Correction of the Gas-Thermometer. 75 



for which experiment and theory both indicate that the 

 variations of the specific heat should be small. But in the 

 case of vapours like steam or C0 2 these variations cannot be 

 neglected ; and it is better to employ the reverse method as 

 in Section 7 above, assuming a convenient type o£ cha- 

 racteristic equation and deducing the corresponding expression 

 for the cooling-effect for comparison with the results of 

 observation. In this case it is easy to take account of the 

 variations of the specific heat by simply inserting the appro- 

 priate value of the specific heat in equation (30) . 



We observe by reference to the differential equations (4) 

 or (5) that the appropriate value of S is that corresponding 

 to the final pressure p" in each experiment, and to the mean 

 temperature (0'+ 0")/2. The variation of the specific heat 

 with temperature can be determined only by experiment. 

 The variation with pressure must be consistent with the 

 characteristic equation chosen, and can be calculated in the 

 following manner. 



Referring to equation (3) for the variation of the total heat, 

 F = E +pv, we have the following values of the partial differ- 

 ential coefficients: — 



{dF/dO) p = $, (dF/dp) 9 = v-0(dvjdd) p = b-(n + rjc, . (50) 



which give for the variation of S with p at constant 0, 



(dS/dp}fi=d 2 F/d0dp=-0(d 2 v[d0\=n(n+l)c/0. . (51) 



Integrating this at constant temperature from to p, we 

 obtain 



S=S o + n(n + l)cp/0, .... (52) 



where S is the limiting value of S at zero pressure and 

 temperature 0. 



This equation enables us to find the complete variation of S, 

 if w r e observe the values of S experimentally at any standard 

 pressure such as 1 atmo, over the required range of 

 temperature. 



Proceeding similarly for the specific heat s at constant 

 volume, we obtain by considering the variation of the 

 intrinsic energy E, 



(dB/d0) v =s, (dE/dv) e =0{dp/d0) v -p, (ds/dv) e =d(d 2 p/dO*) m (5£ 



whence 



s=s 'hn(n—l—nclY)cp/8 (54] 



where s is the limiting value of s at zero pressure, and 

 Y = ]\9/>. This formula is of comparatively little use, 



