The Equipartition Law in a System of Particles. 599 



with the temperature we may also expand our expression 

 for temperature, 



rp _ 1_ r m v 2 ') 



S^Lvi^^Jav. 

 into a series; we obtain 



. . . (22) 



Combining expressions (21) and (22), by subtraction and 

 ansposition we obtain 



K=|«BT+»ifio(g?+g?+^?+...)- ■ (»3) 



For the case of velocities low enough so that v 4 and higher 

 powers can be neglected, this reduces to the familiar expres- 

 sion of Newtonian mechanics K = 3/2?iRT. 



In case we neglect in expression (23) powers higher than 

 v* we have the approximate relation 



Nm v 4 1 (T$m v 2 \ 2 



1 /]Sm v 2 \ 2 



the left-hand term really being the larger, since the average 

 square of a quantity is greater than the square of its average. 



Since ( — 9^—) is approximately equal to l^nRT) , we may 



write the approximation 



K =l" RT+ 2ik6" RT ) 2 ' 



or noting that Nm = M the total mass of the system at the 

 absolute zero, we have* 



If we use the erg as our unit of energy, R will be 8*31 x 10 7 , 



* This equation should be compared with that of Jiittner No. (60-). 

 The possibility here presented of obtaining such a relation by elementary 

 methods without the use of the Hankel cylinder functions should be 

 noted. Attention should also be called to the fact that the results of 

 this article show that the equation applies to a mixture of particles as 

 well as to a system of only one kind of particles. 



