his 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 [int]' Tc/T for the
value of T/p.dp/dT. Following the isentropic curve the value of T/p
dp/dT is equal to k/(k - 1). If k/(k - 1) < [int]'Tc/T, the isentropic
curve rises more steeply than the border-curve. If we take f' = 7 and
choose the value of Tc/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, T3 of one of the gases lies above Tc of the
next.

Linde's apparatus.

A third method of condensing the permanent gases is applied in C. P. G.
Linde's apparatus for liquefying air. Under a high pressure p1 a current
of gas is conducted through a narrow spiral, returning through another
spiral which 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 p2 by passing through a tap with a fine orifice. On
account of the expansion resulting from this sudden decrease of
pressure, the temperature of the gas, and consequently of the two
spirals, falls sensibly. If this process is repeated with another
current of gas, this current, having been cooled in the inner spiral,
will be cooled still further, and the temperature of the two spirals
will become still lower. If the pressures p1 and p2 remain constant the
cooling will increase with the lowering of the temperature. In Linde's
apparatus this cycle is repeated over and over again, and after some
time (about two or three hours) it becomes possible to draw off liquid
air.

The cooling which is the consequence of such a decrease of pressure was
experimentally determined in 1854 by Lord Kelvin (then Professor W.
Thomson) and Joule, who represent the result of their experiments in the