101 



we can woik out an average of evei-j arbilrarj (inanlily fof the 

 group of systems wliicli after a lime t lias developed from llie group 

 considered at first. The value of the (piantity considered varies for (he 

 different systems of our group (pari ensenible\ because Ihe systems 

 wiiere at t := i) the velocity of the particle is ?f„, ma^-- still show 

 considerable differences, so that e.g. the impulses which the particle 

 gets will be widely different. I shall call this average the crz.9é?-rti'é?;v/(/é? 

 u)Uh a given initial velocity. Moreover the velocity u^ may be varied 

 and again the case-average may be worked out and then by making 

 u^ run through all possible values all systems are taken into con- 

 sideration in determining the average at the time t. If now the 

 case-average of a quantity g{u) for ?/„ is g{u^) and if Ihe number 

 of systems of the group is N {u^) then — if K represents the total 

 number of systems in the ensemble — Ihe quantity 



N 



is the case-average for Ihe entire ensemble. 



However, as the ensemble is stationary the case average for a 

 quantity is equal to the average of the corresponding quantity 

 in the ensemble. If in particular g{u) for every n is equal to zero, 

 the average in the ensemble is also zero. I shall now demonstrate 

 that if we start from Einstein's formula the case average of Ku 

 for everj' initial velocity u^ is zero, and from this it follows imme- 

 diately that Einstein's formula does not come into conflict with (he 

 theory of the ensembles, more particularly that 7\u'^ = 0, ( — e 

 means determining the average of an ensemble, which — is used every- 

 where here for the case average). 



Einstein's equation comes into conflict with the theory of the 

 ensembles if we select at ^ =r a group of particles with a given 

 velocity u^ from the ensemble. For if we determine the average of 



the equation it yields m. — = K =: — urn,, whilst according to the 



dt 



theory of ensembles K is independent of the velocity. If however 

 we leave the selected group to itself and if we a|)ply to its motion 

 Einstein's equation which is not right in the first moment, it is 

 evident that in the long run Ihe group moves in such a way that 

 in the long run Einstein's equation can be applied to it. Moreover 

 from the group with particles with a given velocity ?<„ those systems 

 can be selected to which (1) applies. From what follows it becomes 

 apparent that for this group in the long run the usual relations 



