

The Equipartition Law in a System of Particles. 591 



The probability that a particle m a will fall in the region 

 tlx dy dz i/r 2 sin# d6 dcf> dijr will be 



a a e " * V r+<Z dx dy dz ^ 2 sin<9 dO d<j> <ty, 



and since each particle must fall somewhere in the space 

 xyz'tyxtyy-tyz we shall have corresponding to (11) the 

 relation 



v( v r r u ae -* ^^^^ ain <9 ^ ^ a+= 1 1 



Jo J 



Corresponding to equation (12), we also see that the average 

 value of any quantity A, which is a property of the molecules 

 of mass m a , will be given by the expression 



A=47rVf *• ~* ^ +m '^r^ ■ ■ ■ (U) 



Jo 



The Law of Partition. 



We may now obtain a law which corresponds to that o( 

 the equipartition of vis viva in the classical mechanics. 

 Considering equation (13) let us integrate by parts ; we 

 obtain 



' >J ^=Ojo 3 y/if + mj r 



Substituting the limits into the first term we find that it 

 becomes zero and may write 



4,rvf° *.* -" V *^ , * 3 rd+= I ■ 



Jo v/^"+»'«" " 



But by equation (14) the left-hand side of this relation is 

 the average value of ^j^/yfr' 2 + w * 2 tor tne particles of mass 

 w . We have 



r r i =?. 



Introducing equation (8), which defines i/r 2 , we may trans- 

 form this expression into 



r )u " r ' 2 i - ;>> sik\ 



L/w-l.-/' ri5) 



