1049 
probability P should first be defined. In the calculus of probability 
it is customary 
number of cases of equal chance favourable to the event and ihe 
to define the probability as: the ratio between the 
total number of cases of equal chance. If there is not a finite number 
of such eases to be indicated, we get instead the ratio between the 
region of the favourable cases and the total region (either with a 
variable weight or without one). 
When we wish to determine the probability of a certain state for 
a gas mass (and henceforth we shall only consider perfect gases) 
we must understand by this region tle region of the possible phases 
of the gas-mass on definite suppositions of energy and volume. The 
region of the favourable cases is then the phase-extension for which 
the state exists, the probability of which is to be determined. 
When following the example of BOLTZMANN and GrBBs we consider 
the weight-function at a definite energy as a constant (or constant 
between # and 4 dl) we get for the probability simply the 
relation between two regions in the I-space, of 6N —1 or 6.N 
dimensions, according as we suppose definite ME, or variable F 
between EZ, and 2, + db. *) 
The former is the more natural supposition when with Esysrrin 
we think of a time-ensemble, so that the probability is equal to the 
ratio between the time in which the system is in the definite state 
and the total time. The second is more appropriate when with Gipps 
we think of possibilities existing simultaneously, so that we can 
imagine reality to be formed by a random choice from an ensemble. 
The result is of course the same in these two cases. 
If we choose the former method and ihink the state determined 
by the series of the values ,, #,, ete.. which represent the numbers 
of molecules, the p's and q’s of which lie between definite limits, 
the molecule-phase-points, therefore, in definite elements y of the 
u-space (phase-extension of the single molecule), then for definite 
E, the probability of this state is equal to the fraction: 
part of the surface E(pq)— bo in the Z-star of the T-space determined by that state 
total area of the surface. 
If we follow the second method, the denominator becomes the 
contents of a thin shell and the numerator that part of the shell 
lying inside the star in question. For a perfect gas surface and 
shell are spherical. 
Now the question suggests itself*): is this fraction also equal of the 
1) Here and in what follows ideas and names are made use of given in the 
Encyclopaed. article by P. and T. Eurenresv. 
2) Cf. Enrenrest. |. c. § 12 and § 13. 
68* 
