carnot's theory of the motive power of heat. 549 



only remains for us to evaluate this area, that may determine the total mechani- 

 cal effect gained in a complete cycle of operations. Now, from experimental 

 data, at present nearly complete, as will be explained below, we may determine 

 the length of the line A A, for the given temperature S, and a given absorption 

 H, of heat, during the first operation ; and the length of A, A3 for the given lower 

 temperature T, and the evolution of the same quantity of heat during the fourth 

 operation : and the curves A, PA„ A, P'A may be draw as graphical representa- 

 tions of actual observations * The figure being thus constructed, its area may be 

 measured, and we are, therefore, in possession of a graphical method of determin- 

 ing the amount of mechanical effect to be obtained from any given thermal agency. 

 As, however, it is merely the area of the figure which it is required to determine, it 

 will not be necessary to be able to describe each of the curves A, PA., A3 P'A, but 

 it will be sufficient to know the difference of the abscissas corresponding to any 

 equal ordinates in the two ; and the following analytical method of completing 

 the problem is the most convenient for leading to the actual numerical results. 



20. Draw any line P P' parallel to OX, meeting the curviUneal sides of the 

 quadrilateral in P and F. Let f denote the length of this line, and p its distance 

 from X. The area of the figure, according to the integral calculus, will be de- 

 noted by the expression 



/; 



p ^^dp, 



where;?!, andp, (the limits of integration indicated according to Foueiee's nota- 

 tion) denote the lines A, and N3 A3, which represent respectively the pressures 

 during the fii-st and third operations. Now, by referring to the construction de- 

 scribed above, we see that f is the difi-erence of the volumes below the piston at 

 corresponding instants of the second and fourth operations, or instants at which 

 the saturated steam and the water in the cylinder have the same pressure ^9, and, 

 consequently, the same temperature which we may denote by t. Again, through- 

 out the second operation the entire contents of the cylinder possess a greater 

 amount of heat by H units than during the fourth ; and, therefore, at any histant 

 of the second operation there is as much more steam as contains H units of latent 

 heat, than at the corresponding instant of the fourth operation. H ence, if k de- 

 note the latent heat in a unit of saturated steam at the temperature t, the volume 

 of the steam at the two corresponding instants must differ by ^' Now, if <r de- 

 note the ratio of the density of the steam to that of the water, the volume— of 



k 



TT 



Steam will be formed from the volume ^y of water ; and, consequently, we have 



* See Note at the end of this Paper. 

 VOL. XVI. PAKT V. 7 p 



