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 8, and a given absorption 
H, of heat, during the first operation ; and the length of A, A; 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 drawn 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, P A, A; 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 O X, meeting the curvilineal sides of the 
quadrilateral in P and P’. Let ~ denote the length of this line, and p its distance 
from OX. The area of the figure, according to the integral calculus, will be de- 
noted by the expression 
Pi 
ae 3 $ ae; 
where p,, and p; (the limits of integration indicated according to Fourrer’s nota- 
tion) denote the lines O A, and N; A;, which represent respectively the pressures 
during the first and third operations. Now, by referring to the construction de- 
scribed above, we see that < is the difference 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 p, and, 
consequently, the same temperature which we may denote by#. 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 instant 
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 ¢, the volume 
of the steam at the two corresponding instants must differ by = Now, if « de- 
note the ratio of the density of the steam to that of the water, the volume * of 
: H 
steam will be formed from the volume *7 of water ; and, consequently, we have 
* See Note at the end of this Paper. 
VOL. XVI. PART V. 7 Oo 
