892 
iu”. In Fig. 15 on the other hand the “action” (energy X time) 
of the repulsive forces will be very much smaller than that of the 
attractive forces, with this result that now the time-average descends 
but little below w,’. 
With decreasing values of wu, (lower temperatures) the relations 
of Fig. la will more and more shift in the direction of Fig. 16 in 
consequence of the continual increase of , so that c, will descend 
already to a smaller value from the limiting value 6, before the 
temperature has fallen to such a low value, that w,* is in inverse 
logarithmic dependence to w,* (see below) — in other words before 
the region of quanta proper has been entered. 
b. At low temperatures p will appear to be great; i.e. on the 
l—o_ 2f 
supposition that in @=—V / the quantity /—o does not approach 
u 
5 m 
O to the same degree as w,, but much more slowly, so that (l—o) : u, 
will approach oo. It is even probable that /—o does not become 
=0 even at 7'=0, but approaches to a certain small limiting 
value. This is in agreement with the permanent decrease of the 
expansibility at very low temperatures, and with the remaining of 
a certain finite zero-point energy A= ?/, Nf(/—o)? at T=0. 
Our equation (c) now becomes: 
gt hn VI) Me 
AES fog Ep Flap) == st oe : aa » (¢,) 
(eps # yp’ den El glt een 
log2p 8 log2p € 
which at very low temperatures, at which p approaches to 0, will 
become nearer and nearer to 
p pens) 
7 = 2 — ‘ee itt er we ' 
Uy log CD x ( + Dik V 4 (very low temp), . (c‚) 
because the finite term /og 2 can then also be omitted by the side of 
log (29? +1) in log (4p* + 2) = log (2p* + 1) — log 2. But the 
Ut 
factor 1 + 3 aye can be omitted only when « is very large with 
é 
respect to f (strong collisions), which is, however, not very probable 
in view of what was found at high temperatures — unless at high 
temperatures p is so small, that notwithstanding « is very much 
1 ef 
Jr . 
greater than f, the quantity Is Ve would yet remain compara- 
€ 
Pp 
tively great. 
