Joule-Kelvin Inversion Temperature. 145 
which, when the pressures differ only by a small amount, 
becomes 
and, when the pressures are themselves small, reduces to 
*=*.%£ \f^l (42) 
From Reinganum's equation 
R^ 3 a 
P=Jj^Sf-?> (43) 
where a and b are constants, we get in like manner the 
equations 
L _2a ( v --b)( t/-b) (y-byjv'-by 
b vv' '{2t<J-b(v+v')}{2v 2 v ,2 -2bvv'{v + v')+l>\v* + v' 2 )\' 
.... (44) 
giving, for nearly equal pressures, 
**=U^l (45 > 
and when the pressures also are small 
m =m < 46 > 
The critical temperature, on Reinganum's equation, is 
Berthelot * in two great memoirs has discussed the theory 
of the equation of state, especially for low pressures, in a 
thoroughly exhaustive manner. His comparison of theory 
with experiment has led him to formulate a modified form of 
van der Waals's equation for moderate pressures, namely, 
R£ a /AO , 
p = 7--T5, ( 48 ) 
1 v — b tv 
but with the understanding that instead of the critical con- 
stants being derived from a, b, and R, the constants to be 
chosen are t c , p c , and R, with the relations 
7 1 27 RV , 9 R* c '. 
b= -7 v., a= -=-. , and v = tjt, . . (49) 
4 c 64 p c c 62 p c v ; 
* 'Livre Jubilaire dedie a H. A. Lorentz,' 1900, p. 417. "Sur les 
thermometres a Gaz," Trav. et Mem. du Bur. Intern, des Poids et 
Mesures, 1903. 
Phil. Mag. S. 6. Vol. 15. No. 85. Jan. 1908. L 
