( 859 ) 
are not identical; the following considerations mutatis mutandis may, 
however, be applied to both. I shall therefore in the following pages 
only take into account the real energy-space ensemble. Constructing 
several times a real energy space ensemble, we shall find that the 
number of systems lyipg in a given element of the space Ey xy, 
can differ in those cases. How great this number will be, cannot 
be said, if one does not know anything about the way in which 
the energy is supplied to the systems. If, however, we proceed 
without any scheme, the distribution of systems over the space 
Ey, will differ very little in the majority of possible cases. The 
distribution occurring in the majority of the possible cases must be 
stationary. The most simple stationary ensemble is the energy space 
ensemble discussed in § 2. *). 
I shall now introduce the hypothesis that the real ensemble is 
identical with an energy space ensemble. 
If we had supposed the energy of the considered systems to have 
a value between ¢ and ¢ + de, we should have found another kind 
of real ensembles which we can indicate by the term of real micro- 
canonical ensembles. The most frequently occurring and stationary 
ensemble is the ensemble with a homogeneous distribution. (Comp. 
Gipss Chap. XI and XII). *) 
The introduced hypothesis enables us to deduce the properties of 
a real system with the help of the corvesponding mean value in the 
energy space or the microcanonical ensemble. An arbitrary system 
can be obtained by choosing a system from a real ensemble; this 
real ensemble is an energy-space or microcanonical ensemble; the 
1) The ensembles having the constant A different for the strips are also sta- 
tionary. These must be taken into account if we know something more concerning 
the constants of integration. 
2) The distribution of systems in a real ensemble can be chan sed by the motion 
of the representing points, if it is not identical with the energy space-ensemble. 
It is impossible that in consequence of this motion an arbitrary real ensemble 
changes to an energy space ensemble, if the distribution for the strips of § 3 
deviates from that in the energy space ensemble. Suchlike ensembles are, 
however, very rare among all the ensembles, built up of a given number of 
systems in the space H2,—1. If the distribution over the strips agrees with that 
in an energy space ensemble, but is different from this inside the strips themselves, 
the ensemble will, by the motion of the systems, take states in which it deviates 
very little from an energy space ensemble but periodically it will again differ 
‘more from it. Also this kind of deviating ensembles is very rare. As for a real 
microcanonical ensemble, which shows a distribution different from the homo- 
geneous, the distribution will differ after a long time as little as we like from the 
homogeneous in fixed elements of the space E21 which are not too small, 
(Comp. GrBBs Chap. XII.) 
