Joule-Kelvin Inversion Temperature. 127 
Kelvin's formula, governed by the conditions of the Joule- 
Kelvin experiment, is, in the usual notation, 
pdv + pv—p'tf-t\ £dv, . . . (1) 
where Q is the quantity of heat lost by the gas through 
traversing the porous plug from a higher to a lower pressure. 
There are two other forms in which this may be written, 
namely, 
H '(**-*)* (2) 
Q = (XF+jAO--(U+p*), -.-. (3) 
each being, on occasion, more suitable than the other two 
for a particular discussion. In the practical application of 
the results of their experiments, Joule and Kelvin employed 
a formula sent them by Rankine *, which accorded with 
their experimental results, showing that the fall of tempera- 
ture varied inversely as the square of the absolute temperature. 
After this followed Andrews's epoch-making experiments on 
the continuity of state ; and a little later, van der Waals's 
Thesis, which, leaving the Boyle-Charles laws as a first 
approximation to the equation of state of a gas, gave distinct 
and permanent advance to our theoretical knowledge of the 
properties of a gas by means of the well-known equation 
(p+ £)(•-*) = B| (*) 
In his thesis he discussed Andrews's experiments, and showed 
that this equation produced the form of the isothermal which 
James Thomson had suggested as probable. He also dis- 
cussed the Joule-Kelvin porous-plug experiment, and by 
means of the same equation deduced for it a theoretical 
relation, when the pressure is not too high, which he 
expressed (in his notation) by the equation f 
i Ue( t l - t )=*^(^- b y±=p.. . . (5 ) 
Vi J l"29dd \l + at J p Q 
The accuracy of this equation for air was remarkable ; for, 
by calculation, he found that the fall of temperature per 
atmosphere was 0*265, while the average of seven results 
quoted from Joule and Kelvin's work, and ranging from 
* Kelvin, Collected Papers, vol. i. pp. 376, 391. 
f English translation, p. 444. 
