The Equipartition Law in a System of Particles. 597' 



The Partition of Energy. 



We have seen that our new equipartition law precludes 

 the possibility o£ an exact equipartition of energy. It 

 becomes very important to see what the average energy of a 

 particle o£ a given mass does become at any temperature. 



Equation (14) provides a general method of determining 

 the average value of any property of the particles. For the 

 average value of the energy ^/^ 2 + ??? 2 of particles of 

 mass m Q (see equation (9)) we shall have 



i 



h Vi// 2 4-«i 



The unknown constant « may be eliminated with the help of 

 the relation (13) 



Jo 

 and for h we may substitute the value given by (18) which) 

 gives us the desired equation 



/cT 



6 n/^ 2 + m o 2 ir' 2 ify 



«-— aw ■■ ■ ■ ■ <19) 



r 



Ic'£ 



Partition of Energy for Zero Mass. 



Unfortunately, no general method for the evaluation of 

 this expression seems to be available. For the particular 

 case that the mass m of the particles approaches zero, the 

 expression reduces to 



€ kT ^ 3 df 







f- 



yjr-dyjr 



in terms of integral whose values are known. Evaluating, 

 we obtain 



E=3&T. 



