66 Dr. W. F. G. Swann on Eqiupartition of 



or relations infinitely nearly the same as these, hold for all 

 but an infinitesimal fraction of the generalized space corre- 

 sponding to the region in question. In these formulae Bn p 

 represents the number of the coordinates of the p group 

 having values between f and f + df, E^ is the contribution 

 to E for the particular value of f by a coordinate of the 

 p group, similar remarks applying to the quantities with 

 suffixes q, r, &c. Further, the quantities A p , A g , &c, and h 

 are determined by 



1 Sn p = p, I 8n q = q, 



|e^h + J 



&c. 



E q tSn q + . . . &c.= E. 



The important characteristic of equations (2), (3), &c. is 

 that h is the same in all. 



If we now go one step farther and restrict E l5 E 2 , &c, to 

 be proportional to the squares of the generalized coordinates, 

 it follows from (2), (3), &c, that the average value of the 

 E's for the p terms is the same as that for the q terms or for 

 the r terms, the average value in question being 1/2 h *. 

 The theorem at this stage contains all the mathematical law 

 involved in the theorem of equipartition of energy, except 

 that so far there has been no mention of energy or of 

 dynamics. We may give the name energy to the quan- 

 tities E 1? E 2 , &c, and of course nothing will be altered, 

 and it remains to inquire what part is played by the 

 Hamiltonian equations, for though the Hamiltonian equa- 

 tions contain the principle of the conservation of energy, 

 the principle of the conservation of energy does not neces- 

 sitate the Hamiltonian equations. 



It will be remembered that if in the generalized space we 

 take the points distributed with uniform density initially, 

 then the law represented by (2), (3), &c, which sym- 

 bolizes what is called the " Normal State/' is found to 

 apply for all but an infinitely small proportion of the points 

 in the domain of the space under consideration. This would 

 obviously have been true if the points had not been chosen 

 uniformly distributed unless in some of the exceptional 



* In Maxwell's proof of the theorem, the average is taken not merely 

 over all regions of the generalized space corresponding to the normal 

 state, but over the whole space corresponding to the prescribed energy 

 limits. The reason that the two methods give the same result is of 

 course that so much of the space concerned corresponds to the normal 

 state that the inclusion of the remainder of the space makes no difference 

 to the result. 



