§ 3. To get an idea of the probability of different states of the 
gas we can imagine that the stale is determined by a lottery in 
which slips of paper. with different numbers are drawn from an urn. 
This can be arranged in such a way that a slip is drawn for each 
molecule successively, the number on the slip indicating the place 
and the state of motion of the molecule. If for each molecule we 
take the coordinates of the centre and the components of the momentum 
as the coordinates in a six-dimensional space R,, the slip will indicate 
the point in this extension which represents the position and the state 
of motion of the molecule or, as we may say, the place of the 
molecule in &,. 
Now Pranck') has introduced the fundamental conception of the 
theory of quanta by imagining that the space /è, is divided into 
equal finite elements of a definite magnitude G and that only the 
question in which of these elements the molecule bas to be placed is 
decided by the lottery. Whether the molecule will lie at one point 
of the element or at another is not determined in his iheory by a 
consideration of probabilities. Instead of this PLaxck supposes that 
the molecules lying in the same element of space G are uniformly 
distributed over its extension. On these suppositions he tinds an 
expression which tie considers, not only as proportional to the pro- 
bability but as equal to it and which leads to a formula for the 
entropy containing uo indefinite additive constant. 
We need not repeat here these calculations of PLanck. I[t suffices 
to remark that the extension-in-phase Rgy, which we mentioned 
in $ 2, may be regarded as composed of MN extensions-in-phase 
Rs each of which belongs to one molecule and that a division of 
each Zi: into elements of magnitude G involves a division of Rey 
into elements of magnitude GY. Puancx’s final result is found 
if the layer corresponding to dl) (§ 2) is expressed in the domain 
G\ as unity, and if the value of 2 thus found is considered as 
the numerical value of IV. 
Instead of (8) we now get 
(2% Em)*/2N—1 | 2, moN 
ms 
r( x) GN 
2 
If we substitute this expression for W in BorrzmanN’s formula 
and again omit all terms not containing N as a factor we find 
(4) 
1) PLANCK, Vorlesungen über die Theorie der Wärmestrahlung, 2. Aufl. (1913), 
125; Vorträge über die kinetische Theorie der Materie und der Elektrizität 
(Wolfskehl-Kongress, 1913), p. 1. 
47* 
