212-215] General Dynamical Theory 181 



equations (439) at every point, the motion of these points will be "steady 

 motion " in the hydrodynamical sense. To obtain such a motion, then, we 

 may assign arbitrary densities over any one of the loci E= constant. The 

 density of distribution of points is then uniquely determined throughout the 

 whole space, and is given by equation (439). The space, filled with fluid in 

 this way, supplies what we require, namely, a basis of investigation which is 

 independent of the time. 



By distributing the arbitrary densities in different ways, we shall obtain 

 the various distributions of points in the imaginary space which are 

 appropriate to the discussion of different problems, i.e., to the discussion of 

 gases about which various data are given. For instance, suppose we know 

 that a gas has cooled, undisturbed by external agencies, from some higher 

 temperature at which its energy was E , and that nothing further as to its 

 past history is known, then we may appropriately suppose the imaginary 

 space filled with a distribution of points such that the density over E = E is 

 constant, say equal to unity. Let the present energy be E 1} then what we 

 wish to know is the density of points over E = E lt and this by equation (439) 

 is given by 



f AF 



p-a-Uv. .............................. (440). 



Application to a system with an infinite number of Degrees of Freedom. 



215. We now introduce the simplifying assumption that the system 

 consists of a very great number of parts (e.g., molecules) so that n, the 

 number of momenta, is very great. Let p 2 denote the mean value of 



i) 2 , n a /M 2 



PI > P% ' Pn 



at any point in the imaginary space. Then in general E is of the same 

 order of magnitude as np* while AF/2F is obviously of the order of l/p 2 . 

 Hence we can write 



(441), 



and <t> will remain finite whea n becomes infinite, being, as we have seen, 

 always positive. 



The distribution of density in the region E = E l is now, by 

 equation (440), 



p = e n<i> (442). 



There is therefore a point (or series of points) of maximum density obtained 

 by making <I> a maximum, and the density here is infinitely greater than that 

 elsewhere. We have now to examine in what way the density falls off as we 

 recede from such a point, so as to determine whether the whole aggregate of 

 points for which E = E l may be regarded as being concentrated about this 

 point (or series of points). 



