Fig. 2 represents a hillside rising from the sea.  Some distance up there is a lake, L, fed by streams coming down from a still higher level.  Lower down on the slope is a millpond, P, the tail race from which falls into the sea.  At the millpond is established a factory, the turbine driving which is supplied with water by a pipe descending from the lake, L. Datum is the mean sea level; the level of the lake is T, and of the millpond t.  Q is the weight of water falling through the turbine per minute.  The mean sea level is the lowest level to which the water can possibly fall; hence its greatest potential energy, that of its position in the lake, = QT = H. The water is working between the absolute levels, T and t; hence, according to Carnot, the maximum effect, W, to be expected is—­

/ T — t \
W = H( ------- )
\   T   /
/ T — t \
but H = QT [therefore]  W = Q T( ------- )
\   T   /

W = Q (T — t),

that is to say, the greatest amount of work which can be expected is found by multiplying the weight of water into the clear fall, which is, of course, self-evident.

Now, how can the quantity of work to be got out of a given weight of water be increased without in any way improving the efficiency of the turbine?  In two ways: 

1.  By collecting the water higher up the mountain, and by that means increasing T.

2.  By placing the turbine lower down, nearer the sea, and by that means reducing t.

Now, the sea level corresponds to the absolute zero of temperature, and the heights T and t to the maximum and minimum temperatures between which the substance is working; therefore similarly, the way to increase the efficiency of a heat engine, such as a boiler, is to raise the temperature of the furnace to the utmost, and reduce the heat of the smoke to the lowest possible point.  It should be noted, in addition, that it is immaterial what liquid there may be in the lake; whether water, oil, mercury, or what not, the law will equally apply, and so in a heat engine, the nature of the working substance, provided that it does not change its physical state during a cycle, does not affect the question of efficiency with which the heat being expended is so utilized.  To make this matter clearer, and give it a practical bearing, I will give the symbols a numerical value, and for this purpose I will, for the sake of simplicity, suppose that the fuel used is pure carbon, such as coke or charcoal, the heat of combustion of which is 14,544 units, that the specific heat of air, and of the products of combustion at constant pressure, is 0.238, that only sufficient air is passed through the fire to supply the quantity of oxygen theoretically required for the combustion of the carbon, and that the temperature of the air is at 60 deg.  Fahrenheit = 520 deg. absolute.  The symbol T represents the absolute temperature of the furnace, a value which is easily calculated in the following manner:  1 lb. of carbon requires 2-2/3 lb. of oxygen to convert it into carbonic acid, and this quantity is furnished by 12.2 lb. of air, the result being 13.2 lb. of gases, heated by 14,544 units of heat due to the energy of combustion; therefore: