sa 
connected with the use of equation (60)*): it is generally assumed 
as self-evident, that the entropy of a gas is to be taken twice as 
large, if the number of molecules and the volume are both doubled. 
Now it is certainly true, that the increase of the entropy in a given 
process in a gas of twice the number of molecules is twice as large 
as the corresponding increase in the original gas. But what is the 
meaning of taking the entropy itself twice as large and thereby 
settling the entropy-difference between the doubled and the original 
gas? By what reversible process is the double quantity of gas to 
be generated from the original quantity? Without that the entropy- 
d 
difference f = cannot be clearly defined. On account of equation 
(60) one is then confronted with the difficult problem of choosing 
the definitions in such a manner that the “thermodynamic proba- 
bility, of the double quantity of a gas is equal to the square of 
the “thermodynamic probability of the single quantity. ?) 
In order to remove this obscurity it is necessary to return to 
BoLTZMANN’s equation (59) and to apply it to a reversible process 
in which the numbers of the molecules change. 
We shall now go a little more fully into the relation in which 
our theory stands to others which are closely allied to it. ® Special 
interest attaches to the manner, in which in the various theories 
the terms MNilog N are produced. In our theory they originate in 
the combinatory factor: 
See hy NIE 61) 
: NANOS oo Nf 0 GE) A Ge REN ae i 
If instead of a gas-mixture, as in our case, a single gas of mon- 
atomic *) molecules is considered, this factor ) reduces to 
== aes a 
X/ 
(62) 
1) O. SrerN, quite recently remarks: “The difficulty in this deduction lies in 
the introduction of the quantity N, which is done in a very arbitrary manner”. 
(Z. f. Elektroch. 25 (1919), p. 79 at the top on the right). 
2) Comp. our remarks in notes (1) and (2) § 5 with regard to the quantities 
Q and I, which in our theory occur in the entropy and in log} y §. 
3) As regards the theories of Lenz (Vorträge der Wolfskehl-Stiftung 1913 in 
Göttingen, Teubner 1914, p. 125) and Kressom (Phys. Ztschr. 14 (1918), p.212), 
who apply Desue’s method for solids to gases, we may refer to papers by H. 
A. Lorentz (Versl. Kon. Ak. v. Wet. Amst. 23 (1) (1914) p. 515, § 6 — Proceedings 
Amsterdam 19, (1917) p. 737) and O. Stern (Ztschr. für Elektrochemie, 25 (1919), 
79 section C towards the end), where these theories are discussed. 
The same holds for a gas with more atoms in the molecule, if ¢ = 1. 
12 
Proceedings Royal Acad. Amsterdam. Vol. XXIII. 
