Joule- Kelvin Inversion Temperature. 131 
study of the properties of air, published a memoir * on the 
behaviour of this gas in the Joule- Kelvin experiment ; but, 
instead of basing his investigation on any theoretical equation, 
he employed his own experimental results, involving them in 
the thermodynamic equation by the method of quadratures. 
Taking U for the internal energy, he calculated the values of 
the function U + pv for the pressures and temperatures within 
the range of his experiments. With these he plotted a 
series of curves of constant pressure, the coordinates being 
temperature and the excess of the function U -\-pv for 
pressure p above its value for one atmosphere. From these 
curves he constructed curves for constant values of U+/>i\ 
with pressure and temperature as coordinates. The range of 
these curves being from about 130° to 273° (abs.), and 
therefore not including the (so-called) inversion-temperature, 
he pointed out that it was possible, if they had extended 
farther, to have found the slope of the XJ+pv curves 
changing sign, and hence to have determined this tem- 
perature. Thus to Witkowski is due the proof of the 
fact, based on accurate experiment, that for the deter- 
mination of any such temperature it is necessary to take 
the pressure also into consideration. 
In 1906, after the publication of Olszewski's results on the 
release of hydrogen from high pressure, Porter t examined 
the question, using also the function XJ+pv. He plotted 
curves of constant pressure (such curves may be called 
isopiestics), using as coordinates the temperature and the 
variable part, independent of temperature, of the indefinite 
function U+jw. But instead of employing experimental 
results, he used various theoretical equations, and in par- 
ticular van der Waals's equation of state. Equation (7) inav 
be written 
<-, by (4), Q = ( pV ^^(^_|) ; j 
and we have also 
U+/w = <£(*)--+^t7, .... (14) 
where <f>(t) is an unknown function of the temperature. 
Porter took pv — - for his abscissa (the minus appears as 
* Anzeiaer d. Akad. d. Wiss. in Krakau, 1898, pp. 282-295. 
f Phil. Mag. April 1906, vol. xi. pp. 55^-668. 
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