1182 
which latter can namely be formed from gases in a continuous way. 
They would also have to hold for solid substances, when these were 
considered in the same way; as it is however customary to consider 
the molecules in this case as not interchangeable, N/ resp. the term 
with log (N/) must be omitted. This is namely the case when the 
solid substance is imagined as a system of fixed ‘‘oscillators”’. 
Though properly speaking we have only generally proved the 
formulae on the assumption that the system can pass into an ideal 
solid body without loss of degrees of freedom, it yet seems plausible 
that a general validity may be ascribed to it. 
We have, namely, seen in § 5 that it may also be derived in 
another way for a definite case, and the conclusion suggests itself 
that this will also be possible in other cases. We have, however, 
at the same time learnt to know the probable limits of the validity. 
If we want to drop the supposition that no indistinguishable 
atoms occur in a molecule, we shall have to add still a term 
kN log p to (32), when p is the number of different ways, in 
which a molecule can be made to cover itself. 
We can finally give still another form to (32). When the integra- 
tion is replaced by a summation, and when in this dG = h/* is 
always put, we get: 
S=—k > f dG; log (f; d Gi) — klog (N!) —klog(pN). . (35) 
When a canonical ensemble consists of so great a number M of 
systems that the number W;= Mfid(r; lying in an elementary 
region dG; is a large number, we can write: 
MS = — kM fi dGjlog (Mf, dG;) 4+- kM log M—kM log (N!) — kM log (p\)= 
; M! | 
0 SUN (MOMMY, ae 
This being the entropy of a system of J/N molecules, the ex- 
pression must only depend on the product MN, independent of the 
way in which this has been separated into factors. This may be 
seen still better as follows. 
When in (85) N is replaced by MN, we get according to (23): 
MN cr ; | ae 4 Pee 
PynN = Vv uN [7(7)] MN — (Mvy) MN [/(T)] MN — MMN PMN, 
Hence 
BEE M 
e kT 7 iN 
JiMN = 5 ari —. Mn 
because when corresponding elementary regions are compared, NV = N. 
Further dGimn = hiMN = (dGin). 
