586 Prof. P. C. Tolman on the Relativity Theory : 



point is equally likely to be in any one of the little elements 

 dV. In other words, the different states of a system, which we 

 can specify by stating the region dfadfadfa . . . d^r^^d^ • • • 

 in which the values of the coordinates and momenta of the 

 system fall, are all equally likely to occur*. In making use 

 of this principle we must not forget that to agree with 

 relativity mechanics the momentum corresponding to the 

 coordinate (^ must be defined by the relation 



^ 1 =-^-=-^-2m (l-v/l-^). ... (2) 



A System of Particles. 



Consider now a system containing N a particles which have 

 the mass m a when at rest, N& particles which have the mass 

 mi, N c particles which have the mass m c , &c. If at any 

 given instant we specify the particular differential element 

 dx dy dzd^r x d^ d^ z which contains the coordinates x, y, z for 

 each particle and the corresponding momenta ty x , yjr^, yfr z , we 

 shall thereby completely determine what Planck f has well 

 called the microscopic state of the system, and by the previous 

 paragraph any microscopic state of the system in which we 

 thus specify the six-dimensional position of each particle is 

 just as likely to occur as any other microscopic state. 



It must be noticed, however, that many of the possible 

 microscopic states which are determined by specifying the 

 six-dimensional position of each individual particle are in 

 reality completely identical, since if all the particles having 

 a given mass m a are alike among themselves, it makes no 

 difference which particular one of the various available 

 identical particles we pick out to put into a specified range 

 dx dy dz d^r x d^ y d-^r z . 



For this reason we shall usually be interested in specifying 

 the macroscopic state X °f the system, for which purpose 

 we shall merely state the number of particles of a given 



* The criterion here used for determining whether or not the states 

 are equally liable to occur is obviously a necessary requirement, although 

 it is uot so evident that it is a sufficient requirement for equal 

 probability. 



f Planck, ' Warmestrahlung,' Leipzig, 1913. 



£ In using the words microscopic and macroscopic in these con- 

 siderations, it must not be supposed that the latter indicates any less 

 minute observation of the system. The only difference in describing the 

 microscopic and macroscopic states of a system is that in the latter case 

 we are not interested in which particular one of the available particles 

 are used for supplying the necessary quota to a given element dx dy dz 

 d-fyx dtyy dty z . 



