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 (1-22 )av 
ee eee 
(3) {ri eat hae 
(4.) (p—#4) 9 
for the amounts gained during the first and second, and spent during the third and 
fourth operations; and hence, by addition and subtraction, we find 
wdv—p o2, or (v w—p9@) = 
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 of dq and 7, on which 
the given thermal agency depends. For this purpose, we remark that 9 and » are 
alterations of volume and pressure which take place along with a change of tem- 
perature 7, 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 v,, the product p, v, will, by 
the law of compressibility, remain constant; and, if the temperature be elevated 
from 0 to ¢+7, 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+ E (¢+7), 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 REGNAULT 
and of Macnus on the expansion of gases by heat. If, now, the volume be altered 
arbitrarily with the temperature continually at ¢+7, the product of the pressure 
and volume will remain constant; and, therefore, we have 
PV=Po Yo {1+E (¢+1)}. 
Similarly (p—4) (V+ 9)=Po % {1+ Es}. 
Hence, by subtraction, we have 
; va—pot+wo=p, v, Er, 
or, neglecting the product » 9, 
vo—po=p, v, Er. 
* We might also investigate another relation, to express the fact that there 1s 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 g and w in terms of r. 
cD 
VOL. XVI. PART Vi. 
