k Ts 
it ee ne 
1 Pet (vi ae eae es (11) 
(in which only those y,.’s are to be taken into account that differ 
from zero). I will represent the factor (y, ..yn-.Y,) by s,. 
Now in order to examine which of all systems is the most fre- 
quently occurring in the ensemble, which therefore is the system in 
equilibrium, we have to consider for which values of the n,. z or 
log z, i.e. 
k 
= n, (log I,—log nm, + 1—logs,) . . . . . (12) 
1 
is a maximum, (2/ being developed here according to the formula 
of SriRLING). The variations to which the numbers », are submitted 
are «ar, in Which @ is a positive or negative whole number. The 
condition of equilibrium that is reached in this way is 
k 
= (log logen bog Oren Se (12) 
| 
Introducing 
3 
LI, = (22m, 0)? Vy 
Ip, we get 
k 
jh S 
TT EEn ieee eerie Cols 
As x contains still terms that depend on 7’, this formula cannot 
yet be compared to that of Dr. Hornen; however, in many regards 
ee R 
it is already analogous to it. Now, applying the theorem that ne log w, 
vi 
an which w is the probability of a state) is identical with the entropy, 
we find the entropy 3 of an arbitrarily chosen state to be given by 
Die 
>) nz log I,—log nm, + 1—logs, . . . . (14) 
This quantity therefore must agree with the entropy of a non- 
equilibrium state as defined by Dr. Horren. As appears from what 
is mentioned above, it possesses the quality of being a maximum 
in the state of equilibrium. 
Now I will first use the result we obtained to calculate Y and 
through means of it the equation of state. Developing z with respect 
to a and summing up, we find for 4” 
72 
Proceedings Royal Acad. Amsterdam. Vol. XV. 
