160 THOMSON ON CARNOT'S 



TT 



the cycle of operations described above, -=- times, 



we have M = -^- . Hr ..... (3) 



vdq/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 8 to T during the second, the 

 amount of work obtained would have been found 

 to be expressed by means of a double definite in- 

 tegral thus :* 



*= far #..&%. 



i/o ijT vdq/dv 



or 



this second form being sometimes more convenient. 



* 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 

 Clapeyron's graphical construction, would be found to 

 represent the entire mechanical effect gained in the cycle 

 of operations of the air-engine. It is not necessary, how- 

 ever, 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 as clear a view of the reasoning on 

 which it is founded as could 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. 



