( 860 ° 
properties of a real system are therefore those of a system chosen 
arbitrarily from one of those ensembles. 
If we know that the state of a system is stationary, the properties 
of the system will agree with those of the most frequently occurring 
system of the ensemble; after a sufficiently long time every system 
will come to this state just for the very reason one would say that 
it can be thought to belong to a real ensemble. The idea of probability 
of a real system, which strictly speaking has only sense in relation 
to systems lying on the same path, can now be extended in the 
following manner: the system is produced by a construction which 
when repeated many times will lead to a real ensemble, the latter 
is identified with an energy-space (or microcanonical) ensemble; the 
probability that a real system is in a given state is therefore equal 
to the probability of the same state in the energy space or micro- 
canonical ensemble *). 
§ 5. In the following I shall consider the canonical ensembles. 
lt is generally affirmed that these ensembles have no physical meaning 
and that their introduction is only justified because of the simplifica- 
tions, which they allow when used in the calculations; also Hertz 
adheres to this opinion *). I think, however, that by changing a little 
the considerations which enabled us to ascribe a physical meaning 
to the microcanonical ensemble, i.e. by relating them to the real 
ensembles, we can attribute im the same sense a physical meaning to 
the canonical ensembles. If we know that in natare by the action of 
exactly determined causes a system of precisely the energy €, would 
be formed, it is obvious to presume that in consequence of the small 
1) By the following considerations we can avoid the mentioned hypothesis. Sup- 
pose that a real ensemble has been constructed % times; in each construction we 
take N times a point at haphazard in the space H2,—1 and unite the chosen points 
to an ensemble (or we proceed in the same way for the layer between = and 
e 4de). Each possible real ensemble appears a certain number of times among 
the YX ensembles constructed. The probability W. of a given ensemble can be 
defined, as this number divided by the total number of ensembles %. If w: represents 
% 
the probability of a given state in the ensemble under consideration then sw, W, 
l 
can be taken as the definition of the probability of a phase, the sum has to be 
extended over all the % ensembles. The hypotheses mentioned above means that 
we put the probability for the energy-space ensemble equal to 1 and take for w 
the probability in their ensemble. 
2) This simplification is often not so very important; most questions which can 
be solved by means of the canonical ensembles can be treated in a like manner 
without much complication, also by means of the micro-canonical ensembles. 
