554 PROFESSOR WILLIAM THOMSON'S ACCOUNT OF 



Hence, the preceding expression for mechanical effect, gained in the cycle of ope- 

 rations, becomes 



p^v„.hr , 



V 



Or, as we may otherwise express it, 



do 

 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 



XT 



operations described above, j- tunes, we have 



M=^^.H. .... (3) 

 a a 



dv 



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 ;* — 



^=r\fdt.^^ 



d V 



TT C 



^=Y.p,vJ r I'i^.dtdq: 

 J J iv V dq I 



or ~ 



(4). 



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 Mariotte and Boyle, and 

 the law of Dalton 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 nutatiun of the integral calculus 

 to express the area of the curvilinear i|uadrilateral, which, according to Clapeyron's graphical con- 

 struction, would he found to represent the entire mechanical effect gained in the cycle of operations 

 of the air-engine. It is not necessai7, however, to enter into the details of tliis investigation, as the 

 formula (3), and the consequences derived from it, include the whole theoi7 of the air-engine, in 

 the best practical form ; and the investigation of it w^hich I have given in the te.xt, will prcjbably give 

 as clear a view of the reasoning on which it is founded, as c(nild be obtained by the graphical method, 

 which, in this case, is not so valuable as it is from its simplicity in the case of the steam-engine. 



