Joule- Kelvin Inversion Temperature. 
137 
and that the 9-isopiestic touches the envelope in two co- 
incident points at L, which we may call the final point of 
contact, its coordinates being a?=2/3, y = 3. Thus the part 
of the parabola which constitutes the envelope is contained 
between M and N, and deals only with pressures within the 
range a=0...9. Further, a Joule-Kelvin experiment, in 
which the pressures are nearly equal, and as governed by 
van der Waals's equation, cannot involve pressures higher 
than nine times the critical pressure. This limitation has been 
noticed by Dewar, Berthelot, and others ; and it agrees with 
Porter's full-line curve (p. 556 of his paper), for its equation 
may be put in the form 
(f+12,-45/=144(9-|), 
(30) 
where f is reduced pressure, and rj is reduced temperature, 
and real solutions of this equation require f not to exceed 9. 
Equation (30) is that of a parabola, cutting the axis of tem- 
perature at the reduced temperatures 3/4 and 27/4, the latter 
result agreeing with equation (11). 
From equations (29) we get the positions of the points M, 
N, namely, M is (-§, £),' N is (2, 6|). The following- 
Table will help to show how these points of contact follow 
each other, (# l5 y^) being the lower, and (# 2 , t y 2 ) the higher 
point of contact for any one isopiestic. For values of a. 
a. 
>, 
Uv 
x 2 . u % . 

1 1 
I ::::: 
4 
5 
61 
7 
8 
! 
9 
-0-67 
-0-59 
-0 22 
o- 
6-22 
0-75 
0-84 
1-33 
1-69 
2-08 
2-00 0-75 
1-92 6-49 
1 : 56 5 : 33 
1-33 4-69 
I : il 4-08 
#=0-67, 2f=300. 
greater than 9, no isopiestic touches the envelope ; that is, 
there is no inversion-point for a Joule-Kelvin experiment, 
the pressures differing by only a small amount, when these 
pressures exceed nine times the critical pressure. But there 
are inversion-points for finite differences of pressure within 
certain limits, as we shall find later. 
If we seek for the general circumstances in which a given 
temperature may be an inversion-temperature, we find that 
