250 
2 
a= Aw 2, 
a Nen = Const, ede tot ee 
dv dw dv N dv, 
1 
in which uw, = ae (§,? + 4,7 + 52) represents the kinetic energy of 
translation of a molecule whose velocity lies in dw, . 
The condition for a maximum, in conjunction with (7) and 
n = const.*) gives 
2aw n, Se 
hat - ———] + log. ¢ == Qin (8) 
„Re 
n°dv, 
4 
— TO 
/ 3 
— loge n,, — 1, 
dv, 
in which A4 and c are constants. A few reductions lead to 
n 
es 
v 
and (9, 
nf(hm lr — huw, 
Oran ee dv, dw,, 
A 
v\2a 
the well known conditions for equilibrium: macroscopically uniform 
distribution throughout the space, and. Maxwerr’s distribution of 
velocities with the same constant h for each macro-volume-element. 
This constant can be found by obtaining an expression for the 
energy u 
Datta ts ae 
pia, CTR Sy sp Bie eet ae (10) 
From (6) and (9) we obtain for the state of equilibrium 
loge W =n loge v — pele A hth uy — ae n En Ge 
; 2 pi 2 v 3 
in which w,, represents the total kinetic energy, and certain constants 
are omitted. In conjunction with (3) this gives 
3 
JE kpn loge v — En ky nloge h+ kp huw— 5 2 poe „(ll 
On eliminating A between this equation and (10) one obtains a 
fundamental equation of state expressing w as a function of s and 
v,or s as a function of w and v, which Pranck: calls the canonical 
equation of state. On keeping v constant and differentiating (10) and 
Ou 
(11) with respect to /, since de one easily obtains 
u 
8 
ik. 
TSE ate. Se aa ee eee 
ky h : (12) 
1) It will be seen that in the case of the most probable distribution the total 
momentum and the total moment of momentum vanish for each macro-volume- 
element. If one wished to evaluate the entropy for states in which these magni- 
tudes were not zero one should have to introduce here suitable conditions to allow 
for them. 
