Thermodynamics of Fluids. 



313 



is independent of the sliape of tlie circuit, let us suppose the area 

 ABODE (fig. 1) divided uj) by an infinite number of isometi'ics v ^v ^^ 

 v^v^, etc., with equal differ- pj^ ^ 



ences of volume dv, and an 

 infinite number of isopiostics 

 P1P11P2P21 ^t<5., with equal dif- 

 ferences of pressure dp.' Now 

 from the principle of continuity, 

 as the whole figure is infinitely 

 small, the ratio of the area of 

 one of the small quadrilaterals 

 into which the figure is divided 

 to the work done in passing 

 around it is ap]>roximately the 

 same for all the diffei-ent quad- 

 rilaterals. Therefore the area 

 of the figure composed of all the complete quadrilaterals which fall 

 Avithin the given circuit has to the work done in circumscribing this 

 figure the same ratio, which we will call y. But the area of this 

 figure is approximately the same as that of the giA^en circuit, and the 

 work done in describing this figure is a})proximately the same as that 

 done in rlescribing the given circuit, (eq. 5). Therefore the area 

 of the given circuit has to the Avork done or heat received in that 

 circuit this ratio ;/, AA'hich is independent of the shape of the 

 circuit. 



Now if Ave imagine the systems of equidiffei-ent isometrics and 

 isopiestics, which have just been spoken of, extended OA^er the Avhole 

 diagram, the Avork done in circumscribing one of the small quadri- 

 laterals, so that the increase of pressure directly precedes the increase 

 of volume, Avill haA^e in every part of the diagram a constant A^aluc, 

 viz., the product of the differences of volume and pressure [dvX.dp), 

 as may easily be proved by applying e({uation (2) successively to its 

 four sides. But the area of one of these quadrilaterals, Avhich we 

 could consider as constant within the limits of the infinitely small cir- 

 cuit, may vary for different parts of the diagram, and will indicate 

 proportionally the value of y, which is equal to the area divided by 

 dv X dp. 



In like manner, if we imagine systems of isentropics and isother- 

 mals draAvn throughout the diagram for equal differences rZ// and dt, 

 the heat received in passing around one of the small quadrilaterals, 

 so that the increase of t shall directly preceed that of ?/, will be the 

 constant product di/Xdt, as may be proved by equation (3), and the 



