THERMODYNAMICS OF FLUIDS. 5 



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 approxi- 

 mately the same for all the different quadrilaterals. Therefore 

 the area of the figure composed of all the complete quadrilaterals 

 which fall within the given circuit has to the work done in circum- 

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

 area of this figure is approximately the same as that of the given 

 circuit, and the work done in describing this figure is approximately 

 the same as that done in describing the given circuit (eq. 5). There- 

 fore the area of the given circuit has to the work done or heat received 

 in that circuit this ratio y, which is independent of the shape of 

 the circuit. 



Now if we imagine the systems of equidifferent isometrics and 

 isopiestics, which have just been spoken of, extended over the whole 

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

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

 of volume, will have in every part of the diagram a constant value, 

 viz., the product of the differences of volume and pressure (dv x dp), 

 as may easily be proved by applying equation (2) successively to its 

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

 could consider as constant within the limits of the infinitely small 

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

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

 dvxdp. 



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

 mals drawn throughout the diagram for equal differences drj and dt, 

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

 so that the increase of t shall directly precede that of q, will be the 

 constant product dr\ X dt, as may be proved by equation (3), and the 

 value of y, which is equal to the area divided by the heat, will be 

 indicated proportionally by the areas.* 



* The indication of the value of y by systems of equidifferent isometrics and isopies- 

 tics, or isentropics and isothermals, is explained above, because it seems in accordance 

 with the spirit of the graphical method, and because it avoids the extraneous consider- 

 ation of the co-ordinates. If, however, it is desired to have analytical expressions for 

 the value of y based upon the relations between the co-ordinates of the point and the 

 state of the body, it is easy to deduce such expressions as the following, in which a; 

 and y are the rectangular co-ordinates, and it is supposed that the sign of an area is 

 determined in accordance with the equation A = fydjx : 



l_dv dp dp rfv _ C/T; ^_^& &H 

 y~ dx dy dx' dy~ dx ' dy dx dy 



where x and y are regarded as the independent variables ; or 



_dx dy dy dx 



dv dp dv dp' 



