56 
m, denoting the mass of a molecule of the kind x. We now can 
ask, for which values of the numbers 7, this frequency is a maxi- 
mum. In this way we find for the # conditions to which the den- 
sities in the most frequently occurring system are submitted : 
d log wy 
l 
— log nj + & (n„j) Er tt OS J, ie 
1 Ny) ) 
x from 1 to 4. These conditions can be satisfied by means of a 
homogeneous distribution of each of the x kinds over the volume 
V. Further the second variation of £ or of log & has to be negative. 
If -we denote by 7, the values in the most frequently occurring 
system, then the frequency Sa of the system in which these num- 
bers have the values 7,, + 1,, can be represented by 
—Q 
by ye a OS ee 
The quantity Q is a homogeneous quadratic function of the numbers 
t,,. Taking the sum of &, with respect to all possible values of 
these numbers i.e. from — oo to + oo, we obtain > & = JN, from 
which ¥ can be calculated. 
Proceeding in this way we find 
U Bn, 
e/a k ake 
0 2 
e a Lek (ain) fwoln, . Dy. . ng), 2 A) 
1 
In calculating YW, which is equivalent to the free energy, we 
must neglect a factor of the order of unity. However, the formula 
is rigorously exact, the above-mentioned being a mere verification 
of the equation (3). For keeping in mind the definition of Gisps, 
we have for ¥ 
— Ze + m,r,” 
OG 20 ; 
e —— we hs an OF, ah 
and therefore 
3 el 
e = Tt (2rOm,) e de eden, 
1 
and we see that according to the definition of the function w, the 
formula given for ¥ holds exactly *). 
If we would have as a separate system of volume V; the n,,.. 7,5 … 
1», molecules being now in the volume V3, then the free energy 
of this system would be given by the formula 
1) Comp. also my dissertation p. 56, 112, 126. 
