536 Mr, E. Buckingham on the Thermodynamic 
6. Relation of the Constant-Pressure and Constant- Volume 
Scales. 
It is impossible to treat the theory of the constant-volume 
thermometer by means of the Joule-Thomson effect, without 
making some assumption regarding the form of the equation 
of state for the low pressures concerned in gas-thermometry. 
But if the equation of state be known, the relation of the 
constant-pressure and constant-volume scales may be found 
directly. There is then no object in integrating the general 
equation of the constant-volume thermometer, for the 
thermodynamic corrections of the constant-volume scale 
may be found from those of the constant-pressure scale 
already computed. 
Let us assume that the isothermal lines pv=f(p) are 
sensibly straight at low pressures, as experiment shows them 
to be for the more nearly ideal gases. Let the departures 
of the gas from Boyle's law at low pressures be represented 
by the equation 
p 2 v 2 =p 1 v 1 ll + K(p 2 -p 1 )'], .... (19) 
in which K is small and nearly constant. Then if a and ft 
are the coefficients of expansion and of pressure, we may 
easily deduce the equation 
fr-fifefe* (20) 
in which t p and t v are the numerical values of any given 
temperature on the centigrade constant-pressure and centi- 
grade constant-volume scales, respectively, and ir, the constant 
pressure in the one case, is the same as p , the initial pressure 
at the ice-point, in the other. 
If we assume that the behaviour of the gas at low and 
moderate pressures is represented by the equation of Clausius, 
fr + *(^>- 6 >= B * • • • • (21) 
we find that as the pressure approaches zero, the coefficient 
K of equation (19) approaches the limit 
*->(*-*•?> m 
The values of b and of ^ mav be found from Chappuis's * 
XV 
* Trav. et Mem. Bur. Int. xiii. (1903). 
