Intelligence and Miscellaneous Articles. 77 



tity of heat is in itself proportional to the work as soon as the initial 

 temperature is regained, while the pressure and volume may have 

 other values ; as pure loss of heat it appears, it is true, only after a 

 complete restoration of the original condition. 



Let us now suppose that neither Mariotte's law nor the law of the 

 invariability of the capacities is accurately true. If c and c' are sub- 

 ject to any small changes in consequence of changes of pressure or 

 temperature, we may consider them as functions of p and v, and 

 write equations (2) and (3) as follows : — 



m / c'dr= — ( /c'pdvi- /c'vdp], 

 whence by subtraction, 



a 



^—m / c'dr= — / (c—'c')pdv. 

 J act, J 



If now during the change there is only work consumed or work 

 performed, if consequently J9c?u undergoes no change of sign, we may, 

 by a known property of definite integrals, place the factors c and 

 <?--c' before the sign of integration, when the last equation will cor- 

 respond with equation (4) : only we now understand by c' and 

 c — c' certain definite mean values between the greatest and least of 

 those which these functions assume during the entire change. The 

 validity of the relation between work and heat is consequently not 

 changed by small variations in the capacities for heat. The propor- 

 tional number itself is of course subject to simultaneous variations, 

 which however are smaller than the variation in the capacities. 



If the gas be restored to its original volume, so that at any time 

 pdv must change its sign, the proportional number is no longer 



necessarily a mean value of — (c— c') ; yet we see, by representing 



(X OL 



specially the heat conveyed during positive and negative work, that 

 this number can differ but little from the values of its expression as 

 long as the excess is not too small. If, however, there remains from 

 a large amount of work only a small positive excess, it might be dif- 

 ficult to show that the proportional number could not differ consi- 

 derably from the values of its expression. 



Finally, if Mariotte's law be not strictly accurate, we may put 

 pv-\-^ ior pv, and consider jO as a small magnitude depending on p 

 and V, which at the begirming of the motion is zero. In this case, 

 in place oi pdv and vdp, we shall have relatively 



pdv-{- -^ dv ; and vdp + — dp. 

 dv dp 



The last magnitude is cancelled, and does not occur in the resulting 



equation. On the other hand, q now becomes 



dv ^-'-•~: -'<■'••' ':':-' 



'*/ 



