584 Prof. R. C. Tolman on the Relativity Theory : 



partial consideration of the actual partition of energy attained 

 at equilibrium. 



The Equations of Motion. 

 It has been shown in a previous article * that the Hamil- 

 tonian equations of motion 



#!___ BE ~1 

 dt ' <tyi I 



d$i M_ K ...... (1) 



dt B-v/ri I 



&c, j 



will hold in relativity mechanics provided we define the 

 generalized momenta ^ lt i/r 2 , &c, not as the differential of 

 the kinetic energy with respect to the generalized velocities 



$i> ^25 & c -> but as ^ ne differential with respect to fa, fa, &c, 

 of a function T=2m (l — \/l — v 2 ), where m is the mass of 

 a particle having the velocity v and the summation 2 extends 

 over all the particles of the system. 



Representation in Generalized Space. 



Consider now a system defined by the n generalized 



coordinates fa, fa, <t>& fa and the corresponding momenta 



^i? ^2> ^35 • • • ty** Employing the methods so successfully 

 used by Jeans f, we may think of the state of the system at 

 any instant as determined by the position of a point plotted 

 in a 2n dimensional space. Suppose now : we had a large 

 number of systems of the same structure but differing in 

 state, then for each system we should have at any instant a 

 corresponding point in our 2n dimensional space, and as the 

 systems changed their state, in the manner required by the 

 laws of motion, the points would describe stream lines in 

 this space. 



Liouville's Theorem. 



Suppose now that the points were originally distributed in 

 the generalized space with the uniform density p. Then it 

 can be shown by familiar methods that Liouville's theorem 

 holds, just as in the classical mechanics, that is, the density 

 of distribution remains uniform. 



Take, for example, some particular cubical element of 

 our generalized space dfa dfa d<f>% . . . d^ dyjr 2 dyfr z .... The 



* This Journal, ante, pp. 572-582. 



t Jeans, 'The Dynamical Theory of Gases,' Cambridge, 1904. 



