590 Prof. R. 0. Tolman on the Relativity Theory : 



By simple transformations. and the introduction of equa- 

 tion (7) we obtain the desired relation 



E a = ^^+^? ( 9 > 



We may now rewrite equations (6) in the form 



ac n 6 



&C. J 



From these equations we see that the probability that a 

 oiven particle of mass m a will fall in a given region 

 dx dy dz dty x dty y dty z is independent of the position x, y, z *,. 

 but dependent on the momentum yjr = x /yfr x 2 + -^r/ + ^Jr z 2 . 



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

 dx dy dz difr x dyjr^ d-^r z is 



a a e a dxdydzd^ x d^yd^r z \. 



Since every particle must have components of momentum 

 lying between zero and infinity, and lie somewhere in the 

 whole volume V occupied by the mixture, we have the 

 relation 



vrrr^'^^^f,*^-!. . en) 



Jo Jo Jo 

 It is further evident that the average value of any quantity 

 A which is a property of the particles is given by the 

 expression 



,-»oo /- 00 ,-»oo , / f v _|_ 2 



A = V « a e"' Vt " + " , ""A^rff y ^- ! . (12) 



Jo Jo Jo 

 where A is some function of ^, ^r y , and \jr z . 



Polar Coordinates. 



We may express relations corresponding to (11) and (12) 

 more simply if we make use of polar coordinates. Consider 

 instead of the elementary volume dfa, d^ 7J , dyjr z , the volume 

 -*|r 2 sin Odddfydty expressed in polar coordinates. 



* This is only true when, as assumed, no external field of force is 

 acting. 



f For the case that only one kind of particles is present the law of the 

 distribution of velocities in relativity mechanics has already been worked 

 out by Jiittner, he. cit., and is a special case of the law here presented 

 when particles of different masses are present. 



