57 
4 3 
aa 
k é 
=|[e 21 Om,) bongs npe © Nyy) Vy} > 
1 
The function Y may be used to transform the formula for the 
frequence §. For, applying the theorem of STIRLING, we can write § 
in the form 
RA hd 
—— dU 
0 4 2 4 A deve ) Vy x 
oes Ne |] (2 x @Q Mz) he by (Dn) Va Mxd 
1 l My), 
and therefore, introducing %, we obtain for - 
O n aS rt ace 
GN Gea ek. II}. ie E 
1 xd 
For the further discussion we shall not use the set energy W, 
but a function y),'), closely connected with it, and being defined 
by the equation 
!) We can somewhat more closely explain the introduction of the function ~, (comp. 
also my dissertation p. 52 s.). We shall compare the free energy of the system 
considered above to the free energy of the same system in gaseous state and in 
a volume so great that it can be considered as an ideal gas. We now can easily 
show the free energy of the mixture in the gaseous state to be equal to the sum 
of free energies of the components, if each of them occupies the same volume as 
their mixture. Further we can suppose that the volume of each of the substances 
(which now occur as simple substances in k separate volumes), is changed in such 
a way, that the number of particles pro unit of volume which is to be taken very 
great, amounts to v (arbitrarily chosen) for all k systems. The volume occupied 
Nz), ; Ny) IL 
by the «4 component now amounts to —. In this state | — will be so great 
YP vp 
that («(v)n%*) may be put equal to unity. 
We therefore find for the free energy of each of the components, originating 
from the element a 
yr, 3 
(0) 2 x) xe 
en En E = (2 7 Om,) (ey : 
yv 
And for their total free energy : 
k 
B eae 
1 
bo| oo 
ye k 
Sn, 
1 
5) OQ ; : na 
es e EEG = (2 1 Om,) 
For the difference between the free energy in the state a which we started 
and that in the zero-state considered we find 
