230 MILNE ART. F 



temperature ^, then the external work performed, A'VF, satisfies 

 the inequahty 



^'W ^AW - {t - h) dv, 



the integral being taken along the path in the (rj, t) diagram 

 actually traversed by the system and AT^'" having the same 

 meaning as above." 



For, along any path whatever, if the differentials which 

 follow denote positive increments, 



d'Q = d'e -\-d'W 

 and 



d'Q denoting the heat given up at t. Hence 



d'W ^ - d'€-\-t d'v. 



Since d'e and d'r] are the actual increments in the functions e and 

 7j along the path, we may replace them by de and dt^. Now 



Hence 



Integrating, 



d\J/ = de — tdt] — rjdt, 

 d'W ^ -# - 7}dL 



A'W ^ Arp - vdt 



^ A\}/ -\- (tim — ti-qi) — I tdt} 

 ^ AW + t2 (rji - m) - / 'tdr, 



on using the result of the first part. Hence 

 A'W ^AW- \t - k) dr,. 



