, if the initial temperature
was equal to Tc, or but little below it, and we can even predict with
certainty that this will not be the case for all substances. It is
possible, too, that long before the triple point is reached the whole
liquid will have evaporated. The most favourable conditions will, of
course, be attained when the influx of heat is reduced to a minimum. As
a limiting case we imagine the process to be isentropic. Now the
question has become, Will an isentropic line, which starts from a point
of the border-curve on the side of the liquid not far from the
critical-point, remain throughout its descending course in the
heterogeneous region, or will it leave the region on the side of the
vapour? As early as 1878 van der Waals (_Verslagen Kon. Akad.
Amsterdam_) pointed out that the former may be expected to be the case
only for substances for which c_p/c_v is large, and the latter for those
for which it is small; in other words, the former will take place for
substances the molecules of which contain few atoms, and the latter for
substances the molecules of which contain many atoms. Ether is an
example of the latter class, and if we say that the quantity h (specific
heat of the saturated vapour) for ether is found to be positive, we
state the same thing in other words. It is not necessary to prove this
theorem further here, as the molecules of the gases under consideration
contain only two atoms and the total evaporation of the liquid is not to
be feared.

In the practical application of this cascade-method some variation is
found in the gases chosen for the successive stages. Thus methyl
chloride, ethylene and oxygen are used in the cryogenic laboratory of
Leiden, while Sir James Dewar has used air as the last term. Carbonic
acid is not to be recommended on account of the comparatively high value
of T3. In order to prevent loss of gas a system of "circulation" is
employed. This method of obtaining low temperatures is decidedly
laborious, and requires very intricate apparatus, but it has the great
advantage that very _constant_ low temperatures may be obtained, and can
be regulated arbitrarily within pretty wide limits.

Cooling by expansion.

In order to lower the temperature of a substance down to T3, it is not
always necessary to convert it first into the liquid state by means of
another substance, as was assumed in the last method for obtaining low
temperatures. Its own expansion is sufficient, pro