and the Laws regarding the Nature of Heat. 11 



In our case, however, instead of an expansion, a compression 

 has taken place ; hence this last expression must be introduced 

 with the negative sign. During the expansion from of to og, 

 and the compression from oh to oe, heat has been neither 

 received nor given away -, the amount of heat which the gas has 

 received over and above that which it has communicated, or, in 

 other words, the quantity of heat consumed, will therefore be 



The quantities Sv and d'v must now be eliminated ; a conside- 

 ration of the figure furnishes us with the following equation : 

 dv + B'v =Bv + d'v. 



During its compression from oh to oe, consequently during its 

 expansion under the same circumstances from oe to oh, and 

 during the expansion from of to og, both of which cause a de- 

 crease of temperature dt, the gas neither receives nor communi- 

 cates heat : from this we derive the equations 



[(f)-i(§W»'-[(f)+£(§)*>»; 



From these three equations and equation (2.) the quantities 

 d'v, 8v and B'v, may be eliminated ; neglecting during the pro- 

 cess all diiferentials of a higher order than the second, we obtain 



nekeate^en,ed=[^(§)-i{§)],.d, . (3.) 



Turning now to our maxim, which asserts that the production 

 of a certain quantity of work necessitates the expenditure of a 

 proportionate amount of heat, we may express this in the form 

 of an equation, thus : 



The heat expended __ , ,.. 



The work produced "~ ^ 

 where A denotes a constant which expresses the equivalent of heat 

 for the unit of work. The expressions (1.) and (3.) being intro- 

 duced into this equation, we obtain 



a(2)-£(g)]"- _. 



B^.dvdt 



or 



dt 



^/^Q\ d /dQ\^ A.R ,jj. 



it\dv) dv\dt) v ^ *^ 



