76 Intelligence and Miscellaneous Articles, 



be measured by the increments of volume of the gas itself, a will be 

 absolutely constant, and Mariotte's law will be the only fact resting 

 upon experience. 



Let ^ denote the quantity of heat which the mass m of the gaS 

 must receive from without, in order to produce any changes whatever 

 in p, V and r. Then considering p and v as independent variables, 

 and r as a function of both, we can make 



d^ dr d^ ,dr 



=mc—', — z=:mc'—. 



dv dv dp dp 



The magnitudes c and c', defined in this manner, express the capa- 

 cities for heat at a constant volume and at a constant pressure, and 

 may for the present be regarded as constant. Substituting in these 

 expressions the values of the partial differential coefficients of r as 

 obtained from (1), namely, 



dr ^P . ^^ __ ^P 



dv"^ maa* dp ~ ma a 

 we have 



dd __?cp ^ 6/^ __ ^c'v 



dv a a * dp a a* 

 and hence for the total differential of .& 



dd= — (cpdv-\-c'vdp). 

 a a, 



If now the gas pass from one state to another, so that^ and v change 

 according to any definite law, p, v, and ^ become functions of each 

 other, and we have 



^^I-fc/^dv-{-c'/^dp\ (2) 



If tq denote the initial temperature, then from the equation (1) 

 we have 



dr = (pdv + vdp) ; 



ma a 



and if we integrate between the same limits as those to which i& is 

 referred, and multiply by mc', we have 



tnc'ir- Tq) -^c'( fp^^ + /»*). ... (3) 



which, subtracted from equation (2), gives 



^-.W(r-ro)=^(c-c')j, (4) 



where q= fpdv expresses the work done in the change of state. 



The result is therefore as follows : — " The quantity of heat commu- 

 nicated to a gas during any change of volume and pressure consists 

 of two parts, one of which expresses the heat necessary to raise the 

 temperature at a constant volume, while the other is a constant 

 multiple of the work done." In particular we infer that this quan- 



