Joule-Kelvin Inversion Temperature. 129 
arises another, What is meant by the Joule-Kelvin inversion- 
temperature ? , 
For some time the idea has held ground that by merely 
dealing with the Joule-Kelvin experiment at higher or lower 
temperature, one temperature would be found at which the 
escaping gas would show neither heating nor cooling. This 
is correct provided the temperature does not exceed a certain 
limit. But the further extension has been made, that the 
same would still hold true if a great difference of pressures 
was employed. Here, however, the conditions of the Joule- 
Kelvin experiment may be infringed, and without formal 
prouf we are not at liberty to make this assumption. The 
properties of a gas depend in general on two of the three 
quantities, pressure, volume, and temperature, because we 
have an equation of state connecting the three. If a certain 
property depends on the alteration of one of these quantities 
only, some condition must in general have been assumed, 
even although tacitly. In the first idea of the inversion- 
temperature, the tacit assumption was (at least) that the 
difference of pressures was small ; perhaps there was also 
the idea that the pressures themselves were small. But in 
any case, Kelvin's formula fully provided for any pressures 
so long as the conditions of the experiment were maintained. 
The Boyle-Charles laws were originally employed in the 
numerical reduction of the Joule-Kelvin observations ; later, 
a formula due to Rankine was used ; today we may employ 
van der Waals's relation as a step nearer the actual condition 
of a gas. 
I shall use equation (1) for my purpose, and substitute in 
it from equation (4), noting that in general undashed letters 
refer to the initial state (the state of higher pressure) and 
dashed letters to the final state. Thus we get 
q=M^-^)-H^> • • • (7) 
whence, on putting Q = 0, the inversion-temperature U is 
given by 
R <-?K)K> w 
or, if t c is the critical temperature, 
<-¥<.H)K) « 
Phil. Mag. S, 6. Vol. 15. So. 85. Jan. 1908. K 
