Energy and Radiation Theory. 67 



regions where the normal state did not apply the density 

 of the points happened to be infinitely large *. If we 

 imagine the coordinates in the generalized space to repre- 

 sent the coordinates and momenta of some dynamical 

 system, each point corresponding to a particular state of 

 the system, then, as time progresses, the points will move 

 through the generalized space. The Hamiltonian equations, 

 however, insure that if the density of the points was origin- 

 ally constant, it will, during the motion, remain constant so 

 that there will be no tendency for the points to congregate 

 in those abnormal regions of the space where the normal 

 state does not hold f. This is generally taken to prove that 

 in practice the chance of a system being in an abnormal 

 state is infinitesimal, and it is thus in linking up the mathe- 

 matical problem with the physical one that the Hamiltonian 

 equations play their part. 



The Application of the Theorem. 



Arguments of the type summarized above seem to have 

 been accepted largely without question as indicating that 

 the state represented by (2), (3), &c. is infinitely probable 

 in the practical sense. While it would I think b© easy to 

 deny the necessity for this conclusion on purely logical 

 grounds, it is perhaps well to supplement the discussion by 

 keeping our minds centred on a particular case so that we 

 may the more readily judge whether objections which we 

 raise against the above conclusions are really pertinent or 

 merely formal. 



Let us suppose that we take all the copper in the universe 

 and imagine it divided up into equal blocks % each of which 

 we shall speak of as a system. Let us choose, for examina- 

 tion, those blocks for which the temperature is such that E 

 lies between E and E + JE. Now each piece of copper is in 

 a different state, and, having decided what we shall choose 

 as generalized coordinates, suppose that we plot, in the space, 

 points corresponding to each block. We may look upon the 



* See J. H. Jeans, " Report on Radiation and the Quantum Theory/ 

 p. 34. 



■f It is worth while observing that the Hamiltonian equations do not 

 necessarily require that the relative configurations of the points shall 

 remain the same throughout the motion. All that is required is that 

 the density shall remain constant. This permits of points becoming 

 •thinned out, for example, along the path of a stream line provided that 

 they are crowded to a corresponding extent perpendicular thereto. 



X More strictly we should .say : divided up into blocks each containing 

 the same number of corresponding" coordinates. 



F2 



