Joule-Kelvin Inversion Temperature. 133 
on " Physical Constants at Low Temperatures," namely, 
* = fc(i+j\/^£)'. 
We can also verify this result directly. For putting 
i-=v' in equation (8), it becomes 
Vvt 
ym m > 
whence, if £ 2 = ~*~~ > yan ^ er Waals's equation (4) becomes 
01* 
a 3Rf /TTlU 
which is equation (17). But equation (15) is not in a suit- 
able form for the determination of t from given values of p 
and p' . Expanding, and arranging it in powers of r, it takes 
the form 
9**+ 16 (w + 3P)r 3 -2{4(w+P)(57r-llP) + 3S 3 }r 2 
-16 {4(7r + P)-(7r-P)-(57r-P)5 2 }r 
+ 16(7r-rP) 4 -8{2(37r-P) L '-(7r+P)-l^ + S 4 = 0; (19) 
which shows at once that the inversion-temperature is a 
complicated function of the initial and final pressures. It 
is interesting, however, to note that it depends, not explicitly 
upon them, but only upon their difference 8, and their 
arithmetic mean P. Further, since 8 occurs only under the 
square and the fourth power, the phenomenon, but for other 
reasons, would be reversible. Putting 8 = 0, we ought to 
recover the equation for the Joule-Kelvin experiment ; and 
this is the case, for the terms without 8 are 
9r 4 + 16(7r + 3P)> 3 — 8(7r + P)(09r-llP)r 3 
-6±(7r + ¥)%>*■ -Y)r + 16{7r -hPj 4 , 
which can be written as the product 
{9r 2 -4(57r-3P)7-f4(7r4-P) 2 }(7r + P-h^) 2 J ■ (20) 
the first of whose factors equated to zero, and with ^9 written 
for P, is equation (17). The second factor cannot vanish, 
since each of its terms is positive. . 
We can now answer the question, What is meant by a 
Joule-Kelvin inversion-temperature ? Gas is allowed to 
expand through a porous plug from a high pressure of any 
