554 PROFESSOR WILLIAM THOMSON’S ACCOUNT OF 
Hence, the preceding expression for mechanical effect. gained in the cycle of ope- 
rations, becomes 
d oe 
Po Vo. Fg 
vo? 
Or, as we may otherwise express it, 
Ep, 
dv 
Hence, if we denote by M the mechanical effect due to H units of heat descending 
through the same interval r, which might be obtained by repeating the cycle of 
operations described above, tie times, we have 
27. If the amplitudes of the operations had been finite, so as to give rise to 
an absorption of H units of heat during the first operation, and a lowering of 
temperature from S to T during the second, the amount of work obtained would 
have been found to be expressed by means of a double definite integral, thus ;*— 
staf, ag fai Beem | 
"do T aciy dupenittes@ car tae Ae 
or ay i PL de. fee 
™ ty vdq 
this second form being sometimes more convenient. 
28. The preceding investigations, being founded on the approximate laws of 
compressibility and expansion (known as the law of Mariorre and Boy e, and 
the law of Datron and Gay-Lussac), would require some slight modifications, to 
adapt them to cases in which the gaseous medium employed is such as to present 
sensible deviations from those laws. REGNAULT’s very accurate experiments 
shew that the deviations are insensible, or very nearly so, for the ordinary gases 
at ordinary pressures; although they may be considerable for a medium, such as 
* This result might have been obtained by applying the usual notation of the integral calculus 
to express the area of the curvilinear quadrilateral, which, according to CLapryron’s graphical con- 
struction, would be found to represent the entire mechanical effect gained in the cycle of operations 
of the air-engine. It is not necessary, however, to enter into the details of this investigation, as the 
formula (3), and the consequences derived from it, include the whole theory of the air-engine, in 
the best practical form ; and the investigation of it which I have given in the text, will probably give — Ps 
as clear a view of the reasoning on which it is founded, as could be obtained by the graphical method, 
which, in this case, 1s not so valuable as it is from its simplicity m the case of the steam-engine. 

