Joule-Kelvin Inversion Temperature. 139 
equations of the inversion-isothermals in the potential-pressure 
plane. The isopotentials correspond to Witkowski's* second 
(derived) curves. Equation (25) rewritten in a more suitable 
form for an isopotential is 
(83/ + 2a) 2 + 216.^ + 27( < r 2 --4 l r-4)«-729^ 2 (A' + l) = 0, (31) 
in which x is to be given any constant value. For different 
constant values of #, this is the equation of a series of para- 
bolas whose axes are parallel and make with the axis of 
temperature an acute angle of 14° 2' [the scales in fig. 2 
reduce this angle to 1° 26']. In fig. 2 these parabolas are 
shown ; they have there been derived directly from measure- 
ments made on fig. 1, and not by means of equation (31). 
The curve ABCD is the curve corresponding to the envelope 
VMLN in fig. 1. Like it, it extends from y = *75 to y = 6*75, 
the greatest pressure associated with any point on it being 
9, when y = Z (corresponding to the point L). This curve 
is the parabola whose equation has already been given in 
(30), and of which another useful form is 
(12z/ + « + 27) 2 = 172%: .... (32) 
it touches the axis of pressure at the negative pressure 
— 27. The axis of temperature in fig. 2 corresponds to the 
zero-isopiestic in fig. 1. The advantage of the new figure 
is now apparent. The inversion-points in fig. 1 were all 
confined within a rather confusing curvilinear triangle ; 
they are now contained within the more open curvilinear 
quadrilateral BDEB' bounded by the portions of the axes 
BD, B'E, and the parabolas BB', DE whose equations are, 
respectively, 
(4#-f«) 2 -f-24y-6a-27 = . . . (33) 
and 
l% 2 -f 27a- 729 = 0. . . . (34) 
The circumstances in which inversion will take place are 
now to be found from a line of constant temperature. Thus 
if we look along the line for which y = 2 we see that inversion 
will take place for a gas expanding from an initial pressure 
of about 24i times the critical pressure into a vacuum, the 
potential being -Q • or between pressures 22 and 1, the 
potential being *5 ; or between pressures 15 and 3| ; the 
potential being *3 ; or finally between two nearly equal 
pressures of about 8 times the critical pressure. Witkowski's 
isopotentials lie within the rectangle bounded by y=0*9, 
y = 2'l, and a = 0, a = 4. 
Instead of representing the isopotentials we may obtain 
perhaps a better display of the circumstances of inversion 
* Loc. cit. 
