18 Mr. S. B. McLaren on Hamilton's Equations and the 



The product on the right-hand side of (5) is known to be 

 a differential invariant. 



Let there he systems all satisfying (1) but starting from 

 all initial conditions. Let the number originally having 

 their coordinates within the limits dp r , dg r , and so on be dS. 

 This distribution is invariant, in consequence of the invariant 

 character of dN. bueh of the systems as have originally 

 identical values of H will always have the same value for it. 



Each of the systems is on the average in a state of 

 temperature equilibrium, and the distribution of energy 

 proper to the temperature exists in it. This distribution of 

 energy can be inferred by averaging over all the systems, 

 since it exists in each. 



The theorem of equipartition requires 



$»%*-$*•'£" ^ 



(6) is to hold for all values of r and s. 



It will be enough to show 



jjVg^ s =jj 7) 



3H , . 



(") 



integrating 



for the other 



From (7) follows (6) b\ 

 ordinates. 



The final result is to apply to systems all of which poss 

 the same total en erg- 

 may b« taken over the 

 H 2 — Hi can be made infinitesimal. 



Y 



co- 



ess 



. In (7), therefore, the integration 

 region H 2 >H>H 1 . and the difference 



Suppose the curves 11= H 2 aU( i H=H X are represented in 

 the plane of coordinates ; p r may be measured along OX 

 and p s along OY. 



