

 energy is the only quantity of which we know, satisfying the conditions in 

 question. 



An exception to this may occur if the vessel containing the gas is a 

 figure of revolution, having its interior surface perfectly smooth. For then 

 there is always a component of momentum which is not changed by a collision 

 between a molecule and the boundary ; namely, that parallel to a tangent to 

 the containing vessel at the point at which the collision takes place. In this 

 case, then, the moment of momentum of the whole gas about the axis of 

 figure of the containing vessel remains constant throughout the motion. 

 It will, however, be convenient to defer the consideration of special cases of 

 this type until Chapter V. 



The Partition of the Generalised Space Positional Coordinates. 



39. We have supposed the volume of the containing vessel to be 1. 

 Let us suppose the vessel divided up in a number n of small " cells " each of 

 the same volume <w, so that nw = 1. These cells will be referred to as cell 1, 

 cell 2, ..., respectively. The different possible configurations of the gas are 

 now to be classified according to the number of molecules of which the 

 centres fall within the different cells. As a typical class, we consider a class 

 such that a-L molecules have their centres in cell 1, 2 in cell 2, and so on. 

 Let this class be referred to as class A. We have to examine what proportion 

 of the whole of the generalised space represents systems of class A. 



