245 
macro-complevion. The definition of the latter complexion follows 
from that of the group macro-complexion by taking account of the 
individuality of the molecules. The number of micro-complexions in 
the individual macro-complexion has to be separately determined for 
each special problem, and this, multiplied by the number of individual 
macro-complexions contained in the group macro-complexion gives 
the number of micro-complexions contained in the group maecro-com- 
plexion. The number of individual macro-complexions contained in 
the group macro-complexion, which is readily obtained from the 
theory of permutations, we shall call the permutability index of the 
macro-complexion *). 
From the value thus obtained for the probability of a group macro- 
complexion one can ascertain which group macro-complexion is the 
most probable in a self-contained system of molecules of given energy 
and volume. According to BOLTZMANN the distribution of molecules 
according to the coordinates ete. determining it, obtained for this 
macro-complexion, corresponds macroscopically to a state of equilibrium 
of the system of molecules. 
BoLTZMANN’s entropy principle can now be formulated in such a 
way that the entropies of different macroscopically determined states 
are, if we omit an arbitrary additive constant, proportional to the 
logarithms of the probabilities of the different group macro-complexions 
corresponding to those macro-states. In this it is understood that these 
macro-complexions are determined with the same limits (equal 
elements of corresponding spaces) for the coordinates etc. 
In the simple case, in which the same number of micro-complexions 
is present in each of the individual macro-complexions, as in the 
deduction of the equation of state for molecules whose dimensions 
and mutual attractions are neglected *), the entropy is then simply 
proportional to the permutability index of the macro-complexion. 
In general we may write 
Siplast re OD 
in which S represents the entropy, and kp == Ry /N where Ry is 
the molecular gas constant and JN is the AvoGapro number (i. e. 
the number of molecules in the gram molecule). We then obtain for 
the entropy in the state of equilibrium of a gas whose molecules are 
regarded as having no dimensions and as exerting no mutually attractive 
forces, a function of volume and temperature which agrees with the 
thermodynamic expression for the entropy. 
1) Differing slightly from L. BoLTZMANN, loc. cit. p. 243 note 4. 
2) Comp. M. PuancK, Wärmestrahlung, p. 140 sqq.; Acht Vorlesungen, Vierte 
Vorlesung. 
