carnot's theory of the motive power of heat. 553 



the expressions for the mean pressures during the four successive operations. 

 Now, the mechanical effect gained or spent, during any of the operations, will be 

 found by multiplying the mean pressure by the increase or diminution of volume 

 which takes places ; and we thus find 



(1.) p(i-i'iiyv 



(2-) {.(l4")-i.}. 



(3.) J^(^l_i^)_^ jrf„ 



(4.) (p-i<^)p 



for the amounts gained during the fii-st and second, and spent during the third and 

 fourth operations ; and hence, by addition and subtraction, we find 



. dv , N dv 



a d v—p <p , or (v u~p <p) — , 



V V 



for the aggregate amount of mechanical effect gained during the cycle of opera- 

 tions. It only remains for us to express this result in terms ofdq and t, on which 

 the given thermal agency depends. For this purpose, we remark that <f and u are 

 alterations of volume and pressure which take place along with a change of tem- 

 perature T, and hence, by the laws of compressibility and expansion, we may 

 establish a relation* between them in the following manner. 



Let p^ be the pressure of the mass of air when reduced to the temperature 

 zero, and confined in a volume v^ ; then, whatever be ?^„, the product p^ v^ will, by 

 the law of compressibility, remain constant ; and, if the temperature be elevated 

 from to i -I- T, and the gas be allowed to expand freely without any change of 

 pressure, its volume will be increased in the ratio of 1 to 1 -t- E (i + t), where E is 

 very nearly equal to -00366 (the centigrade scale of the air-thermometer being re- 

 ferred to), whatever be the gas employed, according to the researches of Re&nault 

 and of Magnus on the expansion of gases by heat. If, now, the volume be altered 

 ai'bitrarily with the temperature continually at «+t, the product of the pressure 

 and volume will remain constant ; and, therefore, we have 



PV=p„ i;„{l+E(^+r)}. 



Similarly (j>- <>>) (« + p)=Po »o {1 + E *}. 



Hence, by subtraction, we have 



or, neglecting the product o> f, 



vu-pf-p^v^^r. 



* We might also investigate another relation, to express the fact that there is no accession or 

 removal of heat during either the second or the fourth operation ; but it will be seen that this will not 

 affect the result in the text ; although it would enable us to determine both <p and a in terms of r. 



VOL. XVI. PART V. 7 D 



