Correction of the Gas- Thermometer. 51 



v/0=f(p), where f(p) is any arbitrary function of p. But 

 if the fluid satisfies Boyle's law at all temperatures, we must 

 have pv=f (0), and the two conditions together are satisfied 

 only by the ideal gas. Similarly Joule's experiment on the 

 expansion of a gas into a vacuum (dE = 0) leads to the 

 condition 0(dp/d0) v =p, if there is no change of temperature, 

 which is satisfied by any fluid possessing the characteristic 

 equation p/0=f(v), where f(v) is any arbitrary function of v. 

 This condition, in conjunction with Boyle's law, again suffices 

 to define the ideal state ; but no one of the three conditions 

 is sufficient by itself. 



3. Application to the Gas-Thermometer. 



In the practical application of the gas-thermometer, we 

 assume an equation of the form ^v = RT, in which T is the 

 temperature by gas-thermometer, and differs from 6 in 

 proportion as the gas in question deviates from the ideal 

 state. In order to apply the results of the porous-plug 

 experiment to the correction of the scale of the gas-ther- 

 mometer, Thomson originally proposed to estimate the 

 difference 0— T approximately by the following method : — 



Suppose the experiment to be performed in a calorimeter 

 at constant temperature, so that the gas after passing the 

 plug is restored to its initial temperature. The heat absorbed 

 in the calorimeter is evidently equal to the amount $d0 

 which would have been required to heat the gas up to the 

 original temperature at constant pressure if the experiment 

 had been performed adiathermally with a fall of temperature 

 cW. But the heat absorbed at constant temperature in the 

 calorimeter is also by the first law equal to the increase 

 of intrinsic energy (dE/dv)frdv of the gas, together with the 

 external work d(pv) d done by the gas. Writing for 

 (dE/dv)Q its value 0(dp/d0) v —p, we have 



-Sd0={0(dp/d0) v -p)dv + d(pv), . . (5) 



which is evidently equivalent to the equation (4) previously 

 given, but with v instead of p as independent variable, since 

 {dpld0) v (dc/dp) e = —(dv/d0)p. Integrating this expression 

 over the range of an experiment from j»V to p"v" at constant 

 temperature, and putting on the left the observed value of 

 the fall of temperature {& — f/ ), we obtain Thomson's original 

 equation, 



S(d'-d") = 0ldW/d6) v -W+p"v''-p'v' i . (6) 



in which W is the work represented by the integral of pdv 



E 2 



