738 
and it certainly is one of the most important equations of modern 
physics. Nevertheless, difficulties may arise when we come to consider 
the rules according to which the value of JV must ‘be determined. 
§ 2. The state of a gas may be defined by the coordinates of the 
NV molecules and the components of their momenta. These parameters 
may be regarded as the coordinates of a point in a 6 .V-dimensional 
space Fgy, the “extension-in-phase”. The part of this space corre- 
sponding to a given value of the volume and to values of the energy 
between KE and H+dk, a part which we may call a thin “layer”, 
will have a definite magnitude proportional to d/. Let this value 
expressed in some properly chosen unit be Qd. By putting W 
proportional to 2 one really finds formula (1) by means of Bor.TAManN’s 
equation. 
Indeed,. if we take as unit of space in Psy a cube, the edges of 
which are parallel to the axes of coordinates and are of the length 
1, we have 
(2aEm)/2N-1 , 2amvN 
recy 
r( v) 
2 
where the mass of a molecule is denoted by m. *) 
Let us now put W = C@, understanding by Ca factor that has 
the same value for all states of the gas. Omitting in the expression 
for k log 2 all terms which do not contain the factor .V, as we may 
do if .V is very large’), we find 
ee 
(3) 
3 3 oh Ce 
SEN | log (2a Em) + log v— — ol ’) En | + klog C, 
es ; END B B 
which is in agreement with (1), if we put 
3 
k N {log (2am) -— log ( x) +1 | + klog C. 
) 
a 
2 
J 
== 
2 
1) The domain @dE in the extension-in-phase may be decomposed into a domain 
in the extension in-configuration and one in the extension-in-velocity. The nume- 
rical values of these two must be multiplied by each other. The first domain is 
14 Med ecb. et 
v-V and for the second we may write TE” if A is the part of the extension-in- 
aL 
velocity, in which the energy has a value below 4. K is a 3N dimensional sphere 
with radius (2Em\/2, so that we have 
er . SN \*2N hk 
2) We may then write aa lore TES N }. 
Ze ya } 
