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considerations in a geometrical form. The state of a given system 
of n degrees of freedom is determined by » general coordinates 
Qi-+-Q-+-Q, and the n corresponding momenta p,...p,.. apne. Lf 
we take these 27 variables as the coordinates of a point in a space 
Rs, (extension in phase) a point of this space will then represent 
the state of a system. 
All the points taken by the representing point for a system left 
to itself, will lie in a (2n—1)-dimensional space /y,—1, of which 
the equation is as follows: 
SON Sn Dti Oe + ai) ae Ca) arg) eC pes (1) 
The form of the function e‚ which represents the energy, depends 
on the kind of the given system. The motion of the representing 
point in the space /._; is determined by 2n differential equations 
of the form 
de 
Po en dq, 
w taken from 1—7) 
de 
QV = Op,” 
and by the 2n initial values of the p’s and the Q's. The point passes 
through a line in the space /,-1, which I shall indicate as the 
trajectory L. Like Einstein’) Dr. Hertz assumes that the trajectory 
L perfectly fills the whole space /%,.-;. Making use of this hypo- 
thesis they then demonstrate that the mean value of a quantity ina 
time-ensemble is identical with that in the microcanonical ensemble. 
For this reason, it is possible, to reduce the study of the properties 
of an arbitrary system to that of a microcanonical ensemble. This 
ensemble consists of a layer between the spaces-¢ and e + de filled 
homogeneously with systems of a Qs, density. If in the limit de is 
taken equal to O and es, to infinite, but in such a way that va, de 
remains finite, we get an ensemble in the space /,,; which is 
filled with a space density @9,—-;; the thus formed ensemble will be 
called an energy-space ensemble. We have to specify the terms mean 
value in an ensemble and probability of a state. Before I pass on 
to that, I sball examine more closely the hypothesis already mentioned 
of EINSTEIN and Hertz. 
It is impossible for a system to pass rigidly in a tinite time (what- 
ever length it may have) through all the possible phases; and in 
calculating a time-average such a finite time must be imagined. 
1) A. Erster, Ann. d. Phys. Bd. 11, p. 170, 1908. 
