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 T_{3}. 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.

In order to lower the temperature of a substance down to T_{3},
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.
Cooling by expansion.
Its own expansion is sufficient, provided the initial
condition be properly chosen, and provided we take care, even
more than in the former method, that there is no influx of heat.
Those conditions being fulfilled, we may, simply by adiabatic
expansion, not only lower the temperature of some substances
down to T_{3}, but also convert them into the liquid state. This
is especially the case with substances the molecules of which
contain few atoms.

Let us imagine the whole net of isothermals for homogeneous
phases drawn in a pv diagram, and in it the border-curve.
Within this border-curve, as in the heterogeneous region, the
theoretical part of every isothermal must be replaced by a straight
line. The isothermals may therefore be divided into two groups,
viz. those which keep outside the heterogeneous region, and
those which cross this region. Hence an isothermal, belonging
to the latter group, enters the heterogeneous region on the liquid
side, and leaves it at the same level on the vapour side. Let us
imagine in the same way all the isentropic curves drawn for
homogeneous states. Their form resembles that of isothermals
in so far as they show a maximum and a minimum, if the entropy-constant
is below a certain value, while if it is above this value,
both the maximum and the minimum disappear, the isentropic
line in a certain point having at the same time *dp**dv* and *d*²*p**dv*² = 0
for this particular value of the constant. This point, which we
might call the critical point of the isentropic lines, lies in the
heterogeneous region, and therefore cannot be realized, since
as soon as an isentropic curve enters this region its theoretical
part will be replaced by an empiric part. If an isentropic curve
crosses the heterogeneous region, the point where it enters this
region must, just as for the isothermals, be connected with the
point where it leaves the region by another curve. When
*c*_{p}/c_{v} = *k* (the limiting value of *c*_{p}/c_{v} for infinite rarefaction is
meant) approaches unity, the isentropic curves approach the
isothermals and vice versa. In the same way the critical point
of the isentropic curves comes nearer to that of the isothermals.
And if *k* is not much greater than 1, *e.g.* *k* < 1.08, the following
property of the isothermals is also preserved, viz. that an
isentropic curve, which enters the heterogeneous region on the
side of the liquid, leaves it again on the side of the vapour, not
of course at the same level, but at a lower point. If, however, *k*
is greater, and particularly if it is so great as it is with molecules
of one or two atoms, an isentropic curve, which enters on the
side of the liquid, however far prolonged, always remains within
the heterogeneous region. But in this case all isentropic curves,
if sufficiently prolonged, will enter the heterogeneous region.
Every isentropic curve has one point of intersection with the
border-curve, but only a small group intersect the border-curve
in three points, two of which are to be found not far from the top
of the border-curve and on the side of the vapour. Whether
the sign of *h* (specific heat of the saturated vapour) is negative
or positive, is closely connected with the preceding facts. For
substances having *k* great, *h* will be negative if T is low, positive
if T rises, while it will change its sign again before T_{c} is reached.
The values of T, at which change of sign takes place, depend
on *k*. The law of corresponding states holds good for this value
of T for all substances which have the same value of *k*.

Now the gases which were considered as permanent are
exactly those for which *k* has a high value. From this it would
follow that every adiabatic expansion, provided it be sufficiently
continued, will bring such substances into the heterogeneous
region, *i.e.* they can be condensed by adiabatic expansion. But
since the final pressure must not fall below a certain limit,
determined by experimental convenience, and since the quantity
which passes into the liquid state must remain a fraction as
large as possible, and since the expansion never can take place
in such a manner that no heat is given out by the walls or the
surroundings, it is best to choose the initial condition in such a
way that the isentropic curve of this point cuts the border-curve
in a point on the side of the liquid, lying as low as possible. The
border-curve being rather broad at the top, there are many
isentropic curves which penetrate the heterogeneous region
under a pressure which differs but little from *p*_{c}. Availing
himself of this property, K. Olszewski has determined *p*_{c} for
hydrogen at 15 atmospheres. Isentropic curves, which lie on
the right and on the left of this group, will show a point of condensation
at a lower pressure. Olszewski has investigated this
for those lying on the right, but not for those on the left.

From the equation of state (*p* + *a**v*²)(*v*−*b*) = RT, the equation
of the isentropic curve follows as (*p* + *a**v*²)(*v* − *b*)^{k} = C, and
from this we may deduce T(*v* − *b*)^{k−1} = C′. This latter relation
shows in how high a degree the cooling depends on the
amount by which *k* surpasses unity, the change in *v* – *b* being
the same.

What has been said concerning the relative position of the
border-curve and the isentropic curve may be easily tested for
points of the border-curve which represent rarefied gaseous states,
in the following way. Following the border-curve we found
before *f* ′T_{c}T for the value of T*p**dp**d*T. Following the isentropic curve
the value of T*p**dp**d*T is equal to *k**k* – 1. If
*k**k* – 1 < *f* ′ T_{c}T, the isentropic
curve rises more steeply than the border-curve. If we take *f* ′ = 7
and choose the value of T_{c}/2 for T—a temperature at which the
saturated vapour may be considered to follow the gas-laws—then
*k*/(*k* – 1) = 14, or *k* = 1.07 would be the limiting value for the two
cases. At any rate *k* = 1.41 is great enough to fulfil the condition,
even for other values of T. Cailletet and Pictet have availed
themselves of this adiabatic expansion for condensing some
permanent gases, and it must also be used when, in the cascade
method, T_{3} of one of the gases lies above T_{c} of the next.

A third method of condensing the permanent gases is applied in C. P. G. Linde’s apparatus for liquefying air. Under a high pressure *p*_{1} a current of gas is conducted through a narrow spiral, returning through another spiral which Linde’s apparatus.surrounds the first. Between the end of the first spiral and the beginning of the second the current of gas is reduced to a much lower pressure *p*_{2} by passing through a tap with a fine