1054 
extension g\ represent cases of equal chance. Not the absolute 
extent of the phase extension would therefore be of importance, but 
the number of finite cases of equal chance contained in it. 
If we assume that a gas, not in equilibrium, can be represented 
by an arbitrary ensemble between narrow energy limits tending to 
a microcanonie ensemble which represents the gas in the state of 
equilibrium, the elements g\ have the practical signification that 
they represent the extent of the parts of the extension with regard 
to which the distribution has become homogeneous, whereas this is 
not the case inside it. Then the elements g\ would, therefore, have 
no fixed extent, but this would depend on the time during which 
the considered gas is left to itself; they can, namely, be taken smaller 
as the “stirring” has continued longer. 
The division by N/ is necessary to ensure that the entropy of 
the whole is equal to the sum of the entropies of the parts (for so 
far as these parts are large with respect to the space element q’), 
or the extension of the whole equal to the product of the extensions 
of the parts. As a justification of this the following explanation may 
be given. If we have & separate quantities of gas, each of  mole- 
cules in a volume v, the total extension of this in the space of the 
distribution of place is (vt =v. If, however, the volumes v are 
not separated from each other, but parts of a larger volume kv, we 
may not take v” for the extension of the parts, because there need 
not be n definite molecules in every volume v, but all the molecules 
of the vessel can get there. It is, however, difficult to say, how 
much every extension changes through the parts not being separated, 
since the extensions are not independent of each other. For the 
extension of all the states, in which there are 7 molecules in the 
first volume v,, is for the greater part the same as that in which 
there are n molecules in v, ete. We can, however, say what change 
must be effected in the total extension in consequence of the absence 
of the partitions. 
We must, namely, take into account that the molecules from the diffe- 
rent (£) volumes v can be interchanged, so that instead of the original 
(en)! ten 
extension (v")* — vin must be taken: of” x je or in approximation: 
nije 
(kn)kn : : : ; 
kn en (hv), which is also the extension which we should 
n cn 
have found by direct calculation of the whole. The equation 
) | 
(kn)! athe i 
Conk > —— = (kv)kr or (vr)k X =(kv)N can now be taken as 
(nl) (n!)* 
basis to obtain an available function, which can take the place of 
