~); (10) 
130 Mr. J. D. Hamilton Dickson on the 
If the initial pressure is great while the final one is small, 
we may neglect the fraction b/v, and take the simpler form 
of the equation, 
?'•(■ > 
and if both pressures are small, we get the limiting equation 
97 
ti = itc (11) 
To show the differences between these equations, let us 
take Olszewski's* highest values for air, namely, p = 160, 
^•=532° abs. Inserting these two values in van der Waals's 
equation, 
(p + ^t) (»- 1' 538 ) = 2-835*, . . . (12) 
in which the values of «, b, R for one atmosphere as unit 
pressure, and a cubic centimetre as unit volume, have been 
deduced from the values a = *0037, 6 = '0026 given by him t 
for the pressure of a metre of mercury as unit pressure, 
and the volume of the gas at 0° C. and under this pressure 
as unit volume, we find the value of v for one gram 
of air to be approximately 10'07 c.c. Hence equation (9) 
gives £ c =93°, while the limiting equation (11) would give 
78°*8. Neither of these values is near the true value, 133°; 
but we see that the effect of the value of b, although it 
is only about 1-J c.c. in comparison with some 773 c.c. as the 
specific volume of air, when associated with a certain high 
pressure, is to give values for the critical temperature which 
differ in the ratio of 6 to 7. Otherwise, equations (9) (for 
two nearly equal pressures at 160 atmos.) (10), (11) make the 
ratio of the temperature of inversion to the critical tempe- 
rature 4*86, 5*72, and 6'75 respectively. From van derWaals's 
constants for air, and his own formula, equation (5), the tem- 
perature of inversion is about 770° abs., or 5*84 x 133°, 133° 
being the critical temperature quoted by Olszewski, Linde, 
and Berthelot. It may be noted that these temperatures 
increase as the initial pressures diminish. 
In 1887 E. Natanson J, by an elaborate series of experi- 
ments on carbonic acid, conducted on Joule and Kelvin's 
lines, showed that the cooling of this gas on passing the 
porous plug was not independent of the pressure. 
In 1898 Witkowski, afier a most careful experimental 
* Phil. Mag. June 1907, vol. xiii. p. 723. 
f English translation, p. 400. 
X Wied. Ann. xxxi. p. 518 (1887). 
