J he Eqidpartition Law in a System of Particles. 589 

 Transforming we Lave 



-« 6 \ W 



w b = et b e J 



«fec. 



as the expressions which determine the chance that a given 

 particle o£ mass m a , m b , &c. will fall in a given region 

 dxdy dzdty x d^yd-ty z , when we have the distribution of 

 maximum probability. It should be noticed that /*, which 

 corresponds to the \ of the preceding equations, is the same 

 constant in all of the equations, while « a , cc b , &c, are dif- 

 ferent constants, depending on the mass of the particles 

 m a , rag, &c. 



The Energy as a Function of the Momentum. 



E a , E&. &c, are of course functions of x, y, z, ^/r x , ^ y , ^-. 

 Let us now obtain an expression for E a in terms of these 

 quantities. If there is no external field of force acting, the 

 energy of a particle E« will be independent of x, y, and z, 

 and will be determined entirely by its velocity and mass. 

 In accordance with the theory of relativity we shall have * 



B -7?b <" 



where m a is the mass of the particle at rest. 



Let us now express E a as a function of yjrs, yfr y , and ^ z . 

 We have by equation (2) 



= ^ m a ( 1 — >x /l — x 2 4- if + z 2 ), 



■d.r 



m a x 



Constructing the similar expressions for i/r y and yjr c we may 

 write the relation 



,2 i_2, I 2 . I 2 ™ a 2 (P + y 2 -\-Z 2 ) W„ L V 



L — I 1 — <•- 



whicli also defines i/r. 



(8) 



* This expression is that for the total energy of the particle including 

 that internal energy w» which, according- to relativity theory, the particle 



has when it is at rest. 



