1052 
value when the fundamental units are made a definite number of 
times larger or smaller’). The differential of the entropy will have 
other dimensions than the entropy itself ete.; im other words the 
entropy will not be a physical quantity in the usual sense’). The 
same objection holds for the denominator. To avoid this the numerator 
or the denominator itself should not be taken, but it should be 
divided by an expression of equal dimensions, e.g. by g. 
Planck takes the numerator divided by gy; at least it comes 
to this when he defines the ‘thermodynamic probability” as 
the number of possible ways in which a certain distribution of 
N! 
— and now 
EMAL os 
state can come about. Thus we get: W= 
S=hklog P is taken again. 
This omission of g\ is an essential point; for the consequence 
of it is that PLANCK’s entropy function contains the element g; the 
original value was independent of g. 
Thus we find the value: 
S=k}N log N —- = flog f do ~- Nlog do} if gm do 
In this expression for S BOLTZMANN had left out the first and 
the third term as constants, through which he arrived at: 
S=—k flog f do 
> flog f do is Bourzmann’s H function. 
In the state of equilibrium 
S=—kWN}3 log (2 am BE) + logv — log (3 N) + 3 — logg} 
follows from PrANcK’s formula, and 
S=kN$3 log (2am E) + loge — 3 log (3 N) + 3 — log NÀ. 
from that of BOLTZMANN. 
In Bontzmann’s formula the dimensions no longer come right; the 
property (3) from $1, that the entropy of the whole is equal to the sum 
of the entropies of the parts, however, does hold, which property 
does not come true in Prarck, if no definite assumption is made about 
y. To make up for this PLaNnck assumes that g is proportional to NM, quite 
arbitrarily in my opinion; the reasoning, namely, by which he tries 
to make plausible that this “elementary region of probability” would 
be proportional to NV, does not bold for a perfect gas *). 
1) Unless also logarithmic dimensions are assigned to k, which neutralise the 
first. Vut this would be very unpractical. 
2) It should be remarked that GrBBs has not hesitated to state this abnormal 
behaviour of the entropy explicitly. Cf. on this § 4. 
3) Cf: “Die gegenwirlige Bedeutung der Quantenhypothese fiir die kinetische 
Gastheorie” in “Vorträge über die kinetische Theorie der Materie und der Electri- 
zität, von Max Pranck u a.” Leipzig u. Berlin 1914. 
