It follows that the work performed by the transformation of the energy of heat, during any indefinitely small variation of the state of a substance as respects temperature, is measured by the product of the absolute temperature into the variation of a “function,” which function is the rate of variation of the work so done with temperature. This function is the quantity called by Rankine the “heat-potential” of the substance for the given kind of work. A similar function, which comprehends the total heat-variation, including both heat transformed and heat needed to effect accompanying physical changes, is called the “thermo-dynamic function.” Rankine’s expression for the general equation of thermo-dynamics includes the latter, and is given by him as follows:
Jdh = dH = kdτ + τdF = τdφ,
in which J is Joule’s equivalent, dh the variation of total heat in the substance, kdτ the product of the “dynamic specific heat” into the variation of temperature, or the total heat demanded to produce other changes than a transformation of energy, and τdF is the work done by the transformation of heat-energy, or the product of the absolute temperature, τ, into the differential of the heat-potential. φ is the thermo-dynamic function, and τdφ measures the whole heat needed to produce, simultaneously, a certain amount of work or of mechanical energy, and, at the same time, to change the temperature of the working substance.
Studying the behavior of gases and vapors, it is found that the work done when they are used, like steam, in heat-engines, consists of three parts:
(a.) The change effected in the total actual heat-motion of the fluid.
(b.) That heat which is expended in the production of internal work.
(c.) That heat which is expended in doing the external work of expansion.
In any case in which the total heat expended exceeds that due the production of work on external bodies, the excess so supplied is so much added to the intrinsic energy of the substance absorbing it.
The application of these laws to the working of steam in the engine is a comparatively recent step in the philosophy of the steam-engine, and we are indebted to Rankine for the first, and as yet only, extended and in any respect complete treatise embodying these now accepted principles.
It was fifteen years after the publication of the first logical theory of the steam-engine, by Pambour,[114] before Rankine, in 1859, issued the most valuable of all his works, “The Steam-Engine and other Prime Movers.” The work is far too abstruse for the general reader, and is even difficult reading for many accomplished engineers. It is excellent beyond praise, however, as a treatise on the thermo-dynamics of heat-engines. It will be for his successors the work of years to extend the application of the laws which he has worked out, and to place the results of his labors before students in a readily comprehended form.