The Steam Engine Explained and Illustrated (Seventh Edition) / With an Account of Its Invention and Progressive Improvement, and Its Application to Navigation and Railways; Including Also a Memoir of Watt

by Dionysius Lardner

E = a − bS.(11.)

Let a cubic foot of water be evaporated under the pressure P′, and let it produce a volume of steam S′ of that pressure. Let this steam afterwards be allowed to expand to the increased volume S and the diminished pressure P; and let it be required to determine the mechanical effect produced during the expansion of the steam from the volume S′ to the volume S.

Let

E′ = the mechanical effect produced by the evaporation of the water under the pressure P′ without expansion.

E″ = the mechanical effect produced during the expansion of the steam.

E = the mechanical effect which would be produced by the evaporation under the pressure P without expansion.

E = the total mechanical effect produced by the evaporation under the pressure P′ and subsequent expansion.

Thus we have

E = E′ + E″.

Let s be any volume of the steam during the process of expansion, p the corresponding pressure, and e″ the mechanical effect produced by the expansion of the steam. We have then by (10.)

p =	a	− b;
	s
∵ de″ =	ads	− bds.
	s

Hence by integrating we obtain

e″ = a log. s − bs + C;